NASA / TMm2001-210958 AIAA-2001-2252

Near Field Screech Noise Computation

for an Underexpanded Supersonic

Jet by the CE/SE Method

Ching Y. Loh

Taitech, Inc., Cleveland, Ohio

Lennart S. Hultgren and Philip C.E. Jorgenson

Glenn Research Center, Cleveland, Ohio

August 2001

https://ntrs.nasa.gov/search.jsp?R=20010092202 2018-05-14T14:46:09+00:00Z
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NASA / TM--2001-210958 AIAA-2001-2252

Near Field Screech Noise Computation

for an Underexpanded Supersonic

Jet by the CE/SE Method

Ching Y. Loh

Taitech, Inc., Cleveland, Ohio

Lemlart S. Hultgren and Philip C.E. Jorgenson

Glenn Research Center, Cleveland, Ohio

Prepared for the

Seventh Aeroacoustics Conference

cosponsored by the American Institute of Aeronautics and Astronautics

and the Confederation of European Aerospace Societies

Maastricht, The Netherlands, May 28-30, 2001

National Aeronautics and

Space Administration

Glenn Research Center

August 2001
Acknowledgments

The authors wish to thank Dr. J. Panda for fruitful discussions about the jet-screech phenomena, and for kindly

providing his detailed experimental data. This work received support front the Supersonic

Propulsion Technology Project Office of NASA Glenn Research Center.

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at ht .tp://gltrs.grc.nasa.gov/GLTRS
NEAR FIELD SCREECH NOISE COMPUTATION FOR AN UNDEREXPANDED

SUPERSONIC JET BY THE CE/SE METHOD

Ching Y. Loh

Taitech, Inc.

Cleveland, Ohio 44135

Lennart S. Hultgren and Philip C.E. Jorgenson

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

The space-time conservation element and solution element (CE/SE) method is employed to

numerically study the near-field axisymmetric

screech-tone noise of a typical underexpanded

circular jet issuing from a sonic nozzle. For

the computed cases, corresponding to fully expanded Mach numbers of 1.10, 1.15 and 1.19,

the self-sustained feedback loop is automatically established. The computed shock-cell

structure, acoustic wave length, screech tone

frequencies, and sound pressure levels are in

good agreement with experimental results.

1 Introduction

An underexpanded supersonic jet radiates mixing

noise, broadband shock-associated noise, as well as

screech tones under certain conditions. These complicated and technologically important physical phenomena

have been the topic of many experimental and theoretical

investigations, see the review papers [I-4] for a comprehensive list of references and discussion. Simply put, the

mixing noise is directly associated with large-scale structures, or instability waves, in the .jet shear layer; whereas,

the broadband shock-associated noise and screech tones

are associated with the interaction of these waves with

the shock-cell structure in the jet core. The screech tones

arise due to a feedback loop, i.e., part of the acoustic waves generated by the wave/shock-cell interaction

propagate upstream and re-generate the instability waves

at, or in the vicinity of, the nozzle lip. Screech tones are

of particular interest not only because of general noisereduction concerns, but also because of potentially destructive structural interaction(s) leading to sonic fatigue.

The feedback loop leading to screech tones is sensitive

to small changes in the system conditions, and the understanding of the phenomena is to date mostly based on

experimental observations [5-9].

Reliable direct numerical simulation of jet screech

noise has up to quite recently not been feasible. The

numerical scheme is required, on one hand, to resolve

the acoustic waves without too much dispersion error

and numerical dissipation, while on the other hand, it is

required to capture shocks, or other nonlinear phenomena, near or inside the jet correctly. In addition, nonreflecting boundary conditions must be implemented,

which is more difficult to accomplish in the near field

than in the far field.

Shen and Tam 110,11] obtained excellent results in a

direct numerical simulation of screech for circular jets,

using the well-known DRP scheme. A medium size computational domain of 35D long in the streamwise direction and 17D in radial direction is employed in their

computation, where D is the nozzle diameter.

The space-time conservation element and solution element method (CE/SE) [12-15] is another scheme that

meets the above requirements. As demonstrated in

our previous papers, the CE/SE scheme is well suited

for computing waves on compressible shear flows [16]

as well as vorticity/shock interactions [171, both being

cornerstones of the jet-noise phenomena. Because of

the CE/SE non-reflecting boundary conditions (NRBC),

which are based on flux balance, a much smaller near

field computational domain (12D z 8D) can be used in

the present numerical simulation.

