AIAA-2002-3889

COMPUTING AXISYMMETRIC JET SCREECH TONES USING UNSTRUCTUREDGRIDS

Philip C. E. Jorgenson and Ching Y. Loh ∗

National Aeronautics and Space AdministrationGlenn Research Center at Lewis Field

Brookpark, Ohio 44135, USA

AbstractThe space-time conservation element and solu-tion element (CE/SE) method is used to solvethe conservation law form of the compressibleaxisymmetric Navier-Stokes equations. Theequations are time marched to predict the un-steady flow and the near-field screech tonenoise issuing from an underexpanded circularjet. The CE/SE method uses an unstructuredgrid based data structure. The unstructuredgrids for these calculations are generated basedon the method of Delaunay triangulation.

The purpose of this paper is to show that anacoustics solution with a feedback loop can beobtained using truly unstructured grid technol-ogy. Numerical results are presented for twodifferent nozzle geometries. The first is con-sidered to have a thin nozzle lip and the secondhas a thick nozzle lip.

Comparisons with available experimental dataare shown for flows corresponding to severaldifferent jet Mach numbers. Generally goodagreement is obtained in terms of flow physics,screech tone frequency, and sound pressurelevel.

1 IntroductionIt is desirable to develop a general computer code that

can predict the complex flow structure around compli-cated geometries and the associated aeroacoustics. Manycomputational fluid dynamics methods have been devel-oped that take advantage of an inherent grid structurewhich permits flow solutions to be obtained efficiently.

∗Taitech Inc., Member AIAA

Copyright 2002 by the American Institute of Aeronautics andAstronautics, Inc. No copyright is asserted in the United States underTitle 17, U.S. Code. The U.S. Government has a royalty-free license toexercise all rights under the copyright claimed herein for governmentpurposes. All other rights are reserved by the copyright owner.

For computational aeroacoustics (CAA), another obvi-ous advantage of the structured grid is its well-controlledgrid quality (size and shape) that helps to maintain thehigh quality of the computed acoustic waves. This struc-ture, which makes the solver so efficient, often makesit difficult to generate grids about complex geometrieswithout an additional complication of blocked or oversetgrid techniques.

An unstructured grid flow solver can alleviate many ofthe problems associated with structured grids. Unlike astructured grid, the unstructured grid neighborhood mustbe provided to the solver explicitly. The simplest geo-metric figure that can cover a two-dimensional domain isthe triangle. The advantage of the triangle is that it caneasily be used to generate a grid over an arbitrary domainwithout overlapping. It also has the advantage of cluster-ing in high flow gradient regions without the concern ofthe surrounding cell structure. The main disadvantage ofthe unstructured grid is the additional memory require-ments in the flow solver to store the grid. More of theburden is placed on the computer code, thus the user in-put is reduced both during grid generation and flow so-lution such that more configurations can be tested in ashorter period of time. For CAA, the grid quality of anunstructured grid is less controllable and it may deterio-rate the acoustic wave computations. The situation maybe even more critical if a complicated acoustic feedbackloop is present, which is the problem type considered inthis paper.

The problems under consideration in this paper are cir-cular underexpanded supersonic jets [1-8]. These jets ra-diate a combination of mixing noise, broadband shock-associated noise, and under certain conditions screechtones. Mixing noise from jets can be directly associ-ated with large-scale structures, or instability waves inthe shear layer. The broadband shock-associated noiseand screech tones are associated with the interaction ofthese instability waves with shock cell structure in thejet core. The screech tones arise due to a feedback loop,i.e., part of the acoustic waves generated by the instabil-

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38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit7-10 July 2002, Indianapolis, Indiana

AIAA 2002-3889

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code.The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.All other rights are reserved by the copyright owner.

ity wave/shock-cell interaction propagate upstream andre-generate the instability waves at, or in the vicinity of,the nozzle lip. The desire to control screech noise isnot only due to general noise reduction concerns. It canalso cause damage to the structure due to sonic fatigue.The feedback loop is sensitive to small changes such aslip thickness and other phenomena. Most of this under-standing comes from experimental observations. Numer-ical methods that can predict screech tones will help inthe understanding and the eventual control of this noisesource.

A direct numerical simulation was computed by Shenand Tam [9,10] for screech tones issuing from a circularjet, based on the well-known DRP scheme, with a struc-tured grid. A large computational domain of35D longin the streamwise direction and17D in radial directionwas used in their computation, withD being the nozzlediameter.

