AIAA 98-3699Characterization of the Laminar BoundaryLayer in Solid Rocket Motors J. MajdalaniMarquette UniversityMilwaukee, WI 53233

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AIAA-98-3699

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CHARACTERIZATION OF THE LAMINAR BOUNDARY LAYERIN SOLID ROCKET MOTORS

J. Majdalani*Marquette University, Milwaukee, WI 53233

Abstract!

This paper investigates the structure of theboundary layer in cylindrical rocket motors inlight of two recent analytical solutions to the time-dependent axisymmetric flowfield that have beenshown to agree with numerical and experimentalpredictions in the forward portions of the motorwhere the flow remains laminar. To that end,closed form expressions that define the characterof the oscillatory boundary layer are obtained inorder to bring physical details into focus. Theshort flowfield solution published recently(Majdalani, J., and Van Moorhem, W.K.,“Improved Time-Dependent Flowfield Solutionfor Solid Rocket Motors,” AIAA Journal, Vol. 36,No. 2, 1998, pp. 241-248) makes it possible toarrive at analytical expressions that elucidate theintricate features of the boundary-layer zone; thelatter is found to encompass a relatively largeportion of the combustion chamber in mostrockets for low acoustic modes. The depth ofpenetration is found to depend on the size of thepenetration number, the acoustic mode, and thedistance from the head-end. An assessment of thelocation and size of the Richardson overshoot isalso pursued. Closed form expressions areprovided for the penetration depth, speed ofpropagation, wavelength, amplitude and phaserelation between unsteady velocity and pressurecomponents. Increasing viscosity is found toreduce the size of the rotational region. Bycomparison to the acoustic boundary layerassumed in one-dimensional acoustic theory, theactual character of the rotational region is quitedissimilar. Finally, analytical results are verifiednumerically against a modern and reliable,compressible Navier-Stokes solver.

! *Assistant Professor, Department of Mechanical andIndustrial Engineering. Member AIAA.Copyright © 1998 by J. Majdalani. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., withpermission.

Nomenclaturea0 = stagnation speed of sound, " #p0 0/fm = dimensional frequency for mode m, Hzkm = wave number, m R L R a$ %/ /& 0 0L = internal chamber lengthMb = wall injection Mach number, V ab / 0p0 = mean chamber pressure, # "0 0

2a /

p = dimensionless pressure, p p' / 0r = radial position, r R y' & (/ 1a fR = dimensional effective radiusRek = kinetic Reynolds number, % )0

20R /

Sr = Strouhal number, % 0 R V k Mb m b/ /&t = dimensionless time, t a R' 0 /Ur = radial mean flow velocity, (

(r 1 sin*u ( )1 = total unsteady velocity, u a'( ) /1 0Vb = radial injection speed at the wally = distance from the wall, y R r' & (/ 1a fz = axial distance from the head-end, z R' /+ w = normalized pressure wave amplitude" = mean ratio of specific heats, = spatial wavelength of rotational waves) 0 = chamber fluid mean kinematic viscosity* = characteristic variable, 2$r / 2# 0 = chamber mean density% m = dimensionless frequency, m R L$ /% 0 = dimensional frequency, m a L$ 0 /- = viscous parameter, S R Vp b

( &1 02

03% ) /

Subscriptsm = refers to a maximum or a mode numberp = refers to a depth of penetrationw = refers to the wallb = refers to blowing or burning at the wall

Superscripts* = asterisk denotes a dimensional quantity~ = tilde denotes vortical oscillations

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IntroductionIn recent years, the boundary-layer structure in

solid rocket motors has received much attention inthe rocket combustion stability community. Thismight be attributed to the important role that itplays in understanding a number of combustionmechanisms that occur in the vicinity of theburning surface. Since understanding thestructure of the boundary layer can helpunderstand pressure coupling, velocity coupling,transition to turbulence, and the flame zoneinteraction with the internal flowfield, severalresearchers have undertaken analytical,1-9

numerical,10-19 and experimental investigations20-21

aimed at elucidating intricate field interactionsnear the propellant surface.

The main focus of this paper will be to analyzethe boundary-layer structure resulting from tworecent analytical models for the flowfield thathave been shown to agree very favorably withavailable numerical and experimental data.3 Thefirst model was derived by Flandro7 using thevorticity transport equation and regularperturbations. The second was derived byMajdalani and Van Moorhem2-3 using a novel,composite-scale perturbation technique. Bothmodels have been shown recently to concur over alarge range of physical parameters despite theirdissimilar analytical formulations. One appealingfeature of the composite-scale model3 is that itoffers a short expression for the velocity fieldwhich allows extracting information about theboundary layer in closed analytical form. Thecurrent paper will exploit this feature to elucidatethe character of the oscillatory boundary layer andexplain the influence of various flow variables onits structure. In the process, several related issueswill be addressed individually. These include theboundary-layer thickness or penetration depth ofthe rotational region, the peculiar Richardsonovershoot,22 the spatial wavelength and speed ofpropagation, and the controversial phasedifference between oscillatory pressure andvelocity. Since the present analysis is onlyapplicable to laminar fields, it is hoped that theinformation provided here will be used indeveloping a working analytical theory forturbulence, which recent work by Yang and co-workers has shown to appear in the aft portion ofthe rocket chamber.23 Finally, analytical results

are further validated through comparisons drawnagainst reliable computational data acquired froma modern Navier-Stokes solver developed by Rohand co-workers.23

AnalysisWave Characteristics

Using the exact same notation as previously, thecurrent analysis begins by considering the totaltime-dependent velocity obtained from Ref. 3 [Eq.(63)], which is known to the order of the Machnumber:

u r z t k z k tw m m(1)

Irrotational part

, , sin sina f b g b g&L

NMMM

+"

! "### $###

( .O

QPPP

sin sin sin exp sin* * /k z k tm mb g b gWave amplitude Propagation

Rotational part

% ### '#### % # '##

! "####### $#######0 (1)

where

/ -1 *( ) ( ) cscr r r& 3 3 , 0( ) ln tanrSr

&$

*2

(1a)

and

1 r y cy yr c rca f d i& ( . (( (1 1 1ln , c & 3 2/ (1b)

The radial velocity component has beendeliberately ignored in Eq. (1), being smaller inmagnitude than the axial component. The totaltime-dependent velocity consists of a linearjuxtaposition of inviscid-acoustic-irrotational andviscous-solenoidal-rotational fields. Therotational component represents a harmonic wavetraveling radially toward the centerline; this wavesuffers from exponential damping with increasingdistance from the wall. From Eq. (1) it can beinferred that the vortical wave amplitude iscontrolled by two terms: 1) an exponentiallydecaying term (made possible by inclusion ofviscous effects) that diminishes with increasingdistance from the wall, and 2) a sinusoidal term(made possible by inclusion of downstreamconvection of unsteady vorticity by the meanflow) which, in addition to its monotonic decreasewith increasing distance from the wall, varies

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harmonically with the distance from the head-end.Since the exponentially decaying wave amplitudeterm depends directly on - % )& 0

20

3R Vb/ , it is clearthat large viscosity causes the amplitude to decaymore rapidly. Viscosity is hence identified to bean attenuation factor whose role is to impede theinward penetration of vorticity.

Equation 1 also indicates that the axial variationin the wave amplitude along the centerline iscontrolled exclusively by the acoustic field, whilethe radial variation is prescribed by the rotationalfield which plays a crucial role in the accurateassessment of the boundary layer envelope. On aseparate note, recalling that the phase of therotational wave is uniform along lines wherek tm .0b g is constant, Eq. (1) allows solving for the

radial speed of wave propagation which isdetermined to be equal to Culick’s radial meanflow velocity.24 This reassuring result is evidencethat the solution exhibits the correct couplingbetween mean and time-dependent elements andthat the time-dependent field is indeed driven bythe mean flow. More details are furnished below.

Unsteady Axial Velocity ProfileResults from the regular perturbation solution

by Flandro7 and the composite-scale technique(CST)3 are found to concur substantially with thenumerical solution which is achieved with a highorder of accuracy (using a 9-stage Runge-Kuttascheme, and a step size of 10-6, with an associatedglobal error of order seven).25 It follows that theagreement between analytical and numericalpredictions is so remarkable that graphical resultsare visually undiscernible. The periodic velocitydistribution at evenly spaced times is shown inFig. 1, for one full cycle of oscillations, and forthe first four acoustic oscillation modes. Thecontrol parameters are chosen from typical valuesassociated with a tactical rocket motor, asclassified by Flandro (Sr & 51m, Rek &2.1mx10

6

from Table 1 in Ref. 7). The profiles aredisplayed at the axial position corresponding tothe location from the head-end of the first (Figs.1a-d) and last (Figs. 1e-g) acoustic velocityantinode. A key feature captured remarkably bythe analytical solution is that of the rotationalvelocity amplitude vanishing m times at the mth

velocity antinode. As shown in Figs. 1e-g, the

rotational amplitude decays prematurely to zerosomewhere between the wall and the centerlinecorresponding to lines of zero unsteady vorticity.This peculiar effect, which is attributable to thedownstream convection of zero unsteady vorticitylines by Culick’s mean flow,24 is further evidencethat the influence of the mean flow on the time-dependent field has been correctly incorporated.

Boundary Layer Thickness or Penetration DepthIn recognition of the fact that both regular

perturbation and CST models exhibit similarvelocity profiles, their penetration depths areexpected to be similar as well. A typicalcomparison obtained from the aforementionedmodels is drawn in Fig. 2 at two axial locations,for a large range of dynamic similarity parameters,Sr and Rek . Remarkably, the entire family ofcurves shown in Fig. 2 collapses into a singlecurve per axial location, when plotted versus thepenetration number, Sp &

(- 1 , revealed by theanalytical derivation. This appealing discoveryallows us to represent the complete solution forthe boundary-layer thickness on one single graphper oscillation mode. As shown in Fig. 3,characteristic curves of penetration depths atseveral axial locations spanning the length of thechamber are conveniently depicted for thefundamental oscillation mode. Having collapsedthe results onto a single graph provides numerousadvantages, including concrete means to explainand interpret the boundary layer structure.

As could be inferred from Fig. 3, thedependence of the penetration depth on the axiallocation z is minute in the forward half of thechamber, and becomes more pronounced in the afthalf. The increased sensitivity of the boundarylayer thickness to z with increasing axial distancefrom the head-end is attributed to vorticalintensification in the streamwise direction. Forfirst mode oscillations, the axial dependence isfound to be only important in the aft-half of thechamber, when z becomes relatively large. For arange of penetration numbers, the depth ofpenetration is found to be dependent only on thepenetration number and, to a lesser extent, on theaxial location. For small penetration numbers, thepenetration depth is found to be directlyproportional to the penetration number,

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independently of the axial location. This takesplace when the mean flow injection speed is verysmall, resulting in insignificant vorticalintensification in the streamwise direction.Evidently, this range does not correspond torockets characterized by sizeable penetrationnumbers and relatively large penetration depths,especially for fundamental oscillatory modes.

The sensitivity of the penetration depth tovariations in the penetration number decreases athigher values of the penetration numbercorresponding to frictionless flows. As thepenetration number becomes large, say exceeding100, the value of the penetration depth becomesindependent of the penetration number, and can beestimated from an asymptotic solution to theinviscid formulation. This maximum possiblepenetration depth y pm that can occur at any axial

location is shown in Fig. 4 for the first fouroscillation modes. Clearly, the maximumpenetration depth increases with the axial locationand the mode number. The axial increase is notmonotone, since y pm reaches a maximum at the

acoustic velocity nodes where the boundary layerfills the entire chamber. The numerical andanalytical results shown in Fig. 4 are obtainedfrom Eq. (4) and Eq. (5), respectively. Theseequations are derived below.

Boundary-Layer EnvelopeThe outer envelope of the time-dependent

boundary layer depends on the rate of decay of thewave amplitude. From Eq. (1), the waveamplitude that controls the evolution of the outerenvelope of the rotational velocity is easilyrecognized to be

~ sin sin sin exp csc( )u k zS

rw mp

1 3 3&FHG

IKJ

+"

* *1

*b g (2)

The point directly above the wall where thisamplitude reaches 1% of its irrotationalcounterpart in Eq. (1) defines the edge of theboundary layer. In this case, the point must becalculated by finding the root rp of

sin sin sin$ $2 2

2 2r k z rp m pFHIK

FHIK

LNM

OQP

2 FHIK

LNMM

OQPP ( &exp

( )csc sin

1 $3

r

Sr r k z

p

pp p m

3 3 2

20b g (3)

where 3 & 0 01. defines the 99% based boundary-layer thickness. In general, this penetration depthwill depend on the penetration number, the modenumber, and the axial location in the chamber.The larger the penetration number, the larger thepenetration depth will be due to a smallerargument in the exponentially decaying termarising in Eq. (3). This establishes the role ofviscosity, discussed earlier, as an agent thatattenuates the strength and penetration of vorticalwaves. Obviously, the smaller the viscosity, thelarger the penetration depth will be. The upperlimit on the boundary-layer thickness cantherefore be determined from the inviscidformulation of the penetration depth. Setting theviscosity equal to zero in Eq. (3), the maximumpenetration depth is found to be a sole function ofthe axial location and mode numbers:

sin sin sin sin$ $

32 2

02 2r k z r k zpm m pm mFH

IK

FH

IK

LNM

OQP ( &b g

(4)

Inviscid Boundary-Layer EnvelopeEquation (4) can be manipulated algebraically to

reveal a closed form asymptotic expansion for themaximum penetration depth. This is madepossible by taking advantage of the fact that rpm is

smaller than unity. The 99% inviscid thicknesscan be evaluated either numerically or from a one-term expansion of order rpm

6 , extracted from Eq.

(4). This expansion formula is

yk z

k zO rpm

m

mpm& (

LNMM

OQPP .1

42

1 4

6

$3

sin/b g d i (5)

Since the minimum possible y pm is 74.8% at z & 0,

rpm cannot exceed a value of 0.252. The

maximum error associated with Eq. (5) can hencebe calculated to be 0.000259, which is an order ofmagnitude smaller than the Mach number. Thismaximum error can only affect the depth ofpenetration in the third or fourth decimal places, apractically negligible contribution, which also

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explains the excellent agreement in Fig. 4 betweenanalytical and numerical predictions.

Unsteady Velocity OvershootThe phase difference between rotational and

irrotational solutions causes a periodic overshootof the total velocity that can reach almost twicethe irrotational wave amplitude. This overshoot isa well known effect that is characteristic ofoscillatory flows. It was first discovered inexperiments on sound waves in resonators byRichardson22 who first realized that maximumvelocities occurred in the vicinity of the wall.Theoretical verifications of this peculiarphenomenon were carried out by Sexl,26 andadditional confirmatory experiments wereconducted by Richardson and Tyler27 onreciprocating flows subject to pure periodicmotions without mean fluid injection.

The problem at hand is quite original in thesense that it involves injection of a mean flow atthe wall. In this case, the magnitude and thedistance ymax from the wall to the point wheremaximum overshooting occurs can be determinednumerically. The so-called Richardson effect22 ofa velocity overshoot is clearly observed in bothanalytical models to be much more intense thanfor the hardwall case.

Plots of velocity overshoot and loci of thesevelocity extrema are almost indistinguishablefrom corresponding numerical predictions. Notethat the loci are independent of Rek (i.e.,viscosity), and only depend on Sr . For the regularperturbation model of O Sr( / )1 ,7 numerical andanalytical results become discernible when Srdrops below 20. Figure 5 summarizes theobserved trends which, in turn, indicate that theovershoot increases with decreasing kinematicviscosity and frequency. As one would expect,the overshoot occurs in the vicinity of the wall,roughly, in the lower 25% of the solution domain,corresponding, indubitably, to the most sensitiveregion. Since this overshoot is not captured bythe one-dimensional model currently in use, theneed to incorporate the multidimensional field,described here, becomes even more important,especially when proper coupling with combustionis desired near the propellant surface.

Spatial Wavelength and Speed of PropagationNear the wall, the speed of propagation of the

vortical wave can be determined from

k t k t ySr j jm m. 4 ( & &0b g b g 2$ , 1,2,( (6)

or, in dimensional form,

m a

Lt

y

RSr j

$$0 2'

'

(FHG

IKJ & (7a)

ady

dt Srm a

R

LVw b& & &

'

'1

0$ (7b)

As expected, the speed of propagation near thewall is determined by the mean flow velocity. In asimilar fashion, the dimensional wavelength ofpropagation can be calculated to be:

, $%w

w

m

w ba

f

a V

maL& & &2

2

0 0

(8)

Away from the wall, the speed of propagation willnot be a constant anymore. It will decrease withthe radial mean flow velocity. Using the exactexpression for the phase angle, the dimensionalspeed of propagation of the rotational wave in theradial direction is found to be exactly equal to theradial mean flow velocity:

ady

dt Srm a

r

L

r

R

r

RV Uw b r& &

FHGIKJFHGIKJ & (

'

'

' ( '1

200

1

$$

sin

(9)

Having determined the speed of propagation, thespatial wavelength of rotational waves can bededuced easily. Written in nondimensional form,the result is

,$%

$%

$w w br rR

a

R

V

RU

SrU& & ( & (2 2

2

0 0

(10)

