AIAA JOURNAL

Vol 41 No 3 March 2003

Aeroacoustic Instability in Rockets

Gary A Flandrocurren

University of Tennessee Space Institute Tullahoma Tennessee 37388and

Joseph Majdalanidagger

Marquette University Milwaukee Wisconsin 53233

Current solid-propellant rocket instability calculations (eg Standard Stability Prediction Program) accountonly for the evolution of acoustic energy with time However the acoustic component represents only part of thetotal unsteady system energy additional kinetic energy resides in the shear waves that naturally accompany theacoustic oscillations Because most solid-rocket motor combustion chamber con gurations support gas oscillationsparallel to the propellant grain an acoustic representation of the ow does not satisfy physically correct boundaryconditions It is necessary to incorporate corrections to the acoustic wave structure arising from generation ofvorticity at the chamber boundaries Modi cations of the classical acoustic stability analysis have been proposedthat partially correct this defect by incorporating energy sourcesink terms arising from rotational ow effectsOne of these is Culickrsquos ow-turning stability integral related terms that are not found in the acoustic stabilityalgorithm appear A more complete representation of the linearized motor aeroacoustics is utilized to determinethe growth or decay of the system energy with rotational ow effects accounted for already Signi cant changes inthe motor energy gainloss balance result these help to explain experimental ndings that are not accounted forin the present acoustic stability assessment methodology

NomenclatureAr

b = admittance of burning surfacesAr

N = nozzle admittanceAr

S = admittance of nonburning surfacesa0 = mean speed of soundE = time-averaged oscillatory system energyE2

m = normalization constant for mode me = oscillatory energy densityer emicro ez = unit vectors in r micro and z directionskm = wave number for axial mode m mfrac14R=LL = chamber lengthMb = reference Mach number at burning surfacem = mode numbern = outward-pointingunit normal vectorP0 = mean chamber pressurep = oscillatory pressureR = chamber radiusr = radial positionS = Strouhal number km =Mb

t = timeUr Uz = mean ow velocity componentsu = oscillatory velocity vectory = radial distance from the wall 1 iexcl rz = axial positionreg = growth rate (dimensional siexcl1)deg = ratio of speci c heatsplusmn = inverse square root of the acoustic

Reynolds numberp

ordm=a0 R = wave amplitudeordm = kinematic viscosity sup1=frac12

Presented as Paper 2001-3868 at the AIAA 37th Joint PropulsionConference Salt Lake City UT 8ndash11 July 2001 received 22 January 2002revision received 8 July 2002 accepted for publication 31 October 2002Copyright cdeg 2002 by Gary A Flandro and Joseph Majdalani Publishedby the American Institute of Aeronautics and Astronautics Inc with per-mission Copies of this paper may be made for personal or internal useon condition that the copier pay the $1000 per-copy fee to the CopyrightClearance Center Inc 222 Rosewood Drive Danvers MA 01923 includethe code 0001-145203 $1000 in correspondence with the CCC

currenBoling Chair Professor of Advanced Propulsion Mechanical andAerospace Engineering Associate Fellow AIAA

daggerAssistant Professor Department of Mechanical and Industrial Engineer-ing Member AIAA

frac12 = densityAacuter = function de ned in Eq (13)Atilder = exponential argument Eq (11)Auml = mean vorticity amplitude = unsteady vorticity amplitude

Subscripts

b = combustion zonem = mode number

Superscripts

(r) (i ) = realimaginary or cosinesine partscurren = dimensional quantityraquo = vortical (rotational) part

O = acoustic (irrotational)part

I Introduction

C ULICKrsquoS paperson combustioninstability1iexcl5 publishedin theearly 1970s are the foundationfor stabilitypredictionmethods

now in use67 His method is based on three crucial assumptions1) small-amplitude pressure uctuations superimposed on a low-speed mean ow 2) thin chemically reacting surface layer withmass addition and 3) oscillatory ow eld represented by chamberacoustic modes

The rst assumption allows linearization of the governingequationsboth in the wave amplitude and the surface Mach numberof the mean injected ow The second causes all surface reactioneffects includingcombustion to collapse to simple acoustic admit-tanceboundaryconditionsimposedat thechambersurfacesThe lastassumptionoversimpli es theoscillatorygasdynamicsby suppress-ing all unsteady rotational ow effects the acoustic representationis strictly irrotational Concern for this omission was partially ad-dressedby Culick in a paper in which he introducedhis well-knownrotationalmean ow model2 Stabilitycalculationsbasedon this im-proved mean ow representation produced no signi cant changesin the system stability characteristics On this basis it has sincebeen generally assumed that all vorticity (rotational ow effects)including the unsteadypart has negligible in uence on combustioninstability growth rate calculations

Considerableprogress has been made in the last decade in under-standing both the precise source of the vorticity and the resulting

485

486 FLANDRO AND MAJDALANI

changes in the oscillatory ow eld Approximate analytical8iexcl16

numerical17iexcl22 and experimental investigations23iexcl26 have demon-strated that rotational ow effects play an important role in the un-steady gas motions in solid-rocketmotors Much effort has been di-rected to constructingthe requiredcorrectionsto theacousticmodelThis has culminated in a comprehensivepicture of the unsteadymo-tions that agrees with experimental measurements8iexcl10 as well asnumerical simulations11

These modelswere used in carryingout three-dimensionalsystemstability calculations89 in a rst attempt to account for rotational ow effects by correcting the acoustic instability algorithm In thisprocess one discovers the origin and the three-dimensional formof the classical ow-turning correction related terms appear thatare not accounted for in the Standard Stability Prediction Programalgorithm In particular a rotational correction term is identi edthat cancels the ow-turning energy loss in a full-lengthcylindricalgrain However all of these results must now be questionedbecausethey are founded on an incomplete representation of the systemenergy balance

Culickrsquos stability estimation procedure is based on calculatingthe exponential growth (or decay) of an irrotational acoustic wavethe results are equivalent to energy balance models used earlier byCantrell and Hart27 In all of these calculations the system energyis represented by the classical Kirchoff (acoustic) energy densityConsequentlyit does not representthe full unsteady eld includingboth acoustic and rotational ow effects Kinetic energy carried bythe vorticity waves is ignored It will be demonstrated in this paperthat the actual average unsteady energy contained in the system ata given time is about 25 larger than the acoustic energy aloneFurthermore representation of the energy sources and sinks thatdetermine the stability characteristics of the motor chamber mustalso be modi ed Attempts to correct the acousticgrowth rate modelby retention of rotational ow source terms only89 preclude a fullrepresentationof the effects of vorticity generation and coupling

In fact there is a convincing body of evidence pointing to theexistence of other aeroacoustic coupling mechanisms that are notincorporated in the current acoustic-stability theory For examplethe so-called parietal or surface vortex shedding (PVS) has beenidenti ed some years ago as a source of instability that eludes clas-sic theory28 The corresponding phenomenon was rst detected bynumerical simulations of 15 subscale models of the French ArianeV booster29iexcl32 This P230 Moteur a Propergol Solide (MPS) wasknown to exhibit large amplitudeoscillations33 This new type of in-stability was especially important in long segmented rocket motorssuch as the Japanese H-II vehicle34 the Titan 34D solid rocket mo-tor (SRM)35 the Titan IV SRMSRMU (upgrade)36iexcl39 the shuttlerocket booster (SRB)40 and other motors with length-to-portradiusratios ranging from 15 to 25

To compensate for the inability of classic theory3541iexcl48 toexplain the large pressure oscillations driven by so-called ldquocrawl-ingrdquo vortices29 a number of studies have been carried out withthe aim of improving our basic understanding of the suspectedmechanism28 Credit should be given in that regard to Vuil-lot Avalon Casalis Griffond Lupoglazoff Traineau DupaysPineau Tissier Ugurtas and coworkers who have employedexperimental2849iexcl52 numerical53iexcl59 and theoretical avenues60iexcl63

to elucidate the origin of PVS coupling In Refs 60ndash63 CasalisAvalon Pineau and Griffond have based their recent theoreticalstudy on linear instability theory introduced in 1969 by Varapaevand Yagodkin64 Their efforts have provided an alternate sourceof instability whose omission in classic analyses has led them toassociate some of the unforeseenexperimentaland numerical insta-bilities to the hydrodynamic evolution and inception of mean owturbulence6061

At the conclusion of these studies30iexcl3250iexcl52 speculations thatresonance-like pressure ampli cations were caused only by vor-tex shedding at annular restrictors or inhibitor rings were laid torest when similarly intense vorticity-generated oscillations wereobserved in unsegmented rocket motors As noted by Ugurtaset al65 two-dimensional compressible ow simulations of theNavierndashStokes equationsby Lupoglazoffand Vuillot29iexcl31 have con- rmed themeasurementsacquiredfromsubscale ringsas such the

collection of all available data has pointed out to the existence of apowerful vorticity-drivencoupling irrespectiveof whether inhibitorrings or other surface anomalies are present29iexcl31

The studies just revieweddemonstrate that currentmotor stabilityassessmentsbased on irrotational ow effectsalonecannot properlyrepresent the observed system behavior The main objective of thispaper is the construction of a more complete model for the stabil-ity of rocket motor ow elds that accounts for interactions that arelost in the irrotational (acoustic unsteady ow) assumption Stabil-ity terms must be identi ed that will establish the correct energypathways between the rotational and the irrotational parts of thegas oscillations This is best accomplished by applying the energybalance approach because this method enhances the physical un-derstanding of the many gas dynamic interactions involved in theproblem

Considerable work is needed to make these improvements Theoutcome is a stability algorithm that accounts for both acoustic andvortical ow interactionsAlthoughno attempt is made in this paperto explore all possible unsteady ow eld con gurations stabilityterms are deduced that will allow such ow effects to be includedin the stability assessmentFor example whereas turbulenceeffectsand surface vortices (as in the parietal vortex shedding mechanism)are not addresseddirectly the terms in the stabilitybalancein whichdetailedrepresentationsof theseadditionalvortical oweffectsenterthe problem will be identi ed

Signi cant differences between these new results and those cur-rently utilized for motor stability computations are demonstratedThe new model is tested by applying it to several solid-propellantrocket motor con gurations to demonstrate important scaling andgeometricaleffectsA new energy source term is identi ed that rep-resents at least one way in which unexplained instabilities in largesolid propellant booster motors arise This energy source arisingfrom productionof unsteady vorticity at the propellantburning sur-face can be comparable in size to the key combustion pressurecoupling itself It might also be related to velocity coupling ef-fectswhich cannotbe fully representedin the contextof irrotationalacoustic instability theory This same term when evaluated usingvortex shedding ow solutions yields an improved description ofthis energy source

Despite major changes in the mathematical formulation it is notnecessary to discard the current linear stability estimation method-ology required modi cations are readily accomplished by addingthe new stability integrals to those already in place in the currentstability codes It is important to note that all of the original termsidenti ed in the earlier irrotationalmodels are recoveredin the morecomplete results described in this paper

II Unsteady Flow AnalysisThe goal of this section is the constructionof an improved model

of the unsteady ow eld in a rocket chamber that realistically ac-counts for vortical (rotational) as well as acoustical (irrotational)effects In particular close attention is paid to properly satisfyingthe correct boundary conditions on all chamber boundaries includ-ing both inert and reactive surfaces

The validity of the stability calculations depends critically on asuf ciently detailedand physically correct representationof the un-steady velocity eld The assumption of an irrotational unsteady ow as used in most stability calculations must be discarded Theanalytical methods and notation employed here closely follow ear-lier papers8iexcl10 Dimensionless variables are the same as those usedin the classical combustion instability analyses velocities are nor-malized by dividing by the average sound speed and pressure isnormalized by dividing by the product of the mean chamber pres-sure and the speci c heat ratio deg Other thermodynamic propertiesare made dimensionless by dividing them by their mean chamberreference values Lengths are referenced to the chamber radius RFigure 1 illustrates the geometry and coordinate systems to be em-ployed in the analysis

The analysisstarts by assuming small-amplitudeunsteadypertur-bations on a mean ow described by the vector MbU In this paperCulickrsquos mean ow model2 for a cylindrical burning port

U D Ur er C Uzez D iexclriexcl1 sin xer C frac14 z cos xez (1)

FLANDRO AND MAJDALANI 487

Fig 1 Motor geometry and coordinate system used

will be used Note that the combination

x acute 12 frac14r 2 (2)

appears frequently in the analysisIsentropic conditions are assumed but viscous forces both shear

and dilatationalare retainedThe linearizedcontinuityand momen-tum equations then become

frac12 1

tC r cent u1 D iexclMbU cent rfrac12 1 (3)

u1

tC r p1 D Mb

copyiexclr

poundU cent u1

currenC u1 pound r pound U C U pound r pound u1

ordf

C plusmn2

raquosup32 C

cedil

sup1

acuter

poundr cent u1

curreniexcl r pound r pound u1

frac14(4)

This dimensionless set of equations is the starting point for all pre-ceding combustion instability models The superscript (1) denotesthe rst-order terms in a perturbation parameter proportional tothe amplitude of the unsteady pressure uctuation A second smallparameter the mean ow Mach number at the burning surface Mb is also employed in the linearizationprocess only terms to the rstorder in Mb are retained as in the classical combustion instabilityanalysesThe relative size of the viscous forces is set by the param-eter plusmn2 which is the inverse of the acoustic Reynolds number basedon the chamber radius R Viscous terms are usually dropped butare retained here for completeness by assuming that plusmn2 D OMb

The main variables will be represented as combinations of irro-tational and rotational components

8lt

p1 D Op C Qpfrac12 1 D Opu1 D Ou C Qu (5)

The acoustic eld is irrotational (r pound Ou D 0) and the superimposedvortical component is incompressible (r cent Qu D 0) Because of theassumptionof constant entropy the density and the thermodynamic(acoustic) pressure are interchangeableas indicated in Eq (5) Thedensity in the continuity equation is routinely replaced in the clas-sical analysis by the pressure variable The reader is cautioned that

care must be taken in interpreting the unsteady pressure p1 If itappearsin thecontinuityequationthisparameterwill then representa density uctuation This distinction is not important if the ow isirrotational but it is vitally important when rotational effects are tobe incorporated

If irrotational ow is assumed Eqs (3) and (4) yield the usualacoustic wave solutions For axial motions in a rocket chamber thesolution is the standing plane wave

Op D eiexclikt coskm z C OMb (6)

Ou D ieiexclikt sinkm zez C OMb (7)

For axial modes satisfaction of closed-end boundary conditionsrequires that

km D mfrac14 R=L (8)

As Eqs (6) and (7) indicate there exist corrections to the acoustic eld of the order of the mean Mach number These were describedin Ref 8 but are not required in the stability calculationsOnly the rst-order velocity and pressure (in Mb) are needed in evaluatingthe stability integrals

It is immediately apparent that Eq (7) cannot satisfy physicallycorrect boundary conditions either at the edge of the combustionzone or on inert side walls The axial velocity component mustapproach zero at the wall to satisfy the no-slip condition Thus theproductionof axialacousticoscillationsgives rise to correctionsthatmust come from the rotational ow effects Flandro solved Eqs (3)and (4) for the vortical (rotational incompressible) eld by twodifferent methods The inviscid case was treated in Ref 8 a morecomplete analysis was presented in Ref 9 in which viscous shearwave damping was includedAnother improvementin the latter wasthe introductionof an equivalentclosed-formsolutionin placeof thein nite seriesused in Ref 8 Additional improvementsweremade inRef 10 in which turbulent mean ow correctionswere introducedFor clarity we will display only the laminar results for the simplestmotor geometry and acoustic mode structure namely a cylindricalport with axial oscillations other geometries and transverse modesof oscillation can be handled similarly Other investigators such asMajdalani and Roh16 have veri ed the basic results to be used hereby quitedifferentanalyticalapproachesand bydirect uid dynamicscomputations11iexcl19

For rotationalpressure and velocity correctionsthat must accom-pany the assumed acoustic solution of Eqs (6) and (7) one nds

Qp D i Mbeiexclikt sin2xeAacute C iAtildeiexcl

12 frac14 z

centsin[sinxkm z] C O

iexclM 2

b

cent

(9)

Qu D iCeiexcliktrUr eAacute C iAtilde sin[sinxkm z]ez C OMb (10)

An undetermined complex constant C is shown in Eq (10) In theearlier work this constantwas evaluated by assuming that for a thincombustion layer the physical boundary at the solid surface andthe edge of the combustion zone are nearly coincident8iexcl10 Thenif one applies the no-slip condition to the composite unsteady ax-ial velocity [u1 D Ou C Qu] at the edge of the combustion zone one nds that C D 1 This result has been questionedbecause the actualsolid surface lies within a region of nonuniformity the combustionzone (Brown RS private communication Tullahoma TennesseeSept 2000) It is a straightforwardprocess to show that the originalvalue is correct the calculationis too lengthyto displayhereThis isdone by constructing a two-dimensional model for the combustionzone including all gas phase effects The axial momentum equationin the ame zone is solved numerically to account for the gradientsin temperature and density caused by combustionThe resulting in-ner solution is then matched asymptotically to the outer solutionexpressed by Eq (10) Standard methods of singular perturbationtheory apply The simplest approach is to determine solutions inan intermediate region in which both the outer and inner represen-tations are valid Then the matching is readily accomplished Theoutcome con rms that the original result C D 1 is correct unlessthe oscillationfrequencyis very high In that case the rotational ow

488 FLANDRO AND MAJDALANI

effects become entirely buried within the combustion zone and theouter solution collapses to the simple plane wave acoustic model

Flandro9 derived these expressionsdirectly from Eqs (3) and (4)without recourse to the splitting theorem applied in earlier pa-pers Critics have not understood that splitting Eqs (3) and (4)into acoustical and vortical parts (following the method of Chuand Kovasznay66 and Morse and Ingard67) is a simplifying butnot a necessary step in arriving at the solutions displayed (Brownprivate communication) Another misunderstanding centers on thepressure correction Qp sometimes called the pseudopressure It isimportant to understand that this was not simply set to zero by as-sumption in the earlier works Careful solution of the momentumequation (4) shows that it is an additional correction of the order ofthe mean Mach number the result is shown in Eq (9) Correctionsof similar size appear in the irrotational (acoustic) pressure solu-tion Eq (6) Although Qp constitutes a negligible contribution to theoverall pressure its incorporationis helpful in the acoustic stabilitycalculations for reasons that will be described later It is also neces-sary to point out that the pseudopressuremust not be inserted in thecontinuity(3) as part of the density variable as suggestedby Brownin his 2000 private communication

The complex exponential factor expAacute C iAtilde in the solution ex-hibited in Eq (10) suggests that the vortical motion can be inter-pretedas a damped travelingshearwave Both Aacute and Atilde are functionsonly of radial position The imaginary part of the exponential argu-ment namely

Atilder D iexcl[km =frac14 Mb] taniexcl

12 x

cent(11)

sets the wavelength and spatial frequency of the shear wave Forlater use note that the derivative of Atilde with respect to the radius issimply

dAtilde

drD

km

MbUr(12)

This is proportionalto the reciprocalof the radialmean ow velocityThe real part of the argument Aacuter is given by

Aacuter Draquo

frac14 2

micro1 iexcl

1

sinxiexcl x

cosx

sin2xC I x iexcl I

sup31

2frac14

acutepara

raquo acutek2

m plusmn2

M3b

DS2plusmn2

Mb S acute

km

Mb(13)

I x D x C1

18x3 C

7

1800x5 C

31

105840x7 C cent cent cent (14)

where raquo is a parameter of O1 that re ects the relative importanceof the viscous damping its physical description and veri cationare detailed by Majdalani and Roh16 Note that the damping ofthe shear wave in the radial direction increases quadratically asthe frequency of the acoustic oscillation increases Similarly thewavelength of the shear wave decreases as frequency increasesForthe fundamental acousticmode and the lowest order harmonics theshear wave might ll the entire chamber for high-order acousticmodes the shear oscillations become con ned to a thin acousticboundarylayerwhich can lie entirelywithin the combustionzone13

The composite axial velocity solution found by superimposingEqs (7) and (10) has been shown8iexcl1012 to agree quite closely withthe experimental ndingsofRefs 12 25 26 and 68 Figures2 and 3showcomparisonsof theoscillationamplitudeand phaseangle fromthe theory just discussed superimposed on some of Brownrsquos earliercold ow work2469 It is important to understandthat no curve ttinghas been used here the differences between the experimental andtheoretical results arise both from unavoidable experimental errors(orientation and positioning of the hot lm sensor data reductionetc) and from the assumptionset used in the analysisNevertheless

Fig 2 Comparison of theoretical and experimental amplitudes ofunsteady axial velocities Here S = 513m zcurrenL = 0106 Mb = 00018LR = 34 km = 00924m raquo = 0539m2 and m = 1 Triangles andcircles cor-respond to experimental data acquired with a pressure wave amplitudeof a) 00005 and b) 00039

Fig 3 Comparison of theoretical and experimental phase angles(degrees)

adequate agreement is shown Because nitrogen was the working uid at measured temperature and pressure all parameters charac-terizing the ow eld are well known

Numerical solutions based on the full NavierndashStokes representa-tion of the unsteady eld also indicate excellentagreementwith thetheory reviewed here Furthermore other analyticalmethods basedon Wentzel Kramers and Brillouin (WKB) and multiple-scalethe-ories have been applied by Majdalani and Van Moorhem70 Theirshavedemonstratedbasedon theprincipleof least singularitytheex-istence of nonlinear scales underlying the physics of the problem71

In additionto providinga rigorousmathematical treatment that doesnot require guesswork their approach has led to the exact form ofthe characteristic length scales as function of Culickrsquos mean owsolution Particularly their solutions have yielded results that arevirtually indistinguishablefrom those displayed here11iexcl16

In previousanalyses of the rocket motor rotational ow eld spe-cial attentionwas givento the radialcomponentof velocity8911 Thiswas done because in classical instability theory the main source ofacoustic energy is the radial velocity uctuation produced by inter-action of the pressurewave with the combustion processes and heattransfer in the burning zone It was shown in Refs 8 and 9 that inaddition to the radial acoustic velocity there appears a rotationalcorrection that does not vanish at the edge of the combustion zoneThe presence of this additional unsteady radial velocity componentimplies that by analogy to pressure coupling there is an additionalsource of unsteady mass ux at the combustion zone interface Forwant of better terminology this was called the rotational ow cor-rection as rst displayed in Eq (89) of Ref 8 The new term whichdoes not appear in any previous analysis has given rise to consid-erable controversy

By way of providing a precise description of the source of therotational ow correction a detailed physical interpretation and afull resolution of the controversywill appear in the next section ofthispaperThe answers are foundby means of a carefulderivationofthe system stability energy balance It is not necessary to calculatethe vortical radial velocity correction in determining the systemstability The surface acoustic admittance function will be utilized

FLANDRO AND MAJDALANI 489

in the standard fashion to represent the pressure coupling of theoscillatory ow with the combustion processesThen experimentalmeasurements (usuallysecuredwith a T-burneror similar device)ofthe surface response can be utilized as in the acceptedmethodologyfor accounting for this important source of oscillatory energy

Earlier attempts to incorporate vortical ow corrections in thestability calculations were based on the idea that one must re-store rotational source terms which were dropped in the classicalanalyses For example Brown et al24 had suggested that reten-tion of the rotational convective acceleration term U pound r pound u1

on the right side of Eq (4) might have important stability im-plications This term was rst evaluated by Flandro89 whoshowed that this term yields a damping effect that is indistin-guishable from the classical ow-turning loss Additional termsthat cannot be found in the standard stability computations ap-pear in this analysis One of these the rotational ow correc-tion has led to much debate because it exactly cancels the ow-turning effect in some chamber con gurations Brownrsquos as-sertion that the rotational ow correction is just another wayto represent ow turning can be quite readily refuted In fact allmisunderstandings are resolved when a consistent energy methodis used to deduce motor stability characteristics The aw in theearlier approach is that it represented the system by a model thataccounts for growth or decay of the acoustic energy alone Simplyretaining the rotational convective source terms24 in the perturbedwave equation following Culickrsquos method does not address thisimportant defect

To describe the evolution of the oscillatory ow in a rocket mo-tor correctly all unsteady energy must be accounted for rst Themethod used here follows the known energy balance approach de-scribed by Kirchoff72 and used extensively in rocket stability calcu-lationsby Cantrell and Hart27 and Hart and coworkers41iexcl43 It is nownecessary to account for the entire kinetic energy uctuationTo ac-complish this multiply the continuity equation (3) by the acousticpressure (or density) and add this to the momentum equation (4)multiplied by the composite unsteady velocity vector

u1 D Ou C Qu (15)

This operation constitutes the main departure from the precedingapproach Isolating the terms differentiatedwith respect to time andretaining only those terms that are linear in the Mach number Mb one nds

t

micro12

Op2 C12

u1 cent u1

paraDiexcl

poundOpr cent Ou C u1 cent r Op C Qp

curren

iexcl Mb

raquo1

2U centr Op2 C u1 centr

poundU cent u1

curren

iexclu1 centpoundu1 pound r pound U C U pound r pound Qu

currenfrac14

C plusmn2u1 centmicrosup3

2 Ccedil

sup1

acuterr cent Ou iexcl r pound r pound Qu

para(16)

