-
AIAA 98-3699Characterization of the Laminar BoundaryLayer in
Solid Rocket Motors J. MajdalaniMarquette UniversityMilwaukee, WI
53233
For permission to copy or republish, contact the American
Institute of Aeronautics and Astronautics1801 Alexander Bell Drive,
Suite 500, Reston, Virginia 20191-4344
34th AIAA/ASME/SAE/ASEEJoint Propulsion Conference &
Exhibit
July 13–15, 1998 / Cleveland, OH
-
AIAA-98-3699
1American Institute of Aeronautics and Astronautics
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
-
AIAA-98-3699
2American Institute of Aeronautics and Astronautics
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
-
AIAA-98-3699
3American Institute of Aeronautics and Astronautics
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,
-
AIAA-98-3699
4American Institute of Aeronautics and Astronautics
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
-
AIAA-98-3699
5American Institute of Aeronautics and Astronautics
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
,$%
$%
$w w br rR
a
R
V
RU
SrU& & ( & (2 2
2
0 0
(10)
Clearly, the higher the Strouhal number, theshorter the
wavelength, and the steeper the wavecrests will be. Also, as the
centerline isapproached, the spatial wavelength diminishes indirect
proportion with the radial mean flowvelocity. This explains the
larger number ofreversals per unit of traveled distance for a
fluid
-
AIAA-98-3699
6American Institute of Aeronautics and Astronautics
particle in approach of the centerline. Theanalytical expression
for the spatial wavelengthcaptures very accurately the physical
detailsdictated in most part by Culick’s mean flowfield.24
Unsteady Pressure Phase LeadHere 0 is the phase angle of the
vortical
velocity component with respect to the acousticcounterpart at
any radial position within thechamber. This function is
proportional to Sr andcontrols the propagation speed of the
rotationalwave. The angle 5 m by which the sinusoidaltime-dependent
pressure wave leads the time-dependent velocity can be determined
as follows.First, the time-dependent pressure and velocitiesare
written as harmonic functions of time
pk t k z k t k z
wm m m m
( )
cos cos sin cos1
2+$
& & .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
-
AIAA-98-3699
7American Institute of Aeronautics and Astronautics
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
-
AIAA-98-3699
8American Institute of Aeronautics and Astronautics
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
-
AIAA-98-3699
9American Institute of Aeronautics and Astronautics
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,
-
AIAA-98-3699
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: