NASA Technical Reports Server (NTRS) 19980073427: Numerical Simulation of the Generation of Axisymmetric Mode Jet Screech Tones

51 Pages·3.2 MB·English

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AIAA 98-0283
Numerical Simulation of the Generation of
Screech Tones
Axisymmetric Mode Jet
H. Shen and C. K. W. Tam
Florida State University
Tallahassee, FL
36th Aerospace Sciences
Meeting & Exhibit
January 12-15, 1998 / Reno, NV
For permission to copy or republish, contact the American Institute ofAeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, Virginia 20191-4344
NUMERICAL SIMULATION OF THE GENERATION OF
AXISYMMETRIC MODE JET SCREECH TONES t
Hao Shen* and Christopher K.W. Tam**
Florida State University
Tallahassee, FL 32306-3027
Abstract
back loop is driven by the instability waves of the
An imperfectly expanded supersonic jet, invari- jet flow. In the plume of an imperfectly expanded
ably, radiates both broadband noise and discrete fre- jet is a quasi-periodic shock cell structure. Figure
quency sound called screech tones. Screech tones are 1 shows schematically the feedback loop. Near the
known to be generated by a feedback loop driven nozzle lip where the jet mixing layer is thin and
by the large scale instability waves of the jet flow. most receptive to external excitation, acoustic dis-
Inside the jet plume is a quasi-periodic shock cell turbances impinging on this area excite the instabil-
structure. The interaction of the instability waves ity waves. The excited instability waves, extracting
and the shock cell structure, as the former propa- energy from the mean flow, grow rapidly as they
gates through the latter, is responsible for the gen- propagate downstream. After propagating a dis-
eration of the tones. Presently, there are formu- tance of four to five shock cells, the instability wave
las that can predict the tone frequency fairly ac- having acquired a large enough amplitude interacts
curately. However, there is no known way to pre- with the quasi-periodic shock cells in the jet plume.
dict the screech tone intensity. In this work, the The unsteady interaction generates acoustic radia-
screech phenomenon of an axisymmetric jet at low tion, part of which propagates upstream outside the
supersonic Mach number is reproduced by numeri- jet. Upon reaching the nozzle lip region, they ex-
cal simulation. The computed mean velocity profiles cite the mixing layer of the jet. This leads to the
and the shock cell pressure distribution of the jet generation of new instability waves. In this way, the
are found to be in good agreement with experimen- feedback loop is closed.
tal measurements. The same is true with the sim- At the present time, there are reliable screech tone
ulated screech frequency. Calculated screech tone frequency prediction formulas l'a. However, there is
intensity and directivity at selected jet Math num- no known way to predict tone intensity and direc-
ber are reported in this paper. The present results tivity; even if it is entirely empirical. This is not
demonstrate that numerical simulation using com- surprising for the tone intensity is determined by
putational aeroacoustics methods offers not only a the nonlinearities of the feedback loop.
reliable way to determine the screech tone intensity The principle objective of the present work is to
and directivity but also an opportunity to study the simulate the screech phenomenon numerically for
physics and detailed mechanisms of the phenomenon low supersonic Mach number jets. It will be shown
by an entirely new approach. that numerical simulation is an accurate method
for screech tone intensity and directivity prediction.
1. Introduction
Numerical simulation of jet noise generation is not a
Supersonic jet noise consists of three princi- straightforward undertaking. Tam 4had earlier dis-
pal components 1. They are the turbulent mixing cussed some of the major computational difficulties
noise, the broadband shock associated noise and the anticipated in such an effort. First of all, the prob-
screech tones. Screech tones are discrete frequency lem is characterized by very disparate length scales.
sound. At low supersonic Math number, the screech For instance, the acoustic wavelength of the screech
tones are associated with the axisymmetric oscilla- tone is over 20 times larger than the initial thick-
tions of the jet and radiate principally in the up- ness of the jet mixing layer that supports the insta-
stream direction. It has been known since the early bility waves. Further, there is also a large disparity
work of Powell 2that screech tones are generated by a between the magnitude of the velocity of the radi-
feedback loop. Recent works 1suggest that the feed- ated sound and that of the jet flow. Typically, they
are five to six orders different. To be able to com-
! Copyright _)1998 by H. Shen & C.K.W. Tam. Published by the pute accurately the instability waves and the radi-
American Institute of Aeronautics and Astronautics, Inc. with
ated sound, a highly accurate computational aeroa-
permission.
coustics algorithm with shock capturing capability
* Graduate student, Department of Mathematics.
as well as a set of high quality numerical boundary
** Distinguished Research Professor, Department of Mathemat-
ics. Associate Fellow AIAA. conditions are required.
Therestof
this paper is as follows. In Section 2, density), pressure scale = p_a_, temperature scale
the mathematical model, the computation algorithm = Too (ambient gas temperature); scales for k, e
and grid design are discussed. Section 3 describes and v_ are a_, -_- and a_ D, respectively. The di-
the various numerical boundary conditions used in
mensionless governing equations in Cartesian tensor
the simulation. Section 4elaborates on the distribu-
notation are,
tion of artificial selective damping incorporated in
the computation algorithm. The artificial selective
damping terms are for the elimination of the high
0-7+ 0x--T = 0 (1)
wavenumber spurious waves. They have no effect
on the low wavenumber component (the physical so-
O_i _ O_ 0
lution) of the computation. They help to maintain 0----i+ v._ (Pa_aJ)- 0x_ 0xi (_nj) (2)
a high quality numerical solution free from contam- O-_E 0 0
ination by spurious waves and numerical instabil- o---7+- _ (-_E%) = -o_-7(_%)
ity. Comparisons between numerical results and ex-
perimental measurements are provided in Section 5.
These include the mean velocity profiles, the shock 00_ (_ n_) + p_(.i1L 1)00_ P v'-_xJ
cell structure, the dependence of the screech tone fre-
quency on jet Mach number and screech tone inten- +_b-_0z_l(__°k) (3)
sity. Excellent agreements with experimental mea-
surements are found. Computed directivities for the
first two harmonics of the dominant screech tone will Op ko_+___i_ (-_k'%)=--_,-,_-O_2gi_,- -_
also be provided. 1o(o )
+ _-k0-_'xj PV'_xj (4)
2. Mathematical Model, Computation
Scheme and Grid Design O-_c 0 e O_i
In this work, we are interested in simulating the
axisymmetric mode jet screech in the jet Mach num-
ber range of 1.0 to 1.25. The axisymmetric mode - (k_+2 + 1 o( a_) (5)
is the dominant screech mode for axisymmetric jets
7p=_ (6)
from convergent nozzles at these Mach numbers. For
this purpose, only two dimensional computation in
the x - r plane, where (r, 0,x) are the cylindrical E - 7(7 - 1) + 2 _ +k (7)
coordinates, are necessary.
2.1. The Mathematical Model rq = _ 6i_ - v, \ Oxj + Oxi 30x_ 60. (8)
For an accurate simulation of jet screech genera- k2 u
tion, it is essential that the feedback loop be mod- v, = C, (_ + _0----+----_5-----D (9)
eled and computed correctly. The important ele-
ments that form the feedback loop are the shock cell where 7 is the ratio of specific heats, v is the molec-
structure, the large scale instability wave, and the ular kinematic viscosity, k0 = 10-6 and e0 = 10-_
feedback acoustic waves. Since turbulence in the jet are small positive numbers to prevent the division
plays only an indirect role in the feedback loop, no by zero. The model constants are taken to be _,
attempt is made here to resolve it computationally.
C, = 0.0874, a} = 0.324, a, = 0.377,
However, turbulence in the mixing layer of the jet is
C_ = 1.4, C,2 = 2.02, Pr = 0.422,
responsible for its spreading and the spreading rate
agv = 1.7 x 10-%
of the jet affects the spatial growth and decay of
the instability wave. To ensure a good simulation It is to be noted that for the range of Mach number
of the spreading rate, the k -e turbulence model is and jet temperature considered the Pope and Sarkar
adopted. In the computation, the modified k -e of corrections often added to the k - e model _ are not
Ref. [5], optimized for jet flows, are used. necessary. Outside the jet flow both k and e are
Figure 2 shows the physical domain to be simu- zero. On neglecting the molecular viscosity terms,
lated. We will use length scale = D (nozzle exit di- the governing equations become the Euler equations.
ameter), velocity scale = ao. (ambient sound speed), In this work, solutions of the above set of equa-
time scale = D_D__d_ensity scale = p_ (ambient gas tions are to be found numerically. For a given jet
_oo '
operatingcondition,thesolutionistoprovidethe by Tam s. For the present problem, several types
shockcellstructureinthejetplume,themeanflow of numerical boundary conditions are required. In
aswellastheinstabilitywavein themixinglayer Figure 2, outflow boundary conditions are necessary
andtheacoustifcieldofthescreecthoneoutsidethe
along boundary AB. Along boundary BCDE, radi-
jet. ation condition with entrainment flow are required.
On the nozzle wall, the solid wall boundary condi-
2.2.Computationschemeandgrid design
tion is imposed. The jet flow is supersonic. So the
Inthiswork,the7-pointstenciDl RPschem4e'6is inflow boundary condition can be prescribed at the
usedtotime-marchthesolutiontoatimeperiodic nozzle exit plane. Finally, the equations of motion in
statecorrespondintogthescreecchycle.Thecoeffi- cylindrical coordinates centered on the x-axis have
cientsoftheschemiencludingthoseofthebackward an apparent singularity at the jet axis (r ---*0). A
differencsetencilsaregivenin Ref[4](thevalueof special treatment is needed to avoid the singular-
ity computationally. Below, a brief description of
a3 should be 0.0208431427703). The DRP scheme
the different boundary conditions used in the simu-
has proven to be nearly nondispersive over a wide
band of wavenumbers. In the acoustic region, the lations is provided.
use of 8 mesh points per wavelength would be ade- 3.1. Radiation Boundary Conditions with
quate. This allows a fairly coarse grid to be used in Entrainment Flow
the entire region outside the jet flow.
For accurate numerical simulation, the numerical
2.3. Grid Design boundary conditions to be imposed along boundary
BCDE must perform three functions. First, it must
The mixing layer of the jet is very thin. The small-
specify the ambient conditions for the entire compu-
est size grid is employed here to provide needed res-
tation. This information is critical to the correct ex-
olution. Figure 3 shows the entire computation do-
pansion of the jet and the development of the shock
main. The domain extends 5 diameters back from
cell structure. Second, it must allow the acoustic
the nozzle exit and 35 diameters long in the x-
waves generated to leave the computation domain
direction. It is 17 diameters in the r-direction. The
with minimal reflection. Third, it must generate the
domain is divided into four blocks (or subdomains)
entrainment flow induced by the jet. The develop-
as far as the grid size is concerned. In Figure 3,
ment of such a set of radiation boundary conditions
these subdomains are separated by black lines. The
with entrainment flow is discussed in Ref [8].
black lines represent buffer regions of 3 mesh spac-
ings. Since acoustic waves propagate with no prefer- 3.2. Outflow Boundary Conditions
ence in direction, square grids are used. The finest
Along the outflow boundary AB, the mean flow
grid in the block right downstream of the nozzle exit
is nonuniform. For this reason, the nonuniform out-
has Az = 6P-_. This block is enclosed by the next
flow boundary conditions of Tam and Dong 9is used.
block with Az = D which, in turn, is enclosed by However, as the outflow boundary is only 30 jet
another block with Az = D. The outer most block diameters downstream, the instability wave ampli-
has Az = D__ In Figure 3, the dotted curve rep- tude, although damped at such a far distance, re-
8"
resents more or less the edge of the jet flow. This mains quite large. To allow for weak nonlinearities,
is well inside the lightly shaded region in which the we nonlinearized the outflow boundary conditions
governing equations are the k - _ model turbulent of Tam and Dong by replacing the linear terms by
flow equations. The full Euler equations are used in their nonlinear counterparts. In cylindrical coordi-
the unshaded region. nates, the complete set of outflow boundary condi-
The buffer region is a narrow region around the tions used are,
boundaries of a computation subdomain of uniform
size mesh. The change in the mesh size takes place
in the buffer region. The basic design of the buffer Oop-7+ Op + ,,Oop,.- .,1,2(Op + Op + Op)(lO )
region can be found in Ref [7]. In this work, a slightly
c3u Ou c3u 10p (11)
improved version of the basic design is used. o--?+ + = oz
3. Numerical Boundary Conditions
cgv cgv Ov 10p (12)
p,9,.