If the full unsteady velocity vector u1 were to be replaced at thisstage by the acoustic part Ou then the nal result of the stability cal-culation would be exactly that described in earlier papers89 If allrotational terms on the right of Eq (16) were also to be removedby assuming that Qu D 0 then the classical three-dimensional sta-bility result would be reproduced Assumptions such as those justdescribed are motivated mainly by a desire to keep the analysissimple However as we shall now demonstrate much physical sub-stance is lost in such simpli cations We choose not to follow thetraditional approach heremdashmost rotational effects are retained asEq (16) shows the kinetic energy per unit mass on the left-handside re ects all laminar unsteady motions in the chamber Parietal-vortex sheddingand turbulentcorrectionswill require specialatten-tion and future re nementsdiscussionsof turbulent ow effects arefound in the numerical studies by Flandro et al10 Roh et al18 Yangand Roh19 Apte and Yang73 Beddini and Roberts7475 and Vuillotand coworkers5758 So that the analysis is not overly complicated

effectsof solidparticlesin the ow are notdisplayedeither requiredmodi cations are to be incorporated later as needed

Following the standard approach72 one de nes the oscillatoryenergy density

e D 12

poundOp2 C u1 cent u1

curren(17)

and the time-averagedoscillatoryenergy residing in the chamber atany instant as

E DZ Z

V

Zhei dV D

1

2

Z Z

V

Z laquoOp2 C u1 cent u1

notdV (18)

where triangular brackets denote the time average of the enclosedfunction Then from Eq (16) it is clear that the evolution of thesystem energy is controlled by

E

tD

Z Z

V

Zirrotationalz | frac12

iexclr cent Op Ou iexcl12

MbU cent r Op2 iexcl Mb[Ou cent rU cent Ou]

C plusmn2d Ou cent rr cent Ou C Mb[ Ou cent Ou pound shy C Ou cent U pound ]

rotationalz |

iexclQu cent r Op iexcl Mb

microQu cent rU cent Ou C Ou cent rU cent Qu C Qu cent rU cent Qu

iexclQu cent U pound iexcl Qu cent Qu pound shy

para

C plusmn2d Qu cent rr cent Ou iexcl plusmn2[Ou cent r pound C Qu cent r pound ]

iexcl Ou cent r Qp iexcl Qu cent r Qpfrac34

dV (19)

where the irrotational and rotational contributions to the energyrate of change have been partitioned for clarity The last twoldquoirrotationalrdquo terms are in reality caused by rotational effectsThey are placed with the irrotational terms to conform to Culickrsquosmethodology247 The last two ldquorotationalrdquo terms are caused by thepseudopressure a quantity that has often been ignored in previousstudies The mean and unsteady vorticity vectors are represented inEq (19) by shy D r pound U and D r pound Qu respectively

The dilatationalviscous term is simpli ed by de ning

plusmn2d acute plusmn22 C cedil=sup1 (20)

where cedil is the second coef cient of viscosity Several terms cancelFor example

iexclOu cent Qu pound shy iexcl Qu cent Ou pound shy D 0 (21)

Further simpli cations can be made For example several termscan be expressed as integrals over the surface bounding the con-trol volume It is rst necessary to review the protocol needed inevaluating the stability integrals

The terms on the right-hand side of Eq (19) control the rate atwhich the systemenergy changesFrom this informationone can es-timate the growth or decay rate for a given motor con gurationThecomplex wave number (or frequency)k in the assumed exponentialtime dependence is written as

k D km C m C iregm C OiexclM2

b

cent(22)

In the multidimensionalcase m can consist of three mode integersWe will restrict the evaluation to axial modes for clarity then a sin-gle integer m identi es the mode being investigatedMore complexmodes can be handled in a similar way The result for a simple planewave axial mode is shown in Eq (8) The dimensionlesscorrectionsm C iregm are of the order of the mean ow Mach number In theconventionalenergy-balanceprocedurethe frequencycorrectionm

is not evaluated For simplicity we follow this approach The fre-quencycorrectionis implicitlyaccountedfor in thewavenumberkm

490 FLANDRO AND MAJDALANI

Because quadraticcombinationsof the variablesappear it is rstnecessary to take the real parts of the pressure and velocities to beinserted in the equationsThus one writes

Op D Opm expregm t coskm t (23)

Ou D Oum expregm t sinkm t (24)

Qu D expregm tpound

Qurm coskm t C Qui

m sinkm tcurren

(25)

where superscripts (r) and (i ) refer to the cosine and sine multipli-ers stemming from complex-to-real conversion of real and imagi-nary partsThis notationis necessarybecausethe rotationalvelocityvector contains the spatial exponential term exp(Aacute C iAtilde ) Insertingthese expressions into Eq (17) yields the time-averaged energydensity

hei D 14 exp2regm t

irrotationalz | poundOp2

m C Oum cent Oum

C

rotationalz | 2Oum cent Qui

m C Qur m cent Qur

m C Quim cent Qui

m

curren(26)

which has been partitioned to emphasize the new terms This re-sult should be compared to the classical expression for the energydensity which consists of only the rst two terms in Eq (26) Theadditional three terms represent the kinetic energy residing in theunsteady vorticity Inserting these expressions into Eq (18) carry-ing out the time averaging and differentiatingwith respect to timeone nds

E

tD regm exp2regm tE2

m (27)

where the energy normalization function

E2m D

12

Z Z

V

Z pound Op0

m2 C Oum cent Oum C 2 Oum cent Quim

C Qurm cent Qur

m C Qui m cent Qui

m

currendV (28)

is expressed here by the same symbol used in earlier stability anal-yses It is written as a squared quantity to indicate that it is positivede nite

It is interestingto compare values from Eq (28) to those found inearlier computationsThe mode shapes are needed for this purposeEquations (6) (7) and (10) give the required information for theassumed axial oscillations to zeroth order in Mb

Opm D coskm z (29)

Oum D sinkm zez (30)

Qurm D sinx expAacute sinAtilde sin[sinxkm z]ez (31)

Quim D iexclsinx expAacute cosAtilde sin[sinxkm z]ez (32)

The acoustic pressure and velocity mode shapes are related by

Oum D iexclr Opm =km (33)

for later use Figure 4 shows the results of integration of the kineticenergy terms in Eq (28) across the chamber from the centerline tothe wall at a given axial location Note that the term involving thecross product of the acoustic and vortical parts does not contributeto the integrated result because it oscillates in the radial directionand is zero at the upper limit However the rotational correctionshows a net value which is half of the integrated acoustic energydensityClearly the rotationaleffects representa signi cant changein the system energy

When the entire volume integral of Eq (28) is carried out for theaxial mode case using Eqs (29ndash32) one nds that

E2m D 5

8 frac14 L=R (34)

Fig 4 Radial integration of the kinetic energy density components

which is 25 larger than the conventional acoustic value for lon-gitudinal oscillations However the most important changes in thegrowth rate regm appear in the integral terms on the right-handside ofEq (19) We must now attend to this important part of the stabilityanalysis

A Growth Rate CalculationsIn estimating the system growth rate for a given mode of oscil-

lation it can be seen that it consists of the linear superposition of15 volume integrals as shown in Eq (19) The simplest procedure isto consider each of these (or selected groups) to be evaluated indi-vidually One can then represent the net growth rate as a linear sumof gains and losses as

regm D reg1 C reg2 C reg3 C cent cent cent DNX

i D 1

regi (35)

in the usual fashion The dimensionless values of the several termsare given in the following paragraphs The reader is reminded thatin order to compute the corresponding dimensional values it isnecessary to multiply by a0=R the inverse of the characteristictimeused in the formulation Then the growth rates are obtained in thefamiliar units of rads (siexcl1)

B Irrotational Growth Rate ContributionsTo illustrate the computationalprocedureconsider the rst three

irrotational terms

reg1 D1

exp2regm tE2m

Z Z

V

Z frac12iexclr cent

microOp Ou C

1

2MbU Op2

para

iexcl Mb[ Ou cent rU cent Ou]

frac34dV (36)

where the rst two terms have been combined by using the vectoridentity for the divergence of the product of a scalar and a vectorand by noting that r cent U D 0 for continuity of the incompressiblemean ow Thus the rst term can be converted to a surface integralusing Gaussrsquos divergence theorem The second can be written interms of the pressure by means of Eq (33) after some algebra italso reduces to a surface integral Assuming a short quasi-steadynozzle one nds

reg1 DMb

2E2m

8lt

Z Z

Sb

poundAr

b C 1curren

Op2m dS iexcl

Z Z

SN

poundAr

N C Uz

currenOp2

m dS

9=

rsquofrac14 Mb

2E2m

L

R

copypoundAr

b C 1curren

iexcl [deg C 1]ordf

(37)

The rst term in Eq (37) is of course the classical pressure cou-pling effect except that it is now somewhat reduced in size becauseof the larger energy normalization integral The entire surface in-tegral has not been represented in Eq (37) Nonburning chambersurfaces can be accommodated in the surface integrals if required

FLANDRO AND MAJDALANI 491

The several admittance functionsare taken to be quantitiesof O1the mean ow Mach number has been factored out as in most tra-ditional computations The admittance for nonburning surfaces As

is usually neglected although rough chamber walls and insulationlayers can display substantial (usually negative) admittance

The fourth term the volume (dilatational)acousticviscousenergyloss is usually ignored in standard combustion stability computa-tions it is discussed extensively in the acoustic literature becauseit represents a main source of decay of acoustic waves in a varietyof applications It is readily evaluated for the present case with theresult that

reg2 D1

E2m exp2regm t

Z Z

V

Z laquoplusmn2

d Ou cent rr cent OunotdV rsquo iexcl

plusmn2dfrac14k2

m

4E2m

L

R

(38)

where Eq (33) has again been used Note that this damping effectmight not be negligible when turbulence is present then the trans-port properties are modi ed and the effective viscosity coef cientmight be much larger than the laminar values over a substantialvol-ume of the chamber10 Other irrotational growth rate terms are notdisplayed here These include effects of aluminum particulates andresidual combustion76iexcl78 Because the system is linear they can besuperposed later as required

Two of the rotational terms are traditionally included with thestrictly irrotational growth rate contributions just evaluated Theseare the effects of the mean ow rotationality and the ow-turningeffect To account for the rotational mean ow one writes

reg3 D1

E2m exp2regm t

Z Z

V

ZhMb[Oucent Oupoundshy ]i dV D O

iexclM 2

b

cent(39)

as evaluatedby Culick2 in his rotationalmean ow paper and shownby him to represent a negligible correction

Before proceeding to the next term in Eq (19) the reader isreminded that Culick rst identi ed the ow-turning loss in hisone-dimensional acoustic stability analysis44779 As shown byFlandro89 this damping effect appears because by requiring theunsteady ow to enter the duct in a direction perpendicular to theburning surface Culick effectively invoked the no-slip conditionAlthough this term cannot arise in the multidimensionalirrotationalstabilityanalysisCulick insistedthat it be ldquopatchedrdquoonto the acous-tic growth results If we follow this dictum we must add

reg4 D reg ow turning DEiexcl2

m

exp2regm t

Z Z Z

V

hMb Ou cent U pound i dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Op cent U pound tankm ti dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Opm expregm t cent U pound sinkm ti dV

(40)

to the other irrotational terms This is not justi ed in reality becausethe unsteady vorticity must be used in the evaluation of Eq (40)Inclusion of this single rotational term in the energy balance to theexclusion of all of the others is a not a mathematically legitimatestep Neverthelesswe place it in Eq (19) with the irrotational termsin accordance with accepted practice In evaluating Eq (40) it isnecessary to remember that the unsteady vorticity is

D r pound Qu Dsup3

Qur

ziexcl

Quz

r

acuteemicro D iexcl

Quz

remicro C OMb (41)

The derivative of the radial velocity component with respect to z isof the orderof the mean Mach number and can thereforebe droppedUsing Eq (25) for the axial rotationalvelocity the amplitude of thevorticity vector becomes

D iexclexpregm t

micro Qur

m

rcoskm t C

Qui m

rsinkm t

para(42)

Taking the cross product with the mean ow vector yields

U pound D iexclUzer C Ur ez (43)

For longitudinal modes the pressure gradient is in the z directionso the ow-turning integral reduces to

reg4 D iexclMb

2E2m

Z Z Z

V

Ur Qui

m

rsinkm z dV (44)

Flandro8 showed that the volume integralof Eq (44) can be reducedto a surface integral that is identicalto Culickrsquos original ow-turningexpression (Secs VB and VC of Ref 8) Because the ow-turningintegral is considered to be a key damping effect it is appropriateto review its evaluation in detail for the axial mode case

First consider the derivative of the axial rotational velocity (thevorticity) appearing in Eq (44)

Qui m

rD iexcl sin x exp Aacute

cosAtilde

rsinkm z sin x C cent cent cent (45)

Only the leadingterm in this derivativeis shown becauseit is severalordersof magnitudelarger than theother terms resultingfromchain-rule differentiationof factors such as sin x or exp Aacute The reason forthis expansion becomes obvious when it is remembered that thederivativeof Atilde with respect to r is proportionalto the inverse of themean ow Mach number as shown in Eq (12) Then

Qui m

rD

km

MbUrsin x exp Aacute sinAtilde sinkm z sin x C O1 (46)

and the volume integralof Eq (44) can be evaluatedfor the assumedcylindrical chamber as

reg4 D iexcl12

km Eiexcl2m

Z 2frac14

0

dmicro

Z L=R

0

Z 1

0

r sin x exp Aacute sin Atilde

pound sinkm z sin x sinkm z dr dz (47)

To nd the exact value it is necessary to use numerical integrationcaused by the complicated radial dependenceFollowing the obser-vation made in Ref 8 (Sec VB) that the integrandoscillatesaroundzero from the chamber axis to the surface the value of the integraldependsonly on its behaviornear the upper limit Then to very goodapproximation

reg4 D iexclfrac14km

E2m

Z L=R

0

sin2km z dz

Z 1

0

r sin x exp Aacute sin Atilde dr

D iexclfrac14km

2E2m

L

R

Z 1

0

r sin x exp Aacute sinAtilde dr rsquo iexclfrac14 Mb

2E2m

L

R(48)

The nal numerical value of this damping effect is the same as inCulickrsquos original calculation if the acoustic form of the normal-ization parameter (E2

m D 12 frac14 L=R is inserted Then reg4 D iexclMb in

dimensionless form and the dimensional value becomes

regcurren4 D iexclMba0=Rsiexcl1 (49)

C Rotational Flow CorrectionConsidernow the rst rotational ow stability integral in Eq (19)

reg5 D1

E2m exp2regm t

Z Z Z

V

hiexclQu cent r Opi dV (50)

A companion term the product of the acoustic velocity with thepressuregradient term in the momentum equationhas alreadybeenevaluated and shown to give rise to the pressure coupling effectof central importance The physical interpretation of Eq (50) isapparent It represents the rate of work done on the rotational partof the unsteady ow by theoscillatorypressureforcesIn this respectit is analogous to the pressure coupling which as already showncollapses to a surface integral and is then interpreted as the p dVwork done on the incoming ow1 Obviously this term vanishes in

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

486 FLANDRO AND MAJDALANI

changes in the oscillatory ow eld Approximate analytical8iexcl16

numerical17iexcl22 and experimental investigations23iexcl26 have demon-strated that rotational ow effects play an important role in the un-steady gas motions in solid-rocketmotors Much effort has been di-rected to constructingthe requiredcorrectionsto theacousticmodelThis has culminated in a comprehensivepicture of the unsteadymo-tions that agrees with experimental measurements8iexcl10 as well asnumerical simulations11

These modelswere used in carryingout three-dimensionalsystemstability calculations89 in a rst attempt to account for rotational ow effects by correcting the acoustic instability algorithm In thisprocess one discovers the origin and the three-dimensional formof the classical ow-turning correction related terms appear thatare not accounted for in the Standard Stability Prediction Programalgorithm In particular a rotational correction term is identi edthat cancels the ow-turning energy loss in a full-lengthcylindricalgrain However all of these results must now be questionedbecausethey are founded on an incomplete representation of the systemenergy balance

Culickrsquos stability estimation procedure is based on calculatingthe exponential growth (or decay) of an irrotational acoustic wavethe results are equivalent to energy balance models used earlier byCantrell and Hart27 In all of these calculations the system energyis represented by the classical Kirchoff (acoustic) energy densityConsequentlyit does not representthe full unsteady eld includingboth acoustic and rotational ow effects Kinetic energy carried bythe vorticity waves is ignored It will be demonstrated in this paperthat the actual average unsteady energy contained in the system ata given time is about 25 larger than the acoustic energy aloneFurthermore representation of the energy sources and sinks thatdetermine the stability characteristics of the motor chamber mustalso be modi ed Attempts to correct the acousticgrowth rate modelby retention of rotational ow source terms only89 preclude a fullrepresentationof the effects of vorticity generation and coupling

In fact there is a convincing body of evidence pointing to theexistence of other aeroacoustic coupling mechanisms that are notincorporated in the current acoustic-stability theory For examplethe so-called parietal or surface vortex shedding (PVS) has beenidenti ed some years ago as a source of instability that eludes clas-sic theory28 The corresponding phenomenon was rst detected bynumerical simulations of 15 subscale models of the French ArianeV booster29iexcl32 This P230 Moteur a Propergol Solide (MPS) wasknown to exhibit large amplitudeoscillations33 This new type of in-stability was especially important in long segmented rocket motorssuch as the Japanese H-II vehicle34 the Titan 34D solid rocket mo-tor (SRM)35 the Titan IV SRMSRMU (upgrade)36iexcl39 the shuttlerocket booster (SRB)40 and other motors with length-to-portradiusratios ranging from 15 to 25

To compensate for the inability of classic theory3541iexcl48 toexplain the large pressure oscillations driven by so-called ldquocrawl-ingrdquo vortices29 a number of studies have been carried out withthe aim of improving our basic understanding of the suspectedmechanism28 Credit should be given in that regard to Vuil-lot Avalon Casalis Griffond Lupoglazoff Traineau DupaysPineau Tissier Ugurtas and coworkers who have employedexperimental2849iexcl52 numerical53iexcl59 and theoretical avenues60iexcl63

to elucidate the origin of PVS coupling In Refs 60ndash63 CasalisAvalon Pineau and Griffond have based their recent theoreticalstudy on linear instability theory introduced in 1969 by Varapaevand Yagodkin64 Their efforts have provided an alternate sourceof instability whose omission in classic analyses has led them toassociate some of the unforeseenexperimentaland numerical insta-bilities to the hydrodynamic evolution and inception of mean owturbulence6061

At the conclusion of these studies30iexcl3250iexcl52 speculations thatresonance-like pressure ampli cations were caused only by vor-tex shedding at annular restrictors or inhibitor rings were laid torest when similarly intense vorticity-generated oscillations wereobserved in unsegmented rocket motors As noted by Ugurtaset al65 two-dimensional compressible ow simulations of theNavierndashStokes equationsby Lupoglazoffand Vuillot29iexcl31 have con- rmed themeasurementsacquiredfromsubscale ringsas such the

collection of all available data has pointed out to the existence of apowerful vorticity-drivencoupling irrespectiveof whether inhibitorrings or other surface anomalies are present29iexcl31

The studies just revieweddemonstrate that currentmotor stabilityassessmentsbased on irrotational ow effectsalonecannot properlyrepresent the observed system behavior The main objective of thispaper is the construction of a more complete model for the stabil-ity of rocket motor ow elds that accounts for interactions that arelost in the irrotational (acoustic unsteady ow) assumption Stabil-ity terms must be identi ed that will establish the correct energypathways between the rotational and the irrotational parts of thegas oscillations This is best accomplished by applying the energybalance approach because this method enhances the physical un-derstanding of the many gas dynamic interactions involved in theproblem

Considerable work is needed to make these improvements Theoutcome is a stability algorithm that accounts for both acoustic andvortical ow interactionsAlthoughno attempt is made in this paperto explore all possible unsteady ow eld con gurations stabilityterms are deduced that will allow such ow effects to be includedin the stability assessmentFor example whereas turbulenceeffectsand surface vortices (as in the parietal vortex shedding mechanism)are not addresseddirectly the terms in the stabilitybalancein whichdetailedrepresentationsof theseadditionalvortical oweffectsenterthe problem will be identi ed

Signi cant differences between these new results and those cur-rently utilized for motor stability computations are demonstratedThe new model is tested by applying it to several solid-propellantrocket motor con gurations to demonstrate important scaling andgeometricaleffectsA new energy source term is identi ed that rep-resents at least one way in which unexplained instabilities in largesolid propellant booster motors arise This energy source arisingfrom productionof unsteady vorticity at the propellantburning sur-face can be comparable in size to the key combustion pressurecoupling itself It might also be related to velocity coupling ef-fectswhich cannotbe fully representedin the contextof irrotationalacoustic instability theory This same term when evaluated usingvortex shedding ow solutions yields an improved description ofthis energy source

Despite major changes in the mathematical formulation it is notnecessary to discard the current linear stability estimation method-ology required modi cations are readily accomplished by addingthe new stability integrals to those already in place in the currentstability codes It is important to note that all of the original termsidenti ed in the earlier irrotationalmodels are recoveredin the morecomplete results described in this paper

II Unsteady Flow AnalysisThe goal of this section is the constructionof an improved model

of the unsteady ow eld in a rocket chamber that realistically ac-counts for vortical (rotational) as well as acoustical (irrotational)effects In particular close attention is paid to properly satisfyingthe correct boundary conditions on all chamber boundaries includ-ing both inert and reactive surfaces

The validity of the stability calculations depends critically on asuf ciently detailedand physically correct representationof the un-steady velocity eld The assumption of an irrotational unsteady ow as used in most stability calculations must be discarded Theanalytical methods and notation employed here closely follow ear-lier papers8iexcl10 Dimensionless variables are the same as those usedin the classical combustion instability analyses velocities are nor-malized by dividing by the average sound speed and pressure isnormalized by dividing by the product of the mean chamber pres-sure and the speci c heat ratio deg Other thermodynamic propertiesare made dimensionless by dividing them by their mean chamberreference values Lengths are referenced to the chamber radius RFigure 1 illustrates the geometry and coordinate systems to be em-ployed in the analysis

The analysisstarts by assuming small-amplitudeunsteadypertur-bations on a mean ow described by the vector MbU In this paperCulickrsquos mean ow model2 for a cylindrical burning port

U D Ur er C Uzez D iexclriexcl1 sin xer C frac14 z cos xez (1)

FLANDRO AND MAJDALANI 487

Fig 1 Motor geometry and coordinate system used

will be used Note that the combination

x acute 12 frac14r 2 (2)

appears frequently in the analysisIsentropic conditions are assumed but viscous forces both shear

and dilatationalare retainedThe linearizedcontinuityand momen-tum equations then become

frac12 1

tC r cent u1 D iexclMbU cent rfrac12 1 (3)

u1

tC r p1 D Mb

copyiexclr

poundU cent u1

currenC u1 pound r pound U C U pound r pound u1

ordf

C plusmn2

raquosup32 C

cedil

sup1

acuter

poundr cent u1

curreniexcl r pound r pound u1

frac14(4)

This dimensionless set of equations is the starting point for all pre-ceding combustion instability models The superscript (1) denotesthe rst-order terms in a perturbation parameter proportional tothe amplitude of the unsteady pressure uctuation A second smallparameter the mean ow Mach number at the burning surface Mb is also employed in the linearizationprocess only terms to the rstorder in Mb are retained as in the classical combustion instabilityanalysesThe relative size of the viscous forces is set by the param-eter plusmn2 which is the inverse of the acoustic Reynolds number basedon the chamber radius R Viscous terms are usually dropped butare retained here for completeness by assuming that plusmn2 D OMb

The main variables will be represented as combinations of irro-tational and rotational components

8lt

p1 D Op C Qpfrac12 1 D Opu1 D Ou C Qu (5)

The acoustic eld is irrotational (r pound Ou D 0) and the superimposedvortical component is incompressible (r cent Qu D 0) Because of theassumptionof constant entropy the density and the thermodynamic(acoustic) pressure are interchangeableas indicated in Eq (5) Thedensity in the continuity equation is routinely replaced in the clas-sical analysis by the pressure variable The reader is cautioned that

care must be taken in interpreting the unsteady pressure p1 If itappearsin thecontinuityequationthisparameterwill then representa density uctuation This distinction is not important if the ow isirrotational but it is vitally important when rotational effects are tobe incorporated

If irrotational ow is assumed Eqs (3) and (4) yield the usualacoustic wave solutions For axial motions in a rocket chamber thesolution is the standing plane wave

Op D eiexclikt coskm z C OMb (6)

Ou D ieiexclikt sinkm zez C OMb (7)

For axial modes satisfaction of closed-end boundary conditionsrequires that

km D mfrac14 R=L (8)

As Eqs (6) and (7) indicate there exist corrections to the acoustic eld of the order of the mean Mach number These were describedin Ref 8 but are not required in the stability calculationsOnly the rst-order velocity and pressure (in Mb) are needed in evaluatingthe stability integrals

It is immediately apparent that Eq (7) cannot satisfy physicallycorrect boundary conditions either at the edge of the combustionzone or on inert side walls The axial velocity component mustapproach zero at the wall to satisfy the no-slip condition Thus theproductionof axialacousticoscillationsgives rise to correctionsthatmust come from the rotational ow effects Flandro solved Eqs (3)and (4) for the vortical (rotational incompressible) eld by twodifferent methods The inviscid case was treated in Ref 8 a morecomplete analysis was presented in Ref 9 in which viscous shearwave damping was includedAnother improvementin the latter wasthe introductionof an equivalentclosed-formsolutionin placeof thein nite seriesused in Ref 8 Additional improvementsweremade inRef 10 in which turbulent mean ow correctionswere introducedFor clarity we will display only the laminar results for the simplestmotor geometry and acoustic mode structure namely a cylindricalport with axial oscillations other geometries and transverse modesof oscillation can be handled similarly Other investigators such asMajdalani and Roh16 have veri ed the basic results to be used hereby quitedifferentanalyticalapproachesand bydirect uid dynamicscomputations11iexcl19

For rotationalpressure and velocity correctionsthat must accom-pany the assumed acoustic solution of Eqs (6) and (7) one nds

Qp D i Mbeiexclikt sin2xeAacute C iAtildeiexcl

12 frac14 z

centsin[sinxkm z] C O

iexclM 2

b

cent

(9)

Qu D iCeiexcliktrUr eAacute C iAtilde sin[sinxkm z]ez C OMb (10)

An undetermined complex constant C is shown in Eq (10) In theearlier work this constantwas evaluated by assuming that for a thincombustion layer the physical boundary at the solid surface andthe edge of the combustion zone are nearly coincident8iexcl10 Thenif one applies the no-slip condition to the composite unsteady ax-ial velocity [u1 D Ou C Qu] at the edge of the combustion zone one nds that C D 1 This result has been questionedbecause the actualsolid surface lies within a region of nonuniformity the combustionzone (Brown RS private communication Tullahoma TennesseeSept 2000) It is a straightforwardprocess to show that the originalvalue is correct the calculationis too lengthyto displayhereThis isdone by constructing a two-dimensional model for the combustionzone including all gas phase effects The axial momentum equationin the ame zone is solved numerically to account for the gradientsin temperature and density caused by combustionThe resulting in-ner solution is then matched asymptotically to the outer solutionexpressed by Eq (10) Standard methods of singular perturbationtheory apply The simplest approach is to determine solutions inan intermediate region in which both the outer and inner represen-tations are valid Then the matching is readily accomplished Theoutcome con rms that the original result C D 1 is correct unlessthe oscillationfrequencyis very high In that case the rotational ow

488 FLANDRO AND MAJDALANI

effects become entirely buried within the combustion zone and theouter solution collapses to the simple plane wave acoustic model

Flandro9 derived these expressionsdirectly from Eqs (3) and (4)without recourse to the splitting theorem applied in earlier pa-pers Critics have not understood that splitting Eqs (3) and (4)into acoustical and vortical parts (following the method of Chuand Kovasznay66 and Morse and Ingard67) is a simplifying butnot a necessary step in arriving at the solutions displayed (Brownprivate communication) Another misunderstanding centers on thepressure correction Qp sometimes called the pseudopressure It isimportant to understand that this was not simply set to zero by as-sumption in the earlier works Careful solution of the momentumequation (4) shows that it is an additional correction of the order ofthe mean Mach number the result is shown in Eq (9) Correctionsof similar size appear in the irrotational (acoustic) pressure solu-tion Eq (6) Although Qp constitutes a negligible contribution to theoverall pressure its incorporationis helpful in the acoustic stabilitycalculations for reasons that will be described later It is also neces-sary to point out that the pseudopressuremust not be inserted in thecontinuity(3) as part of the density variable as suggestedby Brownin his 2000 private communication

The complex exponential factor expAacute C iAtilde in the solution ex-hibited in Eq (10) suggests that the vortical motion can be inter-pretedas a damped travelingshearwave Both Aacute and Atilde are functionsonly of radial position The imaginary part of the exponential argu-ment namely

Atilder D iexcl[km =frac14 Mb] taniexcl

12 x

cent(11)

sets the wavelength and spatial frequency of the shear wave Forlater use note that the derivative of Atilde with respect to the radius issimply

dAtilde

drD

km

MbUr(12)

This is proportionalto the reciprocalof the radialmean ow velocityThe real part of the argument Aacuter is given by

Aacuter Draquo

frac14 2

micro1 iexcl

1

sinxiexcl x

cosx

sin2xC I x iexcl I

sup31

2frac14

acutepara

raquo acutek2

m plusmn2

M3b

DS2plusmn2

Mb S acute

km

Mb(13)

I x D x C1

18x3 C

7

1800x5 C

31

105840x7 C cent cent cent (14)

where raquo is a parameter of O1 that re ects the relative importanceof the viscous damping its physical description and veri cationare detailed by Majdalani and Roh16 Note that the damping ofthe shear wave in the radial direction increases quadratically asthe frequency of the acoustic oscillation increases Similarly thewavelength of the shear wave decreases as frequency increasesForthe fundamental acousticmode and the lowest order harmonics theshear wave might ll the entire chamber for high-order acousticmodes the shear oscillations become con ned to a thin acousticboundarylayerwhich can lie entirelywithin the combustionzone13

The composite axial velocity solution found by superimposingEqs (7) and (10) has been shown8iexcl1012 to agree quite closely withthe experimental ndingsofRefs 12 25 26 and 68 Figures2 and 3showcomparisonsof theoscillationamplitudeand phaseangle fromthe theory just discussed superimposed on some of Brownrsquos earliercold ow work2469 It is important to understandthat no curve ttinghas been used here the differences between the experimental andtheoretical results arise both from unavoidable experimental errors(orientation and positioning of the hot lm sensor data reductionetc) and from the assumptionset used in the analysisNevertheless

Fig 2 Comparison of theoretical and experimental amplitudes ofunsteady axial velocities Here S = 513m zcurrenL = 0106 Mb = 00018LR = 34 km = 00924m raquo = 0539m2 and m = 1 Triangles andcircles cor-respond to experimental data acquired with a pressure wave amplitudeof a) 00005 and b) 00039

Fig 3 Comparison of theoretical and experimental phase angles(degrees)

adequate agreement is shown Because nitrogen was the working uid at measured temperature and pressure all parameters charac-terizing the ow eld are well known

Numerical solutions based on the full NavierndashStokes representa-tion of the unsteady eld also indicate excellentagreementwith thetheory reviewed here Furthermore other analyticalmethods basedon Wentzel Kramers and Brillouin (WKB) and multiple-scalethe-ories have been applied by Majdalani and Van Moorhem70 Theirshavedemonstratedbasedon theprincipleof least singularitytheex-istence of nonlinear scales underlying the physics of the problem71

In additionto providinga rigorousmathematical treatment that doesnot require guesswork their approach has led to the exact form ofthe characteristic length scales as function of Culickrsquos mean owsolution Particularly their solutions have yielded results that arevirtually indistinguishablefrom those displayed here11iexcl16

In previousanalyses of the rocket motor rotational ow eld spe-cial attentionwas givento the radialcomponentof velocity8911 Thiswas done because in classical instability theory the main source ofacoustic energy is the radial velocity uctuation produced by inter-action of the pressurewave with the combustion processes and heattransfer in the burning zone It was shown in Refs 8 and 9 that inaddition to the radial acoustic velocity there appears a rotationalcorrection that does not vanish at the edge of the combustion zoneThe presence of this additional unsteady radial velocity componentimplies that by analogy to pressure coupling there is an additionalsource of unsteady mass ux at the combustion zone interface Forwant of better terminology this was called the rotational ow cor-rection as rst displayed in Eq (89) of Ref 8 The new term whichdoes not appear in any previous analysis has given rise to consid-erable controversy

By way of providing a precise description of the source of therotational ow correction a detailed physical interpretation and afull resolution of the controversywill appear in the next section ofthispaperThe answers are foundby means of a carefulderivationofthe system stability energy balance It is not necessary to calculatethe vortical radial velocity correction in determining the systemstability The surface acoustic admittance function will be utilized

FLANDRO AND MAJDALANI 489

in the standard fashion to represent the pressure coupling of theoscillatory ow with the combustion processesThen experimentalmeasurements (usuallysecuredwith a T-burneror similar device)ofthe surface response can be utilized as in the acceptedmethodologyfor accounting for this important source of oscillatory energy

Earlier attempts to incorporate vortical ow corrections in thestability calculations were based on the idea that one must re-store rotational source terms which were dropped in the classicalanalyses For example Brown et al24 had suggested that reten-tion of the rotational convective acceleration term U pound r pound u1

on the right side of Eq (4) might have important stability im-plications This term was rst evaluated by Flandro89 whoshowed that this term yields a damping effect that is indistin-guishable from the classical ow-turning loss Additional termsthat cannot be found in the standard stability computations ap-pear in this analysis One of these the rotational ow correc-tion has led to much debate because it exactly cancels the ow-turning effect in some chamber con gurations Brownrsquos as-sertion that the rotational ow correction is just another wayto represent ow turning can be quite readily refuted In fact allmisunderstandings are resolved when a consistent energy methodis used to deduce motor stability characteristics The aw in theearlier approach is that it represented the system by a model thataccounts for growth or decay of the acoustic energy alone Simplyretaining the rotational convective source terms24 in the perturbedwave equation following Culickrsquos method does not address thisimportant defect

To describe the evolution of the oscillatory ow in a rocket mo-tor correctly all unsteady energy must be accounted for rst Themethod used here follows the known energy balance approach de-scribed by Kirchoff72 and used extensively in rocket stability calcu-lationsby Cantrell and Hart27 and Hart and coworkers41iexcl43 It is nownecessary to account for the entire kinetic energy uctuationTo ac-complish this multiply the continuity equation (3) by the acousticpressure (or density) and add this to the momentum equation (4)multiplied by the composite unsteady velocity vector

u1 D Ou C Qu (15)

This operation constitutes the main departure from the precedingapproach Isolating the terms differentiatedwith respect to time andretaining only those terms that are linear in the Mach number Mb one nds

t

micro12

Op2 C12

u1 cent u1

paraDiexcl

poundOpr cent Ou C u1 cent r Op C Qp

curren

iexcl Mb

raquo1

2U centr Op2 C u1 centr

poundU cent u1

curren

iexclu1 centpoundu1 pound r pound U C U pound r pound Qu

currenfrac14

C plusmn2u1 centmicrosup3

2 Ccedil

sup1

acuterr cent Ou iexcl r pound r pound Qu

para(16)

If the full unsteady velocity vector u1 were to be replaced at thisstage by the acoustic part Ou then the nal result of the stability cal-culation would be exactly that described in earlier papers89 If allrotational terms on the right of Eq (16) were also to be removedby assuming that Qu D 0 then the classical three-dimensional sta-bility result would be reproduced Assumptions such as those justdescribed are motivated mainly by a desire to keep the analysissimple However as we shall now demonstrate much physical sub-stance is lost in such simpli cations We choose not to follow thetraditional approach heremdashmost rotational effects are retained asEq (16) shows the kinetic energy per unit mass on the left-handside re ects all laminar unsteady motions in the chamber Parietal-vortex sheddingand turbulentcorrectionswill require specialatten-tion and future re nementsdiscussionsof turbulent ow effects arefound in the numerical studies by Flandro et al10 Roh et al18 Yangand Roh19 Apte and Yang73 Beddini and Roberts7475 and Vuillotand coworkers5758 So that the analysis is not overly complicated

effectsof solidparticlesin the ow are notdisplayedeither requiredmodi cations are to be incorporated later as needed

Following the standard approach72 one de nes the oscillatoryenergy density

e D 12

poundOp2 C u1 cent u1

curren(17)

and the time-averagedoscillatoryenergy residing in the chamber atany instant as

E DZ Z

V

Zhei dV D

1

2

Z Z

V

Z laquoOp2 C u1 cent u1

notdV (18)

where triangular brackets denote the time average of the enclosedfunction Then from Eq (16) it is clear that the evolution of thesystem energy is controlled by

E

tD

Z Z

V

Zirrotationalz | frac12

iexclr cent Op Ou iexcl12

MbU cent r Op2 iexcl Mb[Ou cent rU cent Ou]

C plusmn2d Ou cent rr cent Ou C Mb[ Ou cent Ou pound shy C Ou cent U pound ]

rotationalz |

iexclQu cent r Op iexcl Mb

microQu cent rU cent Ou C Ou cent rU cent Qu C Qu cent rU cent Qu

iexclQu cent U pound iexcl Qu cent Qu pound shy

para

C plusmn2d Qu cent rr cent Ou iexcl plusmn2[Ou cent r pound C Qu cent r pound ]

iexcl Ou cent r Qp iexcl Qu cent r Qpfrac34

dV (19)

where the irrotational and rotational contributions to the energyrate of change have been partitioned for clarity The last twoldquoirrotationalrdquo terms are in reality caused by rotational effectsThey are placed with the irrotational terms to conform to Culickrsquosmethodology247 The last two ldquorotationalrdquo terms are caused by thepseudopressure a quantity that has often been ignored in previousstudies The mean and unsteady vorticity vectors are represented inEq (19) by shy D r pound U and D r pound Qu respectively

The dilatationalviscous term is simpli ed by de ning

plusmn2d acute plusmn22 C cedil=sup1 (20)

where cedil is the second coef cient of viscosity Several terms cancelFor example

iexclOu cent Qu pound shy iexcl Qu cent Ou pound shy D 0 (21)

Further simpli cations can be made For example several termscan be expressed as integrals over the surface bounding the con-trol volume It is rst necessary to review the protocol needed inevaluating the stability integrals

The terms on the right-hand side of Eq (19) control the rate atwhich the systemenergy changesFrom this informationone can es-timate the growth or decay rate for a given motor con gurationThecomplex wave number (or frequency)k in the assumed exponentialtime dependence is written as

k D km C m C iregm C OiexclM2

b

cent(22)

In the multidimensionalcase m can consist of three mode integersWe will restrict the evaluation to axial modes for clarity then a sin-gle integer m identi es the mode being investigatedMore complexmodes can be handled in a similar way The result for a simple planewave axial mode is shown in Eq (8) The dimensionlesscorrectionsm C iregm are of the order of the mean ow Mach number In theconventionalenergy-balanceprocedurethe frequencycorrectionm

is not evaluated For simplicity we follow this approach The fre-quencycorrectionis implicitlyaccountedfor in thewavenumberkm

490 FLANDRO AND MAJDALANI

Because quadraticcombinationsof the variablesappear it is rstnecessary to take the real parts of the pressure and velocities to beinserted in the equationsThus one writes

Op D Opm expregm t coskm t (23)

Ou D Oum expregm t sinkm t (24)

Qu D expregm tpound

Qurm coskm t C Qui

m sinkm tcurren

(25)

where superscripts (r) and (i ) refer to the cosine and sine multipli-ers stemming from complex-to-real conversion of real and imagi-nary partsThis notationis necessarybecausethe rotationalvelocityvector contains the spatial exponential term exp(Aacute C iAtilde ) Insertingthese expressions into Eq (17) yields the time-averaged energydensity

hei D 14 exp2regm t

irrotationalz | poundOp2

m C Oum cent Oum

C

rotationalz | 2Oum cent Qui

m C Qur m cent Qur

m C Quim cent Qui

m

curren(26)

which has been partitioned to emphasize the new terms This re-sult should be compared to the classical expression for the energydensity which consists of only the rst two terms in Eq (26) Theadditional three terms represent the kinetic energy residing in theunsteady vorticity Inserting these expressions into Eq (18) carry-ing out the time averaging and differentiatingwith respect to timeone nds

E

tD regm exp2regm tE2

m (27)

where the energy normalization function

E2m D

12

Z Z

V

Z pound Op0

m2 C Oum cent Oum C 2 Oum cent Quim

C Qurm cent Qur

m C Qui m cent Qui

m

currendV (28)

is expressed here by the same symbol used in earlier stability anal-yses It is written as a squared quantity to indicate that it is positivede nite

It is interestingto compare values from Eq (28) to those found inearlier computationsThe mode shapes are needed for this purposeEquations (6) (7) and (10) give the required information for theassumed axial oscillations to zeroth order in Mb

Opm D coskm z (29)

Oum D sinkm zez (30)

Qurm D sinx expAacute sinAtilde sin[sinxkm z]ez (31)

Quim D iexclsinx expAacute cosAtilde sin[sinxkm z]ez (32)

The acoustic pressure and velocity mode shapes are related by

Oum D iexclr Opm =km (33)

for later use Figure 4 shows the results of integration of the kineticenergy terms in Eq (28) across the chamber from the centerline tothe wall at a given axial location Note that the term involving thecross product of the acoustic and vortical parts does not contributeto the integrated result because it oscillates in the radial directionand is zero at the upper limit However the rotational correctionshows a net value which is half of the integrated acoustic energydensityClearly the rotationaleffects representa signi cant changein the system energy

When the entire volume integral of Eq (28) is carried out for theaxial mode case using Eqs (29ndash32) one nds that

E2m D 5

8 frac14 L=R (34)

Fig 4 Radial integration of the kinetic energy density components

which is 25 larger than the conventional acoustic value for lon-gitudinal oscillations However the most important changes in thegrowth rate regm appear in the integral terms on the right-handside ofEq (19) We must now attend to this important part of the stabilityanalysis

A Growth Rate CalculationsIn estimating the system growth rate for a given mode of oscil-

lation it can be seen that it consists of the linear superposition of15 volume integrals as shown in Eq (19) The simplest procedure isto consider each of these (or selected groups) to be evaluated indi-vidually One can then represent the net growth rate as a linear sumof gains and losses as

regm D reg1 C reg2 C reg3 C cent cent cent DNX

i D 1

regi (35)

in the usual fashion The dimensionless values of the several termsare given in the following paragraphs The reader is reminded thatin order to compute the corresponding dimensional values it isnecessary to multiply by a0=R the inverse of the characteristictimeused in the formulation Then the growth rates are obtained in thefamiliar units of rads (siexcl1)

B Irrotational Growth Rate ContributionsTo illustrate the computationalprocedureconsider the rst three

irrotational terms

reg1 D1

exp2regm tE2m

Z Z

V

Z frac12iexclr cent

microOp Ou C

1

2MbU Op2

para

iexcl Mb[ Ou cent rU cent Ou]

frac34dV (36)

where the rst two terms have been combined by using the vectoridentity for the divergence of the product of a scalar and a vectorand by noting that r cent U D 0 for continuity of the incompressiblemean ow Thus the rst term can be converted to a surface integralusing Gaussrsquos divergence theorem The second can be written interms of the pressure by means of Eq (33) after some algebra italso reduces to a surface integral Assuming a short quasi-steadynozzle one nds

reg1 DMb

2E2m

8lt

Z Z

Sb

poundAr

b C 1curren

Op2m dS iexcl

Z Z

SN

poundAr

N C Uz

currenOp2

m dS

9=

rsquofrac14 Mb

2E2m

L

R

copypoundAr

b C 1curren

iexcl [deg C 1]ordf

(37)

The rst term in Eq (37) is of course the classical pressure cou-pling effect except that it is now somewhat reduced in size becauseof the larger energy normalization integral The entire surface in-tegral has not been represented in Eq (37) Nonburning chambersurfaces can be accommodated in the surface integrals if required

FLANDRO AND MAJDALANI 491

The several admittance functionsare taken to be quantitiesof O1the mean ow Mach number has been factored out as in most tra-ditional computations The admittance for nonburning surfaces As

is usually neglected although rough chamber walls and insulationlayers can display substantial (usually negative) admittance

The fourth term the volume (dilatational)acousticviscousenergyloss is usually ignored in standard combustion stability computa-tions it is discussed extensively in the acoustic literature becauseit represents a main source of decay of acoustic waves in a varietyof applications It is readily evaluated for the present case with theresult that

reg2 D1

E2m exp2regm t

Z Z

V

Z laquoplusmn2

d Ou cent rr cent OunotdV rsquo iexcl

plusmn2dfrac14k2

m

4E2m

L

R

(38)

where Eq (33) has again been used Note that this damping effectmight not be negligible when turbulence is present then the trans-port properties are modi ed and the effective viscosity coef cientmight be much larger than the laminar values over a substantialvol-ume of the chamber10 Other irrotational growth rate terms are notdisplayed here These include effects of aluminum particulates andresidual combustion76iexcl78 Because the system is linear they can besuperposed later as required

Two of the rotational terms are traditionally included with thestrictly irrotational growth rate contributions just evaluated Theseare the effects of the mean ow rotationality and the ow-turningeffect To account for the rotational mean ow one writes

reg3 D1

E2m exp2regm t

Z Z

V

ZhMb[Oucent Oupoundshy ]i dV D O

iexclM 2

b

cent(39)

as evaluatedby Culick2 in his rotationalmean ow paper and shownby him to represent a negligible correction

Before proceeding to the next term in Eq (19) the reader isreminded that Culick rst identi ed the ow-turning loss in hisone-dimensional acoustic stability analysis44779 As shown byFlandro89 this damping effect appears because by requiring theunsteady ow to enter the duct in a direction perpendicular to theburning surface Culick effectively invoked the no-slip conditionAlthough this term cannot arise in the multidimensionalirrotationalstabilityanalysisCulick insistedthat it be ldquopatchedrdquoonto the acous-tic growth results If we follow this dictum we must add

reg4 D reg ow turning DEiexcl2

m

exp2regm t

Z Z Z

V

hMb Ou cent U pound i dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Op cent U pound tankm ti dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Opm expregm t cent U pound sinkm ti dV

(40)

to the other irrotational terms This is not justi ed in reality becausethe unsteady vorticity must be used in the evaluation of Eq (40)Inclusion of this single rotational term in the energy balance to theexclusion of all of the others is a not a mathematically legitimatestep Neverthelesswe place it in Eq (19) with the irrotational termsin accordance with accepted practice In evaluating Eq (40) it isnecessary to remember that the unsteady vorticity is

D r pound Qu Dsup3

Qur

ziexcl

Quz

r

acuteemicro D iexcl

Quz

remicro C OMb (41)

The derivative of the radial velocity component with respect to z isof the orderof the mean Mach number and can thereforebe droppedUsing Eq (25) for the axial rotationalvelocity the amplitude of thevorticity vector becomes

D iexclexpregm t

micro Qur

m

rcoskm t C

Qui m

rsinkm t

para(42)

Taking the cross product with the mean ow vector yields

U pound D iexclUzer C Ur ez (43)

For longitudinal modes the pressure gradient is in the z directionso the ow-turning integral reduces to

reg4 D iexclMb

2E2m

Z Z Z

V

Ur Qui

m

rsinkm z dV (44)

Flandro8 showed that the volume integralof Eq (44) can be reducedto a surface integral that is identicalto Culickrsquos original ow-turningexpression (Secs VB and VC of Ref 8) Because the ow-turningintegral is considered to be a key damping effect it is appropriateto review its evaluation in detail for the axial mode case

First consider the derivative of the axial rotational velocity (thevorticity) appearing in Eq (44)

Qui m

rD iexcl sin x exp Aacute

cosAtilde

rsinkm z sin x C cent cent cent (45)

Only the leadingterm in this derivativeis shown becauseit is severalordersof magnitudelarger than theother terms resultingfromchain-rule differentiationof factors such as sin x or exp Aacute The reason forthis expansion becomes obvious when it is remembered that thederivativeof Atilde with respect to r is proportionalto the inverse of themean ow Mach number as shown in Eq (12) Then

Qui m

rD

km

MbUrsin x exp Aacute sinAtilde sinkm z sin x C O1 (46)

and the volume integralof Eq (44) can be evaluatedfor the assumedcylindrical chamber as

reg4 D iexcl12

km Eiexcl2m

Z 2frac14

0

dmicro

Z L=R

0

Z 1

0

r sin x exp Aacute sin Atilde

pound sinkm z sin x sinkm z dr dz (47)

To nd the exact value it is necessary to use numerical integrationcaused by the complicated radial dependenceFollowing the obser-vation made in Ref 8 (Sec VB) that the integrandoscillatesaroundzero from the chamber axis to the surface the value of the integraldependsonly on its behaviornear the upper limit Then to very goodapproximation

reg4 D iexclfrac14km

E2m

Z L=R

0

sin2km z dz

Z 1

0

r sin x exp Aacute sin Atilde dr

D iexclfrac14km

2E2m

L

R

Z 1

0

r sin x exp Aacute sinAtilde dr rsquo iexclfrac14 Mb

2E2m

L

R(48)

The nal numerical value of this damping effect is the same as inCulickrsquos original calculation if the acoustic form of the normal-ization parameter (E2

m D 12 frac14 L=R is inserted Then reg4 D iexclMb in

dimensionless form and the dimensional value becomes

regcurren4 D iexclMba0=Rsiexcl1 (49)

C Rotational Flow CorrectionConsidernow the rst rotational ow stability integral in Eq (19)

reg5 D1

E2m exp2regm t

Z Z Z

V

hiexclQu cent r Opi dV (50)

A companion term the product of the acoustic velocity with thepressuregradient term in the momentum equationhas alreadybeenevaluated and shown to give rise to the pressure coupling effectof central importance The physical interpretation of Eq (50) isapparent It represents the rate of work done on the rotational partof the unsteady ow by theoscillatorypressureforcesIn this respectit is analogous to the pressure coupling which as already showncollapses to a surface integral and is then interpreted as the p dVwork done on the incoming ow1 Obviously this term vanishes in

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

FLANDRO AND MAJDALANI 487

Fig 1 Motor geometry and coordinate system used

will be used Note that the combination

x acute 12 frac14r 2 (2)

appears frequently in the analysisIsentropic conditions are assumed but viscous forces both shear

and dilatationalare retainedThe linearizedcontinuityand momen-tum equations then become

frac12 1

tC r cent u1 D iexclMbU cent rfrac12 1 (3)

u1

tC r p1 D Mb

copyiexclr

poundU cent u1

currenC u1 pound r pound U C U pound r pound u1

ordf

C plusmn2

raquosup32 C

cedil

sup1

acuter

poundr cent u1

curreniexcl r pound r pound u1

frac14(4)

This dimensionless set of equations is the starting point for all pre-ceding combustion instability models The superscript (1) denotesthe rst-order terms in a perturbation parameter proportional tothe amplitude of the unsteady pressure uctuation A second smallparameter the mean ow Mach number at the burning surface Mb is also employed in the linearizationprocess only terms to the rstorder in Mb are retained as in the classical combustion instabilityanalysesThe relative size of the viscous forces is set by the param-eter plusmn2 which is the inverse of the acoustic Reynolds number basedon the chamber radius R Viscous terms are usually dropped butare retained here for completeness by assuming that plusmn2 D OMb

The main variables will be represented as combinations of irro-tational and rotational components

8lt

p1 D Op C Qpfrac12 1 D Opu1 D Ou C Qu (5)

The acoustic eld is irrotational (r pound Ou D 0) and the superimposedvortical component is incompressible (r cent Qu D 0) Because of theassumptionof constant entropy the density and the thermodynamic(acoustic) pressure are interchangeableas indicated in Eq (5) Thedensity in the continuity equation is routinely replaced in the clas-sical analysis by the pressure variable The reader is cautioned that

care must be taken in interpreting the unsteady pressure p1 If itappearsin thecontinuityequationthisparameterwill then representa density uctuation This distinction is not important if the ow isirrotational but it is vitally important when rotational effects are tobe incorporated

If irrotational ow is assumed Eqs (3) and (4) yield the usualacoustic wave solutions For axial motions in a rocket chamber thesolution is the standing plane wave

Op D eiexclikt coskm z C OMb (6)

Ou D ieiexclikt sinkm zez C OMb (7)

For axial modes satisfaction of closed-end boundary conditionsrequires that

km D mfrac14 R=L (8)

As Eqs (6) and (7) indicate there exist corrections to the acoustic eld of the order of the mean Mach number These were describedin Ref 8 but are not required in the stability calculationsOnly the rst-order velocity and pressure (in Mb) are needed in evaluatingthe stability integrals

It is immediately apparent that Eq (7) cannot satisfy physicallycorrect boundary conditions either at the edge of the combustionzone or on inert side walls The axial velocity component mustapproach zero at the wall to satisfy the no-slip condition Thus theproductionof axialacousticoscillationsgives rise to correctionsthatmust come from the rotational ow effects Flandro solved Eqs (3)and (4) for the vortical (rotational incompressible) eld by twodifferent methods The inviscid case was treated in Ref 8 a morecomplete analysis was presented in Ref 9 in which viscous shearwave damping was includedAnother improvementin the latter wasthe introductionof an equivalentclosed-formsolutionin placeof thein nite seriesused in Ref 8 Additional improvementsweremade inRef 10 in which turbulent mean ow correctionswere introducedFor clarity we will display only the laminar results for the simplestmotor geometry and acoustic mode structure namely a cylindricalport with axial oscillations other geometries and transverse modesof oscillation can be handled similarly Other investigators such asMajdalani and Roh16 have veri ed the basic results to be used hereby quitedifferentanalyticalapproachesand bydirect uid dynamicscomputations11iexcl19

For rotationalpressure and velocity correctionsthat must accom-pany the assumed acoustic solution of Eqs (6) and (7) one nds

Qp D i Mbeiexclikt sin2xeAacute C iAtildeiexcl

12 frac14 z

centsin[sinxkm z] C O

iexclM 2

b

cent

(9)

Qu D iCeiexcliktrUr eAacute C iAtilde sin[sinxkm z]ez C OMb (10)

An undetermined complex constant C is shown in Eq (10) In theearlier work this constantwas evaluated by assuming that for a thincombustion layer the physical boundary at the solid surface andthe edge of the combustion zone are nearly coincident8iexcl10 Thenif one applies the no-slip condition to the composite unsteady ax-ial velocity [u1 D Ou C Qu] at the edge of the combustion zone one nds that C D 1 This result has been questionedbecause the actualsolid surface lies within a region of nonuniformity the combustionzone (Brown RS private communication Tullahoma TennesseeSept 2000) It is a straightforwardprocess to show that the originalvalue is correct the calculationis too lengthyto displayhereThis isdone by constructing a two-dimensional model for the combustionzone including all gas phase effects The axial momentum equationin the ame zone is solved numerically to account for the gradientsin temperature and density caused by combustionThe resulting in-ner solution is then matched asymptotically to the outer solutionexpressed by Eq (10) Standard methods of singular perturbationtheory apply The simplest approach is to determine solutions inan intermediate region in which both the outer and inner represen-tations are valid Then the matching is readily accomplished Theoutcome con rms that the original result C D 1 is correct unlessthe oscillationfrequencyis very high In that case the rotational ow

488 FLANDRO AND MAJDALANI

effects become entirely buried within the combustion zone and theouter solution collapses to the simple plane wave acoustic model

Flandro9 derived these expressionsdirectly from Eqs (3) and (4)without recourse to the splitting theorem applied in earlier pa-pers Critics have not understood that splitting Eqs (3) and (4)into acoustical and vortical parts (following the method of Chuand Kovasznay66 and Morse and Ingard67) is a simplifying butnot a necessary step in arriving at the solutions displayed (Brownprivate communication) Another misunderstanding centers on thepressure correction Qp sometimes called the pseudopressure It isimportant to understand that this was not simply set to zero by as-sumption in the earlier works Careful solution of the momentumequation (4) shows that it is an additional correction of the order ofthe mean Mach number the result is shown in Eq (9) Correctionsof similar size appear in the irrotational (acoustic) pressure solu-tion Eq (6) Although Qp constitutes a negligible contribution to theoverall pressure its incorporationis helpful in the acoustic stabilitycalculations for reasons that will be described later It is also neces-sary to point out that the pseudopressuremust not be inserted in thecontinuity(3) as part of the density variable as suggestedby Brownin his 2000 private communication

The complex exponential factor expAacute C iAtilde in the solution ex-hibited in Eq (10) suggests that the vortical motion can be inter-pretedas a damped travelingshearwave Both Aacute and Atilde are functionsonly of radial position The imaginary part of the exponential argu-ment namely

Atilder D iexcl[km =frac14 Mb] taniexcl

12 x

cent(11)

sets the wavelength and spatial frequency of the shear wave Forlater use note that the derivative of Atilde with respect to the radius issimply

dAtilde

drD

km

MbUr(12)

This is proportionalto the reciprocalof the radialmean ow velocityThe real part of the argument Aacuter is given by

Aacuter Draquo

frac14 2

micro1 iexcl

1

sinxiexcl x

cosx

sin2xC I x iexcl I

sup31

2frac14

acutepara

raquo acutek2

m plusmn2

M3b

DS2plusmn2

Mb S acute

km

Mb(13)

I x D x C1

18x3 C

7

1800x5 C

31

105840x7 C cent cent cent (14)

where raquo is a parameter of O1 that re ects the relative importanceof the viscous damping its physical description and veri cationare detailed by Majdalani and Roh16 Note that the damping ofthe shear wave in the radial direction increases quadratically asthe frequency of the acoustic oscillation increases Similarly thewavelength of the shear wave decreases as frequency increasesForthe fundamental acousticmode and the lowest order harmonics theshear wave might ll the entire chamber for high-order acousticmodes the shear oscillations become con ned to a thin acousticboundarylayerwhich can lie entirelywithin the combustionzone13

The composite axial velocity solution found by superimposingEqs (7) and (10) has been shown8iexcl1012 to agree quite closely withthe experimental ndingsofRefs 12 25 26 and 68 Figures2 and 3showcomparisonsof theoscillationamplitudeand phaseangle fromthe theory just discussed superimposed on some of Brownrsquos earliercold ow work2469 It is important to understandthat no curve ttinghas been used here the differences between the experimental andtheoretical results arise both from unavoidable experimental errors(orientation and positioning of the hot lm sensor data reductionetc) and from the assumptionset used in the analysisNevertheless

Fig 2 Comparison of theoretical and experimental amplitudes ofunsteady axial velocities Here S = 513m zcurrenL = 0106 Mb = 00018LR = 34 km = 00924m raquo = 0539m2 and m = 1 Triangles andcircles cor-respond to experimental data acquired with a pressure wave amplitudeof a) 00005 and b) 00039

Fig 3 Comparison of theoretical and experimental phase angles(degrees)

adequate agreement is shown Because nitrogen was the working uid at measured temperature and pressure all parameters charac-terizing the ow eld are well known

Numerical solutions based on the full NavierndashStokes representa-tion of the unsteady eld also indicate excellentagreementwith thetheory reviewed here Furthermore other analyticalmethods basedon Wentzel Kramers and Brillouin (WKB) and multiple-scalethe-ories have been applied by Majdalani and Van Moorhem70 Theirshavedemonstratedbasedon theprincipleof least singularitytheex-istence of nonlinear scales underlying the physics of the problem71

In additionto providinga rigorousmathematical treatment that doesnot require guesswork their approach has led to the exact form ofthe characteristic length scales as function of Culickrsquos mean owsolution Particularly their solutions have yielded results that arevirtually indistinguishablefrom those displayed here11iexcl16

In previousanalyses of the rocket motor rotational ow eld spe-cial attentionwas givento the radialcomponentof velocity8911 Thiswas done because in classical instability theory the main source ofacoustic energy is the radial velocity uctuation produced by inter-action of the pressurewave with the combustion processes and heattransfer in the burning zone It was shown in Refs 8 and 9 that inaddition to the radial acoustic velocity there appears a rotationalcorrection that does not vanish at the edge of the combustion zoneThe presence of this additional unsteady radial velocity componentimplies that by analogy to pressure coupling there is an additionalsource of unsteady mass ux at the combustion zone interface Forwant of better terminology this was called the rotational ow cor-rection as rst displayed in Eq (89) of Ref 8 The new term whichdoes not appear in any previous analysis has given rise to consid-erable controversy

By way of providing a precise description of the source of therotational ow correction a detailed physical interpretation and afull resolution of the controversywill appear in the next section ofthispaperThe answers are foundby means of a carefulderivationofthe system stability energy balance It is not necessary to calculatethe vortical radial velocity correction in determining the systemstability The surface acoustic admittance function will be utilized

FLANDRO AND MAJDALANI 489

in the standard fashion to represent the pressure coupling of theoscillatory ow with the combustion processesThen experimentalmeasurements (usuallysecuredwith a T-burneror similar device)ofthe surface response can be utilized as in the acceptedmethodologyfor accounting for this important source of oscillatory energy

Earlier attempts to incorporate vortical ow corrections in thestability calculations were based on the idea that one must re-store rotational source terms which were dropped in the classicalanalyses For example Brown et al24 had suggested that reten-tion of the rotational convective acceleration term U pound r pound u1

on the right side of Eq (4) might have important stability im-plications This term was rst evaluated by Flandro89 whoshowed that this term yields a damping effect that is indistin-guishable from the classical ow-turning loss Additional termsthat cannot be found in the standard stability computations ap-pear in this analysis One of these the rotational ow correc-tion has led to much debate because it exactly cancels the ow-turning effect in some chamber con gurations Brownrsquos as-sertion that the rotational ow correction is just another wayto represent ow turning can be quite readily refuted In fact allmisunderstandings are resolved when a consistent energy methodis used to deduce motor stability characteristics The aw in theearlier approach is that it represented the system by a model thataccounts for growth or decay of the acoustic energy alone Simplyretaining the rotational convective source terms24 in the perturbedwave equation following Culickrsquos method does not address thisimportant defect

To describe the evolution of the oscillatory ow in a rocket mo-tor correctly all unsteady energy must be accounted for rst Themethod used here follows the known energy balance approach de-scribed by Kirchoff72 and used extensively in rocket stability calcu-lationsby Cantrell and Hart27 and Hart and coworkers41iexcl43 It is nownecessary to account for the entire kinetic energy uctuationTo ac-complish this multiply the continuity equation (3) by the acousticpressure (or density) and add this to the momentum equation (4)multiplied by the composite unsteady velocity vector

u1 D Ou C Qu (15)

This operation constitutes the main departure from the precedingapproach Isolating the terms differentiatedwith respect to time andretaining only those terms that are linear in the Mach number Mb one nds

t

micro12

Op2 C12

u1 cent u1

paraDiexcl

poundOpr cent Ou C u1 cent r Op C Qp

curren

iexcl Mb

raquo1

2U centr Op2 C u1 centr

poundU cent u1

curren

iexclu1 centpoundu1 pound r pound U C U pound r pound Qu

currenfrac14

C plusmn2u1 centmicrosup3

2 Ccedil

sup1

acuterr cent Ou iexcl r pound r pound Qu

para(16)

If the full unsteady velocity vector u1 were to be replaced at thisstage by the acoustic part Ou then the nal result of the stability cal-culation would be exactly that described in earlier papers89 If allrotational terms on the right of Eq (16) were also to be removedby assuming that Qu D 0 then the classical three-dimensional sta-bility result would be reproduced Assumptions such as those justdescribed are motivated mainly by a desire to keep the analysissimple However as we shall now demonstrate much physical sub-stance is lost in such simpli cations We choose not to follow thetraditional approach heremdashmost rotational effects are retained asEq (16) shows the kinetic energy per unit mass on the left-handside re ects all laminar unsteady motions in the chamber Parietal-vortex sheddingand turbulentcorrectionswill require specialatten-tion and future re nementsdiscussionsof turbulent ow effects arefound in the numerical studies by Flandro et al10 Roh et al18 Yangand Roh19 Apte and Yang73 Beddini and Roberts7475 and Vuillotand coworkers5758 So that the analysis is not overly complicated

effectsof solidparticlesin the ow are notdisplayedeither requiredmodi cations are to be incorporated later as needed

Following the standard approach72 one de nes the oscillatoryenergy density

e D 12

poundOp2 C u1 cent u1

curren(17)

and the time-averagedoscillatoryenergy residing in the chamber atany instant as

E DZ Z

V

Zhei dV D

1

2

Z Z

V

Z laquoOp2 C u1 cent u1

notdV (18)

where triangular brackets denote the time average of the enclosedfunction Then from Eq (16) it is clear that the evolution of thesystem energy is controlled by

E

tD

Z Z

V

Zirrotationalz | frac12

iexclr cent Op Ou iexcl12

MbU cent r Op2 iexcl Mb[Ou cent rU cent Ou]

C plusmn2d Ou cent rr cent Ou C Mb[ Ou cent Ou pound shy C Ou cent U pound ]

rotationalz |

iexclQu cent r Op iexcl Mb

microQu cent rU cent Ou C Ou cent rU cent Qu C Qu cent rU cent Qu

iexclQu cent U pound iexcl Qu cent Qu pound shy

para

C plusmn2d Qu cent rr cent Ou iexcl plusmn2[Ou cent r pound C Qu cent r pound ]

iexcl Ou cent r Qp iexcl Qu cent r Qpfrac34

dV (19)

where the irrotational and rotational contributions to the energyrate of change have been partitioned for clarity The last twoldquoirrotationalrdquo terms are in reality caused by rotational effectsThey are placed with the irrotational terms to conform to Culickrsquosmethodology247 The last two ldquorotationalrdquo terms are caused by thepseudopressure a quantity that has often been ignored in previousstudies The mean and unsteady vorticity vectors are represented inEq (19) by shy D r pound U and D r pound Qu respectively

The dilatationalviscous term is simpli ed by de ning

plusmn2d acute plusmn22 C cedil=sup1 (20)

where cedil is the second coef cient of viscosity Several terms cancelFor example

iexclOu cent Qu pound shy iexcl Qu cent Ou pound shy D 0 (21)

Further simpli cations can be made For example several termscan be expressed as integrals over the surface bounding the con-trol volume It is rst necessary to review the protocol needed inevaluating the stability integrals

The terms on the right-hand side of Eq (19) control the rate atwhich the systemenergy changesFrom this informationone can es-timate the growth or decay rate for a given motor con gurationThecomplex wave number (or frequency)k in the assumed exponentialtime dependence is written as

k D km C m C iregm C OiexclM2

b

cent(22)

In the multidimensionalcase m can consist of three mode integersWe will restrict the evaluation to axial modes for clarity then a sin-gle integer m identi es the mode being investigatedMore complexmodes can be handled in a similar way The result for a simple planewave axial mode is shown in Eq (8) The dimensionlesscorrectionsm C iregm are of the order of the mean ow Mach number In theconventionalenergy-balanceprocedurethe frequencycorrectionm

is not evaluated For simplicity we follow this approach The fre-quencycorrectionis implicitlyaccountedfor in thewavenumberkm

490 FLANDRO AND MAJDALANI

Because quadraticcombinationsof the variablesappear it is rstnecessary to take the real parts of the pressure and velocities to beinserted in the equationsThus one writes

Op D Opm expregm t coskm t (23)

Ou D Oum expregm t sinkm t (24)

Qu D expregm tpound

Qurm coskm t C Qui

m sinkm tcurren

(25)

where superscripts (r) and (i ) refer to the cosine and sine multipli-ers stemming from complex-to-real conversion of real and imagi-nary partsThis notationis necessarybecausethe rotationalvelocityvector contains the spatial exponential term exp(Aacute C iAtilde ) Insertingthese expressions into Eq (17) yields the time-averaged energydensity

hei D 14 exp2regm t

irrotationalz | poundOp2

m C Oum cent Oum

C

rotationalz | 2Oum cent Qui

m C Qur m cent Qur

m C Quim cent Qui

m

curren(26)

which has been partitioned to emphasize the new terms This re-sult should be compared to the classical expression for the energydensity which consists of only the rst two terms in Eq (26) Theadditional three terms represent the kinetic energy residing in theunsteady vorticity Inserting these expressions into Eq (18) carry-ing out the time averaging and differentiatingwith respect to timeone nds

E

tD regm exp2regm tE2

m (27)

where the energy normalization function

E2m D

12

Z Z

V

Z pound Op0

m2 C Oum cent Oum C 2 Oum cent Quim

C Qurm cent Qur

m C Qui m cent Qui

m

currendV (28)

is expressed here by the same symbol used in earlier stability anal-yses It is written as a squared quantity to indicate that it is positivede nite

It is interestingto compare values from Eq (28) to those found inearlier computationsThe mode shapes are needed for this purposeEquations (6) (7) and (10) give the required information for theassumed axial oscillations to zeroth order in Mb

Opm D coskm z (29)

Oum D sinkm zez (30)

Qurm D sinx expAacute sinAtilde sin[sinxkm z]ez (31)

Quim D iexclsinx expAacute cosAtilde sin[sinxkm z]ez (32)

The acoustic pressure and velocity mode shapes are related by

Oum D iexclr Opm =km (33)

for later use Figure 4 shows the results of integration of the kineticenergy terms in Eq (28) across the chamber from the centerline tothe wall at a given axial location Note that the term involving thecross product of the acoustic and vortical parts does not contributeto the integrated result because it oscillates in the radial directionand is zero at the upper limit However the rotational correctionshows a net value which is half of the integrated acoustic energydensityClearly the rotationaleffects representa signi cant changein the system energy

When the entire volume integral of Eq (28) is carried out for theaxial mode case using Eqs (29ndash32) one nds that

E2m D 5

8 frac14 L=R (34)

Fig 4 Radial integration of the kinetic energy density components

which is 25 larger than the conventional acoustic value for lon-gitudinal oscillations However the most important changes in thegrowth rate regm appear in the integral terms on the right-handside ofEq (19) We must now attend to this important part of the stabilityanalysis

A Growth Rate CalculationsIn estimating the system growth rate for a given mode of oscil-

lation it can be seen that it consists of the linear superposition of15 volume integrals as shown in Eq (19) The simplest procedure isto consider each of these (or selected groups) to be evaluated indi-vidually One can then represent the net growth rate as a linear sumof gains and losses as

regm D reg1 C reg2 C reg3 C cent cent cent DNX

i D 1

regi (35)

in the usual fashion The dimensionless values of the several termsare given in the following paragraphs The reader is reminded thatin order to compute the corresponding dimensional values it isnecessary to multiply by a0=R the inverse of the characteristictimeused in the formulation Then the growth rates are obtained in thefamiliar units of rads (siexcl1)

B Irrotational Growth Rate ContributionsTo illustrate the computationalprocedureconsider the rst three

irrotational terms

reg1 D1

exp2regm tE2m

Z Z

V

Z frac12iexclr cent

microOp Ou C

1

2MbU Op2

para

iexcl Mb[ Ou cent rU cent Ou]

frac34dV (36)

where the rst two terms have been combined by using the vectoridentity for the divergence of the product of a scalar and a vectorand by noting that r cent U D 0 for continuity of the incompressiblemean ow Thus the rst term can be converted to a surface integralusing Gaussrsquos divergence theorem The second can be written interms of the pressure by means of Eq (33) after some algebra italso reduces to a surface integral Assuming a short quasi-steadynozzle one nds

reg1 DMb

2E2m

8lt

Z Z

Sb

poundAr

b C 1curren

Op2m dS iexcl

Z Z

SN

poundAr

N C Uz

currenOp2

m dS

9=

rsquofrac14 Mb

2E2m

L

R

copypoundAr

b C 1curren

iexcl [deg C 1]ordf

(37)

The rst term in Eq (37) is of course the classical pressure cou-pling effect except that it is now somewhat reduced in size becauseof the larger energy normalization integral The entire surface in-tegral has not been represented in Eq (37) Nonburning chambersurfaces can be accommodated in the surface integrals if required

FLANDRO AND MAJDALANI 491

The several admittance functionsare taken to be quantitiesof O1the mean ow Mach number has been factored out as in most tra-ditional computations The admittance for nonburning surfaces As

is usually neglected although rough chamber walls and insulationlayers can display substantial (usually negative) admittance

The fourth term the volume (dilatational)acousticviscousenergyloss is usually ignored in standard combustion stability computa-tions it is discussed extensively in the acoustic literature becauseit represents a main source of decay of acoustic waves in a varietyof applications It is readily evaluated for the present case with theresult that

reg2 D1

E2m exp2regm t

Z Z

V

Z laquoplusmn2

d Ou cent rr cent OunotdV rsquo iexcl

plusmn2dfrac14k2

m

4E2m

L

R

(38)

where Eq (33) has again been used Note that this damping effectmight not be negligible when turbulence is present then the trans-port properties are modi ed and the effective viscosity coef cientmight be much larger than the laminar values over a substantialvol-ume of the chamber10 Other irrotational growth rate terms are notdisplayed here These include effects of aluminum particulates andresidual combustion76iexcl78 Because the system is linear they can besuperposed later as required

Two of the rotational terms are traditionally included with thestrictly irrotational growth rate contributions just evaluated Theseare the effects of the mean ow rotationality and the ow-turningeffect To account for the rotational mean ow one writes

reg3 D1

E2m exp2regm t

Z Z

V

ZhMb[Oucent Oupoundshy ]i dV D O

iexclM 2

b

cent(39)

as evaluatedby Culick2 in his rotationalmean ow paper and shownby him to represent a negligible correction

Before proceeding to the next term in Eq (19) the reader isreminded that Culick rst identi ed the ow-turning loss in hisone-dimensional acoustic stability analysis44779 As shown byFlandro89 this damping effect appears because by requiring theunsteady ow to enter the duct in a direction perpendicular to theburning surface Culick effectively invoked the no-slip conditionAlthough this term cannot arise in the multidimensionalirrotationalstabilityanalysisCulick insistedthat it be ldquopatchedrdquoonto the acous-tic growth results If we follow this dictum we must add

reg4 D reg ow turning DEiexcl2

m

exp2regm t

Z Z Z

V

hMb Ou cent U pound i dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Op cent U pound tankm ti dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Opm expregm t cent U pound sinkm ti dV

(40)

to the other irrotational terms This is not justi ed in reality becausethe unsteady vorticity must be used in the evaluation of Eq (40)Inclusion of this single rotational term in the energy balance to theexclusion of all of the others is a not a mathematically legitimatestep Neverthelesswe place it in Eq (19) with the irrotational termsin accordance with accepted practice In evaluating Eq (40) it isnecessary to remember that the unsteady vorticity is

D r pound Qu Dsup3

Qur

ziexcl

Quz

r

acuteemicro D iexcl

Quz

remicro C OMb (41)

The derivative of the radial velocity component with respect to z isof the orderof the mean Mach number and can thereforebe droppedUsing Eq (25) for the axial rotationalvelocity the amplitude of thevorticity vector becomes