Clearly, the higher the Strouhal number, theshorter the wavelength, and the steeper the wavecrests will be. Also, as the centerline isapproached, the spatial wavelength diminishes indirect proportion with the radial mean flowvelocity. This explains the larger number ofreversals per unit of traveled distance for a fluid

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particle in approach of the centerline. Theanalytical expression for the spatial wavelengthcaptures very accurately the physical detailsdictated in most part by Culick’s mean flowfield.24

Unsteady Pressure Phase LeadHere 0 is the phase angle of the vortical

velocity component with respect to the acousticcounterpart at any radial position within thechamber. This function is proportional to Sr andcontrols the propagation speed of the rotationalwave. The angle 5 m by which the sinusoidaltime-dependent pressure wave leads the time-dependent velocity can be determined as follows.First, the time-dependent pressure and velocitiesare written as harmonic functions of time

pk t k z k t k z

wm m m m

( )

cos cos sin cos1

2+$

& & .FHIKb g b g b g

(11)

u A Aw m m( ) cos sin1

2 21& ( .

+"

0 0b g b g2 .sin sink t k zm m m6b g b g (12)

where, from Eq. (1),

tansin

cos6 m

m

m

A

A&

((

001

(13)

A r k z k z rm m m&FHIK

FHIK

LNM

OQP

(sin sin sin sin

$ $2 2

2 1 2b g

2 FHIK

LNM

OQPexp ( ) csc-1

$r r r3 3 2

2(14)

Then, for any axial location, the angle by whichthe pressure leads the velocity is simply

5$

6m m& (2(15)

Near the wall, the angle 0 is written in a Taylorseries form expanded about y & 0 :

0 rSr

rSr

y y ya f & FHIK & ( . (LNM$

$$

$$ $

ln tan2 2 6

2 23

3

. (.

.O

QPP 4 (

$ $ $3 42 3

5 6

4

3

24y y O y ySrd i d i (16)

The effective composite scale that appears in Eq.(14) also exhibits an asymptotic form near thewall.2,3 At y & 0 , the effective composite scalebecomes

1 r ya f & ( (17)

wherefore the vortical velocity amplitude given byEq. (14) simplifies to

A r ym & & (exp ( ) exp-1 -b g (18)

and the angle 6 m , given by Eq. (13), becomes

tanexp sin

exp cos6

-

-my

y ySr

y ySr&

( ( (

( ( (&

7

b g a fb g a f1

0

00

(19)

To remove the indeterminate character of Eq.(19), L’Hospital’s rule is invoked. The result is asimple expression for the phase angle at the wall:

lim y

y ySr Sr y ySr

y ySr Sr y ySr7

( ( . ( (

( ( ( ( (0- - -

- - -

exp sin exp cos

exp cos exp sin

b g a f b g a fb g a f b g a f

& & &Sr

SrSp m-6tan (20)

wherefrom 6 m pSrS& arctand i (21)

5$

m pSrS& (2arctand i (22)

This exact analytical limit is common to allrotational models, whether one-dimensional1,4 ortwo-dimensional,2,3,6,7 and whether using purelyanalytical means,4 regular perturbations,6,7 ormultiple-scale techniques.1,2,3 Additionally, thislimit can be verified very rigorously by numericalcomputations. Near the centerline where acousticvelocity is the only nonzero component, therotational velocity vanishes, 6 m vanishes, and 5 mwill be 90 degrees. Thusly, the sinusoidal time-dependent pressure leads the time-dependentvelocity by an angle that varies from a small valueat the wall to 90 degrees at the centerline. Notunlike the velocity profile, there exists a phaseovershoot that can reach 180 degrees or twice thephase difference between pressure and acoustic

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velocity. At the wall, an exact analytical expressionfor the phase angle is successfully extracted. Byinspection of Eq. (22), the phase angle depends onthe product of the Strouhal number and thepenetration number. In dimensional form, thisproduct scales with the convection to diffusionspeed ratio of the rotational disturbances introducedat the wall:

5$

% )$

$ )mb bV V L

m a& (

FHGIKJ & (

FHG

IKJ2 2

2

0 0

2

0 0

arctan arctan

(23)

It follows that lower injections, shorter chambers,higher oscillation modes, higher viscosities, orhigher speeds of sound result in a larger pressureto velocity phase lead at the wall. The largestphase lead will occur, for instance, in a smallSRM. Practically, this angle is only a few degreesor less. Figure 6 shows the phase lead of the time-dependent pressure with respect to the velocity forthe four typical cases defined in Ref. 6, using two-dimensional viscous3,7 and inviscid formulations,6

in addition to the one-dimensional near-wallsolution from Ref. 4. At the wall, the exactexpression for the phase angle given by Eq. (23) isverified to be common to all three models.

Practical Boundary-Layer EquationIn order for the analytical models to match

corresponding numerical predictions, it is notnecessary to retain all the terms in the rotationalmomentum equation that controls the character ofthe oscillatory boundary layer. In reality, of allthe terms appearing in the momentum equationgiven as Eq. (9) in Ref. 3,

8 9mz z zz z r rku u U

u U U ut Sr z r r

: : ::; : : : :? @

) )) )

2 2

2

1 1m z z r r

k

k u u u u

Re r r r z r zr

A B: : : :. . ( (C D: : : ::E F

) ) ) )(24)

only five significant terms need to be retained:

mz z z zr z z

ku u u UU U u

t Sr r z z

: : : :A B& ( . .C D: : : :E F

) ) ) )2

2m z

k

k u

Re r

:.

:)

(25)

These terms contribute to the solution in bothmodels and can be attributed to five physicalmechanisms. All the remaining terms in Eq. (24)may be included, but the corrections that willresult in retaining them will be smaller than theorder of the error in the solution itself. As can beestablished by tracking the leading order termsthat influence the solution, the most importantphysical mechanisms can be associated withunsteady inertial forces and both radial and axialconvection of unsteady vorticity by the mean flow.Second in importance is the viscous diffusion ofvorticity. Third in importance is the convectivecoupling between unsteady velocity and meanflow vorticity. It is the balance of these importantphysical phenomena that controls the oscillatorymotion of gases inside the chamber. In essence,Eq. (25) is the practical, “real world,” time-dependent boundary-layer equation.