More recently the space-time conservation elementand solution element (CE/SE) method with a triangulatedstructured grid [11-14] has been used to predict the near-field screech tones from underexpanded supersonic jets[15]. The near-field results compared well with the ex-perimental data, [4-8]. This work also demonstrated thecapability of the CE/SE nonreflecting boundary condi-tion. This allows the computational domain to be morecompact reducing the overall mesh memory size require-ments. The present work is a companion work of [15].Here a truly unstructured grid, generated by a point in-sertion Delaunay method rather than a triangulated struc-tured grid, is employed. This is done with the idea thatmore complex geometries will require a more flexiblegrid generation technology. The purpose of this paper isto show that an acoustics solution with a feedback loopcan be obtained using unstructured grid technology.

The governing equations are given in Section 2. Herethe nondimensionalized unsteady axisymmetric Navier-Stokes equations in conservation law form are presented.Unstructured grid generation is discussed in Section 3.The requirements of the grid for the flow solver will beexplained. Also the flow code was executed on a parallelcomputer architecture, so the domain decomposition andmessage passing libraries are briefly reviewed. The un-structured based axisymmetric CE/SE method used hereis briefly discussed in Section 4. Section 5 will providethe problem description with the initial and boundarycondition requirements for the particular screech prob-lems examined in this paper. In Section 6 numerical re-sults are presented and and compared to previous numer-ical findings using structured grids and available exper-imental findings [4-8]. Conclusioning remarks will bemade in Section 7.

2 Governing EquationsThe unsteady axisymmetric Navier-Stokes equations

are used to model the viscous problems in this study,since experimentally for the thin lip nozzle [4-8] withMj 6 1.19 the flow is axisymmetric. In nondimension-alized differential form the conservation laws are writtenas:

U t + F x + Gy = Q, (1)

wherex, y, andt are the streamwise and radial coordi-nates and time, respectively. The conservative flow vari-able vector,U , and the flux vectors in the streamwisedirection,F , and radial direction,G, are given by:

U =

U1

U2

U3

U4

, F =

F1

F2

F3

F4

, G =

G1

G2

G3

G4

,

withU1 = ρ, U2 = ρu, U3 = ρv,

U4 = p/(γ − 1) + ρ(u2 + v2)/2.

Hereρ, u, v, p, andγ are the density, streamwise velocitycomponent, radial velocity component, static pressure,and constant specific heat ratio, respectively. The fluxvectors are further split into inviscid and viscous fluxes:

F = F i − F v,G = Gi −Gv,

where the inviscid fluxes are given as:

Fi1 = U2,

Fi2 = (γ − 1)U4 +[(3− γ)U2

2 − (γ − 1)U23

]/2U1,

Fi3 = U2U3/U1,

Fi4 = γU2U4/U1 − (γ − 1)U2

[U2

2 + U23

]/2U2

1 ,

Gi1 = U3, Gi2 = U2U3/U1,

Gi3 = (γ − 1)U4 +[(3− γ)U2

3 − (γ − 1)U22

]/2U1,

Gi4 = γU3U4/U1 − (γ − 1)U3

[U2

2 + U23

]/2U2

1 ,

and the viscous fluxes are:

Fv1 = 0, Fv2 = µ(2ux − 23∇·V),

Fv3 = µ(vx + uy),

Fv4 = µ[2uux + (uy + vx)v − 23 (∇·V)u+

γ

Pr

∂

∂y(U4

U1− u2 + v2

2)],

Gv1 = 0, Gv2 = µ(vx + uy),

Gv3 = µ(2vy − 23∇·V),

2American Institute of Aeronautics and Astronautics

Gv4 = µ[2vvy + (uy + vx)u− 23 (∇·V)v+

γ

Pr

∂

∂y(U4

U1− u2 + v2

2)],

whereu, v, ux, uy, vx, vy are respectively thex and yflow velocity components and their derivatives, whichcan be written in terms of the conservative variablesU1, U2, U3 and U4. The Prandtl number and viscosityarePr andµ, respectively. The velocity divergence is

∇·V = ux + vy + v/y.

The source term,Q, appearing on the right hand sideof the equation, Eq. (1), is a result of the axisymmetricformulation and is given as:

Q =

Q1

Q2

Q3

Q4

,

where

Q1 = −U3/y, Q2 = −U2U3/U1y,

Q3 = −U23 /U1y, Q4 = −G4/y.

Considering (x, y, t) as coordinates of a three-dimensional Euclidean space,E3, the Gauss divergencetheorem can be used to give the following equivalent in-tegral form of the conservation laws, Eq. (1):∮

S(V )

Hm · dS =∫

V

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

whereS(V ) denotes the surface of the volumeV in E3

andHm = (Fm, Gm, Um) represents the total flux leav-ing the boundary of space-time region. The integral formof the conservation laws allows the capture of shocksand other flow discontinuities naturally. This integralform of the equations is solved numerically by the space-time conservation element and solution element methodwhich is described in Section 4.

3 Grid GenerationAn unstructured grid is used to show that aeroacous-

tics with a feedback loop can be predicted in complexdomain shapes. A triangular shaped cell will be usedas the geometric shape to cover a two-dimensional com-putational domain. The use of an unstructured grid re-moves the available implicit reference of the local com-putational neighborhood by incrementing an index as ina structured grid. Instead, the neighborhood is providedexplicitly through a connectivity matrix. This connectiv-ity matrix usually contains cell based information as wellas edge based information.

Figure 1: Triangulation of computational domain withenlargement showing nozzle lip region.

The use of a triangular unstructured grid formulationhas some distinct advantages over a structured grid. Oneadvantage to using triangular cells is that with them it iseasy to generate grids about complex geometries. Thisreduces the amount of time required to generate a suit-able grid. Also grid adaptation can be done locally with-out adding unnecessary cells to other regions of the do-main.

The method used in this work follows a path similar tothat of Holmes and Snyder [16] to triangulate a region.An example of an unstructured grid for the thin lip nozzlegeometry generated in the above manner is shown in Fig.1. The computation is axisymmetric, so only the upperhalf of the domain needs to be considered. An enlarge-ment of the nozzle lip region gives an indication of thegrid resolution used in these aeroacoustic feedback loopproblems.

The flow code requirements dictate the type of outputthat the grid generation scheme must provide. A connec-tivity array must be generated for an unstructured gridso that a cell neighborhood is completely defined for theflow code. The code in the current work is based on acell centered scheme. Here the triangle itself is the con-trol volume used in the finite volume formulation.

Connectivity is determined by cell nodes, cell edges,and edge cells. Cell nodes are the nodes at the verticesof a triangle. The cell edges are the edges of a triangle

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that connect those nodes. The edge cells are the cellsthat lie on either side of a given edge. Typically thex,y coordinates of the cell nodes are the only grid floatingpoint numbers required as input by a flow code. The gridconnectivity is defined through integer arrays.

Boundary condition information must be defined ex-plicitly. Ghost cells are used across boundaries such thatspecific boundary conditions can be applied. A boundarycondition such as solid wall, inlet, exit, or any other userdefined condition can be prescribed along any cell edgethrough the ghost cell.

For aeroacoustic problems, large numbers of cells areneeded to resolve the unsteady waves. The grids aremade more manageable through the use of parallel com-puter architecture. The publically available code METIS[17] is used to decompose the unstructured mesh into apredetermined number of subdomains so that it can takeadvantage of computers with multiple processors. Themesh partitioning code then load balances, minimizes thecommunication between processors, and minimizes themaximum number of subdomains adjacent to any othersubdomain. Each processor then operates on a subsetof the larger domain. It is necessary to communicate thesubdomain boundary information to the neighboring pro-cessor so that the numerical algorithm can perform com-putations that require data from its immediate neighbor.The message passing libraries provided by the standard-ized package MPI [18] is used for point-to-point com-munication between processors. For unstructured gridbased solvers a list must be maintained that provides theboundary cells so that the appropriate data may be sentand received by a processor.

A windowing technique is used to pack points in re-gions of the computational domain where more resolu-tion is desired. To a limited extent the smoothness of thegrid is sacrificed for an increase in resolution using thistechnique. A grid generated by this method will be re-ferred to as windowed Delaunay point insertion (WDPI).

The unstructured grid generator Gridgen [19] is alsoused to generate a grid in the same domain for compari-son purposes. The stretching techniques in this grid gen-erator results in a much smoother grid than that providedby the windowing technique.

4 Discretization TechniqueThe space-time conservation element and solution el-

ement method is a finite volume method where the el-ements include space and time coordinate dimensions.The original two-dimensional CE/SE method was de-rived for triangular space-time grids. To date the gridsused in computing the aeroacoustics from jet flows us-ing this method have been triangulated structured grids.Here the method is applied to more general unstructuredgrids generated by the Delaunay method described in the

A

B

C

A’

B’

C’

x

y

t

D

E

F

D’

E’

F’

O’

O

Figure 2: Conservation elements used for updating solu-tion.

previous section. In this section a general review of theCE/SE method is given as it is applied to jet noise calcu-lations.

There are three conservation elements (CE’s)that areassociated with a given triangular space-time element,Fig. 2. The three conservation elements are shared ge-ometrically by the three adjacent triangular space-timeelements. The geometric center of the triangular facesadjacent to face∆ABC are shown as pointsD, E, andF . Each conservation element has six faces that forma quadrilateral cylinder in space-time. The bottom andtop faces are formed by connecting the centroids of thetriangular bottom and top faces (of the space-time trian-gular element) to the nodes that make up that face, re-spectively. For a given space-time triangular element,this construction produces three quadrilateral faces oneach of two spatial planes (i.e.,A,D,B, O). These arenonoverlapping elements. For a given space-time trian-gular cylinder, the union of the three conservation el-ements forms a hexagonal cylinder of volumeV (thelower face of this hexagonal cylinder is composed ofnodal pointsA,D,B, E, C, F ). The conservation lawEq. (2) is applied to this hexagonal cylinder. The result-ing solution is associated with the center of the hexagonat the new time level (not necessarily coincident withO′).

The solution elements (SE) used here are the inter-faces of the conservation elements that form the hexago-nal cylinder, Fig. 2.

The integral equation, Eq. (2), will provide a solutionat the geometric center of the top hexagonal face. Thisdoes not necessarily coincide with the triangle cell center(the solution element) location,E. Discretization of theintegral conservation laws, Eq. (2) leads to∑

S(V )

H∗m ·∆S = V (Qm)n+1

j , (3)

4American Institute of Aeronautics and Astronautics

for m = 1, 2, 3, 4, where the space-time flux at the sur-face centers ofS(V ), H∗

m = (F ∗m, G∗

m, U∗m), can be ob-

tained. The source term on the right-hand side of Eq. (3)is computed at the center of the hexagonA′D′B′E′C ′F ′

at the new time level.To evaluate the resulting system of algebraic equa-

tions, Eq. (3), the values must be computed at the solu-tion element faces (actually the face centers). These facecentered values are approximated by the linear Taylor se-ries expansions. The unknowns in the integral equation,Eq. (3), are located at the cell center of the hexagon as-sociated withO′. Expanding the integral equation for agiven space-time element results in 12 scalar equationsfor the 12 unknowns atO′. Each of the three conser-vation elements contributes four scalar equations. Moredetails of the solution method can be found in [14].

The simple non-reflecting boundary conditions de-scribed previously [20,21, 15] still work well with a De-launay based unstructured grid. The weighteda − εCE/SE scheme is used in this work. To suppress wig-gles around shocks the flux derivatives use a weightedaverage similar to that of van Albada [22]. Details areprovided in [14].

The source term evaluation was presented in a previ-ous paper [15]. Briefly, the source termQ is a functionof the unknownU ; therefore, a local Newton iterativeprocedure is needed. The procedure takes about 2-3 iter-ations to converge.

A simplified implicit LES procedure similar to thoseused in [23] is adopted here to account for the strongmomentum exchange in the shear layer. In the LES, asimple Smagorinsky subgrid scale model is used for theeddy viscosity:

µt = (Cs∆)2(2SijSij)1/2,

where

Sij = 12 (

∂ui

∂xj+

∂uj

∂xi),

with ∆ = (∆x∆y)1/2 andCs = 0.1. Then,µ + µt

replacesµ in the actual computation. In practice, theLES model was only applied for radii less than1.5D and4D downstream of the nozzle exit plane.

5 Problem DescriptionThe physical domain of interest for the screech prob-

lem is shown in Fig. 3. A slice of the axisymmetric do-main shows the thin and thick lip nozzle geometries andthe near-field boundaries where boundary conditions arespecified in the flow code. The circular jet flow entersthrough the nozzle from the left and expands into the sta-tionary ambient atmosphere. The flow condition speci-fied at the nozzle exit at the start of the computations ischoked.

recessed nozzle exit plane

0.2D thick lip

0.625D thick lip

axis of symmetry

nozzle wall

InletBC

NRBC

OutflowNRBC

0.5D2D 6D

6D

Figure 3: Geometry of the computational domain.

Here we are interested in predicting the source of thenoise rather than its propagation into the far-field. The di-ameter,D, of the jet nozzle is chosen as the length scale.The density,ρ0, speed of sound,a0, and temperature,T0,in the ambient flow are taken as scales for the dependentvariables.

The Delaunay grid generator is used to cluster the tri-angular cells near the nozzle lip to provide good resolu-tion for the shock cells and the acoustic feedback looppath. Cell stretching is used for these computations. Wewill compare flow solutions computed on a Delaunaybased unstructured grid, described in Section 3 of thispaper, with those previously computed on a triangulatedstructured grid [15]. The triangulated structured mesh isconstructed by dividing each rectangular cell of the struc-tured mesh into four triangles. An unstructured connec-tivity matrix is then produced. More specifics about thenumbers of cells and flow code parameters will be pro-vided in the the results section of this paper.