D iexclexpregm t

micro Qur

m

rcoskm t C

Qui m

rsinkm t

para(42)

Taking the cross product with the mean ow vector yields

U pound D iexclUzer C Ur ez (43)

For longitudinal modes the pressure gradient is in the z directionso the ow-turning integral reduces to

reg4 D iexclMb

2E2m

Z Z Z

V

Ur Qui

m

rsinkm z dV (44)

Flandro8 showed that the volume integralof Eq (44) can be reducedto a surface integral that is identicalto Culickrsquos original ow-turningexpression (Secs VB and VC of Ref 8) Because the ow-turningintegral is considered to be a key damping effect it is appropriateto review its evaluation in detail for the axial mode case

First consider the derivative of the axial rotational velocity (thevorticity) appearing in Eq (44)

Qui m

rD iexcl sin x exp Aacute

cosAtilde

rsinkm z sin x C cent cent cent (45)

Only the leadingterm in this derivativeis shown becauseit is severalordersof magnitudelarger than theother terms resultingfromchain-rule differentiationof factors such as sin x or exp Aacute The reason forthis expansion becomes obvious when it is remembered that thederivativeof Atilde with respect to r is proportionalto the inverse of themean ow Mach number as shown in Eq (12) Then

Qui m

rD

km

MbUrsin x exp Aacute sinAtilde sinkm z sin x C O1 (46)

and the volume integralof Eq (44) can be evaluatedfor the assumedcylindrical chamber as

reg4 D iexcl12

km Eiexcl2m

Z 2frac14

0

dmicro

Z L=R

0

Z 1

0

r sin x exp Aacute sin Atilde

pound sinkm z sin x sinkm z dr dz (47)

To nd the exact value it is necessary to use numerical integrationcaused by the complicated radial dependenceFollowing the obser-vation made in Ref 8 (Sec VB) that the integrandoscillatesaroundzero from the chamber axis to the surface the value of the integraldependsonly on its behaviornear the upper limit Then to very goodapproximation

reg4 D iexclfrac14km

E2m

Z L=R

0

sin2km z dz

Z 1

0

r sin x exp Aacute sin Atilde dr

D iexclfrac14km

2E2m

L

R

Z 1

0

r sin x exp Aacute sinAtilde dr rsquo iexclfrac14 Mb

2E2m

L

R(48)

The nal numerical value of this damping effect is the same as inCulickrsquos original calculation if the acoustic form of the normal-ization parameter (E2

m D 12 frac14 L=R is inserted Then reg4 D iexclMb in

dimensionless form and the dimensional value becomes

regcurren4 D iexclMba0=Rsiexcl1 (49)

C Rotational Flow CorrectionConsidernow the rst rotational ow stability integral in Eq (19)

reg5 D1

E2m exp2regm t

Z Z Z

V

hiexclQu cent r Opi dV (50)

A companion term the product of the acoustic velocity with thepressuregradient term in the momentum equationhas alreadybeenevaluated and shown to give rise to the pressure coupling effectof central importance The physical interpretation of Eq (50) isapparent It represents the rate of work done on the rotational partof the unsteady ow by theoscillatorypressureforcesIn this respectit is analogous to the pressure coupling which as already showncollapses to a surface integral and is then interpreted as the p dVwork done on the incoming ow1 Obviously this term vanishes in

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

488 FLANDRO AND MAJDALANI

effects become entirely buried within the combustion zone and theouter solution collapses to the simple plane wave acoustic model

Flandro9 derived these expressionsdirectly from Eqs (3) and (4)without recourse to the splitting theorem applied in earlier pa-pers Critics have not understood that splitting Eqs (3) and (4)into acoustical and vortical parts (following the method of Chuand Kovasznay66 and Morse and Ingard67) is a simplifying butnot a necessary step in arriving at the solutions displayed (Brownprivate communication) Another misunderstanding centers on thepressure correction Qp sometimes called the pseudopressure It isimportant to understand that this was not simply set to zero by as-sumption in the earlier works Careful solution of the momentumequation (4) shows that it is an additional correction of the order ofthe mean Mach number the result is shown in Eq (9) Correctionsof similar size appear in the irrotational (acoustic) pressure solu-tion Eq (6) Although Qp constitutes a negligible contribution to theoverall pressure its incorporationis helpful in the acoustic stabilitycalculations for reasons that will be described later It is also neces-sary to point out that the pseudopressuremust not be inserted in thecontinuity(3) as part of the density variable as suggestedby Brownin his 2000 private communication

The complex exponential factor expAacute C iAtilde in the solution ex-hibited in Eq (10) suggests that the vortical motion can be inter-pretedas a damped travelingshearwave Both Aacute and Atilde are functionsonly of radial position The imaginary part of the exponential argu-ment namely

Atilder D iexcl[km =frac14 Mb] taniexcl

12 x

cent(11)

sets the wavelength and spatial frequency of the shear wave Forlater use note that the derivative of Atilde with respect to the radius issimply

dAtilde

drD

km

MbUr(12)

This is proportionalto the reciprocalof the radialmean ow velocityThe real part of the argument Aacuter is given by

Aacuter Draquo

frac14 2

micro1 iexcl

1

sinxiexcl x

cosx

sin2xC I x iexcl I

sup31

2frac14

acutepara

raquo acutek2

m plusmn2

M3b

DS2plusmn2

Mb S acute

km

Mb(13)

I x D x C1

18x3 C

7

1800x5 C

31

105840x7 C cent cent cent (14)

where raquo is a parameter of O1 that re ects the relative importanceof the viscous damping its physical description and veri cationare detailed by Majdalani and Roh16 Note that the damping ofthe shear wave in the radial direction increases quadratically asthe frequency of the acoustic oscillation increases Similarly thewavelength of the shear wave decreases as frequency increasesForthe fundamental acousticmode and the lowest order harmonics theshear wave might ll the entire chamber for high-order acousticmodes the shear oscillations become con ned to a thin acousticboundarylayerwhich can lie entirelywithin the combustionzone13

The composite axial velocity solution found by superimposingEqs (7) and (10) has been shown8iexcl1012 to agree quite closely withthe experimental ndingsofRefs 12 25 26 and 68 Figures2 and 3showcomparisonsof theoscillationamplitudeand phaseangle fromthe theory just discussed superimposed on some of Brownrsquos earliercold ow work2469 It is important to understandthat no curve ttinghas been used here the differences between the experimental andtheoretical results arise both from unavoidable experimental errors(orientation and positioning of the hot lm sensor data reductionetc) and from the assumptionset used in the analysisNevertheless

Fig 2 Comparison of theoretical and experimental amplitudes ofunsteady axial velocities Here S = 513m zcurrenL = 0106 Mb = 00018LR = 34 km = 00924m raquo = 0539m2 and m = 1 Triangles andcircles cor-respond to experimental data acquired with a pressure wave amplitudeof a) 00005 and b) 00039

Fig 3 Comparison of theoretical and experimental phase angles(degrees)

adequate agreement is shown Because nitrogen was the working uid at measured temperature and pressure all parameters charac-terizing the ow eld are well known

Numerical solutions based on the full NavierndashStokes representa-tion of the unsteady eld also indicate excellentagreementwith thetheory reviewed here Furthermore other analyticalmethods basedon Wentzel Kramers and Brillouin (WKB) and multiple-scalethe-ories have been applied by Majdalani and Van Moorhem70 Theirshavedemonstratedbasedon theprincipleof least singularitytheex-istence of nonlinear scales underlying the physics of the problem71

In additionto providinga rigorousmathematical treatment that doesnot require guesswork their approach has led to the exact form ofthe characteristic length scales as function of Culickrsquos mean owsolution Particularly their solutions have yielded results that arevirtually indistinguishablefrom those displayed here11iexcl16

In previousanalyses of the rocket motor rotational ow eld spe-cial attentionwas givento the radialcomponentof velocity8911 Thiswas done because in classical instability theory the main source ofacoustic energy is the radial velocity uctuation produced by inter-action of the pressurewave with the combustion processes and heattransfer in the burning zone It was shown in Refs 8 and 9 that inaddition to the radial acoustic velocity there appears a rotationalcorrection that does not vanish at the edge of the combustion zoneThe presence of this additional unsteady radial velocity componentimplies that by analogy to pressure coupling there is an additionalsource of unsteady mass ux at the combustion zone interface Forwant of better terminology this was called the rotational ow cor-rection as rst displayed in Eq (89) of Ref 8 The new term whichdoes not appear in any previous analysis has given rise to consid-erable controversy

By way of providing a precise description of the source of therotational ow correction a detailed physical interpretation and afull resolution of the controversywill appear in the next section ofthispaperThe answers are foundby means of a carefulderivationofthe system stability energy balance It is not necessary to calculatethe vortical radial velocity correction in determining the systemstability The surface acoustic admittance function will be utilized

FLANDRO AND MAJDALANI 489

in the standard fashion to represent the pressure coupling of theoscillatory ow with the combustion processesThen experimentalmeasurements (usuallysecuredwith a T-burneror similar device)ofthe surface response can be utilized as in the acceptedmethodologyfor accounting for this important source of oscillatory energy

Earlier attempts to incorporate vortical ow corrections in thestability calculations were based on the idea that one must re-store rotational source terms which were dropped in the classicalanalyses For example Brown et al24 had suggested that reten-tion of the rotational convective acceleration term U pound r pound u1

on the right side of Eq (4) might have important stability im-plications This term was rst evaluated by Flandro89 whoshowed that this term yields a damping effect that is indistin-guishable from the classical ow-turning loss Additional termsthat cannot be found in the standard stability computations ap-pear in this analysis One of these the rotational ow correc-tion has led to much debate because it exactly cancels the ow-turning effect in some chamber con gurations Brownrsquos as-sertion that the rotational ow correction is just another wayto represent ow turning can be quite readily refuted In fact allmisunderstandings are resolved when a consistent energy methodis used to deduce motor stability characteristics The aw in theearlier approach is that it represented the system by a model thataccounts for growth or decay of the acoustic energy alone Simplyretaining the rotational convective source terms24 in the perturbedwave equation following Culickrsquos method does not address thisimportant defect

To describe the evolution of the oscillatory ow in a rocket mo-tor correctly all unsteady energy must be accounted for rst Themethod used here follows the known energy balance approach de-scribed by Kirchoff72 and used extensively in rocket stability calcu-lationsby Cantrell and Hart27 and Hart and coworkers41iexcl43 It is nownecessary to account for the entire kinetic energy uctuationTo ac-complish this multiply the continuity equation (3) by the acousticpressure (or density) and add this to the momentum equation (4)multiplied by the composite unsteady velocity vector

u1 D Ou C Qu (15)

This operation constitutes the main departure from the precedingapproach Isolating the terms differentiatedwith respect to time andretaining only those terms that are linear in the Mach number Mb one nds

t

micro12

Op2 C12

u1 cent u1

paraDiexcl

poundOpr cent Ou C u1 cent r Op C Qp

curren

iexcl Mb

raquo1

2U centr Op2 C u1 centr

poundU cent u1

curren

iexclu1 centpoundu1 pound r pound U C U pound r pound Qu

currenfrac14

C plusmn2u1 centmicrosup3

2 Ccedil

sup1

acuterr cent Ou iexcl r pound r pound Qu

para(16)

If the full unsteady velocity vector u1 were to be replaced at thisstage by the acoustic part Ou then the nal result of the stability cal-culation would be exactly that described in earlier papers89 If allrotational terms on the right of Eq (16) were also to be removedby assuming that Qu D 0 then the classical three-dimensional sta-bility result would be reproduced Assumptions such as those justdescribed are motivated mainly by a desire to keep the analysissimple However as we shall now demonstrate much physical sub-stance is lost in such simpli cations We choose not to follow thetraditional approach heremdashmost rotational effects are retained asEq (16) shows the kinetic energy per unit mass on the left-handside re ects all laminar unsteady motions in the chamber Parietal-vortex sheddingand turbulentcorrectionswill require specialatten-tion and future re nementsdiscussionsof turbulent ow effects arefound in the numerical studies by Flandro et al10 Roh et al18 Yangand Roh19 Apte and Yang73 Beddini and Roberts7475 and Vuillotand coworkers5758 So that the analysis is not overly complicated

effectsof solidparticlesin the ow are notdisplayedeither requiredmodi cations are to be incorporated later as needed

Following the standard approach72 one de nes the oscillatoryenergy density

e D 12

poundOp2 C u1 cent u1

curren(17)

and the time-averagedoscillatoryenergy residing in the chamber atany instant as

E DZ Z

V

Zhei dV D

1

2

Z Z

V

Z laquoOp2 C u1 cent u1

notdV (18)

where triangular brackets denote the time average of the enclosedfunction Then from Eq (16) it is clear that the evolution of thesystem energy is controlled by

E

tD

Z Z

V

Zirrotationalz | frac12

iexclr cent Op Ou iexcl12

MbU cent r Op2 iexcl Mb[Ou cent rU cent Ou]

C plusmn2d Ou cent rr cent Ou C Mb[ Ou cent Ou pound shy C Ou cent U pound ]

rotationalz |

iexclQu cent r Op iexcl Mb

microQu cent rU cent Ou C Ou cent rU cent Qu C Qu cent rU cent Qu

iexclQu cent U pound iexcl Qu cent Qu pound shy

para

C plusmn2d Qu cent rr cent Ou iexcl plusmn2[Ou cent r pound C Qu cent r pound ]

iexcl Ou cent r Qp iexcl Qu cent r Qpfrac34

dV (19)

where the irrotational and rotational contributions to the energyrate of change have been partitioned for clarity The last twoldquoirrotationalrdquo terms are in reality caused by rotational effectsThey are placed with the irrotational terms to conform to Culickrsquosmethodology247 The last two ldquorotationalrdquo terms are caused by thepseudopressure a quantity that has often been ignored in previousstudies The mean and unsteady vorticity vectors are represented inEq (19) by shy D r pound U and D r pound Qu respectively

The dilatationalviscous term is simpli ed by de ning

plusmn2d acute plusmn22 C cedil=sup1 (20)

where cedil is the second coef cient of viscosity Several terms cancelFor example

iexclOu cent Qu pound shy iexcl Qu cent Ou pound shy D 0 (21)

Further simpli cations can be made For example several termscan be expressed as integrals over the surface bounding the con-trol volume It is rst necessary to review the protocol needed inevaluating the stability integrals

The terms on the right-hand side of Eq (19) control the rate atwhich the systemenergy changesFrom this informationone can es-timate the growth or decay rate for a given motor con gurationThecomplex wave number (or frequency)k in the assumed exponentialtime dependence is written as

k D km C m C iregm C OiexclM2

b

cent(22)

In the multidimensionalcase m can consist of three mode integersWe will restrict the evaluation to axial modes for clarity then a sin-gle integer m identi es the mode being investigatedMore complexmodes can be handled in a similar way The result for a simple planewave axial mode is shown in Eq (8) The dimensionlesscorrectionsm C iregm are of the order of the mean ow Mach number In theconventionalenergy-balanceprocedurethe frequencycorrectionm

is not evaluated For simplicity we follow this approach The fre-quencycorrectionis implicitlyaccountedfor in thewavenumberkm

490 FLANDRO AND MAJDALANI

Because quadraticcombinationsof the variablesappear it is rstnecessary to take the real parts of the pressure and velocities to beinserted in the equationsThus one writes

Op D Opm expregm t coskm t (23)

Ou D Oum expregm t sinkm t (24)

Qu D expregm tpound

Qurm coskm t C Qui

m sinkm tcurren

(25)

where superscripts (r) and (i ) refer to the cosine and sine multipli-ers stemming from complex-to-real conversion of real and imagi-nary partsThis notationis necessarybecausethe rotationalvelocityvector contains the spatial exponential term exp(Aacute C iAtilde ) Insertingthese expressions into Eq (17) yields the time-averaged energydensity

hei D 14 exp2regm t

irrotationalz | poundOp2

m C Oum cent Oum

C

rotationalz | 2Oum cent Qui

m C Qur m cent Qur

m C Quim cent Qui

m

curren(26)

which has been partitioned to emphasize the new terms This re-sult should be compared to the classical expression for the energydensity which consists of only the rst two terms in Eq (26) Theadditional three terms represent the kinetic energy residing in theunsteady vorticity Inserting these expressions into Eq (18) carry-ing out the time averaging and differentiatingwith respect to timeone nds

E

tD regm exp2regm tE2

m (27)

where the energy normalization function

E2m D

12

Z Z

V

Z pound Op0

m2 C Oum cent Oum C 2 Oum cent Quim

C Qurm cent Qur

m C Qui m cent Qui

m

currendV (28)

is expressed here by the same symbol used in earlier stability anal-yses It is written as a squared quantity to indicate that it is positivede nite

It is interestingto compare values from Eq (28) to those found inearlier computationsThe mode shapes are needed for this purposeEquations (6) (7) and (10) give the required information for theassumed axial oscillations to zeroth order in Mb

Opm D coskm z (29)

Oum D sinkm zez (30)

Qurm D sinx expAacute sinAtilde sin[sinxkm z]ez (31)

Quim D iexclsinx expAacute cosAtilde sin[sinxkm z]ez (32)

The acoustic pressure and velocity mode shapes are related by

Oum D iexclr Opm =km (33)

for later use Figure 4 shows the results of integration of the kineticenergy terms in Eq (28) across the chamber from the centerline tothe wall at a given axial location Note that the term involving thecross product of the acoustic and vortical parts does not contributeto the integrated result because it oscillates in the radial directionand is zero at the upper limit However the rotational correctionshows a net value which is half of the integrated acoustic energydensityClearly the rotationaleffects representa signi cant changein the system energy

When the entire volume integral of Eq (28) is carried out for theaxial mode case using Eqs (29ndash32) one nds that

E2m D 5

8 frac14 L=R (34)

Fig 4 Radial integration of the kinetic energy density components

which is 25 larger than the conventional acoustic value for lon-gitudinal oscillations However the most important changes in thegrowth rate regm appear in the integral terms on the right-handside ofEq (19) We must now attend to this important part of the stabilityanalysis

A Growth Rate CalculationsIn estimating the system growth rate for a given mode of oscil-

lation it can be seen that it consists of the linear superposition of15 volume integrals as shown in Eq (19) The simplest procedure isto consider each of these (or selected groups) to be evaluated indi-vidually One can then represent the net growth rate as a linear sumof gains and losses as

regm D reg1 C reg2 C reg3 C cent cent cent DNX

i D 1

regi (35)

in the usual fashion The dimensionless values of the several termsare given in the following paragraphs The reader is reminded thatin order to compute the corresponding dimensional values it isnecessary to multiply by a0=R the inverse of the characteristictimeused in the formulation Then the growth rates are obtained in thefamiliar units of rads (siexcl1)

B Irrotational Growth Rate ContributionsTo illustrate the computationalprocedureconsider the rst three

irrotational terms

reg1 D1

exp2regm tE2m

Z Z

V

Z frac12iexclr cent

microOp Ou C

1

2MbU Op2

para

iexcl Mb[ Ou cent rU cent Ou]

frac34dV (36)

where the rst two terms have been combined by using the vectoridentity for the divergence of the product of a scalar and a vectorand by noting that r cent U D 0 for continuity of the incompressiblemean ow Thus the rst term can be converted to a surface integralusing Gaussrsquos divergence theorem The second can be written interms of the pressure by means of Eq (33) after some algebra italso reduces to a surface integral Assuming a short quasi-steadynozzle one nds

reg1 DMb

2E2m

8lt

Z Z

Sb

poundAr

b C 1curren

Op2m dS iexcl

Z Z

SN

poundAr

N C Uz

currenOp2

m dS

9=

rsquofrac14 Mb

2E2m

L

R

copypoundAr

b C 1curren

iexcl [deg C 1]ordf

(37)

The rst term in Eq (37) is of course the classical pressure cou-pling effect except that it is now somewhat reduced in size becauseof the larger energy normalization integral The entire surface in-tegral has not been represented in Eq (37) Nonburning chambersurfaces can be accommodated in the surface integrals if required

FLANDRO AND MAJDALANI 491

The several admittance functionsare taken to be quantitiesof O1the mean ow Mach number has been factored out as in most tra-ditional computations The admittance for nonburning surfaces As

is usually neglected although rough chamber walls and insulationlayers can display substantial (usually negative) admittance

The fourth term the volume (dilatational)acousticviscousenergyloss is usually ignored in standard combustion stability computa-tions it is discussed extensively in the acoustic literature becauseit represents a main source of decay of acoustic waves in a varietyof applications It is readily evaluated for the present case with theresult that

reg2 D1

E2m exp2regm t

Z Z

V

Z laquoplusmn2

d Ou cent rr cent OunotdV rsquo iexcl

plusmn2dfrac14k2

m

4E2m

L

R

(38)

where Eq (33) has again been used Note that this damping effectmight not be negligible when turbulence is present then the trans-port properties are modi ed and the effective viscosity coef cientmight be much larger than the laminar values over a substantialvol-ume of the chamber10 Other irrotational growth rate terms are notdisplayed here These include effects of aluminum particulates andresidual combustion76iexcl78 Because the system is linear they can besuperposed later as required

Two of the rotational terms are traditionally included with thestrictly irrotational growth rate contributions just evaluated Theseare the effects of the mean ow rotationality and the ow-turningeffect To account for the rotational mean ow one writes

reg3 D1

E2m exp2regm t

Z Z

V

ZhMb[Oucent Oupoundshy ]i dV D O

iexclM 2

b

cent(39)

as evaluatedby Culick2 in his rotationalmean ow paper and shownby him to represent a negligible correction

Before proceeding to the next term in Eq (19) the reader isreminded that Culick rst identi ed the ow-turning loss in hisone-dimensional acoustic stability analysis44779 As shown byFlandro89 this damping effect appears because by requiring theunsteady ow to enter the duct in a direction perpendicular to theburning surface Culick effectively invoked the no-slip conditionAlthough this term cannot arise in the multidimensionalirrotationalstabilityanalysisCulick insistedthat it be ldquopatchedrdquoonto the acous-tic growth results If we follow this dictum we must add

reg4 D reg ow turning DEiexcl2

m

exp2regm t

Z Z Z

V

hMb Ou cent U pound i dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Op cent U pound tankm ti dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Opm expregm t cent U pound sinkm ti dV

(40)

to the other irrotational terms This is not justi ed in reality becausethe unsteady vorticity must be used in the evaluation of Eq (40)Inclusion of this single rotational term in the energy balance to theexclusion of all of the others is a not a mathematically legitimatestep Neverthelesswe place it in Eq (19) with the irrotational termsin accordance with accepted practice In evaluating Eq (40) it isnecessary to remember that the unsteady vorticity is

D r pound Qu Dsup3

Qur

ziexcl

Quz

r

acuteemicro D iexcl

Quz

remicro C OMb (41)

The derivative of the radial velocity component with respect to z isof the orderof the mean Mach number and can thereforebe droppedUsing Eq (25) for the axial rotationalvelocity the amplitude of thevorticity vector becomes

D iexclexpregm t

micro Qur

m

rcoskm t C

Qui m

rsinkm t

para(42)

Taking the cross product with the mean ow vector yields

U pound D iexclUzer C Ur ez (43)

For longitudinal modes the pressure gradient is in the z directionso the ow-turning integral reduces to

reg4 D iexclMb

2E2m

Z Z Z

V

Ur Qui

m

rsinkm z dV (44)

Flandro8 showed that the volume integralof Eq (44) can be reducedto a surface integral that is identicalto Culickrsquos original ow-turningexpression (Secs VB and VC of Ref 8) Because the ow-turningintegral is considered to be a key damping effect it is appropriateto review its evaluation in detail for the axial mode case

First consider the derivative of the axial rotational velocity (thevorticity) appearing in Eq (44)

Qui m

rD iexcl sin x exp Aacute

cosAtilde

rsinkm z sin x C cent cent cent (45)

Only the leadingterm in this derivativeis shown becauseit is severalordersof magnitudelarger than theother terms resultingfromchain-rule differentiationof factors such as sin x or exp Aacute The reason forthis expansion becomes obvious when it is remembered that thederivativeof Atilde with respect to r is proportionalto the inverse of themean ow Mach number as shown in Eq (12) Then

Qui m

rD

km

MbUrsin x exp Aacute sinAtilde sinkm z sin x C O1 (46)

and the volume integralof Eq (44) can be evaluatedfor the assumedcylindrical chamber as

reg4 D iexcl12

km Eiexcl2m

Z 2frac14

0

dmicro

Z L=R

0

Z 1

0

r sin x exp Aacute sin Atilde

pound sinkm z sin x sinkm z dr dz (47)

To nd the exact value it is necessary to use numerical integrationcaused by the complicated radial dependenceFollowing the obser-vation made in Ref 8 (Sec VB) that the integrandoscillatesaroundzero from the chamber axis to the surface the value of the integraldependsonly on its behaviornear the upper limit Then to very goodapproximation

reg4 D iexclfrac14km

E2m

Z L=R

0

sin2km z dz

Z 1

0

r sin x exp Aacute sin Atilde dr

D iexclfrac14km

2E2m

L

R

Z 1

0

r sin x exp Aacute sinAtilde dr rsquo iexclfrac14 Mb

2E2m

L

R(48)

The nal numerical value of this damping effect is the same as inCulickrsquos original calculation if the acoustic form of the normal-ization parameter (E2

m D 12 frac14 L=R is inserted Then reg4 D iexclMb in

dimensionless form and the dimensional value becomes

regcurren4 D iexclMba0=Rsiexcl1 (49)

C Rotational Flow CorrectionConsidernow the rst rotational ow stability integral in Eq (19)

reg5 D1

E2m exp2regm t

Z Z Z

V

hiexclQu cent r Opi dV (50)

A companion term the product of the acoustic velocity with thepressuregradient term in the momentum equationhas alreadybeenevaluated and shown to give rise to the pressure coupling effectof central importance The physical interpretation of Eq (50) isapparent It represents the rate of work done on the rotational partof the unsteady ow by theoscillatorypressureforcesIn this respectit is analogous to the pressure coupling which as already showncollapses to a surface integral and is then interpreted as the p dVwork done on the incoming ow1 Obviously this term vanishes in

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

FLANDRO AND MAJDALANI 489

in the standard fashion to represent the pressure coupling of theoscillatory ow with the combustion processesThen experimentalmeasurements (usuallysecuredwith a T-burneror similar device)ofthe surface response can be utilized as in the acceptedmethodologyfor accounting for this important source of oscillatory energy

Earlier attempts to incorporate vortical ow corrections in thestability calculations were based on the idea that one must re-store rotational source terms which were dropped in the classicalanalyses For example Brown et al24 had suggested that reten-tion of the rotational convective acceleration term U pound r pound u1

on the right side of Eq (4) might have important stability im-plications This term was rst evaluated by Flandro89 whoshowed that this term yields a damping effect that is indistin-guishable from the classical ow-turning loss Additional termsthat cannot be found in the standard stability computations ap-pear in this analysis One of these the rotational ow correc-tion has led to much debate because it exactly cancels the ow-turning effect in some chamber con gurations Brownrsquos as-sertion that the rotational ow correction is just another wayto represent ow turning can be quite readily refuted In fact allmisunderstandings are resolved when a consistent energy methodis used to deduce motor stability characteristics The aw in theearlier approach is that it represented the system by a model thataccounts for growth or decay of the acoustic energy alone Simplyretaining the rotational convective source terms24 in the perturbedwave equation following Culickrsquos method does not address thisimportant defect

To describe the evolution of the oscillatory ow in a rocket mo-tor correctly all unsteady energy must be accounted for rst Themethod used here follows the known energy balance approach de-scribed by Kirchoff72 and used extensively in rocket stability calcu-lationsby Cantrell and Hart27 and Hart and coworkers41iexcl43 It is nownecessary to account for the entire kinetic energy uctuationTo ac-complish this multiply the continuity equation (3) by the acousticpressure (or density) and add this to the momentum equation (4)multiplied by the composite unsteady velocity vector

u1 D Ou C Qu (15)

This operation constitutes the main departure from the precedingapproach Isolating the terms differentiatedwith respect to time andretaining only those terms that are linear in the Mach number Mb one nds

t

micro12

Op2 C12

u1 cent u1

paraDiexcl

poundOpr cent Ou C u1 cent r Op C Qp

curren

iexcl Mb

raquo1

2U centr Op2 C u1 centr

poundU cent u1

curren

iexclu1 centpoundu1 pound r pound U C U pound r pound Qu

currenfrac14

C plusmn2u1 centmicrosup3

2 Ccedil

sup1

acuterr cent Ou iexcl r pound r pound Qu

para(16)

If the full unsteady velocity vector u1 were to be replaced at thisstage by the acoustic part Ou then the nal result of the stability cal-culation would be exactly that described in earlier papers89 If allrotational terms on the right of Eq (16) were also to be removedby assuming that Qu D 0 then the classical three-dimensional sta-bility result would be reproduced Assumptions such as those justdescribed are motivated mainly by a desire to keep the analysissimple However as we shall now demonstrate much physical sub-stance is lost in such simpli cations We choose not to follow thetraditional approach heremdashmost rotational effects are retained asEq (16) shows the kinetic energy per unit mass on the left-handside re ects all laminar unsteady motions in the chamber Parietal-vortex sheddingand turbulentcorrectionswill require specialatten-tion and future re nementsdiscussionsof turbulent ow effects arefound in the numerical studies by Flandro et al10 Roh et al18 Yangand Roh19 Apte and Yang73 Beddini and Roberts7475 and Vuillotand coworkers5758 So that the analysis is not overly complicated

effectsof solidparticlesin the ow are notdisplayedeither requiredmodi cations are to be incorporated later as needed

Following the standard approach72 one de nes the oscillatoryenergy density

e D 12

poundOp2 C u1 cent u1

curren(17)

and the time-averagedoscillatoryenergy residing in the chamber atany instant as

E DZ Z

V

Zhei dV D

1

2

Z Z

V

Z laquoOp2 C u1 cent u1

notdV (18)

where triangular brackets denote the time average of the enclosedfunction Then from Eq (16) it is clear that the evolution of thesystem energy is controlled by

E

tD

Z Z

V

Zirrotationalz | frac12

iexclr cent Op Ou iexcl12

MbU cent r Op2 iexcl Mb[Ou cent rU cent Ou]

C plusmn2d Ou cent rr cent Ou C Mb[ Ou cent Ou pound shy C Ou cent U pound ]

rotationalz |

iexclQu cent r Op iexcl Mb

microQu cent rU cent Ou C Ou cent rU cent Qu C Qu cent rU cent Qu

iexclQu cent U pound iexcl Qu cent Qu pound shy

para

C plusmn2d Qu cent rr cent Ou iexcl plusmn2[Ou cent r pound C Qu cent r pound ]

iexcl Ou cent r Qp iexcl Qu cent r Qpfrac34

dV (19)

where the irrotational and rotational contributions to the energyrate of change have been partitioned for clarity The last twoldquoirrotationalrdquo terms are in reality caused by rotational effectsThey are placed with the irrotational terms to conform to Culickrsquosmethodology247 The last two ldquorotationalrdquo terms are caused by thepseudopressure a quantity that has often been ignored in previousstudies The mean and unsteady vorticity vectors are represented inEq (19) by shy D r pound U and D r pound Qu respectively

The dilatationalviscous term is simpli ed by de ning

plusmn2d acute plusmn22 C cedil=sup1 (20)

where cedil is the second coef cient of viscosity Several terms cancelFor example

iexclOu cent Qu pound shy iexcl Qu cent Ou pound shy D 0 (21)

Further simpli cations can be made For example several termscan be expressed as integrals over the surface bounding the con-trol volume It is rst necessary to review the protocol needed inevaluating the stability integrals

The terms on the right-hand side of Eq (19) control the rate atwhich the systemenergy changesFrom this informationone can es-timate the growth or decay rate for a given motor con gurationThecomplex wave number (or frequency)k in the assumed exponentialtime dependence is written as

k D km C m C iregm C OiexclM2

b

cent(22)

In the multidimensionalcase m can consist of three mode integersWe will restrict the evaluation to axial modes for clarity then a sin-gle integer m identi es the mode being investigatedMore complexmodes can be handled in a similar way The result for a simple planewave axial mode is shown in Eq (8) The dimensionlesscorrectionsm C iregm are of the order of the mean ow Mach number In theconventionalenergy-balanceprocedurethe frequencycorrectionm

is not evaluated For simplicity we follow this approach The fre-quencycorrectionis implicitlyaccountedfor in thewavenumberkm

490 FLANDRO AND MAJDALANI

Because quadraticcombinationsof the variablesappear it is rstnecessary to take the real parts of the pressure and velocities to beinserted in the equationsThus one writes

Op D Opm expregm t coskm t (23)

Ou D Oum expregm t sinkm t (24)

Qu D expregm tpound

Qurm coskm t C Qui

m sinkm tcurren

(25)

where superscripts (r) and (i ) refer to the cosine and sine multipli-ers stemming from complex-to-real conversion of real and imagi-nary partsThis notationis necessarybecausethe rotationalvelocityvector contains the spatial exponential term exp(Aacute C iAtilde ) Insertingthese expressions into Eq (17) yields the time-averaged energydensity

hei D 14 exp2regm t

irrotationalz | poundOp2

m C Oum cent Oum

C

rotationalz | 2Oum cent Qui

m C Qur m cent Qur

m C Quim cent Qui

m

curren(26)

which has been partitioned to emphasize the new terms This re-sult should be compared to the classical expression for the energydensity which consists of only the rst two terms in Eq (26) Theadditional three terms represent the kinetic energy residing in theunsteady vorticity Inserting these expressions into Eq (18) carry-ing out the time averaging and differentiatingwith respect to timeone nds

E

tD regm exp2regm tE2

m (27)

where the energy normalization function

E2m D

12

Z Z

V

Z pound Op0

m2 C Oum cent Oum C 2 Oum cent Quim

C Qurm cent Qur

m C Qui m cent Qui

m

currendV (28)

is expressed here by the same symbol used in earlier stability anal-yses It is written as a squared quantity to indicate that it is positivede nite

It is interestingto compare values from Eq (28) to those found inearlier computationsThe mode shapes are needed for this purposeEquations (6) (7) and (10) give the required information for theassumed axial oscillations to zeroth order in Mb

Opm D coskm z (29)

Oum D sinkm zez (30)

Qurm D sinx expAacute sinAtilde sin[sinxkm z]ez (31)

Quim D iexclsinx expAacute cosAtilde sin[sinxkm z]ez (32)

The acoustic pressure and velocity mode shapes are related by

Oum D iexclr Opm =km (33)

for later use Figure 4 shows the results of integration of the kineticenergy terms in Eq (28) across the chamber from the centerline tothe wall at a given axial location Note that the term involving thecross product of the acoustic and vortical parts does not contributeto the integrated result because it oscillates in the radial directionand is zero at the upper limit However the rotational correctionshows a net value which is half of the integrated acoustic energydensityClearly the rotationaleffects representa signi cant changein the system energy

When the entire volume integral of Eq (28) is carried out for theaxial mode case using Eqs (29ndash32) one nds that

E2m D 5

8 frac14 L=R (34)

Fig 4 Radial integration of the kinetic energy density components

which is 25 larger than the conventional acoustic value for lon-gitudinal oscillations However the most important changes in thegrowth rate regm appear in the integral terms on the right-handside ofEq (19) We must now attend to this important part of the stabilityanalysis

A Growth Rate CalculationsIn estimating the system growth rate for a given mode of oscil-

lation it can be seen that it consists of the linear superposition of15 volume integrals as shown in Eq (19) The simplest procedure isto consider each of these (or selected groups) to be evaluated indi-vidually One can then represent the net growth rate as a linear sumof gains and losses as

regm D reg1 C reg2 C reg3 C cent cent cent DNX

i D 1

regi (35)

in the usual fashion The dimensionless values of the several termsare given in the following paragraphs The reader is reminded thatin order to compute the corresponding dimensional values it isnecessary to multiply by a0=R the inverse of the characteristictimeused in the formulation Then the growth rates are obtained in thefamiliar units of rads (siexcl1)

B Irrotational Growth Rate ContributionsTo illustrate the computationalprocedureconsider the rst three

irrotational terms

reg1 D1

exp2regm tE2m

Z Z

V

Z frac12iexclr cent

microOp Ou C

1

2MbU Op2

para

iexcl Mb[ Ou cent rU cent Ou]

frac34dV (36)

where the rst two terms have been combined by using the vectoridentity for the divergence of the product of a scalar and a vectorand by noting that r cent U D 0 for continuity of the incompressiblemean ow Thus the rst term can be converted to a surface integralusing Gaussrsquos divergence theorem The second can be written interms of the pressure by means of Eq (33) after some algebra italso reduces to a surface integral Assuming a short quasi-steadynozzle one nds

reg1 DMb

2E2m

8lt

Z Z

Sb

poundAr

b C 1curren

Op2m dS iexcl

Z Z

SN

poundAr

N C Uz

currenOp2

m dS

9=

rsquofrac14 Mb

2E2m

L

R

copypoundAr

b C 1curren

iexcl [deg C 1]ordf

(37)

The rst term in Eq (37) is of course the classical pressure cou-pling effect except that it is now somewhat reduced in size becauseof the larger energy normalization integral The entire surface in-tegral has not been represented in Eq (37) Nonburning chambersurfaces can be accommodated in the surface integrals if required

FLANDRO AND MAJDALANI 491

The several admittance functionsare taken to be quantitiesof O1the mean ow Mach number has been factored out as in most tra-ditional computations The admittance for nonburning surfaces As

is usually neglected although rough chamber walls and insulationlayers can display substantial (usually negative) admittance

The fourth term the volume (dilatational)acousticviscousenergyloss is usually ignored in standard combustion stability computa-tions it is discussed extensively in the acoustic literature becauseit represents a main source of decay of acoustic waves in a varietyof applications It is readily evaluated for the present case with theresult that

reg2 D1

E2m exp2regm t

Z Z

V

Z laquoplusmn2

d Ou cent rr cent OunotdV rsquo iexcl

plusmn2dfrac14k2

m

4E2m

L

R

(38)

where Eq (33) has again been used Note that this damping effectmight not be negligible when turbulence is present then the trans-port properties are modi ed and the effective viscosity coef cientmight be much larger than the laminar values over a substantialvol-ume of the chamber10 Other irrotational growth rate terms are notdisplayed here These include effects of aluminum particulates andresidual combustion76iexcl78 Because the system is linear they can besuperposed later as required

Two of the rotational terms are traditionally included with thestrictly irrotational growth rate contributions just evaluated Theseare the effects of the mean ow rotationality and the ow-turningeffect To account for the rotational mean ow one writes

reg3 D1

E2m exp2regm t

Z Z

V

ZhMb[Oucent Oupoundshy ]i dV D O

iexclM 2

b

cent(39)

as evaluatedby Culick2 in his rotationalmean ow paper and shownby him to represent a negligible correction

Before proceeding to the next term in Eq (19) the reader isreminded that Culick rst identi ed the ow-turning loss in hisone-dimensional acoustic stability analysis44779 As shown byFlandro89 this damping effect appears because by requiring theunsteady ow to enter the duct in a direction perpendicular to theburning surface Culick effectively invoked the no-slip conditionAlthough this term cannot arise in the multidimensionalirrotationalstabilityanalysisCulick insistedthat it be ldquopatchedrdquoonto the acous-tic growth results If we follow this dictum we must add

reg4 D reg ow turning DEiexcl2

m

exp2regm t

Z Z Z

V

hMb Ou cent U pound i dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Op cent U pound tankm ti dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Opm expregm t cent U pound sinkm ti dV

(40)

to the other irrotational terms This is not justi ed in reality becausethe unsteady vorticity must be used in the evaluation of Eq (40)Inclusion of this single rotational term in the energy balance to theexclusion of all of the others is a not a mathematically legitimatestep Neverthelesswe place it in Eq (19) with the irrotational termsin accordance with accepted practice In evaluating Eq (40) it isnecessary to remember that the unsteady vorticity is

D r pound Qu Dsup3

Qur

ziexcl

Quz

r

acuteemicro D iexcl

Quz

remicro C OMb (41)

The derivative of the radial velocity component with respect to z isof the orderof the mean Mach number and can thereforebe droppedUsing Eq (25) for the axial rotationalvelocity the amplitude of thevorticity vector becomes

D iexclexpregm t

micro Qur

m

rcoskm t C

Qui m

rsinkm t

para(42)

Taking the cross product with the mean ow vector yields

U pound D iexclUzer C Ur ez (43)

For longitudinal modes the pressure gradient is in the z directionso the ow-turning integral reduces to

reg4 D iexclMb

2E2m

Z Z Z

V

Ur Qui

m

rsinkm z dV (44)

Flandro8 showed that the volume integralof Eq (44) can be reducedto a surface integral that is identicalto Culickrsquos original ow-turningexpression (Secs VB and VC of Ref 8) Because the ow-turningintegral is considered to be a key damping effect it is appropriateto review its evaluation in detail for the axial mode case

First consider the derivative of the axial rotational velocity (thevorticity) appearing in Eq (44)

Qui m

rD iexcl sin x exp Aacute

cosAtilde

rsinkm z sin x C cent cent cent (45)

Only the leadingterm in this derivativeis shown becauseit is severalordersof magnitudelarger than theother terms resultingfromchain-rule differentiationof factors such as sin x or exp Aacute The reason forthis expansion becomes obvious when it is remembered that thederivativeof Atilde with respect to r is proportionalto the inverse of themean ow Mach number as shown in Eq (12) Then

Qui m

rD

km

MbUrsin x exp Aacute sinAtilde sinkm z sin x C O1 (46)

and the volume integralof Eq (44) can be evaluatedfor the assumedcylindrical chamber as

reg4 D iexcl12

km Eiexcl2m

Z 2frac14

0

dmicro

Z L=R

0

Z 1

0

r sin x exp Aacute sin Atilde

pound sinkm z sin x sinkm z dr dz (47)

To nd the exact value it is necessary to use numerical integrationcaused by the complicated radial dependenceFollowing the obser-vation made in Ref 8 (Sec VB) that the integrandoscillatesaroundzero from the chamber axis to the surface the value of the integraldependsonly on its behaviornear the upper limit Then to very goodapproximation

reg4 D iexclfrac14km

E2m

Z L=R

0

sin2km z dz

Z 1

0

r sin x exp Aacute sin Atilde dr

D iexclfrac14km

2E2m

L

R

Z 1

0

r sin x exp Aacute sinAtilde dr rsquo iexclfrac14 Mb

2E2m

L

R(48)

The nal numerical value of this damping effect is the same as inCulickrsquos original calculation if the acoustic form of the normal-ization parameter (E2

m D 12 frac14 L=R is inserted Then reg4 D iexclMb in

dimensionless form and the dimensional value becomes

regcurren4 D iexclMba0=Rsiexcl1 (49)

C Rotational Flow CorrectionConsidernow the rst rotational ow stability integral in Eq (19)

reg5 D1

E2m exp2regm t

Z Z Z

V

hiexclQu cent r Opi dV (50)

A companion term the product of the acoustic velocity with thepressuregradient term in the momentum equationhas alreadybeenevaluated and shown to give rise to the pressure coupling effectof central importance The physical interpretation of Eq (50) isapparent It represents the rate of work done on the rotational partof the unsteady ow by theoscillatorypressureforcesIn this respectit is analogous to the pressure coupling which as already showncollapses to a surface integral and is then interpreted as the p dVwork done on the incoming ow1 Obviously this term vanishes in

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

490 FLANDRO AND MAJDALANI

Because quadraticcombinationsof the variablesappear it is rstnecessary to take the real parts of the pressure and velocities to beinserted in the equationsThus one writes

Op D Opm expregm t coskm t (23)

Ou D Oum expregm t sinkm t (24)

Qu D expregm tpound

Qurm coskm t C Qui

m sinkm tcurren

(25)

where superscripts (r) and (i ) refer to the cosine and sine multipli-ers stemming from complex-to-real conversion of real and imagi-nary partsThis notationis necessarybecausethe rotationalvelocityvector contains the spatial exponential term exp(Aacute C iAtilde ) Insertingthese expressions into Eq (17) yields the time-averaged energydensity

hei D 14 exp2regm t

irrotationalz | poundOp2

m C Oum cent Oum

C

rotationalz | 2Oum cent Qui

m C Qur m cent Qur

m C Quim cent Qui

m

curren(26)

which has been partitioned to emphasize the new terms This re-sult should be compared to the classical expression for the energydensity which consists of only the rst two terms in Eq (26) Theadditional three terms represent the kinetic energy residing in theunsteady vorticity Inserting these expressions into Eq (18) carry-ing out the time averaging and differentiatingwith respect to timeone nds

E

tD regm exp2regm tE2

m (27)

where the energy normalization function

E2m D

12

Z Z

V

Z pound Op0

m2 C Oum cent Oum C 2 Oum cent Quim

C Qurm cent Qur

m C Qui m cent Qui

m

currendV (28)

is expressed here by the same symbol used in earlier stability anal-yses It is written as a squared quantity to indicate that it is positivede nite

It is interestingto compare values from Eq (28) to those found inearlier computationsThe mode shapes are needed for this purposeEquations (6) (7) and (10) give the required information for theassumed axial oscillations to zeroth order in Mb

Opm D coskm z (29)

Oum D sinkm zez (30)

Qurm D sinx expAacute sinAtilde sin[sinxkm z]ez (31)

Quim D iexclsinx expAacute cosAtilde sin[sinxkm z]ez (32)

The acoustic pressure and velocity mode shapes are related by

Oum D iexclr Opm =km (33)

for later use Figure 4 shows the results of integration of the kineticenergy terms in Eq (28) across the chamber from the centerline tothe wall at a given axial location Note that the term involving thecross product of the acoustic and vortical parts does not contributeto the integrated result because it oscillates in the radial directionand is zero at the upper limit However the rotational correctionshows a net value which is half of the integrated acoustic energydensityClearly the rotationaleffects representa signi cant changein the system energy

When the entire volume integral of Eq (28) is carried out for theaxial mode case using Eqs (29ndash32) one nds that

E2m D 5

8 frac14 L=R (34)

Fig 4 Radial integration of the kinetic energy density components

which is 25 larger than the conventional acoustic value for lon-gitudinal oscillations However the most important changes in thegrowth rate regm appear in the integral terms on the right-handside ofEq (19) We must now attend to this important part of the stabilityanalysis

A Growth Rate CalculationsIn estimating the system growth rate for a given mode of oscil-

lation it can be seen that it consists of the linear superposition of15 volume integrals as shown in Eq (19) The simplest procedure isto consider each of these (or selected groups) to be evaluated indi-vidually One can then represent the net growth rate as a linear sumof gains and losses as

regm D reg1 C reg2 C reg3 C cent cent cent DNX

i D 1

regi (35)

in the usual fashion The dimensionless values of the several termsare given in the following paragraphs The reader is reminded thatin order to compute the corresponding dimensional values it isnecessary to multiply by a0=R the inverse of the characteristictimeused in the formulation Then the growth rates are obtained in thefamiliar units of rads (siexcl1)

B Irrotational Growth Rate ContributionsTo illustrate the computationalprocedureconsider the rst three

irrotational terms

reg1 D1

exp2regm tE2m

Z Z

V

Z frac12iexclr cent

microOp Ou C

1

2MbU Op2

para

iexcl Mb[ Ou cent rU cent Ou]

frac34dV (36)

where the rst two terms have been combined by using the vectoridentity for the divergence of the product of a scalar and a vectorand by noting that r cent U D 0 for continuity of the incompressiblemean ow Thus the rst term can be converted to a surface integralusing Gaussrsquos divergence theorem The second can be written interms of the pressure by means of Eq (33) after some algebra italso reduces to a surface integral Assuming a short quasi-steadynozzle one nds

reg1 DMb

2E2m

8lt

Z Z

Sb

poundAr

b C 1curren

Op2m dS iexcl

Z Z

SN

poundAr

N C Uz

currenOp2

m dS

9=

rsquofrac14 Mb

2E2m

L

R

copypoundAr

b C 1curren

iexcl [deg C 1]ordf

(37)

The rst term in Eq (37) is of course the classical pressure cou-pling effect except that it is now somewhat reduced in size becauseof the larger energy normalization integral The entire surface in-tegral has not been represented in Eq (37) Nonburning chambersurfaces can be accommodated in the surface integrals if required

FLANDRO AND MAJDALANI 491

The several admittance functionsare taken to be quantitiesof O1the mean ow Mach number has been factored out as in most tra-ditional computations The admittance for nonburning surfaces As

is usually neglected although rough chamber walls and insulationlayers can display substantial (usually negative) admittance

The fourth term the volume (dilatational)acousticviscousenergyloss is usually ignored in standard combustion stability computa-tions it is discussed extensively in the acoustic literature becauseit represents a main source of decay of acoustic waves in a varietyof applications It is readily evaluated for the present case with theresult that

reg2 D1

E2m exp2regm t

Z Z

V

Z laquoplusmn2

d Ou cent rr cent OunotdV rsquo iexcl

plusmn2dfrac14k2

m

4E2m

L

R

(38)

where Eq (33) has again been used Note that this damping effectmight not be negligible when turbulence is present then the trans-port properties are modi ed and the effective viscosity coef cientmight be much larger than the laminar values over a substantialvol-ume of the chamber10 Other irrotational growth rate terms are notdisplayed here These include effects of aluminum particulates andresidual combustion76iexcl78 Because the system is linear they can besuperposed later as required

Two of the rotational terms are traditionally included with thestrictly irrotational growth rate contributions just evaluated Theseare the effects of the mean ow rotationality and the ow-turningeffect To account for the rotational mean ow one writes

reg3 D1

E2m exp2regm t

Z Z

V

ZhMb[Oucent Oupoundshy ]i dV D O

iexclM 2

b

cent(39)

as evaluatedby Culick2 in his rotationalmean ow paper and shownby him to represent a negligible correction

Before proceeding to the next term in Eq (19) the reader isreminded that Culick rst identi ed the ow-turning loss in hisone-dimensional acoustic stability analysis44779 As shown byFlandro89 this damping effect appears because by requiring theunsteady ow to enter the duct in a direction perpendicular to theburning surface Culick effectively invoked the no-slip conditionAlthough this term cannot arise in the multidimensionalirrotationalstabilityanalysisCulick insistedthat it be ldquopatchedrdquoonto the acous-tic growth results If we follow this dictum we must add

reg4 D reg ow turning DEiexcl2

m

exp2regm t

Z Z Z

V

hMb Ou cent U pound i dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Op cent U pound tankm ti dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Opm expregm t cent U pound sinkm ti dV

(40)

to the other irrotational terms This is not justi ed in reality becausethe unsteady vorticity must be used in the evaluation of Eq (40)Inclusion of this single rotational term in the energy balance to theexclusion of all of the others is a not a mathematically legitimatestep Neverthelesswe place it in Eq (19) with the irrotational termsin accordance with accepted practice In evaluating Eq (40) it isnecessary to remember that the unsteady vorticity is

D r pound Qu Dsup3

Qur

ziexcl

Quz

r

acuteemicro D iexcl

Quz

remicro C OMb (41)

The derivative of the radial velocity component with respect to z isof the orderof the mean Mach number and can thereforebe droppedUsing Eq (25) for the axial rotationalvelocity the amplitude of thevorticity vector becomes

D iexclexpregm t

micro Qur

m

rcoskm t C

Qui m

rsinkm t

para(42)

Taking the cross product with the mean ow vector yields

U pound D iexclUzer C Ur ez (43)

For longitudinal modes the pressure gradient is in the z directionso the ow-turning integral reduces to

reg4 D iexclMb

2E2m

Z Z Z

V

Ur Qui

m

rsinkm z dV (44)

Flandro8 showed that the volume integralof Eq (44) can be reducedto a surface integral that is identicalto Culickrsquos original ow-turningexpression (Secs VB and VC of Ref 8) Because the ow-turningintegral is considered to be a key damping effect it is appropriateto review its evaluation in detail for the axial mode case

First consider the derivative of the axial rotational velocity (thevorticity) appearing in Eq (44)

Qui m

rD iexcl sin x exp Aacute

cosAtilde

rsinkm z sin x C cent cent cent (45)

Only the leadingterm in this derivativeis shown becauseit is severalordersof magnitudelarger than theother terms resultingfromchain-rule differentiationof factors such as sin x or exp Aacute The reason forthis expansion becomes obvious when it is remembered that thederivativeof Atilde with respect to r is proportionalto the inverse of themean ow Mach number as shown in Eq (12) Then

Qui m

rD

km

MbUrsin x exp Aacute sinAtilde sinkm z sin x C O1 (46)

and the volume integralof Eq (44) can be evaluatedfor the assumedcylindrical chamber as

reg4 D iexcl12

km Eiexcl2m

Z 2frac14

0

dmicro

Z L=R

0

Z 1

0

r sin x exp Aacute sin Atilde

pound sinkm z sin x sinkm z dr dz (47)

To nd the exact value it is necessary to use numerical integrationcaused by the complicated radial dependenceFollowing the obser-vation made in Ref 8 (Sec VB) that the integrandoscillatesaroundzero from the chamber axis to the surface the value of the integraldependsonly on its behaviornear the upper limit Then to very goodapproximation

reg4 D iexclfrac14km

E2m

Z L=R

0

sin2km z dz

Z 1

0

r sin x exp Aacute sin Atilde dr

D iexclfrac14km

2E2m

L

R

Z 1

0

r sin x exp Aacute sinAtilde dr rsquo iexclfrac14 Mb

2E2m

L

R(48)

The nal numerical value of this damping effect is the same as inCulickrsquos original calculation if the acoustic form of the normal-ization parameter (E2

m D 12 frac14 L=R is inserted Then reg4 D iexclMb in

dimensionless form and the dimensional value becomes

regcurren4 D iexclMba0=Rsiexcl1 (49)

C Rotational Flow CorrectionConsidernow the rst rotational ow stability integral in Eq (19)

reg5 D1

E2m exp2regm t

Z Z Z

V

hiexclQu cent r Opi dV (50)

A companion term the product of the acoustic velocity with thepressuregradient term in the momentum equationhas alreadybeenevaluated and shown to give rise to the pressure coupling effectof central importance The physical interpretation of Eq (50) isapparent It represents the rate of work done on the rotational partof the unsteady ow by theoscillatorypressureforcesIn this respectit is analogous to the pressure coupling which as already showncollapses to a surface integral and is then interpreted as the p dVwork done on the incoming ow1 Obviously this term vanishes in

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

FLANDRO AND MAJDALANI 491

The several admittance functionsare taken to be quantitiesof O1the mean ow Mach number has been factored out as in most tra-ditional computations The admittance for nonburning surfaces As

is usually neglected although rough chamber walls and insulationlayers can display substantial (usually negative) admittance

The fourth term the volume (dilatational)acousticviscousenergyloss is usually ignored in standard combustion stability computa-tions it is discussed extensively in the acoustic literature becauseit represents a main source of decay of acoustic waves in a varietyof applications It is readily evaluated for the present case with theresult that

reg2 D1

E2m exp2regm t

Z Z

V

Z laquoplusmn2

d Ou cent rr cent OunotdV rsquo iexcl

plusmn2dfrac14k2

m

4E2m

L

R

(38)

where Eq (33) has again been used Note that this damping effectmight not be negligible when turbulence is present then the trans-port properties are modi ed and the effective viscosity coef cientmight be much larger than the laminar values over a substantialvol-ume of the chamber10 Other irrotational growth rate terms are notdisplayed here These include effects of aluminum particulates andresidual combustion76iexcl78 Because the system is linear they can besuperposed later as required

Two of the rotational terms are traditionally included with thestrictly irrotational growth rate contributions just evaluated Theseare the effects of the mean ow rotationality and the ow-turningeffect To account for the rotational mean ow one writes

reg3 D1

E2m exp2regm t

Z Z

V

ZhMb[Oucent Oupoundshy ]i dV D O

iexclM 2

b

cent(39)

as evaluatedby Culick2 in his rotationalmean ow paper and shownby him to represent a negligible correction

Before proceeding to the next term in Eq (19) the reader isreminded that Culick rst identi ed the ow-turning loss in hisone-dimensional acoustic stability analysis44779 As shown byFlandro89 this damping effect appears because by requiring theunsteady ow to enter the duct in a direction perpendicular to theburning surface Culick effectively invoked the no-slip conditionAlthough this term cannot arise in the multidimensionalirrotationalstabilityanalysisCulick insistedthat it be ldquopatchedrdquoonto the acous-tic growth results If we follow this dictum we must add

reg4 D reg ow turning DEiexcl2

m

exp2regm t

Z Z Z

V

hMb Ou cent U pound i dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Op cent U pound tankm ti dV

DiexclMb Eiexcl2

m

km exp2regm t

Z Z Z

V

hr Opm expregm t cent U pound sinkm ti dV

(40)

to the other irrotational terms This is not justi ed in reality becausethe unsteady vorticity must be used in the evaluation of Eq (40)Inclusion of this single rotational term in the energy balance to theexclusion of all of the others is a not a mathematically legitimatestep Neverthelesswe place it in Eq (19) with the irrotational termsin accordance with accepted practice In evaluating Eq (40) it isnecessary to remember that the unsteady vorticity is

D r pound Qu Dsup3

Qur

ziexcl

Quz

r

acuteemicro D iexcl

Quz

remicro C OMb (41)

The derivative of the radial velocity component with respect to z isof the orderof the mean Mach number and can thereforebe droppedUsing Eq (25) for the axial rotationalvelocity the amplitude of thevorticity vector becomes

D iexclexpregm t

micro Qur

m

rcoskm t C

Qui m

rsinkm t

para(42)

Taking the cross product with the mean ow vector yields

U pound D iexclUzer C Ur ez (43)

For longitudinal modes the pressure gradient is in the z directionso the ow-turning integral reduces to

reg4 D iexclMb

2E2m

Z Z Z

V

Ur Qui

m

rsinkm z dV (44)

Flandro8 showed that the volume integralof Eq (44) can be reducedto a surface integral that is identicalto Culickrsquos original ow-turningexpression (Secs VB and VC of Ref 8) Because the ow-turningintegral is considered to be a key damping effect it is appropriateto review its evaluation in detail for the axial mode case

First consider the derivative of the axial rotational velocity (thevorticity) appearing in Eq (44)

Qui m

rD iexcl sin x exp Aacute

cosAtilde

rsinkm z sin x C cent cent cent (45)

Only the leadingterm in this derivativeis shown becauseit is severalordersof magnitudelarger than theother terms resultingfromchain-rule differentiationof factors such as sin x or exp Aacute The reason forthis expansion becomes obvious when it is remembered that thederivativeof Atilde with respect to r is proportionalto the inverse of themean ow Mach number as shown in Eq (12) Then

Qui m

rD

km

MbUrsin x exp Aacute sinAtilde sinkm z sin x C O1 (46)

and the volume integralof Eq (44) can be evaluatedfor the assumedcylindrical chamber as

reg4 D iexcl12

km Eiexcl2m

Z 2frac14

0

dmicro

Z L=R

0

Z 1

0

r sin x exp Aacute sin Atilde

pound sinkm z sin x sinkm z dr dz (47)

To nd the exact value it is necessary to use numerical integrationcaused by the complicated radial dependenceFollowing the obser-vation made in Ref 8 (Sec VB) that the integrandoscillatesaroundzero from the chamber axis to the surface the value of the integraldependsonly on its behaviornear the upper limit Then to very goodapproximation

reg4 D iexclfrac14km

E2m

Z L=R

0

sin2km z dz

Z 1

0

r sin x exp Aacute sin Atilde dr

D iexclfrac14km

2E2m

L

R

Z 1

0

r sin x exp Aacute sinAtilde dr rsquo iexclfrac14 Mb

2E2m

L

R(48)

The nal numerical value of this damping effect is the same as inCulickrsquos original calculation if the acoustic form of the normal-ization parameter (E2

m D 12 frac14 L=R is inserted Then reg4 D iexclMb in

dimensionless form and the dimensional value becomes

regcurren4 D iexclMba0=Rsiexcl1 (49)

C Rotational Flow CorrectionConsidernow the rst rotational ow stability integral in Eq (19)

reg5 D1

E2m exp2regm t

Z Z Z

V

hiexclQu cent r Opi dV (50)

A companion term the product of the acoustic velocity with thepressuregradient term in the momentum equationhas alreadybeenevaluated and shown to give rise to the pressure coupling effectof central importance The physical interpretation of Eq (50) isapparent It represents the rate of work done on the rotational partof the unsteady ow by theoscillatorypressureforcesIn this respectit is analogous to the pressure coupling which as already showncollapses to a surface integral and is then interpreted as the p dVwork done on the incoming ow1 Obviously this term vanishes in

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

492 FLANDRO AND MAJDALANI

a strictly irrotational environment hence it does not appear in theusual acoustic stability computationsUsing Eqs (23) and (25) andcarrying out the time averaging Eq (50) reduces to

reg5 D iexcl1

2E 2m

Z Z Z

V

Qurm cent r Opm dV (51)

Notice that for axial modes it is only necessary to know the axialcomponent of the rotational velocity eld because the pressure gra-dient is in the z direction No information regarding the rotationalradial velocity is required in the evaluation of reg5 Equation (51) isreadily evaluated for axial modes in a cylindrical chamber by thesame method described for the ow-turning integral one nds forlow-order modes

reg5 rsquo 12

iexclfrac14 Mb

macrE2

m

centL=R (52)

which is equal to the ow-turning result but opposite in signThe growth rate contribution reg5 has been the source of much

dispute in the rocket combustion instability community (Brownprivate communication) It must therefore be scrutinized here infull detail It appears in Eq (51) in its natural form as an integralover the chamber volume When rst reported89 reg5 arose as a sur-face integral becauseCulickrsquos perturbedwave equationmethod wasused for computing the growth rate To display reg5 in its originalformat convert Eq (51) to surface integral form by application ofGaussrsquos divergence theorem Remembering that the rotational eldis solenoidal r cent Qu D 0 and using the vector identity for the diver-gence of a scalar times a vector one can write

r cent Qu Op D Qu cent r Op C Opr cent Qu D Qu cent r Op (53)

The conversion to a surface integral follows immediately one ndsZ Z

V

Z Qu cent r Op dV D

Z Z

V

Zr cent Qu Op dV D

Z Z

S

n cent Qu Op dS (54)

The growth rate contribution in surface integral form then becomes

reg5 D iexcl1

2E 2m

Z Z

S

n cent Qur m Opm dS (55)

This is precisely the term described in Eq (89) of Ref 8 as therotational ow correction for want of a better name The dif cultywith the surface integral form is that its evaluation implies detailedknowledgeof the normal componentof the rotationalvelocity at thesurface This approach was followed in Refs 8 and 9 and the ra-dial componentn cent Qu D Qur was deducedby integratingthe continuityequation11 It was found that the radial component of the rotationalvelocity Qur is proportionalto the mean ow Mach number does notvanish at the edge of the combustion zone and is proportional tothe unsteady pressure mode shape8911 These ndings lead to anunfortunate interpretationreported in Ref 8 that reg5 can be treatedas a correction to the surface coupling effect deduced in the irro-tational part of the stability calculation This forces one to searchfor new sources of mass ux within the ame zone Critics (Brownprivate communication) misinterpreted this observation to implythat changes in the surface response to pressure uctuations are im-plied The actual source of the new radial mass ux can be readilyidenti ed in the parallel gas motions within the combustion layerBecause there exists a momentum defect at the surface as a resultof the no-slip requirement additional oscillatory mass ux in theradial direction is generated The simple control volume shown inFig 5 describes the two-dimensional eld in the combustion zoneand illustrates these ideas

It is clear thatbecauseof thegradientin axialvelocity uctuationsparallel to the surface there is a net ux of mass into the verticalsidesof thecontrolvolumeThis is re ectedin the netunsteadymass ux at the edge of the combustion zone It is this additional radialmass ux that is accounted for in Eq (52) no modi cation of theoscillatory mass ux created by pressure coupling at the propellantside (ie solidgas interface) of the control volume is implied

Fig 5 Mass balance across the combustion zone

The formal approach just described carried out in detail fullyjusti es the results reported in the earlier papers89 Despite these ndings the combustion instability community is understandablywary of any suggestion indicating that a mechanism other than thetime-honoredpressurecouplingarises in the combustion zone Thispoint of confusion is partly resolved by using the volume integralform Eq (51) in place of the surface integral Eq (55) Both formsrepresent the same destabilizing in uence on system stability butthe volume integral is more easily evaluated and is therefore thepreferred form

It is also important to understand that the rotational ow correc-tion Eq (51) and the ow-turning integral Eq (51) are separateand distinct parts of the interaction of the acoustic eld with vor-ticity production Despite claims to the contrary Eq (51) is not analternative way to represent ow turning Brown has attempted toprove that the sign on reg5 given in Refs 8 and 9 should be reversed(Brown private communication) If he is correct in this assertionthen both contributions to the growth rate have the same sign Be-cause as shown in Ref 8 they have nearly equalmagnitudeBrownthen assumes that they must represent in reality the same physicaleffect He claims that Eq (51) duplicateswhat is already accountedfor by Eq (40) Because this assertion has resulted in confusionin the combustion stability research community it is necessary toresolve the matter here

Consider the combination of the ow turning and the rotational ow correction as represented by Eqs (44) and (51) Their sum is

reg4 C reg5 D1

2E2m

Z Z Z

V

microiexclMbUr

Quim

rC km Qur

m

parasinkm z dV

(56)

If the value for the derivative of the imaginary part from Eq (46)is used and Eq (31) is substituted for the real part of the rotationalaxial velocity then the integrand becomes

iexclMbUr Qui

m

rC km Qur

m D iexclMbUrkm

MbUrsin x exp Aacute sin Atilde

pound sinkm z sin x C km sin x exp Aacute sin Atilde sinkm z sin x D 0

(57)

This provesthat the ow turningis exactlycanceledby the rotational ow correction as asserted in Refs 8 and 9 that is

reg4 C reg5 D 0 (58)

This resultmight seemcontraryto some experimentalresultswhichapparently require that ow turning be included for acceptableagreement with the stability theory80 Contrary opinions have been

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

FLANDRO AND MAJDALANI 493

expressed by many other experimentalistswho nd that removal ofthe ow turning leads to better agreement with motor growth ratedata However several other rotational ow corrections remain tobe evaluated Let us reserve judgment until all of the pieces havebeen nally assembled

D More Rotational Growth Rate ContributionsIt is now necessary to tackle the remaining terms in Eq (19)

a seemingly daunting task None of these new terms has been ac-counted for in previous studies Fortunately some of them do notcontribute signi cantly to the system energy balance Again it isinstructive to examine these stability integrals individually

The second term on the third line in Eq (19) can be converted im-mediatelyto a surfaceintegralbyusingstandardvectoridentitiesandthe fact that the rotationalvelocity eld is solenoidalOne nds thatZ Z Z

V

hiexclMb Qu cent rU cent Oui dV DZ Z

S

hiexclMbn cent QuU cent Oui dS

D OiexclM 2

b

cent(59)

Because as already shown the normal velocity uctuation at thesurface is of order Mb this term is negligible

The third and fourth terms are most easily handled together Thevolume integral can be converted to the sum of a surface integraland a simpler volume integral The result isZ Z

V

ZhiexclMb Ou C Qu cent rU cent Qui dV D iexclMb

Z Z

S

hn cent uU cent Qui dS

C Mb

Z Z

V

ZhU cent Qur cent Oui dV (60)

where the surface integral is again secondorder in the Mach numberbecause the normal velocity uctuation is of the order of Mb Thesurface integral can be expanded into

iexclMb

Z Z

SN

hn cent uU cent Qui dS D iexclMb

Z Z

SN

hiexcl Ouz C QuzUz Quzi dS

D Mb

Z Z

SN

h OuzUz Quzi dS C Mb

Z Z

SN

h QuzUz Quzi dS

D Mb

Z Z

SN

laquoQu2

z Uz

notdS (61)

Both steady and unsteady analytical expressions deteriorate in theproximity of the nozzle where the ow suddenly accelerates be-comes compressible then choked at the throat Evaluating the sur-face integral is further exacerbated by the open area and geometricirregularity associated with different motors In view of these un-certainties Eq (61) has been routinely ignored

The remaining volumetric integral in Eq (60) must be evaluatedUsing the continuity equation with Eq (23) one writes

r cent Qu D iexcl Opt

D km Opm expregm t sinkm t C OMb (62)

and inserting Eq (25) one nds the growth rate to be

km Mb

2E2m

Z Z Z

V

poundU cent Qui

m

currenOpm dV

Dkm Mb

2E2m

(iexcl 2frac14 2

Z 1

0

cos x sin x exp Aacute cosAtilde

pound

Z L=R

0

z sinkm z sin x dz

dr

)(63)

The radial integralyieldsa valueof the orderof the mean Mach num-ber and so this term representsa negligiblegrowth ratecontribution

The fth term is highly interesting because it is the companionof the original ow-turning effect representedby Eq (40) Now wemust evaluate

reg6 D1

E2m exp2regm t

Z Z Z

V

hMb Qu cent U pound i dV (64)

where the rotationalvelocity appears in the place of the acousticve-locityas in the ow-turningintegralUsing Eq (42) for the unsteadyvorticity and dropping terms of the second order one nds

reg6 D iexclMb

4E2m

Z Z Z

V

Ur

r

poundQur cent Qur C Qui cent Qui

currendV (65)

Inserting the expressions for the rotational velocity and integratingover the volume one nds for a cylindrical chamber

reg6 Dfrac14 Mb

4E2m

L

R

Z 1

0

sin x

rsin2 x exp2 Aacute dr rsquo

frac14 Mb

4E2m

L

R(66)

This is anenergysourcerather thana sinkaswas thecasefor the owturning In a sense it representsan unexpected ndingConventionalanalyses indicate that the only energy source in the motor chamberis that produced by the pressure coupled combustion response allother stability integrals are thought to be sinks of energy This newsource term is clearly related to the creation of unsteadyvorticityatthe boundaries even when viscosity is discounted For this reasonreg6 will be referred to as the inviscid vortical correctionNotice thatreg6 has been evaluated here only for the vorticity produced at thesurface as a natural consequence of the no-slip condition in thecase of the axial acoustic mode If for example one takes accountof vortex shedding effects [including the parietal vortex sheddingmechanism (PVS)] this term must be evaluated with appropriatemodi cations to the vorticity vector and rotational unsteady owAdditional contributions to system instability from reg6 must then beexpectedFuture workwill be focusedon thesepotentiallyimportantinteractions

The sixth remaining term involves the volume integralZ Z Z

V

iexclMbhiexclQu cent Ou pound shy i dV (67)

This growth rate contributionis negligiblebecausethe crossproductyields an axial vector component proportional to the radial acousticvelocity Therefore only terms of second order in the mean owMach number are generated

The last two terms are viscousdampingexpressionsIn the classi-cal (irrotational)combustion instability calculationsviscous effectsare ignored completelyon the basis that there are no strong velocitygradients at the surface to give rise to signi cant shearing stressesAcoustic boundary-layer corrections of the usual sort do not prop-erlyaccountfor the shearingstresseswhen there is strongconvectionthrough the surface layer A correction to the dilatational (volumedamping) effect is represented in the seventh rotational term Usingthe same methods used to evaluate the other terms this one can betransformed into a surface integral namely

Z Z Z

V

laquoplusmn2

d Qu cent rr cent OunotdV D iexclplusmn2

d

Z Z

S

frac12n cent Qu

p1

t

frac34dS (68)

which for realistic values of the parametersmust be negligible be-causeboth the dimensionlessviscositycoef cient plusmn2

d and the normalvelocity at the bounding surfaces are of the order of the mean owMach number

Finally the last term represents the viscous damping

reg7 D1

E2m exp2regm t

Z Z Z

V

hiexclplusmn2 Ou C Qu cent r pound i dV (69)

where the composite unsteady velocity appears instead of just theacoustic part as in previous works [see Eq (95) in Ref 8] Aftercarrying out the indicated calculations and inserting the various

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

494 FLANDRO AND MAJDALANI

components of the velocityvectors the viscousgrowth rate reducesto

reg7 Dplusmn2

2E2m

Z Z Z

V

microQur

m

2 Qurm

r 2C Qui

m

2 Qui m

r 2

paradV

D iexclplusmn2

2E2m

sup3km

Mb

acute2 Z Z Z

V

r 2 exp2 Aacute sin2km z dV (70)

where smaller terms have been dropped This expression is easilyevaluated for a full-length cylindrical grain The result is

reg7 rsquo iexclfrac14plusmn2

6E2m

L

R

sup3km

Mb

acute2

D iexclfrac14 Mbraquo

6E2m

L

R(71)

to goodapproximationNotice that the importanceof viscousdamp-ing increasesrapidlywith frequencyBecausethe squareof theMachnumber appears in the denominator this term can be as importantas any of the others retained in the analysis Contrary to earlierassessments we nd that viscous damping must not be discardedespecially in the case of turbulent mean ows Then the transportproperties are modi ed and plusmn2 might be much larger than for thelaminar case To evaluate Eq (70) properly in the turbulent casea comprehensivenumerical algorithm will be needed Work of thissort has already begun101819

E Pseudopressure Growth Rate CorrectionsThe last two terms in Eq (19) are caused by interactionsbetween

the unsteady velocity eld and the pseudopressureassociated withthe vorticity-drivendisturbancesThese two terms need to be care-fully evaluated from

reg8 Diexcl1

E2m exp2regm t

Z Z Z

V

h Ou cent r Qpi dV (72)

reg9 Diexcl1

E2m exp2regm t

Z Z Z

V

h Qu cent r Qpi dV (73)

Using Eq (9) one nds

reg8 Diexclfrac14 Mb

4E2m

Z Z Z

V

sinkm z sin2x expAacute cosAtilde

pound fsin[km z sinx] C km z sinx cos[km z sinx]g dV

rsquoM3

b

E2m k2

m

L

R(74)

Clearly reg8 is small being of order M 3b The remaining pseudo-

rotational correction needs to be evaluated from

reg9 Dfrac14 Mb

4E2m

Z Z Z

V

sin2x sinx exp2Aacutefsin[km z sinx]

C km z sinx cos[km z sinx]g sin[km z sinx]

C km z sin2x sin2x exp2Aacute sin[km z sinx]

pound cos[km z sinx] dV rsquo9frac14 2 Mb

200E2m

L

R(75)

Unlike reg8 reg9 is of the same order as reg6 this new term constitutesanother destabilizing source that needs to be retained

Table 1 Physical parameters for typical motor systems

Motor L m R m Mb plusmn km S raquo Ar

b f Hz

Small motor (Yang and Culick) 060 0025 17iexcl3 549iexcl4 133iexcl1 7687 10309 25 1227Tactical rocket (Typical geometry) 203 0102 31iexcl3 274iexcl4 158iexcl1 5084 00624 12 360Space shuttle SRB 351 0700 23iexcl3 104iexcl4 627iexcl2 2724 00035 10 195

III DiscussionThe system stability is determined by superposition of a set of

stability integrals that includes both the original set found fromthe classical irrotational analysis and several new ones that rep-resent the effects of the rotational unsteady ow We must nowassess the impact of the proposed changes on stability assessmentmethodology

It is useful to compare the theoretical irrotational and rotationalcalculations for typical motor parameters Please note that accountis not taken of other gain loss effects such as particulate drag andresidual combustion that could be important in real rocket motorsThen for the classical model the growth rate is

regstandard D reg1 C reg2 C reg3 C reg4 (76)

which includes the ow-turning correction that is usually assumedand where evaluations are to be made using the irrotational energynormalization value

iexclE2

m

centstandard

D 12 frac14 L=R (77)

for the case of axial modes in a cylindrical combustion chamberFor the new stability calculation including additional rotational

corrections one must use

regcomposite D reg1 C reg2 C reg3 C reg6 C reg7 C reg8 C reg9 (78)

where the cancellation of the ow-turning effect and the rotational ow correction is accounted for furthermore evaluations must bemade using the corrected energy normalization value

iexclE2

m

centcomposite

D 58 frac14 L=R (79)

written again for axial modes in a full-length cylinderNo attempt is made here to compare numerical values to experi-

mental resultspublishedrecentlywhich come from nonlineardecaydata8081 It has been claimed that some measurements demonstratethat the standard acoustic stability results give acceptable predic-tion of the observed behavior It must be remarked that none ofthe experimental points illustrated are taken from sections of thedata where there was rapid growth in the waves only pulse decaydata were utilized The standard stability calculations did not pre-dict linear growth for any of the motors tested Careful study ofthe data reduction process suggests that the reason data points dis-playedshow unusuallyhigh damping is that it was necessaryto lterthe data to eliminate the strong harmonic content the presence ofhigher modes with quite appreciable amplitude accompanying the ltered rstmodedata indicatesnonlinearinteractionsthatcannotbeignoredLinear theory does not apply in the situationdescribed8081

A few simple calculations will demonstrate some features of theresults found here It is quite easy to apply the formulas for a cylin-drical chamber Table 1 from Ref 9 showing values for the keyparameters for representativemotors is reproducedhere for conve-nience Let us focus on large motors of the type described as SRBin the table As described in the Introductionthere have been manyobservationsof longitudinalmode instabilityin largemotors such asthe SRB These oscillationshave most often been attributed to vor-tex shedding phenomena In that model the natural instability of themean ow especiallywhen ow separation is present suggests thatlarge-scalevortex structures are generatedwhich can add energy tothe unsteady eld in the manner of a wind musical instrument

Let us test the result of the analysis given here by applying it toa simpli ed SRB geometry in which the grain is assumed to be along straight and unsegmented cylinder No measurements of thepropellant admittance function need to be attempted for the SRB

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

FLANDRO AND MAJDALANI 495

Table 2 Comparison of stability estimates

Motor Ar

b f Hz regstandard siexcl1 regcomposite siexcl1

Small motor 250 1227 200 131(Yang and Culick)

Tactical rocket 120 360 iexcl489 235(Typical geometry)

Space shuttle SRB 100 195 iexcl585 173

propellant because the natural frequencies in this long motor arelow (195-Hz rst mode notwithstandingthe effect of end cavities)Hence one can assume a typical (dimensionless) value of Ar

b rsquo 1for this situationStandardshortnozzledampingis alsoassumedNoattempt is made to account for particle damping because this effectis negligible for such a low frequencyof oscillationApplication ofthe standard stability code yields for this case

regstandard rsquo iexcl585 siexcl1 (80)

Being comparable to values that were computed during the devel-opment of this motor standard predictions lead to the impressionthat the SRB would be very stable For such large motors growth ordecay rates are always found to be small The result is misleadingbecause in practice signi cant thrust oscillations (raquo20 klbf peakto peak) are observed in static tests of all versions of the SRBsOscillations are also detected in ight with amplitudes sometimesexceeding 2 psi (peak to peak) If the new analysis is used insteadone nds

regcomposite rsquo 173 siexcl1 (81)

The new result reconciles with actual observations indicating thatstrong vorticity waves should be expected in this motor It is in-teresting that the growth rate predictions for the next two or threelongitudinal modes are of the same order of magnitude as the rstmode results shown In the SRB case three or four modes werealways readily discernible in the waterfall data

Table 2 shows comparisons of stability results ( rst axial acous-tic mode) for three of the con gurations de ned in Table 1 Typicaladmittancevalues are used and particulatedamping and other two-phase ow effects are ignored In all cases the new model predictsa less stable system This is due mainly to the effective cancellationof the ow-turningdamping term This greatly improves agreementbetween theory and experimentas demonstrated in Ref 8 In largermotors (eg Shuttle SRB) the new term reg6 also introduces a sig-ni cant change in system stability The new results predict unstablebehavior in agreement with much experimental data The originalSSPP calculationsindicated that the Shuttle and Ariane motors (andsimilar large solid rocket boosters) were not susceptible to oscilla-tory waves

IV ConclusionsThe new stability estimation method described in this paper dis-

plays new energy source terms not found in the acoustic instabil-ity methodology The new results help to explain the growth oflongitudinal oscillations in large solid-propellant motors that haveappeared in many development programs Considerable work liesahead in fully utilizing what we have found as the basis for a pre-dictive algorithm for use in motor design and in data reduction andinterpretation Although the general formulation is not geometrydependent we have relied heavily on the assumption of a simplecylindrical geometry with longitudinal plane wave oscillations toenable the evaluations shown

Clearly much work will be needed to determine the stability char-acteristics of waves in more realistic motor con gurations Exten-sion of the results to partial grains requires further study Complexgeometrical featuresof the burningport such as slots and ns repre-sent dif culties that might require a full numerical treatment of boththe steady and the unsteady ow by application of computational uid dynamics Inclusionof regions of separated ow for exampleat segment interfaces in large motors introduces additional com-plications Full NavierndashStokes solutions might eventually provide

the necessary information It is possible that new numerical simu-lations can be validated by use of the analytical models that havebeen described herein

The newresultsintroducedin thispapercanbe readilytestedusingan experimental apparatus already in place For instance the cold ow models used by French investigatorsin their Ariane work couldbe used to test the new theoretical framework Because only growthrates are predicted in the linearized theory it would be necessaryto modify the test procedure in such a way that transient behaviorcould be assessed

In the latter regard new emphasis on modeling of the nonlinearaspects of unsteady ow with rotational ow corrections is clearlyjusti ed There are no methods available for predicting importantfeatures of real motor operation such as limit-cycle amplitude ortriggering Certainly rotational ow effects will play a central rolein resolving these dif culties

Finally new stability integrals discovered in the present workin particular the inviscid vortical growth rate reg6 and the pseudo-pressure-rotational growth rate reg9 might have a bearing on theunresolved problem of velocity coupling Both growth rate termsrepresent distinct avenues by which velocity uctuations Qu affectthe system stability In fact the inviscid vortical term reg6 can be re-cast in the form of a surface integral and it is then obvious that aldquovelocity couplingrdquo response functioncan be connectedto it Thesematters require further study

AcknowledgmentsThis work was sponsored partly by the University of Tennessee

and partly by the California Institute of Technology Multidis-ciplinary University Research Initiative under Of ce of NavalResearch Grant N00014-95-1-1338 Program Manager JudahGoldwasser The rst author wishes to express appreciation for ad-ditional support from the Edward J and Carolyn P Boling Chairof Excellence in Advanced Propulsion University of TennesseeThe second author wishes to express appreciation for the FacultySeed and Higher Education Awards received from NASA and theWisconsin Space Grant Consortium under the direction and man-agement of R Aileen Yingst Sharon D Brandt Steven I Dutchand Thomas H Achtor

References1CulickFEC ldquoAcoustic Oscillations in SolidPropellantRocketCham-

bersrdquo Acta Astronautica Vol 12 No 2 1966 pp 113ndash1262Culick F E C ldquoRotational Axisymmetric Mean Flow and Damping of

Acoustic Waves in a Solid Propellant Rocketrdquo AIAA Journal Vol 4 No 81966 pp 1462ndash1464

3Culick F E C ldquoStability of Longitudinal Oscillations with Pressureand Velocity Coupling in a Solid Propellant Rocketrdquo Combustion Scienceand Technology Vol 2 No 4 1970 pp 179ndash201

4Culick F E C ldquoThe Stability of One-Dimensional Motions in a RocketMotorrdquo Combustion Science and Technology Vol 7 No 4 1973 pp 165ndash175

5Culick F E C ldquoStability of Three-Dimensional Motions in a RocketMotorrdquoCombustionScience and Technology Vol 10 No 3 1974 pp 109ndash124

6Lovine R L Dudley D P and Waugh R D ldquoStandardized StabilityPrediction Method for Solid Rocket Motorsrdquo Vols 1 2 and 3 Aerojet SolidPropulsion Co Rept AFRPL TR 76-32 San Jose CA May 1976

7Nickerson G R Culick F E C and Dang L G ldquoStandardized Sta-bility Prediction Method for Solid Rocket Motors Axial Mode ComputerProgramrdquo Software and Engineering Associates Inc AFRPL TR-83-017San Jose CA Sept 1983

8Flandro G A ldquoEffects of Vorticity on Rocket Combustion StabilityrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 607ndash625

9Flandro G A ldquoOn Flow Turningrdquo AIAA Paper 95-2530 July 199510Flandro G A Cai W and Yang V ldquoTurbulent Transport in Rocket

Motor Unsteady Flow eldrdquo Solid Propellant Chemistry Combustion andMotor Interior Ballistics edited by V Yang T B Brill and W-Z RenVol 185 Progress in Astronautics and Aeronautics AIAA Reston VA2000 pp 837ndash858

11Majdalani J Flandro G A and Roh T S ldquoConvergence of TwoFlow eld Models Predicting a Destabilizing Agent in Rocket CombustionrdquoJournal of Propulsion and Power Vol 16 No 3 2000 pp 492ndash497

12Majdalani J and Van Moorhem W K ldquoImproved Time-DependentFlow eld Solution for Solid Rocket Motorsrdquo AIAA Journal Vol 36 No 21998 pp 241ndash248

496 FLANDRO AND MAJDALANI

13Majdalani J ldquoBoundary-Layer Structure in Cylindrical Rocket Mo-torsrdquo AIAA Journal Vol 37 No 4 1999 pp 505ndash508

14Majdalani J ldquoVorticity Dynamics in Isobarically Closed PorousChan-nels Part 1 Standard Perturbationsrdquo Journal of Propulsion and PowerVol 17 No 2 2001 pp 355ndash362

15Majdalani J and RohTS ldquoVorticityDynamics in IsobaricallyClosedPorousChannels Part 2Space-ReductivePerturbationsrdquoJournalof Propul-sion and Power Vol 17 No 2 2001 pp 363ndash370

16Majdalani J and Roh T S ldquoThe Oscillatory Channel Flow withLarge Wall Injectionrdquo Proceedings of the Royal Society Series A Vol 456No 1999 2000 pp 1625ndash1657

17Vuillot F and Avalon G ldquoAcoustic Boundary Layer in Large SolidPropellant Rocket Motors Using NavierndashStokes Equationsrdquo Journal ofPropulsion and Power Vol 7 No 2 1991 pp 231ndash239

18Roh T S Tseng I S and Yang V ldquoEffects of Acoustic Oscilla-tions on Flame Dynamics of Homogeneous Propellants in Rocket MotorsrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 640ndash650

19Yang V and Roh T S ldquoTransient Combustion Response of SolidPropellant to Acoustic Disturbance in Rocket Motorsrdquo AIAA Paper 95-0602 Jan 1995

20Vuillot F ldquoNumerical Computation of Acoustic Boundary Layers inLarge Solid Propellant Space Boosterrdquo AIAA Paper 91-0206 Jan 1991

21Baum J D Levine J N and Lovine R L ldquoPulsed Instabilitiesin Rocket Motors A Comparison Between Predictions and ExperimentsrdquoJournal of Propulsion and Power Vol 4 No 4 1988 pp 308ndash316

22Baum J D ldquoInvestigation of Flow Turning Phenomenon Effects ofFrequency and Blowing Raterdquo AIAA Paper 89-0297 Jan 1989

23Glick R L and Renie J P ldquoOn the Nonsteady Flow eld in SolidRocket Motorsrdquo AIAA Paper 84-1475 June 1984

24Brown R S BlacknerA MWilloughbyP G and DunlapR ldquoCou-pling Between Acoustic Velocity Oscillations and Solid PropellantCombus-tionrdquo Journal of Propulsion and Power Vol 2 No 5 1986 pp 428ndash437

25Shaeffer C W and Brown R S ldquoOscillatory Internal Flow FieldStudiesrdquo United Technologies AFOSR CR F04620-90-C-0032San JoseCA 1990

26Shaeffer C W and Brown R S ldquoOscillatory Internal Flow StudiesrdquoChemical Systems Div TR 2060 FR United Technologies San Jose CAAug 1992

27CantrellR H and Hart R W ldquoInteraction Between Soundand FlowinAcoustic Cavities Mass Momentum and Energy Considerationsrdquo Journalof the Acoustical Society of America Vol 36 No 4 1964 pp 697ndash706

28Avalon G Casalis G and Griffond J ldquoFlow Instabilities and Acous-tic Resonance of Channels with Wall Injectionrdquo AIAA Paper 98-3218 July1998

29Lupoglazoff N and Vuillot F ldquoNumerical Simulations of ParietalVortex-Shedding Phenomenon in a Cold-Flow Set-Uprdquo AIAA Paper 98-3220 July 1998

30Lupoglazoff N and Vuillot F ldquoParietal Vortex Shedding as a Causeof Instability for Long Solid Propellant Motors Numerical Simulations andComparisons with Firing Testsrdquo AIAA Paper 96-0761 Jan 1996

31LupoglazoffN and VuillotF ldquoNumerical SimulationofVortex Shed-ding Phenomenon in Two-Dimensional Test Case Solid Rocket MotorsrdquoAIAA Paper 92-0776 Jan 1992

32Prevost M Vuillot F and Traineau J C ldquoVortex-Shedding DrivenOscillations in Subscale Motors for the Ariane 5 MPS SolidRocket MotorsrdquoAIAA Paper 96-3247 July 1996

33Scippa S Pascal P and Zanier F ldquoAriane 5 MPS -Chamber PressureOscillations Full Scale FiringsResults Analysis and Further Studiesrdquo AIAAPaper 94-3068 June 1994

34Fukushima Y ldquoH-Ii Launcher Propulsion Experiencerdquo 5th AAAFSymposium International on Propulsion in Space Transportation AAAFPaper 59-520 May 1996

35Brown R S Dunlap R Young S W and Waugh R C ldquoVortexShedding as a Source of Acoustic Energy in Segmented Solid RocketsrdquoJournal of Spacecraft and Rockets Vol 18 No 4 1981 pp 312ndash319

36Dotson K W Koshigoe S and Pace K K ldquoVortex Shedding in aLarge Solid Rocket Motor Without Inhibitors at the Segment InterfacesrdquoJournal of Propulsion and Power Vol 13 No 2 1997 pp 197ndash206

37Dotson K W Murdock J W and Kamimoto D K ldquoLaunch Vehi-cle Dynamic and Control Effects from Solid Rocket Motor Slag EjectionrdquoJournal of Propulsion and Power Vol 15 No 3 1999 pp 468ndash475

38Dotson K W Baker R L and Sako B H ldquoLaunch Vehicle Self-Sustained Oscillation from Aeroelastic CouplingPart 1 Theoryrdquo Journal ofSpacecraft and Rockets Vol 35 No 3 1998 pp 365ndash373

39Dotson K W Baker R L and Bywater R J ldquoLaunch Vehicle Self-Sustained Oscillation from Aeroelastic Coupling Part 2 Analysisrdquo Journalof Spacecraft and Rockets Vol 35 No 3 1998 pp 374ndash379

40Nesman T ldquoRSRM Chamber Pressure Oscillations Full Scale Groundand Flight Test Summary and Air Flow Test Resultsrdquo 31st AIAA JointPropulsion Conf AIAA Solid Rocket Technical Committee July 1995 pp27ndash48

41Hart R W and McClure F T ldquoCombustion Instability AcousticInteraction with a BurningPropellantSurfacerdquo JournalofChemical PhysicsVol 10 No 6 1959 pp 1501ndash1514

42Hart R W Bird J F Cantrell R H and McClure F T ldquoNonlinearEffects in Instability of Solid Propellant Rocket Motorsrdquo AIAA JournalVol 2 No 7 1964 pp 1270ndash1273

43Hart R W and McClure F T ldquoTheory of Acoustic Instability inSolid Propellant Rocket Combustionrdquo Tenth Symposium (International) onCombustion 1964 pp 1047ndash1066

44Culick F E C ldquoNon-Linear Growth and Limiting Amplitude ofAcoustic Oscillations in Combustion Chambersrdquo Combustion Science andTechnology Vol 3 No 1 1971 pp 1ndash16

45Culick F E C BurnelyV S and SwensonG ldquoPulsed Instabilities inSolid-PropellantRocketsrdquo Journalof Propulsionand Power Vol 11 No 41995 pp 657ndash665

46Culick F E C ldquoSome Recent Results for Nonlinear Acoustics inCombustion Chambersrdquo AIAA Journal Vol 32 No 1 1994 pp 146ndash168

47Culick F E C and Yang V ldquoPrediction of the Stability of UnsteadyMotions in Solid Propellant Rocket Motorsrdquo Nonsteady Burning and Com-bustion Stability of Solid Propellants edited by L De Luca E W Priceand M Summer eld Vol 143 Progress in Astronautics and AeronauticsAIAA Washington DC 1992 pp 719ndash779

48Culick F E C ldquoRotational Axisymmetric Mean Flow and Dampingof Acoustic Waves in a Solid Propellant Rocketrdquo Journal of Propulsion andPower Vol 5 No 6 1989 pp 657ndash664

49Avalon G and Comas P ldquoSimulative Study of the Unsteady FlowInside a Solid Propellant Rocket Motorrdquo AIAA Paper 91-1866 June 1991

50Vuillot F ldquoVortex-Shedding Phenomena in Solid Rocket MotorsrdquoJournal of Propulsion and Power Vol 11 No 4 1995 pp 626ndash639

51Traineau J C Prevost M and Lupoglazoff N ldquoA Subscale TestProgram to Assess the Vortex Shedding Driven Instabilities in SegmentedSolid Rocket Motorsrdquo AIAA Paper 97-3247 July 1997

52Vuillot F Traineau J C Prevost M and Lupoglazoff N ldquoExper-imental Validation of Stability Assessment Methods for Segmented SolidPropellant Motorsrdquo AIAA Paper 93-1883 Jan 1993

53Couton D Doan-Kim S and Vuillot F ldquoNumerical Simulation ofVortex-Shedding Phenomenon in a Channel with Flow Induced ThroughPorous Wallrdquo International Journal of Heat and Fluid Flow Vol 18 No 31997 pp 283ndash296

54Tissier P Y Godfroy F and Jacquemin P ldquoCFD Analysis of VortexShedding Inside a Subscale Segmented Motorrdquo AIAA Paper 94-2781 June1994

55Tissier P Y Godfroy F and Jacquemin P ldquoSimulation of Three Di-mensional Flows Inside Solid Propellant Rocket Motors Using a SecondOrder Finite Volume MethodmdashApplication to the Study of Unstable Phe-nomenardquo AIAA Paper 92-3275 July 1992

56Traineau J C Hervat P and Kuentzmann P ldquoCold-Flow Simulationof a Two-Dimensional Nozzleless Solid-Rocket Motorrdquo AIAA Paper 86-1447 July 1986

57Vuillot F and Lupoglazoff N ldquoCombustion and Turbulent Flow Ef-fects in 2-D Unsteady NavierndashStokes Simulations of Oscillatory RocketMotorsrdquo AIAA Paper 96-0884 Jan 1996

58Vuillot F Dupays J Lupoglazoff N Basset T and Daniel Eldquo2-D NavierndashStokes Stability Computations for Solid Rocket Motors Rota-tionalCombustionandTwo-PhaseFlowEffectsrdquo AIAA Paper97-3326July1997

59Ugurtas B Avalon G Lupoglazoff N and Vuillot F ldquoNumericalComputations of Hydrodynamic Instabilities Inside Channels with WallInjectionrdquo AIAA Paper 99-2505 June 1999

60Casalis G Avalon G and Pineau J-P ldquoSpatial Instability of PlanarChannelFlowwith Fluid Injection ThroughPorousWallsrdquoPhysics ofFluidsVol 10 No 10 1998 pp 2558ndash2568

61Griffond J Casalis G and Pineau J-P ldquoSpatial Instability of Flowin a Semiin nite Cylinder with Fluid Injection Through Its Porous WallsrdquoEuropean Journal of Mechanics BFluids Vol 19 No 1 2000 pp 69ndash87

62Griffond J and Casalis G ldquoOn the Dependence on the Formulationof Some Nonparallel Stability Approaches Applied to the Taylor FlowrdquoPhysics of Fluids Vol 12 No 2 2000 pp 466ndash468

63Griffond J and Casalis G ldquoOn the Nonparallel Stability of theInjectionInducedTwo-DimensionalTaylorFlowrdquoPhysics ofFluids Vol 13No 6 2001 pp 1635ndash1644

64Varapaev V N and Yagodkin V I ldquoFlow Stability in a Channel withPorous Wallsrdquo Fluid Dynamics Vol 4 No 5 1969 pp 91ndash95

65Ugurtas B Avalon G Lupoglazoff N Vuillot F and Casalis GldquoStability and Acoustic Resonance of Internal Flows Generated by Side In-jectionrdquo Solid Propellant Chemistry Combustion and Motor Interior Bal-listics edited by V Yang T B Brill and W-Z Ren Vol 185 Progress inAstronautics and Aeronautics AIAA Reston VA 2000 pp 823ndash836

66Chu B-T and Kovasznay L S G ldquoNon-Linear Interactions in aViscous Heat-Conducting Compressible Gasrdquo Journal of Fluid MechanicsVol 3 No 5 1957 pp 494ndash514

FLANDRO AND MAJDALANI 497

67Morse P M and Ingard K U Theoretical Acoustics McGrawndashHillNew York 1968 pp 278 279

68Barron J Majdalani J and Van Moorhem W K ldquoA Novel Investi-gationof the Oscillatory Field over a Transpiring Surfacerdquo Journalof Soundand Vibration Vol 235 No 2 2000 pp 281ndash297

69Brown R S Blackner A M Willoughby P G and Dunlap RldquoCouplingBetween Velocity Oscillations andSolidPropellantCombustionrdquoAIAA Paper 86-0531 Jan 1986

70Majdalani J and Van Moorhem W K ldquoLaminar Cold-Flow Modelfor the Internal Gas Dynamics of a SlabRocketMotorrdquoJournalof AerospaceScience and Technology Vol 5 No 3 2001 pp 193ndash207

71Majdalani J ldquoVortical and Acoustical Mode Coupling Inside a Two-Dimensional Cavity with Transpiring Wallsrdquo Journal of the Acoustical So-ciety of America Vol 106 No 1 1999 pp 46ndash56

72Kirchoff G Vorlesungen Uber MathematischePhysik Mechanik 2nded Teubner Leibzig 1877 pp 311ndash336

73Apte S and Yang V ldquoEffects of Acoustic Oscillations on TurbulentFlow eld in a Porous Chamber with Surface Transpirationrdquo AIAA Paper98-3219 July 1998

74Beddini R A and Roberts T A ldquoTurbularization of an AcousticBoundary Layer on a Transpiring Surfacerdquo AIAA Journal Vol 26 No 81988 pp 917ndash923

75BeddiniR A and Roberts T A ldquoResponse of PropellantCombustion

to a Turbulent Acoustic Boundary Layerrdquo Journal of Propulsionand PowerVol 8 No 2 1992 pp 290ndash296

76Thomas H D FlandroG A and Flanagan SN ldquoEffects of Vorticityon Particle Dampingrdquo AIAA Paper 95-2736 July 1995

77Dupays J Prevost M Tarrin P and Vuillot F ldquoEffects of ParticulatePhase on Vortex Shedding Driven Oscillations in Solid Rocket MotorsrdquoAIAA Paper 96-3248 July 1996

78Sabnis J S de Jong F J and Gibeling H J ldquoCalculation of ParticleTrajectories in Solid-Rocket Motors with Arbitrary Accelerationrdquo Journalof Propulsion and Power Vol 8 No 5 1992 pp 961ndash967

79Culick F E C ldquoRemarks on Entropy Production in the One-Dimensional Approximation to Unsteady Flow in Combustion Cham-bersrdquo Combustion Science and Technology Vol 15 No 4 1977pp 93ndash97

80Blomshield F S Crump J E Mathes H B Stalnaker R A andBeckstead M W ldquoStability Testing of Full-Scale Tactical Motorsrdquo Journalof Propulsion and Power Vol 13 No 3 1997 pp 349ndash355

81Blomshield F S Mathes H B Crump J E Beiter C A andBeckstead M W ldquoNonlinear Stability Testing of Full-Scale Tactical Mo-torsrdquo Journal of Propulsion and Power Vol 13 No 3 1997 pp 356ndash366

J P GoreAssociate Editor