Unsteady Flow Evolution and Combustion Dynamics of ...web.engr.oregonstate.edu/~sva/archive/apte_candf_SRM03.pdf · A time-resolved numerical analysis of combustion dynamics of double-base

Unsteady Flow Evolution and Combustion Dynamics ofHomogeneous Solid Propellant in a Rocket Motor

SOURABH APTE and VIGOR YANG*Department of Mechanical and Nuclear Engineering, The Pennsylvania State University,

University Park, PA 16802, USA

A time-resolved numerical analysis of combustion dynamics of double-base homogenous solid propellant in arocket motor is performed by means of a Large-Eddy Simulation (LES) technique. The physiochemicalprocesses occurring in the flame zone and their influence on the unsteady flow evolution in the chamber areinvestigated in depth. A five-step reduced reaction mechanism is used to obtain the two-stage flame structureconsisting of a primary flame, a dark zone, and a secondary flame in the gas phase. It is observed that, forhomogeneous solid propellant combustion, the chemical time scale is much greater than the smallest turbulencetime scale, rendering a highly stretched and thickened flame. The chemical reactions proceed at a slower ratethan turbulent mixing, and propellant combustion may be locally treated as a well-stirred reactor. The flowfieldin the chamber consists of three regions of evolution: the upstream laminar regime, the central transitionalsection, and the fully developed turbulent regime further downstream. A theoretical formulation exploring thechamber flow and flame dynamics is established to study the intriguing phenomenon of combustion instability.The work done by Reynolds stresses, vorticity-flame interactions, and coupling between the velocity field andentropy fluctuations may cause resonance effects and excite pressure oscillations leading to self-sustainedunsteady motions within the chamber. © 2002 by The Combustion Institute

NOMENCLATURE

a ! speed of soundCp ! constant-pressure specific heat of gasDij ! binary mass diffusivity

Dim ! effective molecular mass diffusivityof species i

Ei ! activation energye ! specific total energy

es ! specific sensible energyf ! frequency

G ! spatial filterhi ! specific enthalpy of species ihs ! total specific sensible enthalpyI ! turbulence intensity, ! u"u" ! v"v"

L ! chamber lengthm ! mass flow rateN ! number of species in gas-phase reac-

tionsPr ! Prandtl numberp ! pressureq ! rate of heat release per unit volumeR ! chamber radius

Ru ! universal gas constantSc ! Schmidt numberSij ! strain-rate tensor, 1

2(("ui/"xj) # ("uj/" xi)) $ 1

3 #ij("uk/" xk)

St ! Strouhal numbers ! entropy

sL ! laminar flame speedT ! temperaturet ! time

u, v ! bulk velocities in axial and verticaldirections, respectively

u1, u2 ! bulk velocities in axial and verticaldirections, respectively

u1, u2 ! diffusion velocities in axial and ver-tical directions, respectively

u ! velocity vectorvw ! radial injection velocity at propellant

surfaceV ! specific volume

W! mix ! molecular weight of mixtureWi ! molecular weight of species i

x, y ! axial and radial coordinates, respec-tively

x1, x2 ! axial and radial coordinates, respec-tively

Yi ! mass fraction of species i% ! filter width#ij ! Kronecker delta$ ! length scale of small turbulent eddy% ! thermal conductivity& ! density' ! viscous stress( ! dynamic viscosity*Corresponding author. E-mail: [email protected]

COMBUSTION AND FLAME 131:110–131 (2002)0010-2180/02/$–see front matter © 2002 by The Combustion InstitutePII S0010-2180(02)00397-8 Published by Elsevier Science Inc.

) ! kinematic viscosity*i ! rate of production of species i& ! vorticity

Subscripts

0 ! head endc ! centerlinei ! species ip ! propellants ! sensible

Superscripts

a ! periodic fluctuationt ! turbulent fluctuation" ! sum of periodic and turbulent fluctu-

ationsr ! resolved scales ! unresolved (subgrid) scale˜ ! density-weighted quantity! ! time-averaged quantity

INTRODUCTION

This paper extends our previous work on com-bustion dynamics of double-base, homogeneoussolid propellants in a rocket motor [1], to per-form a comprehensive analysis of the resultsobtained from the Large-Eddy Simulation(LES). Emphasis is placed on the gas-phasephysiochemical processes characterizing the mi-croscale motions above the propellant surfaceand their effect on the macroscale motions inthe bulk of the chamber. A theoretical formu-lation exploring the chamber flow and flamedynamics is developed to study the intriguingphenomenon of combustion instability. The re-sponse of the gas-phase flame to unsteady flowevolution in the chamber is addressed in depth.

Combustion instabilities in solid-propellantrocket motors have been explored extensivelyover the past few decades using analytical, ex-perimental, and numerical techniques [2]. Thedynamic coupling between propellant combus-tion and fluctuating chamber flow may lead tooscillations of heat-release distribution. Theturbulence-enhanced heat and mass transferrates modify the flame structure, which in turnaffects the transient combustion response of

propellant. The amplitudes of these distur-bances increase due to the resonance effect andare eventually limited by dissipation to renderlimit-cycle oscillations [3]. Many efforts havebeen made to analyze these driving mechanismsseparately. Investigation of their collective ef-fects in realistic rocket environments, however,has not been attempted.

Most of the earlier work in exploring the flowevolution in a rocket chamber was based oncold-flow studies with injection of inert gasesthrough the sidewalls of the chamber simulatingthe gas influx from the burning propellant. Inthis idealized configuration, Taylor [4] and laterCulick [5] obtained analytical expressions forthe velocity distributions in laminar incompress-ible flows. The study was later extended byBalakrishnan et al. [6] to include the effects ofrotationality and compressibility. Dunlap et al.[7] and Traineau et al. [8] conducted experi-ments for incompressible and compressibleflows, respectively, to quantify the turbulencecharacteristics and mean velocity transitions inthe chamber. Three regimes of flow evolution:the upstream laminar, transitional, and down-stream turbulent regions, were identified. Bed-dini [9] and Sabnis et al. [10] applied turbulenceclosure models to obtain numerical results ofthe flowfield. The acoustic-wave excitation insolid rocket motors by vortex shedding wasaddressed by Flandro [11] and further investi-gated by Vuillot [12] in their cold-flow analyses.Recently, Apte and Yang [13, 14] performedtime-resolved numerical simulations based onLES techniques to investigate the behavior ofunsteady motor flowfields under conditions withand without externally imposed oscillations.The frequency spectra of flow oscillations ob-tained from these simulations were in qualita-tive agreement with the stability analysis per-formed by Casalis et al. [15] and Ugurtas et al.[16] for the parietal vortex-shedding in a chan-nel with side-wall injection. Significant insightinto the energy-transfer mechanisms among themean, turbulent, and periodically oscillatoryflowfields was obtained. The periodic oscilla-tions give rise to energy exchange between theorganized and turbulent motions, in addition tothe well-established interactions between themean and turbulent flowfields [14, 18]. Thisconsequently produces enhanced turbulence in-

111COMBUSTION DYNAMICS OF SOLID PROPELLANT IN ROCKET MOTOR

tensity and leads to earlier laminar-to-turbu-lence transition than that observed in stationaryflows without acoustic forcing. The turbulence-enhanced momentum transport, on the otherhand, promotes effective eddy viscosity andtends to dissipate the vortical wave generatingfrom the injection surface. These LES-basedcold-flow studies demonstrated their effective-ness in exploring unsteady motions in rocketchambers. It was also established that turbu-lence effects can best be modeled using LES forreacting flows to achieve the credibility of pre-dictions from numerical simulations.