Comparisons to Computational PredictionsPreviously in Ref. 3, analytical results were

shown to be in fair agreement with experimentalobservations made by other researchers.Presently, comparisons will be made against areliable numerical code developed totallyindependently by Roh and co-workers.23

Sometimes referred to as the “dual time-stepping”(DTS) code, this compressible Navier-Stokessolver has recently received wide acceptance inthe combustion stability community by virtue ofits established accuracy and reliability.

On that account, DTS data (shown in dashedlines) are compared in Fig. 7 to analyticalpredictions (shown in solid lines) atapproximately the same time intervals for a typicalcase of a cylindrical chamber ( L &2.03 m,R & 0.102 m). The injection speed is held constantat 1.02 m/s (corresponding to a Mach number of0.003), and the kinematic viscosity is taken to be2.612 (10 5 m2/s. The corresponding dynamicsimilarity parameters are calculated to beRek &2.1210

5 m , Sr m& 52 6. , and S mp & 12.44 / .

As shown in Fig. 7, there is a good agreementbetween computational and analytical predictionsfor velocity amplitudes and spatial wavelengthsnear the wall. A strong resemblance in the generalstructure of the boundary layer may be said toexist at higher oscillation modes, as shown in

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Figs. 7b-c. In particular, both approaches predictthe occurrence of m points of zero rotational waveamplitude at the mth velocity antinode attributed tothe downstream convection of zero vorticity linesby the bulk fluid motion. These comparisons werelimited to the first two acoustic modes due to therapidly increasing cost of achieving numericalsolutions at higher modes.

The slight deviation of DTS data from analyticalpredictions can be attributed to unavoidablelimitations in available computational power. Inreality, several sources of numerical uncertaintieshave been identified as possible reasons for theobserved discrepancy.28

First, due to memory resource limitations, itbecomes unaffordable to refine the gridsufficiently enough in regions that are distant fromthe wall where the mean radial velocity becomesvery small. The reason for using very fine gridspacing is necessitated by the need to properlyresolve the vorticity wave whose wavelengthdepends directly on the mean radial velocity. Theanalytical model does not suffer from thislimitation and, as shown in Fig. 7, is capable ofresolving very precisely the spatial wavelengthaway from the wall even when the mean velocitybecomes infinitesimally small. In light of thisargument, a progressive deviation from analyticalpredictions is to be anticipated as the distancefrom the wall is increased, when a slightdeterioration in numerical accuracy becomesunavoidable.

Second, due to the numerical inability to matchexactly the time intervals required forcomparisons during a cycle (i.e., $ / 2 , $ , and3 2$ / ), which happen to be irrational numbers,numerical data is acquired at time intervals thatare closest to the times desired. This restriction inthe numerical approach is caused by the need forfinite time discretization and is obviated inanalytical formulations. In the current analysis,the time period was divided into 100 time steps,making it difficult to match the prescribed timeintervals which, evidently, brings in additionalerrors to DTS data. This explains the slightasymmetry in the numerically generated curves,and their subtle deviation from analytical curvesaway from the wall, in the fully irrotational zone.

Third, due to the reliance of numericalcalculations on artificial dissipation, it canbecome difficult to refine the artificial dissipationsufficiently enough. Needless to say, analyticalmodels are not dependent on artificial dissipation.

Reducing numerical errors, which is expected toimprove substantially the agreement withanalytical predictions, can be accomplishedthrough 1) decreasing artificial dissipation in thenumerical scheme, 2) refining the grid, and 3)decreasing each time step. Unfortunately, theseimprovements can only be implemented at theexpense of increased computational time, cost,and memory allocation which, collectively, canbecome prohibitive. In conclusion, the analyticalmodels described heretofore, being exempt fromcomputational setbacks, appear to capture the keyphysical details furnished by the DTS procedure,for the laminar case, while remaining immune tonumerical restrictions.

Impact and ImplicationsThe unsteady boundary layer in oscillatory

flows with sidewall injection is an interestingaddition to boundary-layer theory in fluidmechanics. It is also of value in the studies ofturbulence in oscillatory flows over transpiringsurfaces. Fortuitously, this solution can beverified analytically to be rigorous since it reducesto Sexl’s solution26 near the wall in the limit of avery small injection velocity (to be addressed inour forthcoming work). In rocket dynamics, itfurnishes a simple yet powerful expressioncapable of elucidating the intricate features of theacoustic boundary layer whose structure has beenthe subject of much controversy in the past.

Importance in Fluid MechanicsA multidimensional analytical solution that

quantifies the Stokes boundary layer in anoscillatory duct flow with sidewall injection isexploited here including the axial dependence. Itappears that this analytical solution, along withFlandro’s model,7 are the only two-dimensionalaxisymmetric expressions pertaining to this typeof flow which have been obtained so far. Bothoffer important steps aimed at a more completeunderstanding of the structure of the Stokes layerover porous surfaces. Such understanding may beneeded to allow improved formulations in

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aerodynamics, gas dynamics, studies of bloodflow in arteries, and other applications. In thestudies of turbulence, the availability of a laminarsolution can be used as a basis for investigatingturbulent behavior, which, up to this time, is notvery well understood. Both experimental andnumerical studies of turbulence in oscillatory ductflows with sidewall injection can benefit from aclosed form solution of the internal flowfield asfurnished here. The analytical methodology itselfmay be applicable to similar physical settingsinvolving oscillatory flows.