The unstructured-grid axisymmetric CE/SE scheme

used here is briefly discussed in Section 2. Section 3

illustrates the initial and boundary conditions for the

screech problem, in particular, the CE/SE NRBC. The

numerical results are presented and compared to available experimental findings [5-9] in Section 4. Conclusions are drawn in Section 5.

2 The Unstructured Axisymmetric CE/SE

Navier-Stokes Solver

In this Section, several new developments of the

CE/SE numerical method are applied: (a) use of an

NA SA/TM--2001-210958 1
unstructuredgrid,whichhelpstofacilitatethenumericalimplementation [15];(b)theNavier-Stokes CE/SE

solver;and(c)theaxisymmetric LES(largeeddysimulation),whichaccountsforthestrongmomentumexchangebetweenthejetandtheambientflowandresults

inamuchimprovednumerical simulation.Thesenew

developments aredescribedinthefollowingsubsections.

2.1 Conservation Form of the Unsteady

Axisymmetric Navier-Stokes Equations

In general, the CE/SE method systematically solves a

set of integral equations derived directly from the physical conservation laws and naturally captures shocks and

other discontinuities in the flow. Both dependent variables and their derivatives are computed simultaneously.

As a consequence, the flow vorticity can be obtained

without reduction in accuracy. Non-reflecting boundary conditions (NRBCs) are also easily implemented because of the flux-conservation formulation.

Consider a dimensionless conservation form of the unsteady axisymmetric Navier-Stokes equations of a perfect gas. Let p, u, v, p, and "_be the density, streamwise

velocity component, radial velocity component, static

pressure, and constant specific heat ratio, respectively.

The axisymmetric Navier-Stokes equations then can be

written in the following vector form:

Ut + Fx + Gy = Q, (I)

where x, y _> 0, and t are the streamwise and radial coordinates and time, respectively. The conservative flow

variable vector U and the flux vectors in the streamwise

and radial directions, F and G, are given by:

U [ U.2 F = _,r) G = G2

U3 ' Fa ' Ga '

\ U_I F,1 G4

with

Ul = p, U2 = pu, U3 = pv,

U4 = P/(7 1) + p(u 2 + v2)/2.

The flux vectors are further split into inviscid and viscous

fluxes:

F = Fi - Fv,G = Gi - Gv,

where the inviscid fluxes are the same as in the Euler

equations:

F,1 = _,

F,2 = (3- 1)U_ + [(3 - 7)U,_ (2,- 1)U:_]/2U1,

Fi3 = U2U..3/U1,

p,_ = _/u_udu, - (,_- a)_:2[u._+ u_] t2u(,

Gil = U3, Gi2 = LSU3/U1,

ai3 = ("t- 1)U4 + [(3 - "7)U:_ - (2'- 1)U_]/2U1,

G,4 _- "_U3Ud/U1 - ("f - 1)U3 [U{ + U_]/2U?,

and the viscous fluxes are:

F_,I = O, Ft,2 = I*(2ux - _v.v), 2

F_,3= u(vx + uy),

Fv4 = #[2uu:r + (Uy + Vx)V- ._(V.V)u+

Pr Oy U1 2 )]'

Gvl=0, G_,2=tt(v_+uu),

G, 3 = #(2vy - _V.V),

Gv4 : _.t[2b/@ + (tly 4-Ux)_ -- 2(V'V)t'-[-

i' a(u4 _2+,2

Pr Oy Ul 2 )]'

where u, v, Ux, uy, Vx, vy are respectively the z- and yflow velocity components and their derivatives, which

can be written in terms of the conservative variables

U1, U2, U3 and U4, with Pr (= 0.72) being the Prandtl

number,/z the viscosity, the velocity divergence

V.V= Ux + % + v/y.

The right hand source term Q is the same as in the

axisymmetric Euler equations [ 18]:

Q'2

Q = Q:_ '

Q_

where

QI = -U3/y, Q2 = -U2Ua/uly,

Q3 = -u_/uly, Q4 = -G4/y.

By considering (x,y,t) as coordinates of a threedimensional Euclidean space, E3, and using Gauss' divergence theorem, it follows that Eq. (I) is equivalent to

the following integral conservation law:

fs'(v) Hm "dS = fv Q,,dV,

m = 1,2,3,4, (2)

where S(V) denotes the surface around a volume V in

E3 and Hm = (F,,,, Gin, U,,).

NASAfI'M--2001-210958 2
time level n+l

y

B'

I ! I ii !

I ! I II I

time level n _[--_ B

a hexagoncylinderAD6ECFA- A'D'B'E'C'F'A'

wihils3CE__AD'B', OBEC-O'B'E'C,

aid OCFA- O'C'F'A'

.c

..... /

(quadrilateraZs)

b. projection of E3 in the x-y plane

Figure 1: CE/SE unstructured grid

2.2 Unstructured Grid for CE/SE

The CE/SE scheme is constructed to take advantage of

an unstructured triangle grid. The unstructured geometry used with the CE/SE scheme is illustrated in Fig. 1.

Here, AABC is a typical triangle cell and D, E, F are

the triangle centers of the neighboring cells. The flow

variables at the previous time step are stored at the triangle cell centers. Three quadrilateral cylinders (conservation elements) are formed by the edges that connect

the vertices and the center of the triangle and its three

neighbors. The projection of these cylinders in the x-y

plane are shown in Fig. l(b) and the CE's are shown in

Fig. I(a). In the space-time E3 space, Eq. (2) is applied

to the hexagon cylinder (of volume V) that consists of

these 3 quadrilateral cylinder CEs, see Fig l(a). The

discrete approximation of Eq. (2) is then

H_,_ .dS = ,,_,,,tj , (3)

* * * 7* for _1_ -- 1, 2, 3, 4, where H,, = (F,,, G,n, _,,). The

right-hand side of Eq. (3), in general, is the volume V

times the source term evaluated at an appropriate Gaussian quadrature node. Here, the Gaussian quadrature

node is the center of the hexagon A'D'B'E'C'F' at the

time level lz + 1, i.e., 0'.

In the CE/SE scheme, the above flux conservation relation, Eq. (3), in space-time is the only mechanism that

transfers information between node points. A conservation element CE Ihere, quadrilateral cylinders) is the finite volume to _hich Eq. (3) is applied. Discontinuities

are allowed to occur in a conservation element. A solution element SE associated with a grid node (e.g., D,

E, or F in Fig. I) is here a set of interface planes in E:_

that passes through this node (e.g. DAA'D', DBB'D',

EBB'E', ECC'E', etc.). Each surface S(CE) is made

up of segments belonging to two neighboring CE's.

Within a given solution element SE(j, n), where j, n are

the node index and time step, respectively, the flow variables are not only considered continuous but are also approximated by the linear Taylor series expansions:

U*(:v,._, t: j,,,) = U)' + (U_)_(.," xj)+

(Uu)_'(y - ',qj) + (Ut)'j'(f - t'_), (4)

r*(:,_, ._,t_.i, ,,) = F'__ + (F_,)'; (.,,- :,.j)+

(Fu))'(y - yj) + (Ft))'(t - t"), (5)

c* (.,.,;j, t_.i.,, ) -- c_' + (c.,)_' (.,. --:,-j)+

(G_)','(u - uj)+ (a,)_'(t - t"), (6)

where j is the node index of D, E or F. The partial

derivatives of F and G can be related to the corresponding one of U by using the chain rule with Ut provided

by Eq. ( I ). The surface flux can be calculated accurately

NAS AFFM--2001-210958 3
and easily by first evaluating the flux vectors at the geometrical center of the surface through the above Taylor

series expansions.

At time level n, the solution variables U, Ux, and

Uy are given at the three nodes D, E, F in Fig. I and

U, Ux and U u at O' at the new time level n + 1 are

to be computed. In principle, each of the 3 CEs provides 4 scalar equations when Eq. (3) is applied to the element. Hence, the 12 scalar equations needed for the 12

scalar unknowns at O' are available. All the unknowns

are computed based on these relations. No extrapolations (interpolations) across a stencil of cells are needed.

In practice, however, the dependent variables are first obtained at the hexagon center at the new time level n + 1

using Eq. (3) and then their values at the center O' of the

triangle ABC can be obtained by a linear Taylor series

expansion. Details can be found in [15].

With an unstructured grid, the CE/SE procedure is

simplified and more easily adapted to complicated geometries. Only a single mesh is needed as compared

to the spatially staggered mesh used in the structuredgrid case and the time-marching is completed in one step

rather than two. Furthermore, the simple non-reflecting

boundary conditions described previously [16-191 still

work well with an unstructured grid. More details about

the unstructured CE/SE method can be found in [15].

The weighted a - e CE/SE scheme is used here.

2.3 Treatment of the Source Term

The treatment is identical to the one used in [18] and is

briefly reiterated here. Since the source term Q itself is a

function of the unknown U, a local iterative procedure is

needed to determine U. The discretized integral equation

(3) reduces to the form

U - Q(U)At - Utt, (7)

where Un is the local homogeneous solution (Q = 0

locally). Note that Un only depends on the solution at

the previous time step, i.e., Uu is obtained using explicit

formulas. A Newton iterative procedure to determine U

is then

u (_+') = u _ - (°_)-'[_(u(')) u.],

OU

where i is the iteration number and

O(U) - U - Q(U)At.

Normally, U at the previous time step is a good initial

guess U (°/ and the procedure takes about 2-3 iterations

to converge. The Jacobian matrix is given by

/1 0 A-! 0/u

-_ 1+ Uamt t-!z_ 0

0(I) u{ _ {,qu U_y

OU - -_ 0 1 + _ ()

A_ A2 A._ 1 +

, 2frl g

where

A1- U_Atbu _ _ (_- 1)(u] + u_)],

U1 _/ U1

U2U_At

A2=-(?-1) U_y '

zxt_ ,_- 1 u._ + 3u_ ,_u_

Aa

-y- L 2 U]2 + --_-] '

The inverse of the Jacobian, i.e., (ff_-) 1 can easily be

derived analytically for this particular case, thus, leading

to a savings in CPU time.

2.4 Large Eddy Simulation (LES)

A simplified LES procedure similar to those used in [20]

is adopted here to account for the strong momentum exchange in the shear layer. In the LES, a simple Smagorinsky subgrid scale model is used for the eddy viscosity:

where

ttt= (CsA)2(2S_iSij) 1/2,

Oui Ouj .

with A = (AxAy) _/'2 and Cs = 0.1. Then, p + P.t

replaces ,a in the actual computation. In practice, the

LES model was only applied for radii less than 1.5D and

4D downstream of the nozzle exit plane.

3 The Jet Screech Noise Problem

Consider a circular jet as sketched in Fig. 2. The flow

at the start of the computations is choked, i.e., nozzle exit

Mach number, M,,, is unity, and the surrounding fluid is

stationary. Three cases, which if lully expanded would

have jet Mach numbers, M_, of I. 10, I.! 5, and 1.19, are

considered. These conditions correspond to the experimental conditions of Panda [5-8]. In these experiments,

it was shown that for Mj _< 1.19, the jet noise field is in

an axisymmetric mode. Hence, the axisymmetric CE/SE

code described above is appropriate for computation at

these conditions.

In the investigation, our attention is focussed on the

near field of the nozzle since this is the noise source region. The diameter, D, of the jet nozzle is chosen as the

length scale. The density, Po, speed of sound, uo, and

temperature To in the ambient flow are taken as scales

for the dependent variables.

In order to clearly display the upstream propagating

screech waves, the computational domain was extended

4D upstream of the nozzle exit. The full computational

domain is a circular cylinder of 12D axial length and 8D

radius. At the nozzle exit, the inflow plane is recessed by

two cells so as not to numerically restrict or influence the

feed-back loop. Ponton et. al.'s [9] straight nozzle lip of

NASAfI'M--2001-210958 4
NRBC

0.5D

Figure 2: Geometry of the computational domain

0.2D in thickness is adopted in the CE/SE computations.

Note that in the experimental setup of Panda [5,61, although the actual nozzle diameter is the same (25.4 ram)

as in [9], the nozzle exit lip is 0.25D thick and beveled.

The unstructured grid currently used is formed by cutting a rectangle cell into 4 triangles. These rectangular cells are non-unilbrm since good grid resolution is

needed in the jet shear layer and their numbers in the .r

and y directions are typically 240 and 130, respectively,

corresponding to about 132000 triangles. The last 10

cells in the streamwise and radial directions have exponentially growing .r and ,q size, respectively, and serve as

butler zones to essentially eliminate any small remaining numerical reflection from the outflow boundary. The

CE/SE scheme is of the o-e type l 13,141 with a weighting

parameter _ = 0 and e = 0.5.

3.1 Initial Conditions

Initially, the flow of the entire domain is set at the ambient flow conditions, i.e., (using nondimensional variables)

1

Po 1, P0 = -, u0 =0, ro ::0.

-}

3.2 Boundary Conditions

At the inlet boundary, the conservative flow variables and

their spatial derivatives are specified to be those of the

ambient flow, except at the nozzle exit, where an elevated

pressure is imposed, i.e., the jet is under-expanded, as in

the physical experiments. By using the ideal gas isentropic relations, it follows that the nondimensional flow

variables at the nozzle exit, with 3L, = 1, are given by

?(_ + 1)p_

2I;,

I

= ( 2Jr ,]1/2

ue \h+lJ ' _'e =0,

where T_ is the reservoir (plenum) temperature. We will

also follow the experimental cold-flow condition where

the reservoir temperature equals the ambient one, i.e.,

1"_,= 1.

At the symmetry axis, i.e., y = 0, a simple reflective

boundary condition is applied. At the top and outflow

boundaries, the Type I and Type II CE/SE non-reflecting

boundary conditions as described in the next subsection

are imposed, respectively. The no-slip boundary condition is applied on the nozzle walls.

3.3 Non-Reflecting Boundary Conditions

In the CE/SE scheme, non-reflecting boundary conditions (NRBC) arc constructed so as to allow fluxes from

the interior domain of a boundary CE to smoothly exit

from the domain. There are various implementations

of the non-reflecting boundary condition and in general

they have proven to be well suited for aeroacoustic problems [18,19]. The following NRBCs are employed in

this paper.

For a grid node (j, n) lying at the outer radius of the

domain the non-reflective boundary condition (Type I)

requires that

= : o,

while U_' is kept fixed at the initial steady boundary

value. At the downstream boundary, where there are

substantial gradients in the radial direction, the nonreflective boundary condition (Type II) requires that

while U_' and (U._j)}' are now defined by simple extrapolation from the nearest interior node f, i.e.,

Uj.... U ''-1/2J, _IUu)"j = (U_)j, i/'.,

As will be observed later, these NRBCs, when combined

with the above buffer zones, are robust enough to allow

a near field computation without disturbing or distorting

the flow and acoustic fields.

4 Numerical Results

In this section, numerical results for the axisymmetric

undercxpanded circular.jet are presented and compared

to experimental rcsults 15-9]. Computations are conducted for three different jet Mach numbers 31j - 1.1,

1.15, and 1.19. With an appropriate time step size, a

large number (minimum 410,000) of time steps are run

for these cases in order to achieve appropriate accuracy

in the Fourier analysis of the time series data.

NA SA/TMI2001-210958 5
It isimportant toemphasizethatnoharmonicforcing

isimposedinthenumerical simulation. Theinitialimpactof theboundaryconditionatthenozzleexitstimulatesthejetshearlayerandtriggersthefeedbackloop

thatgeneratesthescreechwaves.

Inthefollowing,thecomputedjetflowdataandacoustic screechdataarepresentedseparately, andcomparisonsaremadewithavailableexperimental results.

4.1 Time-Averaged Flow and Shock-Cell

Structure

For the typical case of _,lj = 1.19, Fig. 3 shows p, p,

.u contours and a numerical Schlieren picture. They are

time averaged over 40,000 time steps and are obtained

after the start-up transients have moved out of the computational domain. In the p, p, and u contours, the shock

cell structure and the shear layer spreading are clearly

captured. Note that although the LES model is highly

simplified, it still distinctly affects the flow field. Since

the momentum exchange around the shear layer is accounted for through a high local value of the eddy viscosity, the shear layer spreads much more rapidly as compared to our earlier related Euler computations [18], and

the static pressure maximum increases by 15 - 20%.

Experimental results for jets are often documented in

terms of Schlieren pictures. It is straightforward to construct Schlieren graphs (density-gradient modulus) from

the numerical results. Such a numerical plot is shown in

the last panel of Fig. 3. A comparison of the numerical Schlieren contours from the current computations

and the experimental Schlieren photograph in [5] (Fig. 4)

shows good agreement in shock-cell structure. For example, the shock cell width is about 0.SD in the streamwise

direction. In both plots, it is observed that the first two

shock cells appear to be sharp and clear since the shearlayer instability wave is too weak at these locations to

significantly affect the shock cell. However, it interacts

strongly with the shock cells once it has gained a sufficient amplitude through its streamwise growth, which is

evident by the deformation of the third and fourth shock

cells. The delbrmation and movement of the shock cells

are also seen in the experiments [5,6]. This is, as pointed

out by Seiner [2], the mechanism by which the acoustic waves that produce the broadband shock-associated

noise and the screech tones are generated. Furthermore,

the fifth shock cell and those further downstream are so

severely delbrmed that they almost disappear.

4.2 Near-Field Radiating Screech Waves

For the same case of Mj = 1.19, Fig. 5 illustrates the radiating screech waves at the time level of 410,000 steps.

Since no forcing is applied in the numerical simulation,

these waves are a clear indicator of a self-sustained oscillation. For this figure, the very high isobar level contours, corresponding to hydrodynamic waves around the

p (.363-.608)

rho (.695-1

u (-.121-1.257) /--

/-

Numerical Schlieren

Figure 3: Time averaged p, p and u contours and numerical Schlieren plot, Mj = 1.19.

:' i r

Figure 4: Experimental Schlicren plot showing shock

cell structure, Mj = 1.19.

NASA/TM--2001-210958 6
isobars at time step 410,000

0.715

0.714

0.713

_0.712

nozzle

Figure 5: Screech wave radiation, 3Ij = 1.19.

jet core area are 'cut off' and the 'colors' are appropriately adjusted so that the acoustic waves outside of the

jet core are clearly displayed. Furthermore, the corresponding instantaneous numerical Schlieren image is superimposed on the plot in the core region. The screech

wavelength is about 1.6D, well in line with experimental

results (e.g. [5]) and theoretical predictions, e.g. Seiner

[21, Tam [I I.

4.3 Sound Pressure Level and Screech

Frequency

The numerical time history is recorded for a select number of locations in the flow field and later post-processed

to obtain spectral information using Fast Fourier Transform (FFT) techniques. The data at a selected position is recorded every twentieth actual time step, and the

recording begins after an initial time period has elapsed

(80,000 actual time steps). Typically, each time trace

contains 33,000 points and corresponds to more than 0.7

million computational time steps. Figure 6 is a time trace

of the unsteady pressure at a location on the nozzle exit

lip wall (.r = 4.0, .tl 0.642) for .'IIj 1.19. The lime

signal is quite irregular.

Estimates of the sound pressure level (SPI,), in dB are

shown in Figs. 7-9. The abscissas denote frequency in

Hz. In the following, all the spectral analyses are conducted at the nozzle exit lip wall (.r = 4.0, ._I- 0.642).

This is the location used by Ponton et a119] in their measurement, which is somewhat different from the location

used by Panda [5 IFigure 7 displays the SPL lbr the case of Mj = 1.19.

The SPL shows that there are multiple spikes in the

spectrum. The two distinct higher spikes correspond to

screech frequencies of 8890 Hz (SPL= 126 dB, A2 mode)

and 6945 Hz (SPL=131 dB, A1 mode) respectively.

Figure 8 shows the SPL for the case of M 3 - 1.15.

Still, there are multiple spikes in the spectrum. The

two distinct higher spikes correspond to screech frequencies of 9410 Hz _SPL=I25 dB, A2 mode) and 7690 Hz

(SPL=137 dB, Aj mode) respectively. It is observed

that these clearly stand out from the mixing and shockassociated noise, along with spikes at their first harmonics. The above two cases support the possibility that

there could be two screech tone resonances occur at the

same time or that screech-mode switching is present.

Figure 9 demonstrates the SPL for the case of Mj -

1.10. This time, only a single distinct spike stands out

in the spectrum. The spike corresponds to a screech frequency of 9040 Hz (SPL=I39 dB, At mode).

4.4 Acoustics Properties Compared to

Experiments

Since the feedback loop is very sensitive to any change

in the nozzle geometry, the computed screech tone IreNA SA/TM--2001-210958 7
.,P'-'_n= 1oi Q kPa

J

g

-3OO

-900

ooooo oo15o oO3OO oO45O ooeco

amQ (_)

Figure 6: Time history of unsteady pressure at nozzle

exit (z = 4.0,.q = 0.642), Mj = 1.19

loo

2OOOO

Figure 9: SPL at nozzle exit (x = 4.0, y = 0.642), Mj --

1.10, 75 Hz digital binwidth.

• r

o sOc_ toooo ISCOo _ooo

_equency {Hz_

Figure 7: SPL at nozzle exit (x = 4.0, y = 0.642), Mj =

1.19, 75 Hz digital binwidth.

I

, ,i _

5OO0 _oooo I_ 20o00

neq_ncy (Hz)

Figure 8: SPL at nozzle exit (x = 4.0, y = 0._42), Mj =

1.15, 75 Hz digital binwidth.

11oo0

1oooo

oooo

_ooo

"I',_,,7ooo

o>'600o

e4) 5oo0

'4ooo

IL3000

2000

1000

_._8

Frequency vs. Mj

[] Panda (A 1 mode)

0 Panda (A2 mode)

Z_ Ponton et al (A 1 mode)

V Ponton et al (A 2mode)

• CE/SE (A 1mode)

• CEISE (A2 mode)

II I i I I IIi I il a Jl Ii I .... i .... I

1.1 1.12 1.14 1.16 1.18 1.2

Mj

Figure 10: Comparison of screech tone frequencies.

quency and SPL are slightly different from those observed in experiments. Even between the experimental

results of Panda et al. [7] and Ponton et al. [9], there

are differences in the measured results due to general

experimental setup and slight differences in nozzle lip

thickness and shape, although they have the same nozzle

diameters (25.4 mm) and operating Mach number.

The numerical results are summarized in Figs. 10 and

1I, and compared to the experimental data. Figure 10

displays a comparison of the screech tone frequencies between the numerical and experimental results. Both the

experimental data of Panda et al. [7] and Ponton et al. [9]

show distinctly the A1 and A2 screeching modes, and so

does the numerical data. The CE/SE results agree reasonably well with Panda et al.'s [7] data and seem to

have a consistent trends for the A] and A2 modes. Their

NASA/TM--2001-210958 8
_oo[ SPL vs. Mj

16o [ •[__

140 _'

_.12o[ •

_,oo[ [] Panda(A, mode)

8Ol- O Panda (A 2 mode)

I Z_ Ponton et al (A1 mode)

e0_- V Ponton et al (A2 mode)

4ol • CE/SE (A1 mode)

20 • CE/SE (A2 mode)

i I I i I I , I I I , | i I I _.0, ,!1 1._ 1." ,e ,.,, ,._

Mj

Figure I 1: Comparison of screech tone SPL.

difference is less than the difference between the two

experimental data sets of Panda et al. [7] and Ponton

et al. [9]. Figure 11 demonstrates a comparison of the

screech SPL between the numerical and experimental results. The computed SPL results are generally lower than

the experimental ones, but--except for the ),la 1.15

case--are in general agreement with Ponton et al.'s [9]

data. It is not surprising that the numerical SPL agrees

better with Ponton et al.'s [9] data, since their nozzle lip

(thickness=0.2D, no bevel) is used in the computation

and the data is taken at the same location 9 0.6.12 at

the nozzle lip wall, while Panda et al.'s [7] data is taken

at a slightly different location.

5 Concluding Remarks

In this paper, the unstructured CE/SE Navier-Stokes

method is applied for screech tone noise analysis of an

underexpanded supersonic axisymmetric .jet.

Many hydrodynamic and acoustic aspects of the computed results are in good agreement with experimental

findings [5-9], including the variation of screech frequency with mach number for the Al and A2 modes.

LES modeling was used to better capture the .jet spreading but is not believed to be a critical element for the

screech feedback loop.

The advantages of the CE/SE scheme were confirmed

in the present simulation. The implementation did not

require any special treatment or parameter selections to

capture the acoustics. The simple yet robust CE/SE

NRBC in combination with buffer zones virtually eliminates numerical reflections and allows for an effective

near-field computation. The use of an unstructured grid

offers flexibility to deal with more complicated geometries.

References

[l ] Tam, C. K W., "Supersonic Jet Noise," Ann. Rev.

Fluid Mech. vol. 27, pp. 17-43 (1995).

[2] Seiner, J. M., "Advances in High Speed Jet Aeroacoustics," AIAA Paper 84-2275 (I 984).

[3] Tam, C. K. W., "Jet Noise Generated by Large Scale

Coherent Motion," NASA RP-1258, pp. 311-390

(1991).

[4] Raman G., "Advancces in Understanding Supersonic Jet Screech: Review and Perspective," Prog.

Aerospace Sci. vol. 34, pp. 45-106 (1998).

[5] Panda, J., "'Shock Oscillation in Underexpanded

Screeching Jets," J. Fluid Mech., vol. 363, pp. 173-

198 (1998).

[6] Panda, J., "'An Experimental Investigation of

Screech Noise Generation," J. Fluid Mech., vol.

378, pp. 71-96 (1999).

[7] Panda, J, Raman, G. and Zaman, K. B. M. Q.," Underexpanded Screeching Jets from Circular, Rectangular and Elliptic Nozzles", AIAA paper 97-1623

(1997).

[8] Panda, J. and Seasholtz, R.G, "Measurement of

Shock Structure and Shock-Vortex Interaction in

Underexpandcd Jets Using Rayleigh Scattering",

Phys. Fluids, vol. II, pp. 3761-3777 (1999).

191 Ponton, M. K., Seiner, J. M. and Brown, M. C., "

Near Field Pressure Fluctuations in the Exit Plane

of a Choked Axisymmetric Nozzle", NASA TM

113137 (1997).

[10] Shen, H. and Tam, C. K. W., "Numerical Simulation of the Generation of Axisymmetric Mode Jet

Screech Tones," AIAA Paper 98-0283 (1998).

[11] Shen, H. and Tam, C. K. W., " Three -Dimensional

Numerical Simulation of the Jet Screech Phenomenon" A IAA Paper 2001-0820 (2001).

112] Chang, S. C., "The Method of Space-Time Conservation Element and Solution Element--A New

Approach for Solving the Navier-Stokes and Euler

Equations," Journal of Computational Physics. vol.

I 19, pp. 295-324 (1995).

NAS A/TM--2001-210958 9
113]Chang,S.C.,Yu,S.T.,Himansu, A.,Wang,X.

Y.,Chow,C. Y.andLoh,C.Y.,"TheMethod

ofSpace-Time Conservation ElementandSolution

Element--ANewParadigmforNumerical Solution

ofConservationLaws",Computational Fluid Dynamics Review, eds. M. M. Hafez and K. Oshima,

Wiley (1997).

[I 4] Chang, S.-C., Wang, X.-Y. and Chow, C.-Y., "The

Space-Time Conservation Element and Solution

Element Method--A New High Resolution and

Genuinely Multidimensional Paradigm for Solving

Conservation Laws:' J. Comp. Phys. vol. 159, pp.

89-136 (1999).

[15] Wang, X.-Y. and Chang S.-C., " A 2-D Nonsplitting Unstructured Triangular Mesh Euler

Solver Based on the Space-Time Conservation Element and Solution Element Method" C.F.D.J. vol.

8, pp309-325 (1999).

[16] Loh, C. Y., Hultgren, L. S. and Chang S.-C.,

"Computing Waves in Compressible Flow Using

the Space-Time Conservation Element Solution Element Method," AIAA J.. Vol. 39, pp. 794-801

(2001 ).

[171 Loh, C. Y., Hultgren, L. S., Chang, S.-C. and Jorgenson, P. C. E., "Vortex Dynamics Simulation in

Aeroacoustics by the Space-Time Conservation Element Solution Element Method," AIAA Paper 99-

0359 (1999).

[181 Loh, C. Y., Hultgren, L. S., Chang, S.-C. and Jorgenson, P. C. E., "Noise Computation of a Supersonic Shock-Containing Axisymmetric Jet by the

CE/SE Method," AIAA Paper 2000-0475 (2000).

[I 9] Loh, C. Y., Chang, S.-C. and Scott, J. R., "Computational Aeroacoustics via the Space-Time Conservation Element/Solution Element Method", AIAA

Paper 96-1687 (1996).

[20] Bogey, C., Bailly, C. and Juv6, D., "Computation of

the Sound Radiated by a 3-D Jet Using Large Eddy

Simulation," AIAA Paper 2000-2009 (2000).

NASA/TM--200 1-210958 10
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4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Near Field Screech Noise Computation for an Underexpanded

Supersonic Jet by the CE/SE Method

6. AUTHOR(S)

Ching Y. Loh, Lennan S. Huhgren, and Philip C.E. Jorgenson

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administralion

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135-3191

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NASA TM--200 ]-210958

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11. SUPPLEMENTARY NOTES

Prepared for the Seventh Aeroacoustics Conference cosponsored by the American lnsfilute of Aeronat.tics and Astronautics and the

Contederation of European Aerospace Societies, Maastricht. The Netherlands, Ma} 28-30, 2001. Chmg Y. Loh, Tailech. Inc.,

21000 Brookpark Road, Cleveland, Ohio 44135; Lennarl S. Hullgren and Philip C.E. lorgenson, NASA Glenn Resca,ch Center.

Responsible person. Philip C.E. Jorgenson, organization code 5880. 216-433-5386.

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13. ABSTRACT (Maximum 200 words)

The space-time conservation element and solution element (CE/SE) method i_ employed to numerically study the

near-field axisymmetric screech-tone noise of a typical underexpanded circular jet issuing from a sonic nozzle. For

the computed cases, corresponding to fully expanded Math numbers of 1.10, I. 15 and 1.19, the self-sustained

feedback loop is automatically established. The computed shock-cell structure, acoustic wave length, screech tone

frequencies, and sound pressure levels are in good agreement with experimental results.

14. SUBJECT TERMS

Screech noise: CE/SE method

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