To clearly display the upstream propagating screechwaves, the computational domain was extended2D up-stream of the nozzle exit. The full computational do-main is a circular cylinder of8D axial length and6Dradius. At the nozzle exit, the inflow plane is recessedby two cells so as not to numerically restrict or influ-ence the feedback loop. The first nozzle geometry def-inition follows that given by Pontonet al. [8] where astraight nozzle lip of0.2D in thickness is adopted. Thesecond nozzle geometry has the same computational do-main, however the nozzle lip thickness is increased to0.625D. The computational domain is modified by ex-tending the upstream, downstream, and radial boundariesby 2D to determine if the boundary locations has an ef-fect on the solutions. However, no discernible effect isrealized. Only one case was used to test the effect of theextended computational domain.

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5.1 Initial ConditionsInitially, the flow in the computation domain is set to am-bient flow conditions,i.e., (using nondimensional vari-ables)

ρ0 = 1, p0 =1γ

, u0 = 0, v0 = 0.

5.2 Boundary ConditionsAt the inlet boundary, the conservative flow variables andtheir spatial derivatives are specified to be those of theambient flow, except at the nozzle exit, where an elevatedpressure is imposed,i.e., the jet is under-expanded, as inthe physical experiments. By using the ideal gas isen-tropic relations, it follows that the nondimensional flowvariables at the nozzle exit, withMe = 1, are given by

ρe =γ(γ + 1)pe

2Tr,

pe =1γ

[2 + (γ − 1)M2

j

γ + 1

] γγ−1

,

ue =(

2Tr

γ + 1

)1/2

, ve = 0,

whereTr is the reservoir (plenum) temperature. We willalso follow the experimental cold-flow condition wherethe reservoir temperature is set to the ambient tempera-ture,i.e., Tr = 1.

At the symmetry axis,i.e., y = 0, a simple reflectiveboundary condition is applied. At the top and outflowboundaries, the Type I and Type II CE/SE non-reflectingboundary conditions as described in the next subsectionare imposed, respectively. The no-slip boundary condi-tion is applied on the nozzle walls.

5.3 Non-Reflecting Boundary ConditionsIn the CE/SE scheme, non-reflecting boundary condi-tions (NRBC) are constructed to allow fluxes from theinterior domain of a boundaryCE to smoothly exit thedomain. There are various implementations of the non-reflecting boundary condition and in general they haveproven to be well suited for aeroacoustic problems [15].The following NRBCs are employed in this paper.

For a grid node(j, n) lying on the outer radius of thedomain, the non-reflective boundary condition (Type I)requires that

(Ux)nj = (Uy)n

j = 0,

while Unj is kept fixed at its initial steady boundary

value. At the downstream boundary, where there aresubstantial gradients in the radial direction, the non-reflective boundary condition (Type II) requires that

(Ux)nj = 0,

while Unj and(Uy)n

j are now defined by simple extrap-olation from the nearest interior nodej′, i.e.,

Unj = U

n−1/2j′ (Uy)n

j = (Uy)n−1/2j′ .

These are identical boundary condition specificationsused in [15] for the results computed on a triangulatedstructured grid.

6 ResultsThe CE/SE method is applied to the underexpanded

axisymmetric circular jet, with two nozzle lip thick-nesses. The thin lip nozzle has a wall thickness of20%of the nozzle jet exit diameter. This case is used to com-pare the results obtained on a Delaunay based unstruc-tured grid with experimental data [8] as well as with re-sults previously obtained using a triangulated structuredgrid [15]. The flow solver for both grids is identical.The flow data is compared at a jet Mach number,Mj ,of 1.19. Screech frequency and sound pressure levelsover a range of jet Mach numbers are also compared forthis thin lip nozzle configuration. The thick lip nozzlehas a wall thickness of62.5% of the nozzle jet exit di-ameter. This case is compared with expermental results[8] over a range of jet Mach numbers. A large number oftimes steps are required to achieve appropriate accuracyin the Fourier analysis of the time series data. Typically80K time steps are needed to remove the initial tran-sients from the computational domain. Then, another330K time steps are required to record the time seriesdata.

It is important to emphasize that no harmonic forcingis imposed in the numerical simulation. The initial im-pact of the boundary condition at the nozzle exit stim-ulates the jet shear layer and triggers the feedback loopthat generates the screech waves.

In the following, the computed jet flow data and acous-tic screech data are presented separately, and compar-isons are made with available experimental results. Thethin lip nozzle is examined first and comparisons aremade with the experimental and the previously obtainednumerical results. The results of the thick lip nozzle arethen presented and compared with the experimental re-sults.

6.1 Flow and Shock-Cell Structure for the ThinLip Nozzle

For the flow data, the CE/SE method is applied to thethin lip nozzle case at aMj of 1.19. The resulting shockcell structure for the axisymmetric flow calculation af-ter 200K time steps is shown as a Schlieren (density-gradient modulus) contour plot, Fig. 4. The experimen-tal [4] shock cell structure is shown in Fig. 5, and thenumerical results show good agreement with the shock-cell structure in the experiment. For example, the shock

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Figure 4: Instantaneous snapshot of shock cell structurefor thin lip nozzle, (numerical Schlieren)Mj = 1.19.

Figure 5: Experimental Schlieren plot showing shockcell structure,Mj = 1.19.

cell width is about0.8D in the streamwise direction. Inboth plots, it is observed that the first two shock cellsappear to be sharp and clear since the shear-layer insta-bility wave is too weak at these locations to significantlyaffect the shock cell. However, it interacts strongly withthe shock cells once it has gained a sufficient amplitudethrough its streamwise growth, which is evident by thedeformation of the third and fourth shock cells. The de-formation and movement of these shock cells are alsoseen in the experiments [4,5]. This is, as pointed out bySeiner [2], the mechanism by which the acoustic wavesthat produce the broadband shock-associated noise andthe screech tones are generated. Furthermore, the fifthshock cell and those further downstream are so severelydeformed that they almost disappear.

As demonstrated in the previous numerical results[15], the screech wave length of1.6D is predicted in thecomputations. This is in good agreement with the exper-imental results (e.g.[4]) and the theoretical predictions ofSeiner [2] and Tam [1].

The thin lip nozzle flow solution on the WDPI andGridgen based grids compare well with the solution ob-tained previously on the triangulated structured grid [15].The triangulated structured grid used a grid consisting of132K cells. The WDPI grid has167K cells. The Grid-gen has a136K cell grid. Since the flow code is thesame for all of the grids presented here, there is no clearadvantage to using a triangulated structured grid. In fact,a truly unstructured grid should be much more effectivewhen the CE/SE method is applied to complex geome-tries.

1.08 1.1 1.12 1.14 1.16 1.18 1.20

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Panda (A1 mode)

Panda (A2 mode)Ponton et al. (A1 mode)

Ponton et al. (A2 mode)CE/SE triangulated structured (A1 mode)

CE/SE triangulated structured (A2 mode)CE/SE unstructured WDPI (A1 mode)CE/SE unstructured WDPI (A2 mode)CE/SE unstructured Gridgen (A1 mode)CE/SE unstructured Gridgen (A2 mode)

Fre

quen

cy(H

z)

Mj

Thin Lip Nozzle

Figure 6: Frequency of thin lip nozzle over a range ofMj .

6.2 Screech Frequency and Sound PressureLevel

Because the data is unsteady, many of the same tech-niques used to analyze experimental data are also usedto analyze the numerical results. At a point on the nozzleexit lip (x = 2.0, y = 0.642) the time history of pressureis stored over a large number of time steps. This datais then post processed by Fast Fourier Transform (FFT)techniques to obtain spectral information.

The resulting screech frequencies are plotted over arange of jet Mach number in Fig. 6. Two experimentaldata sets are provided by Pandaet al. [6] and Pontonet al. [8]. The nozzles in the experiments have the samediameters and operating Mach number, however they dif-fer slightly in shape and thickness. The geometry used inthe numerical predictions is that of Pontonet al. [8]. TheCE/SE method is used to compute a solution on the samejet geometry but on three different grids. The first seriesof solutions are computed on the triangulated structuredgrid [15]. The second set of numerical data are computedon a grid generated by WDPI. This screech frequencycompares quite well with that in [15]. They slightly un-derpredict the experimental data at the lower jet Machnumbers but they do predict the same trend. At the higherlevels ofMj the screech frequencies compare well withexperiment.

The third set of solutions are computed on a grid gen-erated by Gridgen [19]. The frequencies produced by thesolution on the Gridgen grid are lower than those pro-duced on the the other two grids. At a low jet Mach num-ber,Mj , of 1.09 there are no detectable acoustic waves.

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1.08 1.1 1.12 1.14 1.16 1.18 1.20

20

40

60

80

100

120

140

160

180

Thin Lip Nozzle

Mj

SPL

(dB

)

CE/SE triangulated structured (A1 mode)

Panda (A1 mode)Panda (A2 mode)Ponton et al. (A1 mode)

Ponton et al. (A2 mode)

CE/SE triangulated structured (A2 mode)

CE/SE unstructured WDPI (A1 mode)CE/SE unstructured WDPI (A2 mode)CE/SE unstructured Gridgen (A1 mode)

CE/SE unstructured Gridgen (A2 mode)

Figure 7: Sound pressure level of thin lip nozzle over arange ofMj .

The computations at the other jet Mach numbers under-predict theA1 mode screech frequencies observed in theexperiment, but they do predict the same trend. TheA2

mode screech frequencies give a close match to the ex-perimental data.

The sound pressure level (SPL) is shown in Fig. 7. Thecomputed results are in general lower than the experi-mental data. They are closer to the experimental resultsof Pontonet al. [8]. The solutions on the WDPI gridcompare well with the triangulated structured grid. Thesolutions on the grid generated by Gridgen [19] under-predicts the SPL of theA2 mode significantly. This dif-ference could be due to the lack of resolution in that griddownstream of the nozzle.

In general it is felt that it is necessary to cluster cellsin critical flow areas. The grid generated by WDPI isprobably better in this case since points are forced intothe region of the flow where the interaction between theshock cell and shear layer is strongest. A careful choiceof α and time step is necessary to obtain the appropriateamount of dissipation.

6.3 Flow and Shock-Cell Structure for theThick Lip Nozzle

For the thick lip nozzle case with aMj = 1.19, thecomputed shock cell structure is shown in Fig. 8. Thisis an instantaneous snapshot of isobars combined witha numerical Schlieren. Unlike the thin lip nozzle case,the shock cell location which interacts with the instabil-ity wave begins between the second and third shock cell.This is much closer to the nozzle lip than is observed in

nozzle

shock cells

entrainmentvortex

Figure 8: Instantaneous snapshot of screech wave radia-tion for thick lip nozzle, (isobars and Schlieren)Mj =1.19. Solution shows a strong subharmonic.

the thin lip nozzle case. This is probably due to a strongerfeedback mechanism due to the thick lip. Figure 8 is thelarger computational domain, alluded to earlier in thispaper, to test the effect of boundary location. The nozzleexit location is not changed, so the pressure data is stilltaken atx = 2.0, y = 0.642.

The thick lip case shows a strong fluid flow entrain-ment which produces a counter rotating vortex just up-stream of the nozzle lip, Fig. 8. This entrainment vortexis observed for over500K time steps. The location ofthe vortex practically does not move. The vortex passescleanly through and breaks the acoustic waves as theymove upstream.

A strong subharmonic wave is observed in this calcu-lation. It propagates from a region in the flow where thepairing of vortices occurs and out through the upstreamboundary. The subharmonic wave is also split by the en-trainment vortex. Due to the two-dimensional axisym-metric aspect of the computation, the subharmonic wavemay be much stronger than the physical one.

The total number of grid cells for the WDPI grid is184K cells. The grid is clustered in the region of thenozzle lip using the windowing technique described ear-lier. The Gridgen grid contains140K cells. This grid isalso clustered but with smoother results than that givenby windowing. Figure 8 has250K cells with grid clus-

8American Institute of Aeronautics and Astronautics

tering in the vicinity of the nozzle lip.For this geometry anα of 0.5 is used for the WDPI

solution with a time step of0.0005. Any larger time steprequires a correspondly largerα to the point where it af-fects the acoustic solution. The Gridgen based solutionstill could be run with anα of 1.0 and a time step size of0.002.

6.4 Screech Frequency and Sound PressureLevel

The FFT is used to analyze the unsteady feedback loopthat is generated between the shock cell structure andthe nozzle lip. The pressure history is recorded over alarge number of time steps, Fig. 9. The pressure levelis recorded every twentieth time step after a time periodsuch that the start-up transients have washed out of thecomputational domain. Figure 9 represents the time traceof pressure (nondimensional) at the point on the nozzleexit lip, x = 2.0, y = 0.642 for a Mj = 1.15 using theWDPI grid. This signal is then analyzed by FFT. Theresulting SPL spectrum is shown in Fig. 10, where onlythe frequency range of0 to 20000 Hz is displayed. Thereare two SPL spikes in the spectrum that correspond to es-timated screech frequencies of7195 Hz (SPL = 142dB,A1 mode) and9169 Hz (SPL = 136dB,A2 mode) witha binwidth (or bandwidth) of326 Hz. This indicates thatthere is either two screech tone frequencies occurring atthe same time or that screech mode switching is present.A strong subharmonic is also shown in this case.

The screech frequencies at the lower jet Mach num-bers matched the results of Ponton et al. [8] very well,Fig. 11. At higher jet Mach numbers,Mj = 1.15, 1.17,and1.19, the screech frequencies are overpredicted.

The sound pressure levels for the thick lip nozzle in theexperiment has a higher level for theA2 mode than fortheA1 mode, Fig. 12. This is not seen in the numericalpredictions.

A major advantage of the grid produced by Gridgenis that it requires less input from the user than does theWDPI grid. The grid resolution could be augmented con-siderably in the feedback region of the flow domain withproportionately better results. The smoothness charac-teristics of the Gridgen grid may also play an importantrole in getting a better solution.

The axisymmetric computation may be predicting amuch larger subharmonic frequency than would be seenin an experiment or in a three-dimensional computation.This may be due in part to the fact that the flow is beingartificially confined as axisymmetric.

7 Concluding RemarksIn this paper, the unstructured CE/SE Navier-Stokes

method is applied to underexpanded supersonic axisym-metric jets over a range of jet Mach numbers for thin lip

0.002 0.003 0.004 0.005 0.006 0.0070.7

0.705

0.71

0.715

0.72

0.725

0.73

Time(sec)P

Thick Lip Nozzle

Figure 9: Time history of unsteady nondimensional pres-sure at nozzle exit (x = 2.0, y = 0.642), Mj = 1.15.

0 5000 10000 15000 2000090

100

110

120

130

140

150

Frequency(Hz)

SPL

(dB

)

Thick Lip Nozzle

Figure 10: Sound pressure level at nozzle exit (x =2.0, y = 0.642), Mj = 1.15, 326 Hz binwidth. Notethe strong subharmonics to the left.

9American Institute of Aeronautics and Astronautics

1.08 1.1 1.12 1.14 1.16 1.18 1.20

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Fre

quen

cy(H

z)

Mj

Ponton et al. (A2 mode)

Thick Lip Nozzle

Ponton et al. (A1 mode)

CE/SE unstructured WDPI (A1 mode)

CE/SE unstructured WDPI (A2 mode)CE/SE unstructured Gridgen (A1 mode)

CE/SE unstructured Gridgen (A2 mode) α=1.0, ∆Τ=0.002

α=0.5, ∆Τ=0.0005α=0.5, ∆Τ=0.0005

α=1.0, ∆Τ=0.002

Figure 11: Frequency of thick lip nozzle over a range ofMj .

1.08 1.1 1.12 1.14 1.16 1.18 1.20

20

40

60

80

100

120

140

160

180

Thick Lip Nozzle

Mj

SPL

(dB

)

Ponton et al. (A1 mode)Ponton et al. (A2 mode)

CE/SE unstructured WDPI (A2 mode)

CE/SE unstructured Gridgen (A2 mode)CE/SE unstructured Gridgen (A1 mode)

CE/SE unstructured WDPI (A1 mode)

α=1.0, ∆Τ=0.002

α=0.5, ∆Τ=0.0005α=0.5, ∆Τ=0.0005

α=1.0, ∆Τ=0.002

Figure 12: Sound pressure level of thick lip nozzle overa range ofMj .

and thick lip nozzle geometries. The resulting screechtones are analyzed.

Truly unstructured grids instead of triangulated struc-tured grids are used to show the flexibility of the CE/SEmethod and to test its crucial capability of using unstruc-tured grids to capture the physics in aeroacoustic prob-lems with a feedback loop. The shock cell structure andmany acoustic aspects of the flows compare well with theavailable experimental data, [4-8] and previous numeri-cal results, [15] including the variation of screech fre-quency with jet Mach number for theA1 andA2 modes.

The thick lip case shows a strong fluid flow entrain-ment which produced a counter rotating vortex just up-stream of the nozzle lip. This vortex is not seen in thethin lip case.

The difference in the total number of computationalcells used for a given domain by the solver with a trian-gulated structured grid and the truly unstructured grid isnot significant, so it is possible to consider aeroacousticproblems with more complex geometries using truly un-structured grids. Great care must be exercised in generat-ing an unstructured grid. There were several cases wherelack of appropriate grid resolution in critical regions ofthe computational domain produced a flow solution butno acoustics.

It is necessary to control the grid clustering to cap-ture the aeroacoustics in these jet flows with a feedbackloop. Grid qualities like smoothness, size, and shape,are important for the dissipative aspects of the numericalmethod.

AcknowledgementsThis work was supported by the Supersonic Propul-

sion Technology Program at the NASA John H. GlennResearch Center, Lewis Field.

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11American Institute of Aeronautics and Astronautics