Although there have been several experimen-tal and numerical studies on non-reacting flowsin rocket motors, obtaining quantitative data forburning solid propellant under realistic condi-tions is a formidable challenge because of thecomplexities involved in the flowfield. For dou-ble-base homogeneous propellants, the flamestandoff distance is around 1 mm at 25 atm. anddecreases with increasing chamber pressure,leading to practical difficulties in obtaining ac-curate experimental data. The predictive capa-bility of time-resolved numerical analysis is thusimportant to obtaining deep insight into thecombustion phenomenon. In an effort to studythe detailed coupling between propellant com-bustion dynamics and local flow oscillations inrocket motors, Yang and co-workers [19–22]conducted a series of numerical investigationsinto homogeneous propellant combustion un-der conditions representative of practical pro-pulsion systems. Turbulence closure wasachieved using a two-layer model calibrated fornon-reacting motor flows. Much informationhas been obtained about the flame structure,heat-release mechanism, propellant combustionresponse, and flow development. Results indi-cate that the oscillatory flow characteristicswere significantly altered because of the turbu-lence-enhanced mass and energy transport inthe gas phase. As a consequence, the energyreleased in the flame zone and propellant com-bustion response were modified and causedsignificant changes in the motor stability behav-ior. Apte and Yang [1] recently extended theturbulence model adopted in those analyses byincluding a LES-based treatment. The intricateinteractions among the acoustic, vortical, andentropy waves in a motor were examined.

The present work is a sequel to our previousstudy [1] and establishes a methodology toanalyze the various mechanisms responsible fordriving unsteady motions in a rocket motor. Insubsequent sections, a theoretical formulationincluding chemical kinetics models and govern-ing equations for time-resolved simulation ofsolid-propellant combustion in a cylindricalchamber is established. The interactions be-tween turbulence and chemistry are then de-scribed in detail, followed by a brief summary ofthe numerical technique used in the analysis.Results obtained from the numerical simulationare presented. Finally, the self-sustained flowfluctuations within the chamber are analyzed,with special attention given to the effect ofturbulence on the gas-phase flame dynamics.

THEORETICAL FORMULATION

Figure 1 shows a schematic diagram of thecylindrical combustion chamber analyzedherein, which is loaded with a double-basepropellant grain along the entire azimuth andconnected downstream to a choked convergent-divergent nozzle. The propellant undergoesdegradation in a thin superficial layer under-neath the burning surface. The pyrolysis prod-ucts are injected into the combustion chamberand react to form a multistage flame in the gasphase [20].

Chemical Kinetics Model

Owing to the difficulties in establishing a com-plete chemical kinetics scheme and limitationsof computational resources, a thorough consid-eration of all physical and chemical processes doesnot appear feasible. A reduced reaction mecha-

Fig. 1. Schematic diagram of a solid rocket motor.

112 S. APTE AND V. YANG

nism is therefore used to describe the combustionwave structure in both the gas and condensedphases [20]. The scheme has been well establishedand provides reasonably accurate informationabout the major chemical kinetic pathways. Thechemical behavior of the condensed phase isdescribed by two reactions: solid propellant de-composition (R1) and reaction of NO2 (R2),providing elementary species such as NO2, NO,and aldehydes to maintain the gas-phase flames.

double-base propellant3 2.49 NO2

! 2.36 CH2O ! 1.26 'CHO(2

! 0.17 CO ! minor residuals (R1)

NO2 # 0.56 CH2O # 0.16 (CHO)2

3 NO # 0.38 CO # 0.5 CO2

# 0.5 H2O # 0.22 H2 (R2)

The stoichiometric coefficients in reaction (R1)are determined based on the composition of a“hot” double-base propellant containing 52%nitrocellulose (NC), 43% nitroglycerin (NG),and 5% minor additives. The heat of combus-tion of the propellant is about 1100 cal/g.

The gas-phase chemical kinetics is specifiedby five reactions: two first-order NO2 and NO-carbon reactions, and three second-order reac-tions involving aldehydes and NO. These reac-tions represent the most important and sloweststeps limiting the reaction rates in the gas phase:

NO2 # 0.56 CH2O # 0.16 (CHO)23 NO

# 0.38 CO # 0.5 CO2

# 0.5 H2O # 0.22 H2 (R3)

CH2O # CH2O3 CO # 0.5 C2H4

# H2O (R4)

(CHO)2 # (CHO)23 4CO # 2 H2 (R5)

NO # 0.23 CO # 0.15 C2H4

3 0.5 N2 # 0.53 CO2 # 0.17 H2O

# 0.13 H2 (R6)

C # NO3 CO # 0.5 N2 (R7)

The rates of production of each constituentspecies in the gas and condensed phases and the

physiochemical properties are given in Refs. 17and 20. In the present work, emphasis is placedon the gas-phase combustion dynamics to ex-plore the physiochemical processes occurring nearthe propellant surface and their overall effect onthe complex flow physics within the chamber. Adetailed numerical analysis involving the dynamiccoupling between the condensed and gas phaseshas been studied for both laminar and turbulenceflows using second-order turbulence closure mod-els [21, 22], and can be extended further toperform LES-based computations.

Gas-Phase Governing Equations

The five-step combustion model described aboveis important to predict the two-stage flame struc-ture accurately. The finite-rate chemical kineticsand steep gradients near the burning surface, onthe other hand, require small time-steps for nu-merical stability. Because of the intricacy of theproblem, which involves a wide range of lengthand time scales, and the limitations of computa-tional resources, the analysis deals only with theconservation laws in axisymmetric coordinates.This treatment lacks thevortex-stretchingphenom-enon commonly observed in turbulent flows andmay under-predict the turbulence production anddissipation rates. The model, nonetheless, allowsfor a systematic investigation into the interactionsbetween propellant combustion and motor flowdevelopment. Much useful information can beobtained about the effect of turbulence on flamedynamics, especially in the near-surface region. Arecent study of rocket motor internal flow basedon a comprehensive three-dimensional, LES tech-nique [17, 23] indicates that the two-dimensionalsimulation indeed can capture the salient featuresof turbulent flows and leads to good agreement withexperimental data in terms of mean flow proper-ties and acoustic-wave induced flow oscillations.

A spatial filter G is employed to decomposethe flow variables into a large (resolved) and asubgrid (unresolved) scale

ℑ'x, t( + ℑr'x, t( ! ℑs'x, t(

with ℑr'x, t(

+ "D

G'x , x!, %(ℑ'x!, t( d3x" (1)


where D is the entire domain, % the filtersize which determines the size of the un-resolved scale, and ℑ any flow property:&, u, v, T, or Yi. The superscripts r and srepresent the resolved and unresolved scalesof flow properties, respectively. Favre averag-ing is further used to simplify the governingequations for compressible turbulent-flowsimulation.

ℑr +'&ℑ(r

&r (2)

The filtered conservation equations for axi-symmetric configuration can be written asmass

"' x2&r(

"t!

"' x2&rukr (

" xk+ 0 (3)

axial momentum

"' x2&ru1r (

"t!

"' x2&ru1r u1

r (

" x1!

"' x2&ru1r u2

r (

" x2+ $

"' x2pr(

" x1!