Importance in Rocket DynamicsThe existence of an accurate, yet simple,

analytical expression for the unsteady flowcomponent has a major impact on the internalflowfield modeling strategy and combustionstability assessment in solid rocket motors. Thecurrent standard prediction model that is used toanalyze combustion stability of various rocketsassumes the existence of a one-dimensionalirrotational component of the time-dependent flowand introduces patches to account for three-dimensional effects. The current analysisemphasizes the importance of the rotational flowcomponent in altering the boundary layercharacter. Evidently, the actual structure of theboundary layer is quite different from the “thin”acoustic layer assumed in one-dimensionalmodels. By analogy to Culick’s steady flowsolution,7 the present unsteady solution could beincorporated into existing codes and models toimprove prediction capabilities. Othermechanisms that are associated with combustioninstability could also be revisited in light of thisnew model. For example, the flow turning lossthat is used as a corrective term to patch the one-dimensional imperfection of the model can beshown to be no longer necessary.7 Flandro hasactually shown that, when his formulation isused,7 a term will appear —in the resultingsolution— that is identical to the flow turning loss;the latter being artificially added to the one-dimensional solution. In other areas, the velocitycoupling phenomenon can be quite possiblyimproved by incorporating an accurate, yet simpleformulation of the time-dependent velocity field.The same can be said of studies involving

particulate damping, acoustic streaming, acousticadmittance, erosive burning, turbulence, etc..

Importance of a New Similarity ParameterBy analogy to the Stokes number that governs

the thickness of the boundary layer in oscillatingflows with inert walls, the penetration number isfound to play a similar role in the case when thewalls are made porous. This number

SV

Rp

b& &1 3

02

0- % )(26)

explains what other researchers1-9 have noticedbefore; namely, that the thickness of the boundarylayer will depend mostly on the injection velocity(being elevated to the third power). Thefrequency of oscillations is the second mostimportant parameter. Doubling the frequency ofoscillations decreases the penetration number by afactor of four, which, at sufficiently highfrequencies, reduces the boundary-layer thicknessby a factor of four also (since the penetrationnumber and the penetration depth are directlyproportional in the lower portion of the domain,regardless of axial position). The role of viscosityis finally established as an attenuation factor.This is due to the fact that the penetration numberis inversely proportional to the kinematicviscosity. In contrast to steady boundary layers,or to Stokes boundary layers in oscillatory flowswith imporous walls, the role of viscosity wheninjection is included is to attenuate rather thanpromote the growth of the boundary layer. Thepenetration depth is found to be a measure of therotational region of the flow. Physically,oscillatory vorticity is constantly generated at thewall as a result of the oscillatory pressure gradientwhich is parallel to the solid boundary at theinjection surface. Due to the mean flow motion,vorticity is convected inwardly in an attempt tocontaminate the irrotational fluid with vorticity.The growth of the vortical region results from theconvection and diffusion of vorticity into the innerregions of the domain where convective, diffusive,and inertial acceleration effects stand balanced.The amplitude of the oscillatory vorticity willcease to change when viscous dissipation anddownstream convection of vorticity manage to

AIAA-98-3699

10American Institute of Aeronautics and Astronautics

annihilate the radial propagation effects. Theedge of the boundary layer is hence recognized asa point that is located at a distance y p from the

wall, above which the propagation of vorticity isnegligible. The flowfield above this depth ofpenetration can be said to be irrotational. Theboundary-layer region is, in the context describedhere, a region of highly concentrated vorticity.Finally, the chamber geometry appears to have adirect effect on the penetration number also.Decreasing the motor’s effective radius causes thepenetration depth to grow proportionately larger.This is to be expected because the effect ofblowing becomes more appreciable when thecross-sectional area is reduced.

ConclusionsThe classical concepts of boundary-layer theory

regarding inner, near-wall, and outer, externalregions are almost reversed for the case of anunsteady flow over a transpiring surface. Near thewall, instead of observing the thin, inner, viscouslayer as in unsteady Stokes or steady flows, athick rotational layer is established near the solidboundary when sidewall injection is incorporatedbecause of vorticity convection in the radialdirection. The penetration depth is simply ameasure of the vortical region. The thin layerwhere viscous friction is important is removedfrom the wall to the edge of the Stokes boundarylayer. The penetration depth is a direct functionof a similarity number that is proportional to thecube of the injection speed, inversely proportionalto the square of the frequency, and inverselyproportional to the viscosity and chambereffective radius. This dependence is in totalagreement with empirical observations as well asnumerical analyses. Accordingly, the role ofviscous diffusion is to attenuate the amplitude ofshear waves and to reduce the depth ofpenetration. The role of frequency is similar toviscosity, only twice as important. Injectionvelocity is the most important variable affectingthe boundary-layer thickness. Higher combustiontemperatures in rockets lead to higher kinematicviscosities and, therefore, to smaller penetrationnumbers. Higher oscillation modes (and,therefore, frequencies) have a similar effect. Theaxial location in the chamber also affects the

boundary-layer thickness depending on theacoustic mode. The role of the Strouhal numberas the controlling parameter for the vortical-to-acoustical phase angle has been elucidated. Thecurrent analysis clearly shows that increasing theStrouhal number steepens the vortical wave crestand reduces its wavelength. The pressure-to-velocity phase at the wall is found to be controlledby the ratio of the convection-to-diffusion speedof the vortical waves.

The key elements defining the structure of theboundary layer are accurately captured by the CSTsolution which, unlike computational predictions,does not suffer from limitations imposed on gridresolution, time discretization size, artificialdissipation, and so forth. It is hoped that thistechnique be further explored in relatedcombustion stability research, taking advantage ofthe scaling synthesis verified to be accurate in thisinvestigation, and which can be particularly usefulin pursuing models for turbulence. Analyticaldevelopment of a turbulent flow model can nowevolve from the established knowledge ofsimilarity parameters and agents in control of thelaminar boundary layer.

AcknowledgmentsThe author is indebted to Prof. Vigor Yang and hisco-workers from Pennsylvania State Universityfor their generous contribution of DTS data thatmade analytical comparisons to nonlinearized,compressible Navier-Stokes predictions possible.

References1 Majdalani, J., and Van Moorhem, W.K., “The

Unsteady Boundary Layer in Solid RocketMotors,” Paper No. AIAA-95-2731, July 1995.

2 Majdalani, J., and Van Moorhem, W.K., “AMultiple-Scales Solution to the AcousticBoundary Layer in Solid Rocket Motors,” Journalof Propulsion and Power, Vol. 13, No. 2, 1997,pp. 186-193.

3 Majdalani, J., and Van Moorhem, W.K., “AnImproved Time-Dependent Flowfield Solution forSolid Rocket Motors,” AIAA Journal, Vol. 36, No.2, 1998, pp. 241-248.

4 Flandro, G.A., “Effects of Vorticity Transporton Axial Acoustic Waves in a Solid PropellantRocket Chamber,” Combustion InstabilitiesDriven By Thermo-Chemical Acoustic Sources,

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11American Institute of Aeronautics and Astronautics

NCA Vol. 4, HTD Vol. 128, American Society ofMechanical Engineers, New York, 1989.

5 Smith, T.M., Roach, R.L., and Flandro, G.A.,“Numerical Study of the Unsteady Flow in aSimulated Rocket Motor,” AIAA Paper 93-0112,Jan. 1993.

6 Flandro, G.A., “Effects of Vorticity on RocketCombustion Stability,” Journal of Propulsion andPower, Vol. 11, No. 4, 1995, pp. 607-625.

7 Flandro, G.A., “On Flow Turning,” AIAAPaper 95-2730, July 1995.

8 Beddini, R.A., and Roberts, T.A., “Response ofPropellant Combustion to a Turbulent AcousticBoundary Layer,” AIAA Paper 88-2942, July1988.

9 Beddini, R.A., and Roberts, T.A.,“Turbularization of an Acoustic Boundary-Layeron a Transpiring Surface,” AIAA Paper 86-1448,June 1986.

10 Sabnis, J.S., Giebeling, H.J., and McDonald,H., “Navier-Stokes Analysis of Solid PropellantRocket Motor Internal Flows, Journal ofPropulsion and Power, Vol. 5, No. 6, 1989, pp.657-664.

11 Tissier, P.Y., Godfrey, F., Jacquemin, P.,“Simulation of Three Dimensional Flows InsideSolid Propellant Rocket Motors Using a SecondOrder Finite Volume Method – Application to theStudy of Unstable Phenomena,” AIAA Paper 92-3275, July 1992.

12 Vuillot, F. and Avalon, G., “AcousticBoundary Layer in Large Solid Propellant RocketMotors Using Navier-Stokes Equations,” Journalof Propulsion and Power, Vol. 7, No. 2, 1991, pp.231-239.

13 Vuillot, F. Lupoglazoff, N., “Combustion andTurbulent Flow Effects in 2D Unsteady Navier-Stokes Simulations of Oscillatory RocketMotors”, AIAA Paper 96-0884, Jan. 1996.

14 Vuillot, F., “Acoustic Mode Determination inSolid Rocket Motor Stability Analysis,” Journalof Propulsion and Power, Vol. 3, No. 4, 1987, pp.381-384.

15 Vuillot, F., “Numerical Computation ofAcoustic Boundary Layers in a Large SolidPropellant Space Booster,” AIAA Paper 91-0206,Jan. 1991.

16 Vuillot, F., “Numerical Computation ofAcoustic Boundary Layers in Large Solid

Propellant Space Booster,” AIAA Paper 91-0206,Jan. 1991.

17 Vuillot, F., and Avalon, G., “AcousticBoundary Layer in Large Solid Propellant RocketMotors Using Navier-Stokes Equations,” Journalof Propulsion and Power, Vol. 7, No. 2, 1991, pp.231-239.

18 Vuillot, F., Avalon, G. “Acoustic BoundaryLayers in Solid Propellant Rocket Motors UsingNavier-Stokes Equations,” Journal of Propulsionand Power, Vol. 7, No. 2, 1991, pp. 231-239.

19 Vuillot, F., Dupays, J., Lupoglazoff, N.,Basset, Th., and Daniel, E., “2D Navier-StokesStability Computations for Solid Rocket Motors:Rotational, Combustion and Two-Phase FlowEffects,” AIAA Paper 97-3326, July 1997.

20 Roberts, T.A. and Beddini, R.A., “Responseof Solid Propellant Combustion to the Presence ofa Turbulent Acoustic Boundary Layer,” AIAAPaper 88-2942, July 1988.

21 Shaeffer, C.W., and Brown, R.S., “OscillatoryInternal Flow Studies,” Chemical Systems Div.,Rept. 2060 FR, United Technologies, San Jose,CA, Aug. 1992.

22 Richardson, E. G., “The Amplitude of SoundWaves in Resonators,” Proceedings of thePhysical Society, London, Vol. 40, Dec. 1937-Aug. 1928, pp. 206-220.

23 Roh, T.S., Tseng, I.S., and Yang, V., “Effectsof Acoustic Oscillations on Flame Dynamics ofHomogeneous Propellants in Rocket Motors,”Journal of Propulsion and Power, Vol. 11, No. 4,July 1995, pp. 640-650.

24 Culick, F.E.C., “Rotational AxisymmetricMean Flow and Damping of Acoustic Waves inSolid Propellant Rocket,” AIAA Journal, Vol. 4,No. 8, 1966, pp. 1462-1464.

25 Butcher, J.C., “On Runge-Kutta Processes ofHigher Order,” Journal of the AustralianMathematical Society, Vol. 4, 1964, pp. 179-194.

26 Sexl, T., “Über den von E.G. Richardsonentdeckten ‘Annulareffekt’,” Zeitschrift fürPhysik, Vol. 61, 1930, pp. 349-362.

27 Richardson, E.G., and Tyler, E., “TheTransverse Velocity Gradient Near the Mouths ofPipes in Which an Alternating or ContinuousFlow of Air Is Established,” Proceedings of thePhysical Society, London, Vol. 42, 1929, pp. 1-15.

28 Through personal communication with W. Caifrom Pennsylvania State University.

AIAA-98-3699

12American Institute of Aeronautics and Astronautics

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

a)

z*/L = 1/2

Sp = 16 m = 1

7$/6 11$/6 4$/3 5$/3

0 $ 2$$/6 5$/6 $/3 2$/3 $/2

%ot* = 3$/2

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

b)

z*/L = 1/4S

p = 4 m = 2

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

c)

z*/L = 1/6S

p = 1.8 m = 3

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

d)

z*/L = 1/8S

p = 1.0 m = 4

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

e)

z*/L = 3/4S

p = 4.0 m = 2

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

f)

z*/L = 5/6S

p = 1.8 m = 3

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

g)

z*/L = 7/8S

p = 1.0 m = 4

y

Fig. 1 Velocity evolutions from numerical, regular perturbations,7 and CST3 models shown at 13 evenlyspaced times in a typical tactical rocket motor for the first 4 acoustic modes at the first (a-d) and last (e-g)acoustic velocity antinode.

AIAA-98-3699

13American Institute of Aeronautics and Astronautics

101

102

103

0.0

0.2

0.4

0.6

0.8

1.0

z*/L = 0.5

CST & Numeric Solutionsy

p

Sr0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

a)

Rek = 10

4

107

106

Regular Pert. Solution

105

108

101

102

103

0.0

0.2

0.4

0.6

0.8

1.0

z*/L = 0.95 CST & Numeric Solutionsy

p

Sr0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

b)

Rek = 10

4

107

106

Regular Pert. Solution

105

108

Fig. 2 Penetration depths obtained numerically andfrom two analytical models3,7 for a wide range ofcontrol parameters and two axial locations.

10-2

10-1

100

101

102

103

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Flandro7

z*/L

0.950

0.875

0.750

0.500

0.250

0.125

0.050

0.5

Numeric

z*/L

0.950

0.875

0.750

0.500

0.250

0.125

0.050

CST & Numeric

Solutions

Regular Pert.

0.05

105 < Rek < 10

8

0.5

CST3

z*/L

0.950

0.875

0.750

0.500

0.250

0.125

0.050

z*/L = 0.95 y

p

Sp

Fig. 3 Locus of the boundary-layer thicknessobtained numerically and from two analyticalmodels3,7 for a vast range of control parameters atvarious axial locations.

0.00.0 0.1 0.20.2 0.3 0.40.4 0.5 0.60.6 0.7 0.80.8 0.9 1.01.0

0.750

0.775

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

0.750

0.775

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000m = 2y

pm

a)

m = 1

Numerical

Analytical

0.00.0 0.1 0.20.2 0.3 0.40.4 0.5 0.60.6 0.7 0.80.8 0.9 1.01.0

0.750

0.775

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

0.750

0.775

0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000m = 4y

pm

z*/L

b)

m = 3

Numerical

Analytical

Fig. 4 Trace of the maximum boundary-layerthickness for the first four acoustic modes: (a) m = 1,2 and (b) m = 3, 4.

10 100 10000.0

0.2

1.0

1.2

1.4

1.6

1.8

2.0

Locus of Overshoot

CST & Numeric

Regular Pert.

Loc

us

O

vers

hoot

Fac

tor

Sr

105

106

107

108

Rek = 10

4

Fig. 5 Analytical and numerical predictions for thelocus and magnitude of Richardson’s velocityovershoot at z* = 0.5L and a wide range of controlparameters.

AIAA-98-3699

14American Institute of Aeronautics and Astronautics

0.0 0.2 0.4 0.6 0.8 1.00o

30o

60o

90o

120o

150o

180o Small Research Motor

Sr = 77 Rek = 4.4 105 S

p = 0.97

a)

0.77o 1D Viscous2D Inviscid2D Viscous

y

Pre

ssur

e P

hase

Lea

d

0.0 0.2 0.4 0.6 0.8 1.00o

30o

60o

90o

120o

150o

180o Tactical Rocket Motor

Sr = 51 Rek = 2.1 106

Sp = 16

b)

0.07o

1D Viscous

2D Inviscid2D Viscous

y

Pre

ssur

e P

hase

Lea

d

0.0 0.2 0.4 0.6 0.8 1.00o

30o

60o

90o

120o

150o

180o Cold Flow Experiment

Sr = 28 Rek = 2.5 105

Sp = 11

c)

0.18o

1D Viscous

2D Inviscid2D Viscous

y

Pre

ssur

e P

hase

Lea

d

0.0 0.2 0.4 0.6 0.8 1.00o

30o

60o

90o

120o

150o

180o Shuttle Rocket Booster

Sr = 27 Rek = 5.8 106

Sp = 288

d)

0.007o

1D Viscous

2D Inviscid2D Viscous

y

Pre

ssur

e P

hase

Lea

d

Fig. 6 Unsteady pressure to velocity phase leadusing CST,3 regular perturbations, viscous7 andinviscid6 models, and the one-dimensional model,4 allat m = 1 and chamber midlength; the four typicalcases span the range of solid rocket motors.

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

a)

z*/L = 1/2

Sp = 1.44 m = 1

0 $

$/2

%ot* = 3$/2

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

b)

z*/L = 1/4

Sp = 0.36 m = 2

y

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

c)

z*/L = 3/4

Sp = 0.36 m = 2

y

Fig. 7 Comparison of time-dependent velocityevolutions at 4 evenly spaced times acquired fromanalytical predictions3 (shown by solid lines) andNavier-Stokes data23 (shown by dashed lines) for thefirst 2 modes of oscillations evaluated at the axiallocation of the first (a-b) and last (c) acousticvelocity antinode.

ma3: ma2: ma1: