Ching Y. Loh and Ananda HimansuTaitech, Inc., Beaver Creek, Ohio

Lennart S. HultgrenGlenn Research Center, Cleveland, Ohio

A 3-D CE/SE Navier-Stokes SolverWith Unstructured Hexahedral Gridfor Computation of Near Field JetScreech Noise

NASA/TM—2003-212314

June 2003

AIAA–2003–3207

https://ntrs.nasa.gov/search.jsp?R=20030064175 2020-05-21T00:30:48+00:00Z

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Ching Y. Loh and Ananda HimansuTaitech, Inc., Beaver Creek, Ohio

Lennart S. HultgrenGlenn Research Center, Cleveland, Ohio

A 3-D CE/SE Navier-Stokes SolverWith Unstructured Hexahedral Gridfor Computation of Near Field JetScreech Noise

NASA/TM—2003-212314

June 2003

National Aeronautics andSpace Administration

Glenn Research Center

Prepared for theNineth Aeroacoustics Conference and Exhibitcosponsored by the American Institute of Aeronautics and Astronauticsand the Confederation of European Aerospace SocietiesHilton Head, South Carolina, May 12–14, 2003

AIAA–2003–3207

Acknowledgments

This work received support from the Supersonic Propulsion Technology Project Office atNASA Glenn Research Center.

Available from

NASA Center for Aerospace Information7121 Standard DriveHanover, MD 21076

National Technical Information Service5285 Port Royal RoadSpringfield, VA 22100

This report contains preliminaryfindings, subject to revision as

analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov

The Propulsion and Power Program atNASA Glenn Research Center sponsored this work.

AbstractA 3-D space-time CE/SE Navier-Stokes solverusing an unstructured hexahedral grid is de-scribed and applied to a circular jet screechnoise computation. The present numerical re-sults for an underexpanded jet, correspondingto a fully expanded Mach number of 1.42, cap-ture the dominant and nonaxisymmetric ‘B’screech mode and are generally in good agree-ment with existing experiments.

1 IntroductionJet noise is one of the most important topics in com-

putational aeroacoustics. Many of its aspects are ofprimary practical importance and the associated com-plicated physical phenomena are the topic of many ex-perimental and theoretical investigations. Refs. [1–4]provide a comprehensive discussion and further refer-ences. An over/under-expanded supersonic jet emitsmixing noise, broadband shock-associated noise, as wellas screech tones under certain conditions. The mixingnoise is directly associated with large-scale structures,or instability waves, in the jet shear layer; whereas, thebroadband shock-associated noise and screech tones areassociated with the interaction of these waves with theshock-cell structure in the jet core. The screech tonesarise due to a feedback loop, i.e., some of the acous-tic waves generated by the wave/shock-cell interactionpropagate upstream and regenerate the instability wavesat, or in the vicinity of, the nozzle lip. The feedback loopleading to screech tones is sensitive to small changes inthe system conditions, and the understanding of the phe-nomena is to date based mostly on experimental obser-vations [5–8].

Although substantial progress in numerical computa-tion of jet mixing noise has been made, reliable directnumerical simulation of jet screech noise has up to quiterecently not been feasible. Shen and Tam [9, 10] ob-tained excellent results in direct numerical simulationsof screech for circular jets using the well-known DRPscheme. In their 3-D computation [10], a spectral methodwas adopted in the azimuthal direction, and by using onlya limited number of spectral functions substantial sav-ings of computer memory and CPU time was achieved,without deterioration of accuracy. Other recent compu-tational work includes [11].

The present authors [12–14] successfully computedaxisymmetric near-field screech noise for round jets us-ing the recent CE/SE (space-time conservation-elementand solution-element) method utilizing (unstructured)triangulated grids. Because of the implementation (basedon flux balance) of the non-reflecting boundary con-ditions (NRBC), a much smaller near field computa-tional domain can be used with this method. However,when the CE/SE method is applied to a 3-D rectangu-lar jet screech noise computation [15], a major challengeemerges. Namely, the number of unstructured tetrahedralcells required to achieve a certain resolution could be ashigh as tens of millions, which is currently somewhat be-yond the capability of common parallel computers suchas Linux PC clusters. An alternative is to employ an un-structured hexahedral grid. The number of cells may thenbe greatly reduced without much loss of resolution. The3-D CE/SE Navier-Stokes (N-S) solver needs to be mod-ified to accommodate the unstructured hexahedral grid,however.

A general description of the CE/SE Euler method witha hexahedral grid was first given by Zhang et al [16]. Inthe present paper, the current 3-D CE/SE N-S solver us-ing an unstructured hexahedral grid is described in §2.The parallel computation implementation of the schemeis outlined in §3. A 3-D circular jet screech problemis described in §4. The numerical results are presentedand compared with the available experimental data in §5.Concluding remarks are presented in §6.

1NASA/TM—2003-212314

∗Member AIAA†Associate Fellow AIAA

A 3-D CE/SE NAVIER-STOKES SOLVER WITH UNSTRUCTURED HEXAHEDRALGRID FOR COMPUTATION OF NEAR FIELD JET SCREECH NOISE

Ching Y. Loh* and Ananda HimansuTaitech, Inc.

Beaver Creek, Ohio 45430

Lennart S. Hultgren†

National Aeronautics and Space AdministrationGlenn Research CenterCleveland, Ohio 44135

2 The Numerical SchemeAs stated in §1, for fully 3-D computations of com-

plicated problems such as the jet noise problem, a largenumber of computational cells are needed to provide suf-ficient numerical resolution. This imposes severe mem-ory and speed requirements on the computer system. Ofcourse, the choice of unstructured grid type to be used inthe computation greatly affects these demands. It turnsout that the use of an unstructured hexahedral grid in theCE/SE scheme rather than the heretofore standard tetra-hedral one leads to significant reductions of the memoryand CPU time requirements. Its drawback, of course, isa modest increase in dissipation in the scheme. Zhang etal [16] first presented a 3-D CE/SE Euler method usinghexahedral cells instead of tetrahedral ones and provideda detailed description of the implementation.