Numerical boundary conditions play a crucial role
in the simulation of the jet screech phenomenon. Re- 1 cgp Op c3p p-p 0 (13)
cently an in-depth review of this subject was given V(e----0--)-7+ c°s eb-Tz+ sin ebTr + ----if- =
Ok Ok cOk the purpose of eliminating spurious short waves and
+ + = o (14)
to improve numerical stability, artificial selective
a¢ 0¢ a_ damping terms 11are added to the discretized finite
+ + =o (15)
difference equations.
In the interior region, the seven-point damping
where V(O) = u cos 0+ a(1 - M 2sin_0)½, M = _. a
stencil 4 (with half-width cr = 0.2r) is used. An in-
is the speed of sound and (19,R) are spherical polar
coordinates (the x-axis is the polar axig); the origin verse mesh Reynolds number (R_ 1-- _(a_. txx) where
of R has been taken to be at the end of the potential va is the artificial kinematic viscosity) of 0.05 is pre-
core of the jet. The last two equations above are the scribed over the entire computation domain. This
nonlinearized form of the linear asymptotic k - ¢ is to provide general background damping to elimi-
equations (without sources), p is the static pressure nate possible propagating spurious waves. Near the
calculated by the entrainment flow model at the edge boundaries of the computation domain where a 7
of the jet flow at the outflow boundary. points stencil does not fit, the 5 and 3 points damp-
ing stencils given in Ref. [4] are used.
3.3. Inflow Boundary Conditions
Spurious numerical waves are usually generated
At the nozzle exit plane, the flow variables are at the boundaries of a computation domain. The
taken to be uniform corresponding to those at the boundaries are also favorite sites for the occurrence
exit of a convergent nozzle. In addition, both k and of numerical instability. This is true for both exte-
¢are assumed to be zero. In other words, the mixing rior boundaries as well as internal boundaries such
layer is regarded to be very thin. Thies and Tam 5,in as the nozzle walls and buffer regions where there is
their jet mean flow calculation work, found that this a change in mesh size. To suppress both the gen-
is a reasonably good way to initiate the computation. eration of spurious numerical waves and numerical
For cold jets, the mixing layer evolves rapidly into instability, additional artificial selective damping is
a quasi-similar state resembling that in a physical imposed along these boundaries. Along the inflow
experiment. (radiation) and outflow boundaries, a distribution
of inverse mesh Reynolds number in the form of a
3.4. Boundedness Treatment at the Jet Axis
Gaussian function with a half-width of 4 mesh points
In cylindrical coordinates, the governing equation (normal to the boundary) and a maximum value of
has an apparent singularity at the jet axis (r --+ 0). 0.1 right at the outermost mesh points is incorpo-
Ref [8] discusses two ways to treat this problem. In rated into the time marching scheme. Adjacent to
the present work, the governing equations are not the jet axis, a similar addition of artificial selective
used at r = 0. Instead, the formal limit of these damping is implemented with a maximum value of
equations as r --_ 0 are used. Our experience is that the inverse mesh Reynolds number at the jet axis
this can be implemented in a straightforward man- set equal to 0.35. On the nozzle wall, the use of a
ner by extending the 7-point stencil into the negative maximum value of additional inverse mesh Reynolds
region ofr. The flow variables p, p and u in the r < 0 number of 0.2 has been found to be very satisfactory.
region are determined by symmetric extension about
The two sharp corners of the nozzle lip and the
the jet axis while v is obtained by an antisymmet-
transition point between the use of the outflow and
ric extension. These are the proper extensions for
the radiation boundary condition on the right side of
axisymmetric jet screech oscillations.
the computation domain are locations requiring fur-
3.5. Wall Boundary Conditions ther additional numerical damping. This is done by
adding a Gaussian distribution of damping around
On the nozzle wall, the boundary condition of
these special points.
no through flow is implemented by the ghost point
As shown in figure 3, the four computation sub-
method of Tam and Dong 1°. For the purpose of
domains are separated by buffer regions. Here ad-
eliminating the generation of spurious waves, extra
ditional artificial selective damping is added to the
amounts of artificial selective damping are imposed
finite difference scheme. In the supersonic region
around the nozzle wall region. By judging from the
downstream of the nozzle exit, a shock cell structure
computed results, this is an effective way to avoid
develops in the jet flow. In order to provide the DRP
the generation of short spurious waves.
scheme with shock capturing capability, the variable
4. Artificial Selective Damping stencil Reynolds number method of Tam and Shen 12
The DRP scheme is a central difference scheme is adopted. The jet mixing layer in this region has
and, therefore, has no intrinsic dissipation. For very large velocity gradient in the radial direction.
Becausoefthis,theUstenci I of the variable stencil a function of q* = _ in figure 5 where r0.5 is
X
Reynolds number method is determined by search- the radial distance from the jet axis to the location
ing over the seven-point stencil in the axial direction where the axial velocity is equal to half the fully
and only the two immediately adjacent mesh points expanded jet velocity. Numerous experimental mea-
in the radial direction. An inverse stencil Reynolds surements have shown that the mean velocity profile
number distribution of the form, when plotted as a function of r/* would nearly col-
lapse into a single curve. The single curve is well
Rsten¢il-I= 4.5F(x)G(r) (16) represented by an error function in the form,
where U
-- = 0.511 - erf(ar/*)] (17)
u1
( 1, 0 < x < 9
F(z)
"_ l exp[-_(z- tn 2 9)_], 9 < z where _ is the spreading parameter. Extensive jet
mean flow data had been measured by Lau 16. By in-
terpolating the data of Lau to Mach 1.2, it is found
a(r) = e1x,p[-_(r- tr*2 0.8)_], 00<.8r<0<.8r that experimentally 6ris nearly equal to 17.0. The
empirical fit, formula (17), with cr = 17.0 is also
is used in all numerical simulations. It is possible
plotted in figure 5 (the circles). As can readily be
to show, based on the damping curve (a = 0.3zr)
seen, there is good agreement between the empiri-
that the variable damping has minimal effect on the
cal mean velocity profile and those of the numerical
instability wave of the feedback loop. Also exten-
simulation.
sive numerical experimentations indicate that the
One important component of the screech feedback
method used can, indeed, capture the oscillatory
loop is the shock cell structure inside the jet plume.
shocks in the jet plume and that the time averaged
To ensure that the simulated shock cells are the same
shock cell structure compares favorably with exper-
as those in an actual experiment, we compare the
imental measurements.
time averaged pressure distribution along the cen-
5. Numerical Results and terline of the simulated jet at Mach 1.2 with the ex-
Comparisons with Experiments perimental measurements of Norum and Brown 17
We have been able, using the numerical algorithm Figure 6 is a plot of the simulated and measured
described above, to reproduce the jet screech phe- pressure distribution as a function of downstream
distance. It is clear from this figure that the first five
nomenon computationally. Figure 4 shows the com-
shocks of the simulation are in good agreement with
puted density field in the z- r-plane at one in-
stance after the initial transient disturbances have experimental measurements both in terms of shock
cell spacing and amplitude. Beyond the fifth shock
propagated out of the computational domain. The
screech feedback loop locks itself into a periodic cy- cell, the agreement is less good. At this time, we are
unable to determine the cause of the discrepancy.
cle without external interference. As can be seen,
sound waves of the screech tones are radiated out However, it is known that screech tones are gener-
ated around the fourth shock cell. Therefore, any
in a region around the fourth to fifth shock cells
minor discrepancies downstream of the fifth shock
downstream of the nozzle exit. Most of the promi-
cell would not invalidate our screech tone simula-
nent features of the numerically simulated jet screech
tion.
phenomenon are in good agreement with physical
During a screech cycle, the shock cell is not sta-
experiments 13'14'15.
tionary. In the past, Westley and Woolley 13 had
5.1. Mean Velocity Profiles and Shock Cell made extensive high speed stroboscopic schlieren ob-
Structure servations of the motion of the shock cells and the
To demonstrate that the present numerical simula- disturbances/instability wave in the mixing layer of
tion can actually reproduce the physical experiment, the jet. Figure 7a is their hand sketch of the promi-
we will first compare the mean flow velocity profile of nent features inside the jet plume. Figure 7b is the
the simulated jet with experimental measurements. density field in a plane cutting through the center-
For this purpose, the time averaged velocity profile line of the simulated jet. In comparing figures 7a and
of the axial velocity of a Mach 1.2 jet from 1 diam- 7b one must be aware that the lighting in schlieren
eter downstream of the nozzle exit to 7 diameters observation gives an integrated view of the density
downstream at one diameter interval are measured field across the jet. Despite this inherent difference,
from the numerical simulation. They are shown as there are remarkable similarities between the two fig-
ures.Notonlythegrossfeaturesofthelargetur- 5.3. Directivity
bulencestructure(intheformoftoroidalvortices)
The directivity patterns of the simulated screech
andshockcellsarealike,thedetailedfeatureosfthe
tones have been measured. Typical directivities for
curvedshocksarenearlythesame.Basedonthe
the A1 and A2 modes are shown in figures l0 and
abovecomparisonist,isbelievetdhatthenumerical
11. A search over the literature fails to find directiv-
simulationi,ndeedc,anreproducaelltheimportant
ity measurements for the axisymmetric mode screech
elementosfthescreecphhenomenon.
tones. Validation of these results, therefore, cannot
5.2.ScreechToneFrequencyandIntensity be carried out at this time.
It iswellknownthatatlowsupersonjiect Mach Figure 10a shows the directivity of the A1 screech
numbert,herearetwoaxisymmetrisccreecmhodes; mode (fundamental frequency) at jet Mach number
the 1.18 scaled to a distance of 65D. The directivity of
Al and the As modes. Earlier, Norum 14 had
the second harmonic is given in figure 10b. Those
compared the frequencies of the A1 and As modes
for the A2 mode at Mach 1.2 are shown in figures
measured by a number of investigators. His com-
lla and llb. Overall, the directivity patterns of the
parison indicates that the screech frequencies and
A1 and As modes are similar. However, there are
the Mach number at which the transition from one
differences in detailed features. For the fundamen-
mode to the other takes place (staging) vary slightly
tal tone, the directivity pattern consists of two prin-
from experiment to experiment. It is generally
cipal lobes. One lobe radiates upstream and form
agreed among experimentalists that the screech phe-
part of the screech feedback loop. The other radi-
nomenon is extremely sensitive to minor details of
ates downstream peaked at a relatively small angle
the experimental facility and jet operating condi-
from the jet flow direction. This is not sound gener-
tions.
ated by the interaction of instability wave and shock
In the present numerical simulation, both the A1
cells. It is Mach wave radiation generated directly
and As axisymmetric screech modes are reproduced.
by the instability wave as it propagates down the jet
Figure 8 shows the variation of _, where _ is the
columnl, is.
acoustic wavelength of the tone, with jet Mach num-
ber obtained by the present numerical simulation. The directivity pattern of the second harmonic,
Since AD- -- _(ID)'where f is the screech frequency, figures 10b and llb, also displays two principal
lobes. One lobe peaks around the 90 deg. direction.
this figure essentially provides the frequency Mach
number relationship. Plotted on this figure also are This is the principal lobe. The sound is generated by
the measurements of Ponton and Seiner is. The data the nonlinearities of the source (nonlinear instabil-
from both the numerical simulation and experiment ity wave shock cell interaction). The other lobe is in
the upstream direction. We believe this is generated
fall on the same two curves, one for the A1 mode and
the other for the As mode. This suggests that the by the nonlinear propagation effect of the upstream
propagating feedback acoustic waves. The screech
calculated screech frequencies are in complete agree-
tone intensity is quite high. This leads immediately
ment with experimental measurements; although the
Mach number at which staging takes place is not the to wave steepening and the generation of harmon-
ics. An examination of the waveforms measured at
same.
Ponton and Seiner Is mounted two pressure trans- upstream locations confirms that they are not sinu-
soidal but somewhat distorted. Thus the main lobes
ducers at a radial distance of 0.642D and 0.889D,
of the second harmonic are of entirely different ori-
respectively, on the surface of the nozzle lip in their
gin.
experiment. By means of these transducers, they
were able to measure the intensities of the screech
Concluding Remarks
tones. Their measured values are plotted in figures
9a and 9b. The transducer of figure 9b is closer Recently, rapid advances have been made in the
development of computational aeroacoustic meth-
to the jet axis and hence shows a higher dB level.
ods. In this work, we have demonstrated that it is
Plotted on these figures also are the corresponding
tone intensities measured in the numerical simula- now possible to perform accurate numerical simula-
tion of the jet screech phenomenon by the use of one
tion. The peak levels of both physical and numerical
of these methods, namely, the DRP scheme with ar-
experiments are nearly equal. Thus, except for the
tificial selective damping. Numerical boundary con-
difference in the staging Mach number, the present
ditions are also crucial to the success of the simula-
numerical simulation is, indeed, capable of providing
accurate screech tone intensity prediction as well. tions. At the present time, such numerical boundary
conditions are available in the literature. In a pre-
viousreview4,it waspointedout,unliketraditional 8. Tam, C.K.W., "Advances in Numerical Bound-
computationaflluiddynamicsproblemsn,umerical ary Conditions for Computational Aeroacoustics."
simulatioonfjet noisegeneratioinssubjectetdothe AIAA Paper 97-1774, June 1997 (to appear in the
difficultiesoflargelengthscaledisparityandthe Journal of Computational Acoustics).
needtoresolvethemanyordersofmagnitudedif- 9. Tam, C.K.W. and Dong, Z., "Radiation and Out-
ferenceinssoundandflow.Thisworkindicatetshat flow Boundary Conditions for Direct Computa-
theseproblemcsanbeovercombeyacarefudlesign tion of Acoustic and Flow Disturbances in a
ofthecomputatiognridandtheuseofan optimized Nonuniform Mean Flow," Journal of Computa-
high order finite difference scheme. tional Acoustics, Vol. 4, June 1996, pp. 175-201.
The present work is restricted to the low super- 10. Tam, C.K.W. and Dong, Z., "Wall Boundary Con-
sonic Mach number range for which the screech tones ditions for High-Order Finite Difference Schemes
are axisymmetric. Future plans call for the exten- in Computational Aeroacoustics," Theoretical and
sion of the work to three dimensions to allow the Computational Fluid Dynamics, Vol. 8, 1994, pp.
simulation of flapping modes at higher Mach num- 303-322.
bers. 11. Tam, C.K.W., Webb, J.C., and Dong, Z., "A
Study of the Short Wave Components in Com-
Acknowledgment putational Acoustics," Journal of Computational
This work was supported by Dr. Steven It. Walker Acoustics, Vol. 1, Mar. 1993, pp. 1-30.
at the USAF Wright Laboratory under delivery or- 12. Tam, C.K.W. and Shen, H., "Direct Computation
der F33615-96-D-3011. The second author also of Nonlinear Acoustic Pulses Using High-Order Fi-
wishes to acknowledge partial support from NASA nite Difference Schemes," AIAA Paper 93-4325,
Langley Research Center under grant NAG 1-1776. Oct. 1993.
Supercomputing time was provided by the USAF 13. Westley, R. and Woolley, J.H., "An Investigation
Wright Laboratory, the SP2 computer of the Su- of the Near Noise Fields of a Choked Axisym-
percomputer Research Institute and the SGI Power metric Air Jet," Proceedings of AFOSR-UTIAS
Challenge computer of the Florida State University. Symposium on Aerodynamic Noise, Toronto, May
1968, pp. 147-167.
References
14. Norum, T.D., "Screech Suppression in Supersonic
1. Tam, C.K.W., "Supersonic Jet Noise," Annual Re- Jets," AIAA Journal, Vol. 21, Feb. 1983, pp.
view of Fluid Mechanics, Vol. 27, 1995, pp. 17-43. 235-240.
2. Powell, A., "On the Mechanism of Choked Jet 15. Ponton, M.K. and Seiner, J.M., "The Effects of
Noise," Proceedings Physical Society, London, Nozzle Exit Lip Thickness on Plume Resonance,"
Vol. 66, 1953, pp. 1039-1056. Journal of Sound and Vibration, Vol. 154, No. 3,
3. Tam, C.K.W., Shen, H. and Raman, G., "Screech 1992, pp. 531-549.
Tones of Supersonic Jets from Bevelled Rectangu- 16. Lau, J.C., "Effects of Exit Mach Number and
lar Nozzles," AIAA Journal, Vol. 35, July 1997, Temperatures on Mean-Flow and Turbulence
pp. 1119-1125. Characteristics in Round Jets," Journal of Fluid
4. Tam, C.K.W., "Computational Aeroacoustics: Is- Mechanics, Vol. 105, 1981, pp. 193-218.
sues and Methods," AIAA Journal, Vol. 33, Oct. 17. Norum, T.D. and Brown, M.C, "Simulated High-
1995, pp. 1788-1796. Speed Flight Effects on Supersonic Jet Noise."
5. Thies, A.T. and Tam C.K.W., "Computation AIAA Paper 93-4388, Oct. 1993.
of Turbulent Axisymmetric and Nonaxisymmetric 18. Tam, C.K.W. and Burton, D.E., "Sound Gener-
Jet Flows Using the k - e Model", AIAA Journal, ated by Instability Waves of Supersonic Flows.
Vol. 34, Feb. 1996, pp. 309-316. Part l, Two Dimensional Mixing Layers; Part 2,
6. Tam, C.K.W. and Webb, J.C., "Dispersion- Axisymmetric Jets," Journal of Fluid Mechanics,
Relation-Preserving Finite Difference Schemes for Vol. 135, 1984, pp. 249-295.
Computational Acoustics," Journal of Computa-
tional Physics, Vol. 107, Aug. 1993, pp. 262-281.
7. Dong, Z., Fundamental Problems in Computa-
tional Acoustics. Ph.D. Thesis, 1994, Florida
State University.
j b/c ¢,4
,.z
U
old
tcoi
shock cell structure
instability wave
¢cqi
Figure 1. Schematic diagram of the screech tone feedback loop.
ci
¢_.
-o,15 -o.lo -0.05 o.oo 0.05 o.lo o.15
r_*
D _entrainment flow C
Figure 5. Comparison between mean velocity profiles of numeri-
cal simulation at Mj = 1.2 . --, x/D = 1.0, - - - -, x/D = 2.0,
--. --, x/D =3.0,-- ---, x/D = 4.0,-- ..--, x/D =5.0,--- -
--, x/D = 6.0,. ..... ,x/D -- 7.0, and U/Uj -- 0.511 -er.f(arf )],
sound _ B 0, a = 17from experiment (Lau 1981).
nozzl_2]- ":::::Z.............Z...IIIIII".IZIIY""-""""" " " •-
ow:
0 0
_ instability wav_ _" _"/ /f ":'- i:
0
0 0
a_
Figure 2. The physical domain to be simulated.
0 'i ..... _......... i......... i ..... i......... _......... i....... r ...... t........
0.0 LO 2.0 3.0 4,0 5.0 6.0 7.0 8.0 9.0
x/O
Figure 6. Comparison between calculated tlme-averaged pressure
distribution along the centerline of a Mach 1.2 cold jet and the
measurement of Norum and Brown (1993). -- simulation, O
experiment.
r
...............
nozzle
jet axis *
Figure 3. The computation domain in the r - x plane showing
the different subdomain and mesh sizes.
nozzle shock cells
Figure 4. Density field from numerical simulation showing the generation and
radiation of screech tone associated with a Mach 1.13, cold supersonic jet from a
convergent nozzle.