"' x2'11r (

" x1!

"' x2'12r (

" x2!

"' x2-u1u1(

" x1

!"' x2-u1u2

(

" x2, #2

3"

" x1'(u2(

r$ (4)radial momentum

"' x2&ru2r (

"t!

"' x2&ru2r u1

r (

" x1!

"' x2&ru2r u2

r (

" x2+ $

"' x2pr(

" x2!

"' x2'21r (

" x1!

"' x2'22r (

" x2!

"' x2-u2u1(

" x1

!"' x2-u2u2

(

" x2, #pr ,

43 %(

u2

x2& r

!23 %(

"u1

" x1& r

,23 %u2

"(

" x2& r$ (5)

energy

"' x2&resr(

"t!

"' x2&rhsruk

r (

" xk+ $

"' x2qkr (

" xk,

"' x2-ukh(

" xk,

"' x2-ukulul(

" xk!

"' x2ul'kl(r

" xk, '

i!1

N

hfi

0*ir

! #$23

"

" x1'(u1u2(

r ,23

"

" x2'(u2

2(r$ (6)

species concentration

"' x2&rYir(

"t!

"' x2&rYiruk

r (

" xk+ $

"' x2qikr (

" xk

,"' x2-ukYi

(

" xk! *i

r (7)

where the terms in curly brackets represent theresolved part of the axisymmetric source terms.For a multi-component mixture, the pressurepr, specific sensible internal energy es

r, andspecific sensible enthalpy hs

r follow the defini-tions given below

pr + &rRuTr 'i!1

N Yir

Wi

esr + '

i!1

N "Tref

T

YirCpi dT ,

pr

&r !u1

r 2 ! u2r 2

2(8)

hsr + '

i!1

N "Tref

T

YirCpi dT !

u1r 2 ! u2

r 2

2

Note that instead of using the conventional“total energy” formulation, the energy equationis written with heat of reaction ($¥i!1

N hfi

0*ir) as


a source term, for which accurate experimentaldata is available [20].

The shear and normal stresses are given by

'klr + (%"uk

r

" xl!

"ulr

" xk,

23 %"ui

r

" xi!

u2r

x2#kl&& (9)

The diffusion term qkr consists of contributions

from heat conduction and mass diffusion pro-cesses.

qkr + $%

"Tr

" xk! &r '

i!1

N

'hiYiuk(r (10)

The species diffusion term qik

r is calculated usingFick’s law

qikr + &rYiuk + $&rDim

"Yir

" xk(11)

where Dim is the effective molecular mass-diffusion coefficient for species i.

The terms -ukul, -ukh, and -ukYm

are the sub-grid scale (sgs) stresses, heat fluxes, and speciesmass fluxes, respectively. These terms, along withthe triple correlation -ukulul

, the viscous dissipationrate (ul'kl)

r and the filtered chemical sourceterms, require closure models. A Smagorinskymodel extended to reacting flows is used to com-pute these terms [24]. The sgs-stresses are given as

-ukul+ $&r''ukul(

r , ukr ul

r(

+ 2CR&r%2)S1/ 2% Skl

r ,13

Siir #kl&

,23

CI&r%2)S#kl (12)

where % is the average size of the computationalcell. CR(*0.01) and CI(*0.007) are the modelconstants based on Erlebacher et al. [25]. Thequantities )S, Skl

r are defined, respectively, as

)S+ Sklr Skl

r (13)

Sklr +

12 %"uk

r

" xl!

"ulr

" xk& ,

13

#kl%"uir

" xi& (14)

The sgs heat and species mass fluxes are simi-larly obtained as

-ukh + $&rCpCR

PrT%2)S

1/ 2 "Tr

" xk(15)

-ukYi+ $&r CR

ScT%2)S

1/ 2 "Yir

" xk(16)

The turbulent Prandtl and Schmidt numbers(PrT and ScT) are assigned a value of 0.4 [25].The triple correlation -ukulul

accounts for theinteractions between the resolved velocity vec-tor and the sgs kinetic energy, and is modeled asfollows [26]:

-ukulul+ $ul

r%2CR&r%2)S1/ 2% Skl

r ,13

Siir #kl&&

(17)

Turbulence/Flame-Structure Interaction

The reaction rates *i in the species-concentra-tion equations need to be modeled to capturethe effect of subgrid scales on chemical reac-tions. Peters [27, 28] examined the problem ofturbulence/chemistry interactions for premixedflames in terms of three non-dimensional pa-rameters: turbulent Reynolds number Ret, tur-bulent Damkohler number Da, and turbulentKarlovitz number Ka, as defined below

Ret ()"

sL

!t

!F(18)

Da +tt

tF(

!t/)"

!F/sL(19)

Ka +tF

tK( %)"

sL&3/ 2% !t

!F&$1/ 2

(20)

where tt, tF, and tK represent the turbulent-eddy, characteristic flame, and Kolmogorovtime scales, respectively. The turbulentDamkohler and Karlovitz numbers are ex-pressed in terms of laminar flame thickness !F,flame speed sL + vw, turbulent length scale !t,and turbulence intensity v". Figure 2 (takenfrom Peters [27, 28]), commonly referred to asthe regime diagram for premixed turbulentcombustion, shows the variation of v"/sL with!t/!F. The Damkohler number on this log-logplot corresponds to the inverse slope of a line ascan be seen from Eq. 19. Also shown are twoconstant Karlovitz-number lines, Ka ! 1 and100, where Ka# ! 100 Ka corresponds to theKarlovitz number based on the inner reaction


layer, !# ! #!F. Peters [28] argues that forpremixed flame structures there exist a chemi-cally inert preheat zone with thickness of theorder of the flame thickness and a thin reactionlayer (called inner layer) where the majority ofthe heat release occurs. The inner layer isfollowed by a post-flame oxidation layer wherethe final products are formed. For hydrocarbonpremixed flames, # + 0.1 typically and the thininner layer controls the overall flame structure.

Four different regimes in premixed turbulentcombustion are identified based on the relativemagnitudes of these parameters, as illustratedin Fig. 2. If chemical reactions proceed at rateslower than those of turbulent mixing (i.e., Da ,1), and the flame stretch is strong enough tocause penetration of small eddies into the flamezone (i.e., Ka - 1), then the chemical kineticsbecome sufficiently slow to result in thickflames. For Ret - 1, Ka - 1, and Ka , 100,the inner reaction layer is thin, !F - $ - !#. Inthis situation, the small eddies which cannotpenetrate into the reaction zone can enter thepreheat zone to increase scalar mixing andconsequently destroy the quasi-steady flameletstructure that exists in the corrugated flameletregime (Ka , 1). The flamelet theory can stillbe applied to this regime and agrees well withthe flamelet boundary obtained in numericalstudies of Poinsot et al. [29], where two-dimen-

sional interactions between a laminar premixedflame front and a vortex pair were analyzed. ForKa - 100, the flames become thick and !F -!# - $. Small eddies can now penetrate into thereaction zone, and modify the flame structure. ForKa - 100 and Da , 1, turbulence is weak and nodirect interactions between turbulence and com-bustion occur in this regime. The flame structurecan be locally modeled as a well-stirred reactor.Turbulence, however, modifies the mixing ratethrough eddy viscosity and diffusivity.

In Fig. 2, the Da and Ka numbers are esti-mated for different regimes of the two-stageflame structure of homogeneous solid propel-lant combustion. Accordingly, estimates of thelaminar and turbulent velocity scales are ob-tained in the turbulent regime for the presentsimulation conditions (described later). For thehead-end pressure of 2.5 MPa, the propellantregression rate and density are rb ! 4.8 mm/sand &p ! 1620 kg/m3, respectively. From themass balance across the interface between thecondensed and gaseous phases (&prb ! &gvw),the gas-phase injection velocity (vw) can beobtained. This injection velocity is in fact thelaminar flame speed (sL). Typically, for &g *12.2 kg/m3, the flame speed is +0.8 m/s. Asshown later in the paper, the secondary flamethickness and peak turbulence intensity in thefully turbulent regime are approximately 1 mmand 70 m/s, respectively. Based on these esti-mates, typical Karlovitz (Ka) and Damkohler(Da) numbers are in the range of 100–200,0.2–0.4, respectively. These are represented inthe premixed-flame phase diagram. Tseng andYang [19] also corroborated that for double-basehomogeneous propellants, Ka is large and Da , 1.They “locally” modeled the turbulence/flame-structure interactions as well-stirred.

As shown later in the present study, theturbulent Reynolds, Damkohler, and Karlovitznumbers vary in the axial direction. Their radialdistributions in the turbulent flame region, how-ever, fall in the regime of Da , 1 and Ka -100. Under these conditions, turbulence rapidlypenetrates into the flame zone through en-hanced mass transfer, leaving chemical reac-tions as the rate-controlling processes. Accord-ingly, the closure for reaction rates follows thestandard approximation of resolved reactionrates [17, 24]

Fig. 2. Phase diagram showing different regimes in pre-mixed turbulent combustion.


*ir + *i

r'&, T, Y1, Y2, . . . , YN(

+ *i'&r, T r, Y1r, Y2

r, . . . , YNr ( (21)

The above model allows for fluctuations in therates of species production as functions of in-stantaneous flow properties, and thus is lessstringent than that used in the second-orderturbulence-closure scheme [19].

Boundary Conditions

The present work focuses on gas-phase flamedynamics, with the propellant surface condi-tions specified based on their values at thechamber head-end. The surface properties arefirst obtained by solving the coupled species-concentration equations in the gas and con-densed phases, with proper interfacial boundaryconditions conserving mass and energy at thehead-end of the chamber [17, 20]. Results offlame structure and temperature agree well withexperimental data over a wide range of pres-sure. The burning rate, surface temperature,and species concentrations remain fixedthroughout the entire motor accordingly. Fol-lowing the method of characteristics, the exitplane requires no physical boundary condition,since the flow is supersonic downstream of thenozzle throat. At the upstream boundary, theaxial velocity and pressure gradient, as well asthe radial velocity gradient, are set to zero alongthe solid wall. The last condition is required toprevent the occurrence of a numerically pro-duced recirculating flow at the head end [19].The mass burning rate, total temperature, andspecies mass fluxes are fixed at the propellantsurface. Normal injection of the pyrolysis prod-ucts is enforced by employing the no-slip bound-ary condition. Finally, flow symmetry is assumedat the centerline. In order to perturb the flow-field to obtain transition to turbulence, pseudo-turbulence in the form of white noise is intro-duced at the propellant surface, with spatial andtemporal fluctuations in the mass-flow ratewithin 1% of the mean value.

NUMERICAL METHOD

One characteristic trait of solid-propellantrocket motor internal flowfields is that the Mach

number varies from zero at the head end tounity at the nozzle throat as the flow acceleratesrapidly in the downstream region. In the low-Mach number region, the inviscid compressibleform of the conservation equations is poorlycoupled and stiff [30]. The associated disparityamong the eigenvalues results in significantslowdown in convergence. Chemical reactionsexhibit another category of numerical difficul-ties because of the wide ranges of time andlength scales involved. In regions where reac-tion rates are high, species mass concentrationsmay vary rapidly over a short period of time,and numerical stability may require an enor-mous number of iterations. To avoid this stiff-ness problem, chemical source terms are usuallytreated implicitly, in a method analogous topreconditioning the time derivative terms of thespecies conservation equations so that all chem-ical and convective processes proceed at ap-proximately the same numerical rate.

In the present work, a preconditioning tech-nique along with the dual-time stepping integra-tion procedure described in Refs. 17 and 30 isfirst used to establish a stable flame and elimi-nate the initial transients in the flowfield. Atime-accurate, semi-implicit Runge-Kuttascheme with fourth-order central differencingfor spatial discretization is then employed be-cause the preconditioning method is computa-tionally expensive and difficult to parallelize. Adetailed description of the numerical schemeand parallel implementation is provided in Ref.17.

RESULTS AND DISCUSSION

Figure 1 shows the subject physical model,consisting of an axisymmetric rocket motor con-nected downstream to a choked nozzle. Thechamber measures 55 cm in length and 2 cm indiameter, and is loaded with a double-basehomogeneous solid propellant grain (53% NCand 42% NG and minor additives) over theentire azimuth. The high aspect ratio chosenhere ensures occurrence of turbulence in thedownstream region. The nozzle throat area isset based on a contraction ratio of 1.25, whichleads to high Mach numbers and low pressuresin the downstream region. The head-end pres-


sure is around 2.5 MPa. Although it is lowerthan the usual operating range for solid propel-lant rocket motors, it serves two purposes. First,the lower gas-phase density renders a higherinjection velocity and a shorter chamber fortransition to turbulence in the downstream.Second, the higher injection rate increases theflame standoff distance and thus reduces thedegree of grid-stretching required to resolve theprimary flame. The mass flux at the propellantsurface is 8.1 kg/m2s, and the correspondinginjection Reynolds and Mach numbers are 104

and 1.85 . 103, respectively. The surface tem-perature is fixed at 640 K.

The steep temperature gradient near the pro-pellant surface in the gas and condensed phasesdictates the numerical grid resolution requiredto accurately capture the combustion wavestructure. A detailed grid dependence study wasperformed to obtain the optimal grid spacing inthe radial direction near the surface. To predictthe burning surface conditions by solving thecoupled gas- and condensed-phase conservationequations with appropriate interfacial condi-tions, the smallest grid spacing on the order of 1(m is necessary at the propellant surface [20–22]. This allows accurate computation of thesteep temperature gradient at the surface andprovides correct heat feedback to the con-densed phase to determine the burning surfaceproperties. In the present study, however, wefocus on the gas-phase flame dynamics, and thecondensed-phase effects appear through speci-fied surface parameters as boundary conditions.A coarser grid is found to be sufficient to obtainthe correct behavior of the flame structure [17].Accordingly, 650 . 150 grids are used in theaxial and radial directions, respectively. Thesmallest grid spacing normal to the surface isabout 10 (m. Good resolution in the axialdirection is necessary to resolve the turbulencekinetic-energy spectrum correctly. A time-stepof 50 nanoseconds is employed to achieve nu-merical stability and accurate temporal resolu-tion of the flame dynamics. Statistically mean-ingful turbulence properties are acquired bytime-averaging numerical results over a span of15 ms (about 3–4 flow-through times) afterobtaining a stationary solution for the flamestructure.

Vortical Dynamics

Figure 3 shows the time evolution of the azi-muthal vorticity field in the combustion cham-ber. Only the lower half of the chamber ispresented to facilitate discussion, where y/R ! 1corresponds to the burning surface. Vorticity isproduced at the propellant surface because ofthe no-slip condition [11]. At the injection sur-face, &z ! ("vr/" x $ "ur/" y)w. As the flowaccelerates in the axial direction, pressure anddensity decrease while the injection velocity(vw) increases to keep the mass flow rate (mw)constant. The axial variation of vw, however, isnot significant and the vorticity is mainly dic-tated by the radial gradient of the axial velocity.Near the head-end, the fluctuations in vorticityappear to be small and the flow is mostlylaminar. Transition to turbulence occurs aroundx/R ! 25 and the flow becomes highly turbulentfurther downstream. The magnitude of vorticityat the burning surface increases non-linearlyalong the axial direction and the size of thevortical structures grows in the downstream.The upstream vortical disturbance originatingclose to the wall overtakes a neighboring down-stream disturbance since it is convected at ahigher speed. As these vortical structures travelaway from the wall, their size grows and resultsin large-scale mixing in the downstream.

The evolution of vorticity is governed by theCrocco-Vazsonyi equation which can be derivedby taking the curl of the momentum equation.

D!

Dt+ '! " /(u , !'/ " u( , /V . /p

! )/2! , (/V . '/ . !( !43

(/V

. /'/ " u( (22)

where ! is the vorticity vector defined as / . u,and V ! 1/& the specific volume. In an axisym-metric flow without heat release and viscouseffects, the vorticity of a given fluid particle isconserved. In the present case, the conservationproperty of vorticity is no longer valid becauseof viscous dissipation, $(/(V) . (/ . !) #43 (/(V) . /(/ " u), volume dilatation,$!(/ " u), and baroclinicity, $/V . /p, re-sulting from the misalignment between the den-


Fig. 3. Time evolution of vorticity field.

Fig. 6. Snapshots of temperature, heat-release, and NO mass-fraction fields, t ! 21.7 ms.


sity and pressure gradients. Vorticity created atthe propellant surface is convected by the meanflow and is transported by viscous diffusion(v/2!).

Figure 4 shows the snapshots of the baroclinicand dilatation effects that modify the vorticitytransport within the chamber. In the flame zone,density varies dramatically so that volume dila-tation and baroclinicity become predominantand viscous dissipation gives negligible contri-bution to the vorticity evolution. The decreasein density across the flame zone implies that thevolume dilatation, $!(/ " u), acts as a sinkterm in Eq. 22, decreasing the vorticity level inthe flame zone. Similarly, the baroclinic effectattempts to diminish the vorticity in the flamezone. The present axisymmetric computationlacks the vortex-stretching mechanism, (! " /)u,responsible for transfer of energy from the largeto the small scales through the energy cascadeprocess, and leads to lower dissipation and pro-duction rates. Nevertheless, it provides much use-ful insight into the flow development and itsinteractions with the propellant flame dynamics,which was lacking in previous numerical studiesusing second-order turbulence closure schemes.

Figure 5 shows the power density spectra ofaxial velocity fluctuations (u"), normalized withthe maximum centerline velocity at the chamberexit (u! r), at two radial locations in the fullyturbulent regime, x/R ! 40. Here, the fluctuat-ing variable is obtained by subtracting the time-mean value from its instantaneous quantity. The

peak magnitude of u" increases as the flowaccelerates in the axial direction [17]. The dis-tinct peaks at frequencies of 2.2, 3.1, and 6.1kHz may correspond to the combined effects ofhydrodynamic instability causing vortex-shed-ding and acoustic resonance within the motor.The halving of the dominant frequency awayfrom the surface is indicative of the vortex-pairing mechanism commonly observed in tur-bulent shear flows. The same phenomenon wasfound by Dunlap et al. [7] in their cold-flowexperiments in a porous chamber with surfacemass injection simulating propellant burning.Recently, Casalis et al. [15] and Ugurtas et al.

Fig. 4. Effects of baroclinicity and volume dilatation on evolution of vortical flowfield, t ! 21.7 ms.

Fig. 5. Power density spectra of axial velocity fluctuations attwo radial locations, x/R ! 40.


[16] conducted linear stability analyses and ex-perimental measurements of non-reacting injec-tion-driven flows in porous chambers. Thepower density spectra obtained from cold-flowsimulations by Apte and Yang [13, 23] were inqualitative agreement with these theoreticalpredictions. The Strouhal number, defined asSt ! fR/vw to characterize the parietal vortexshedding in the chamber, was found to bearound 6 for pure hydrodynamic instabilitieswithout acoustic resonance. For the presentreacting-flow simulation, however, the Strouhalnumber is around 10 and varies significantlywithin the combustion chamber because ofsteep density and temperature gradients in theflame zone. The deviation in the Strouhal num-ber may be attributed to the flame-vortex inter-actions modifying the vorticity evolution in theflame regime. The local Reynolds number ischanged significantly by the presence of theflame, which alters the hydrodynamic instabilitycharacteristics. The ability of LES to predictsuch unsteady flowfields must be explored fur-ther in order to reveal important combustioninstability phenomena in rocket chambers. Theenergy spectra are also shown on a log-log scaleand compared with the Kolmogorov-Obukhovspectrum (i.e., the $5/3 law) obtained for ho-mogeneous turbulence at large Reynolds num-bers. The cut-off frequency of the present cal-culation lies in the inertial subrange of theturbulent energy spectrum, further ensuringthat the large-scale motions are correctly re-solved.

Flame Dynamics

The temporal evolution of the temperature fieldand associated heat-release distribution givesdirect insight into the turbulent combustionmechanism. The instantaneous heat-release isgiven by the source term ($¥i!1

N hfi

0*ir) in the

energy equation. Figure 6 shows snapshots ofthe temperature, heat-release, and NO mass-fraction fields. In the laminar regime, the tem-perature increases rapidly within 0.25 mm from640 K at the surface to around 1600 K at the endof the primary flame, which is followed by adark zone of thickness 1.2 mm. The tempera-ture further rises to its final value of 2850 Kdownstream of the secondary flame. The flame

fluctuates significantly in the turbulent regimeowing to the vigorous vortical motions and thedark-zone thickness is substantially reduced.The intensive heat-release contours indicate afluctuating flame front corresponding to thesecondary flame zone, whose thickness is of theorder of 1 mm. The highly exothermic reductionof NO to N2 (R6) provides the major heatsource in the secondary flame zone. NO is firstformed near the surface through the reaction ofNO2 and aldehydes (R2 and R3). Turbulence-enhanced mixing results in rapid depletion ofNO in the dark zone because of increased localtemperature and the release of a large amountof energy. NO and CO then react in the sec-ondary flame zone to give the final products.

To explore the microscale motions above thepropellant surface, a close-up view of the flamestructure and vorticity contours is shown in Fig.7. For the present flow conditions, the temper-ature gradient at the burning surface remainsunaffected even in the turbulent regime. Small-scale fluctuations in vorticity, temperature,heat-release, and NO mass-fraction near thesurface indicate that turbulence does not pene-trate significantly to modify the primary flamestructure. For higher Reynolds numbers, how-ever, turbulence may penetrate into the primaryflame structure and consequently alter the pro-pellant combustion response, a phenomenoncommonly referred to as erosive burning [19].

In the present analysis of homogeneous-pro-pellant combustion dynamics, the reaction-zonethickness was shown to be larger than theKolmogorov length scale and turbulent combus-tion can be locally modeled as a well-stirredreactor (see Fig. 2). To further verify this treat-ment, radial distributions of the turbulent Reyn-olds, Karlovitz, and Damkohler numbers wereobtained from the present simulations. It shouldbe, however, noted that Ret, Ka, and Da num-bers were related to the flow, chemical andturbulent time scales. The chemical time andlength scales may differ from the flame-charac-teristic scales, and thus Ret, Ka, and Da areobtained locally from Ret ! k2/)/, Da !(k//)/(*i/&), and Ka ! (*i/&)0//), where kand / represent the turbulent kinetic energy andits dissipation rate, respectively. These are ob-tained locally from numerical computations andare shown in Fig. 8 for a typical location of


x/R ! 45. The turbulent-eddy and Kolmogorovtime scales are approximately 1 ms and 5 (s,respectively. The magnitudes of the Damkohlerand Karlovitz numbers vary along the axial

direction, however, their radial distributions inthe turbulent regime shows that Da , 1 andKa - 100 throughout the two-stage flame zone.Both numbers approach their limiting values

Fig. 7. Close-up view of vorticity, temperature, heat-release, and NO mass-fraction fields immediately above propellantsurface, t ! 21.7 ms.

Fig. 14. Distributions of mean heat release and normalized Rayleigh’s parameter.


near the end of the secondary flame zone,leading to a highly stretched and thickenedflame. The flame extends considerably into thechamber and thus can not be approximated as athin sheet. It should be noted that the order ofmagnitude of Ka is much higher than the earlierestimate (shown in Fig. 2) owing to the fact thatchemical time scales were used in the presentdefinitions. For high chamber pressures, how-ever, the flame thickness decreases and thewell-stirred reactor assumption may becomequestionable.

Mean Flowfield

The mean flowfield is obtained by performing atime average of the instantaneous quantitiesover 10 to 12 ms after a stationary flame isobtained, to achieve statistically meaningfuldata. Figure 9 shows the contour plots of themean Mach number and pressure. The Mach

number increases along the centerline almostlinearly from zero at the head end to 0.6 at thenozzle entrance, and the flow becomes super-sonic in the divergent section of the nozzle. Theflowfield clearly exhibits a two-dimensionalstructure, with the Mach number near the sur-face having a much smaller value than that atthe centerline. The situation with pressure,however, is substantially different. The pressuredecreases from the head end to the nozzle exitas the flow accelerates as a result of the propel-lant burning. The variations in the radial veloc-ity because of the presence of the flame aresmall and do not affect the radial pressuredistribution. Thus, an almost uniform distribu-tion of the chamber pressure in the radialdirection is obtained [19, 20].

Figure 10 shows the radial velocity profiles atdifferent axial locations, normalized with thesurface injection velocity at the head end (vw,0).The analytical solution obtained by Taylor [4]and Culick [5] for an inviscid, incompressiblecold flow is also provided for comparison. Itfollows a sinusoidal distribution and deviatesdrastically from that in the present reacting flowcase. In the upstream region (e.g., x/R ! 18, 27),as the gas particle leaves the burning surface,the rapid temperature increase in the primaryflame causes the volume to dilate nearly isobari-cally. Consequently, the fluid element acceler-ates in the radial direction to maintain massconservation. The radial velocity further in-creases in the secondary flame zone and thendecreases to zero at the centerline because offlow symmetry. In the highly turbulent region

Fig. 8. Distributions of turbulent Reynolds number,Damkohler number, and Karlovitz number within the flame,x/R ! 45.

Fig. 9. Contour plots of mean Mach number and pressure.

Fig. 10. Radial distributions of radial velocity at variousaxial locations.


(e.g., x/R ! 50), the radial velocity increasesrapidly close to the surface and then decreasesmonotonically towards the centerline. The dif-ference in the flowfield can be attributed to thepartial merging of the primary and secondaryflames in the turbulent regime, elevating thedark-zone temperatures and thus increasing thepeak magnitude of the radial velocity. Thedistributions of the radial velocity in the primaryflame zone are almost identical throughout thechamber. This suggests that despite the strongturbulent flow in the downstream region, turbu-lence does not penetrate into the primary flameand consequently exerts little influence on thenear-surface velocity profile.

The variation of the axial velocity in the radialdirection resembles that of a non-reacting injec-tion-driven flow in a porous chamber [13]. Theflowfield is basically determined from the bal-ance between the inertial force and the pressuregradient arising from mass injection at theburning surface. The viscous stress is negligiblecompared with pressure force, except in a nar-row region where x/y + O(1/Re) near thehead-end of the chamber. As the flow under-goes transition to turbulence, the turbulentshear stress causes the axial velocity profiles togradually steepen near the propellant surfaceand flatten in the core region. The effect ofturbulence on the axial velocity near the burn-ing surface is not significant, further suggestingthat turbulence effects near the burning surfaceare minimal.

Effect of Turbulence on Flame Structure

To study the effect of turbulence-enhancedmixing on the flame structure, variations of themean temperature and species mass fractionswere obtained. They are shown in Figs. 11 and12, respectively. A two-stage flame structureconsisting of the primary-flame, dark, and sec-ondary-flame zones is clearly observed in theupstream laminar region. The flame character-istics such as the thickness of various zones andspecies and temperature distributions agreewell with experimental data as well as theone-dimensional analysis [17]. The overall reac-tion mechanism for the propellant burning canbe globally grouped into the following steps: 1)molecular degradation and ensuing exothermic

reactions of NO2 and aldehydes in the subsur-face layer; 2) generation of NO, CO, CO2, andH2O, as well as removal of NO2 and aldehydes,in the primary flame; and 3) reduction of NO toform the final products, such as N2, CO2, H2,and so forth, in the secondary flame [20]. Thethickness of the dark zone depends on thechemical-reaction time and the flow-convectiontime dictated by the radial velocity distribution.In the turbulent region, the enhanced thermal

Fig. 11. Radial distributions of temperature at various axiallocations.

Fig. 12. Radial distributions of species mass fractions inlaminar and turbulent regimes.


diffusion facilitates NO reduction, which nor-mally occurs at elevated temperatures due to itslarge activation energy. This process leads to arapid temperature rise at the end of the primaryflame and consequently a thinner dark zone.The temperature plateau around 2000 K mayresult from the increased radial velocity, whichessentially reduces the local flow-residence timerequired for completing chemical reactions.

Figure 13 shows the radial variations of theaxial and radial components of the turbulenceintensity at various axial locations. Severalpoints should be noted here. First, the onset ofturbulence occurs in regions away from thepropellant surface because of the blowing effectcreated by mass injection from the burningsurface [13]. The peak of turbulence intensitytends to move away from the primary flamezone along the chamber. One factor contribut-ing to this phenomenon is the intricate couplingamong the flow convection, chemical reaction,and turbulent mixing in the downstream flamezone, as discussed in connection with Figs. 10and 11. Second, because the flow is dominatedby strain rates in the axial direction, the axialturbulence intensity appears to be much greaterthan its radial counterpart. Third, the chamberpressure of 25 atm at the head end in thepresent study is lower than the pressure of 65atm studied in Ref. [19]. The resultant massburning rate and injection Reynolds numberbecome smaller so that the turbulence intensitynear the surface is too weak to exercise anyinfluence on the flame structure. Hence, themean temperature profile in the primary flame

zone remains unchanged even in the turbulentregime.

Effect of Unsteady Heat Release

The classical approach based on Rayleigh’s cri-terion has demonstrated that under the pres-ence of combustion instabilities, the unsteadyfluctuations in heat release are usually in phasewith the pressure fluctuations. This enables thefluctuating component of heat release to con-tribute energy to the amplification of the pres-sure oscillations, thereby, leading to self-sus-tained unsteady motions within the motor.Mathematically, the local Rayleigh parameterR(x) is defined as

R'x( +1- "

t0

t0#r

p"'x, t(q"'x, t( dt (23)

where - is the time period of oscillation, andp"(x, t) and q"(x, t) are the pressure andheat-release fluctuations, respectively. A posi-tive value of R(x) causes local amplification anda negative value leads to damping of oscillatorymotions. Integrating Eq. 23 over the entire flowdomain gives the global Rayleigh parameter R*.

Figure 14 shows the distributions of the meanheat-release and normalized Rayleigh parame-ter R(x), respectively. In the upstream laminarregime, a large amount of heat release occurs inthe primary and secondary flame zones. Theenhanced heat and mass transfer rates becauseof turbulent motions in the downstream, how-ever, cause the primary and secondary flames topartially merge and spread the heat release overa thicker region above the propellant surface.The peak value of heat release in the turbulentregime decreases because of reduced flamestiffness. Similarly, the distribution of the Ray-leigh parameter indicates drastic differencesbetween the laminar and turbulent regions. Thepressure and heat-release fluctuations are posi-tively correlated in the primary flame zonethroughout the motor, rendering a positive dis-tribution close to the propellant surface. In thelaminar region, the Rayleigh parameter pla-teaus in the dark zone. The positive and nega-tive peaks in the secondary flame zone suggest adipole source for driving acoustic instability. In

Fig. 13. Axial and radial components of turbulence intensityat various axial locations.


the transition region (around x/R ! 27), onepositive and two negative peaks are clearlyvisible. The positive peak shifts closer to theprimary flame in the fully turbulent regimeowing to the partial merging of the primary andsecondary flames, causing amplification of pres-sure fluctuations near the burning surface.

Figure 15 shows the axial variation of theRayleigh parameter R( x)/R*, obtained by inte-grating R( x, y) in the radial direction. Twoimportant observations are made. First, themagnitude of the spatially averaged Rayleighparameter is positive throughout the chamber,indicating strong driving mechanisms for un-steady motions within the motor. Second, themagnitude of the Rayleigh parameter decreasesfrom the head end toward the transition region,and becomes almost constant in the fully devel-oped turbulent regime. This may be partiallyattributed to the existence of an acoustic pres-sure node in the transition regime. The pressurefluctuation of the first longitudinal modereaches its minimum in the mid-section of themotor. The enhanced heat and mass transferbecause of turbulence further downstream,however, widens the flame and the associatedheat-release distribution, compared with thelaminar flowfield. The combined effect of pres-sure and heat-release fluctuations is reflected inthe Rayleigh parameter. Accordingly, the Ray-leigh parameter decreases from the laminar tothe turbulent region. The positive distributionof R( x)/R* throughout the motor indicates thesustenance of unsteady motions within thechamber.

Analysis of Unsteady Motions

The unsteady flow motions in the chamber canbe decomposed into three components: the acous-tic, vorticity, and entropy waves. The acousticwave travels at the speed of sound, while thevorticity and entropy waves are convected down-stream at the local fluid velocity. These waves mayinteract with each other to obtain self-sustainedeigenmode oscillations at certain distinct frequen-cies. Our previous numerical studies on the evo-lution of injection driven non-reacting flows [13,14] suggest the existence of several mechanisms ofenergy transfer between the acoustic/vorticalwaves and turbulent fluctuations. In the pres-ence of an unsteady flame, entropy waves aregenerated and modify the above mechanisms.Such interactions are complex and separation ofthese waves is a formidable task. To explorethese mechanisms using the data obtained fromnumerical simulations, a generalized waveequation in terms of pressure fluctuations is firstderived from the conservation equations writtenin the following vector form [3],

&"u"t

! &'u " /(u + $/p ! F (24)

"p"t

! 'u " /( p + $0p'/ " u( ! P (25)

with

F + / " -( (26)

P +RC)

'Q , / " q ! '-( " /(u( (27)

where Q is the rate of heat release because ofchemical reactions, q the heat flux, and -( theviscous stress tensors

-(ij+ (%"ui

" xj!

"uj

" xi& ,

23

(#ij/ " u (28)

Assuming constant properties, a flow variable ℑis decomposed into the mean ℑ! , periodic (i.e.,deterministic or organized) fluctuation ℑa, andturbulent motion ℑt,

ℑ + ℑ! ! ℑa ! ℑt. (29)

The triple decomposition is obtained by defin-ing “long-time” average ℑ! and “short-time”average 1ℑ2 as follows

Fig. 15. Axial variation of Rayleigh’s parameter.


ℑ! +1

T3"

t0

t0#T3

ℑ't ! '( d' (30)

1ℑ2 +1T "

t0

t0#T

ℑ't ! '( d' (31)

where T is much smaller than T3, but muchlarger than the characteristic time scales ofturbulent motions. T3 is much larger than theacoustic time scale. The organized and turbu-lent fluctuations can then be obtained as follows

ℑa + 1ℑ2 , ℑ! (32)

ℑt + ℑ , ℑ! , ℑa (33)

The total time-varying quantity ℑ" is the sum-mation of the periodic and turbulent motions

ℑ" + ℑa ! ℑt (34)

Application of the decomposition defined inEqs. 29 and 34 to the governing Eqs. 24 and 25,leads to the following wave equation:

/2p" ,1a! 2

"2p"

"t2 + ℜ (35)

where a! ! 00RT! is the speed of sound and ℜthe source term involving fluctuations in the

velocity, heat-release, entropy, and pressurefields. By taking the short-time average of Eq.31, the wave equation for organized motions isobtained:

/2pa ,1a! 2

"2pa

"t2 + 1ℜ2. (36)

where the source term 1ℜ2 can be classified intofive major groups of mechanisms affecting pres-sure oscillations in the chamber.

1ℜ2 + I ! II ! III ! IV ! V (37)

with

I + $/ " %&!"ua

"t & !1a! 2

"

"t'u! " /pa(

!1a! 2

"

"t'ua " /p! ( !

0

a! 2"

"t' p! / " ua(

!0

a! 2"

"t' pa/ " u! ( (38)

II + $/ " '&! 'u! " /(ua( , / " %pa

a! 2 'u! " /(u!&! / " %&! sa

Cp'u! " /(u!& (39)

III + $/ " '1&! 'u! " /u!(2( ! / " % &!Cp

'u! " /(1s"u!2& ! / " %)&! s"

Cpu! " /u!*& , / " %)p"

a! 2 'u! " /(u!*&, / " %)p"

a! 2 'u! " /(u!*& (40)

IV + $/ " %)p"

a! 2"u!

"t *& ! / " %)&! s"

Cp

"u!

"t *&!

1a! 2

"

"t1'u! " /( p"2 !

0

a! 2"

"t1' p"/ " u!(2

(41)

V + $1a! 2

"Pa

"t! / " Fa (42)

The entropy fluctuation s" can be determined interms of pressure and density oscillations as follows:

s" + Cp% p"

0p!,

&"

&! & (43)

The first group (I) represents temporal de-velopment of the correlations between the or-ganized and mean flowfields. The three terms ofthe second group (II) represent the effect of themean flowfield on the transport of the periodicmotions representing the acoustic, vortical, andentropy waves, respectively. At higher Machnumbers, the mean convective velocity becomescomparable with the acoustic speed and thetransport of organized motions by the meanflowfield may become critical. The group (III)represents second-order correlations among theoscillatory velocity, pressure, and entropy fields.These terms are negligible for laminar flows, butbecome significant for transitional and fully


developed turbulent flows. The group (IV) rep-resents temporal evolution of second-order cor-relations. Finally, the effects of heat release,thermal diffusion, and viscous dissipation onpressure fluctuations are represented by thegroup (V). The above wave equation can beused to perform systematic analysis of the vari-ous interaction mechanisms leading to unsteadymotions within the chamber.

Figure 16 shows the time histories of fluctu-ating pressure, heat-release, and entropy in theturbulent secondary flame zone (x/R ! 43 andy/R ! 0.7). The pressure fluctuation p" is nor-malized by the mean head-end pressure, whilethe mean heat release at the propellant surfaceis used to normalize q". The magnitude of thepressure oscillation is around 2% of the meanchamber pressure. A broad-band turbulent sig-nal superimposed with peaks corresponding tocertain periodic motions is obtained for pres-sure and heat-release fluctuations. The pressurefluctuations are primarily obtained due to thevolume dilatation effects produced by the oscil-latory flame (i.e., group V). The entropy fluctu-ation s" obtained from Eq. 43 also indicatesstrong correlations with p" and q". Turbulenceplays an important role in exciting a range offrequencies corresponding to the hydrodynamicinstability, and triggering certain eigenmodeoscillations within the chamber (groups III andIV).

Quantitative information regarding the cou-pling mechanisms is obtained from the powerdensity spectra of fluctuations in pressure, axial

velocity, heat-release, and entropy, as shown inFigs. 17 and 18, respectively. The spectra indi-cate a range of frequencies being triggered inthe laminar (x/R ! 8), transitional (x/R ! 27),and fully turbulent regions (x/R ! 43). In theupstream laminar regime, the axial velocity fluc-tuation and turbulence intensity level are notsignificant, but the radial flame oscillation andits ensued heat-release variation lead to self-sustained pressure fluctuations. High-frequencypressure oscillations around 57 kHz (not shownhere), corresponding to the first radial mode ofthe acoustic wave, are also observed because ofthe rapid flame fluctuations caused by the sur-face-generated pseudo turbulence. The similar-ities between the pressure and heat-releasespectra suggest strong interactions among them.

The magnitude of axial velocity fluctuationincreases in the transitional and fully turbulentregimes with the peak frequency of 3.1 kHzcorresponding to the hydrodynamic instability.These vortex-shedding frequencies vary in theaxial direction as the local Reynolds numberincreases. In the downstream turbulent regime,low frequency oscillations (i.e., 850 and 1900Hz) are triggered and resemble the longitudinalacoustic nodes of the motor. Their magnitudes,however, are much smaller than the hydrody-namic oscillation at 3.1 kHz, possibly because ofthe effective dissipation of acoustic waves. Thefirst term on the right hand side of Eq. 40indicates that the correlation / " (&!1u! " /)u!2) inthe fully turbulent regime may sustain large-scale pressure oscillations within the chamber.

Fig. 16. Time histories of fluctuating pressure, heat-release, and entropy at x/R ! 43 and y/R ! 0.7.


The peaks at 2.2 and 3.1 kHz in the spectra ofentropy fluctuations may be attributed to thetransport of hot spots across the flame zone bylarge-scale vortical motions.

Summary of Unsteady Flow Evolution

Figure 19 summarizes the unsteady vorticity,heat-release, entropy, and pressure fields in thecombustion chamber. The vorticity fluctuations

because of turbulent motion are primarily in-duced by the oscillatory flame and are con-vected downstream by the mean flowfield. Themagnitude of vorticity fluctuation grows rapidlyas the flow undergoes transition to turbulence.Fluctuations in heat-release are related to thevariations in convection and chemical timescales. As the velocity field fluctuates, the con-vection time changes rapidly. The chemical re-action time, on the other hand, is almost unaf-

Fig. 17. Power density spectra of pressure and axial-velocity fluctuations at various axial locations, y/R ! 0.7.

Fig. 18. Power density spectra of heat-release and entropy fluctuations at various axial locations, y/R ! 0.7.


fected in the present case as the combustion ofdouble-base propellants is locally treated as awell-stirred reactor. The heat-release fluctua-tion, thus, follows the oscillatory vorticity field.The entropy field is derived from its staterelationship with pressure and density. Varia-tions in density are closely related with the heatrelease in the flame zone through the volumedilatation effect. In the present case of distrib-uted reactions, entropy fluctuations occurring atvarious spatial locations are convected down-stream, and then interact with strong gradientsin the mean velocity field and excite acousticoscillations. As in the mean pressure field, theoscillatory pressure is predominantly one-di-mensional and exhibits large variations in theaxial direction. Turbulence plays an importantrole in triggering and modifying these waves.

The above phenomena were analyzed in detailby introducing a generalized wave equation interms of pressure fluctuations. Triple decomposi-

tion of the instantaneous pressure field into the“long-time” average, acoustic, and turbulent fluc-tuations indicated several source terms involvingcomplex interactions among the acoustic, vortical,and entropy waves as well as turbulent motions. Asystematic data deduction based on power-densityspectra of pressure, heat-release, velocity, andentropy fluctuations was performed to explorecombustion dynamics of homogeneous solid pro-pellants. The present chemical and turbulent modelscan be utilized to perform parametric studies overa range of chamber pressures to identify thefundamental causes of unsteady motions in aSRM. The data set can be used to calibratesimpler RANS-based turbulence closure schemes.

CONCLUSION

The gas-phase flame dynamics of a double-basehomogeneous solid propellant in an axisymmet-

Fig. 19. Snapshots of fluctuating vorticity, heat-release, entropy, and pressure, t ! 21.7 ms.


ric rocket motor has been investigated in depth.Turbulence closure was obtained by means of aLES technique that allows a comprehensiveanalysis of the interactions between propellantcombustion and unsteady flow development. Atheoretical analysis exploring the coupling be-tween turbulence and flame dynamics is per-formed to provide insight into self-sustainedunsteady motions. The effect of turbulence ap-pears mainly in determining flow structure. Theinteractions between turbulence and chemicalreactions are so weak that the propellant com-bustion can be locally treated as a well-stirredreactor. Results indicate that the temperatureand heat-release distributions in the flame zoneexert a significant influence on the flow evolu-tion. The primary flame structure is little af-fected in the present study because the smoothaxial velocity gradient and vertical flow convec-tion prevent turbulence from deeply penetrat-ing into the primary flame. Turbulence-inducedlarge-scale vortical motions, however, may driveflow oscillations with distinct frequencies.

This work was sponsored partly by the Pennsyl-vania State University and partly by the CaliforniaInstitute of Technology Multidisciplinary Univer-sity Research Initiative under ONR Grant No.N00014-95-1-1338. The use of ARSC supercom-puters in Alaska is also appreciated.

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Received 11 January 2002; revised 30 April 2002; accepted 16May 2002


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