In general, the CE/SE method [17, 18] systematicallysolves a set of integral equations derived directly fromthe physical conservation laws and naturally capturesshocks and other discontinuities in the flow. In order tohave a compact cell stencil, both conservative variablesand their derivatives are computed simultaneously as un-knowns. A brief sketch of the 3-D CE/SE scheme withhexahedral cells is given below.

2.1 Conservation Form of the 3-D UnsteadyCompressible Navier-Stokes Equations

Consider a dimensionless conservation form of the un-steady 3-D Navier-Stokes equations of a perfect gas. Letρ, u, v, w, p, and γ be the density, streamwise trans-verse and spanwise velocities, static pressure, and con-stant specific heat ratio, respectively. The 3-D Navier-Stokes equations then can be written in the followingvector form:

U t + F x + Gy + Hz = 0, (1)

where x, y, and z are the streamwise, transverse, andspanwise coordinates, and t is time. The five componentsof the conservative flow variable vector U are given by

U1 = ρ, U2 = ρu, U3 = ρv, U4 = ρw,

U5 = p/(γ − 1) + ρ(u2 + v2 + w2)/2.

The flux vectors in the x, y, and z directions, F , G, andH, respectively, are further split into inviscid and vis-cous fluxes,

F = F i − F v, G = Gi − Gv, H = H i − Hv,

where the inviscid fluxes are

Fi1 = U2,

Fi2 = (γ−1)U5+[

(3 − γ)U22 − (γ − 1)(U2

3 + U24 )

]

/2U1,

Fi3 = U2U3/U1, Fi4 = U2U4/U1,

Fi5 = γU2U5/U1 − (γ − 1)U2

[

U22 + U2

3 + U24

]

/2U21 ;

Gi1 = U3, Gi2 = U2U3/U1,

Gi3 = (γ−1)U5+[

(3 − γ)U23 − (γ − 1)(U2

2 + U24 )

]

/2U1,

Gi4 = U3U4/U1,

Gi5 = γU3U5/U1− (γ−1)U3

[

U22 + U2

3 + U24 )

]

/2U21 ;

Hi1 = U4, Hi2 = U2U4/U1, Hi3 = U3U4/U1,

Hi4 = (γ−1)U5+[

(3 − γ)U24 − (γ − 1)(U2

2 + U23 )

]

/2U1,

Hi5 = γU4U5/U1−(γ−1)U4

[

U22 + U2

3 + U24 )

]

/2U21 ;

and the viscous fluxes are

Fv1 = 0, Fv2 = µ(2ux − 23∆),

Fv3 = µ(uy + vx), Fv4 = µ(uz + wx),

Fv5 = µ[2uux + v(uy + vx) + w(wx + uz) −23u∆+

γ

Pr

∂

∂y(U4

U1−

u2 + v2

2)],

Gv1 = 0, Gv2 = µ(vx + uy),

Gv3 = µ(2vy − 23∆),

Gv4 = µ(vz + wy),

Gv5 = µ[2vvy + u(uy + vx) + w(wy + vx)w − 23v∆+

γ

Pr

∂

∂y(U4

U1−

u2 + v2

2)],

Hv1 = 0, Hv2 = µ(wx + uz),

Hv3 = µ(vz + wy), Hv4 = µ(2wz − 23∆),

Hv5 = µ[2wwz + u(uz + wx) + v(wy + vx) − 23v∆+

γ

Pr

∂

∂y(U4

U1−

u2 + v2

2)],

where u, v, w, ux, uy, uz, etc. are the flow velocitiesand their spatial derivatives, which can be expressed interms of the conservative variables U1, U2, etc. alongwith their gradients. Pr = 0.72 is the Prandtl number, µthe kinematic viscosity, and

∆ = ux + vy + wz

is the velocity divergence.By considering (x, y, z, t) as coordinates of a four-

dimensional Euclidean space, E4, and using Gauss’ di-vergence theorem, it follows that Eq. (1) is equivalent tothe following integral conservation law:

∮

S(V )

Im · dS = 0, m = 1, 2, 3, 4, 5, (2)

where S(V ) denotes the surface around a volume V inE4 and Im = (Fm, Gm, Hm, Um).

2NASA/TM—2003-212314

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1

2

3

4

5

6

7

8

A

B

C

Figure 1: A hexahedral cell with its six neighboring av-erage cell centers (ACs). For clarity, only three of them,A, B, C are shown. Together, they form a polyhedron of24 faces or the base of the CV .

2.2 Unstructured Hexahedral GridFig. 1 illustrates a hexahedron cell 1234 − 5678 in theunstructured grid. Each of the six quadrilateral surfaces(e.g. 1265, 2376, etc.) is associated with a neighbor-ing hexahedral cell. Note that the four nodes of such aquadrilateral may not lie on the same plane. Assume A,B, C,..., are the average centers (ACs) of the respectiveneighboring cells. The coordinates of the average centerof a hexahedron are the simple average of the coordinatesof its eight vertices. The average center may not neces-sarily coincide with the geometrical (gravity) center. Theeight vertices of each hexahedron cell and the six averagecenters (A, B, C..., only three are shown in Fig. 1) of theneighboring cells form a polyhedron of 24 faces, which,when incorporating the time t direction, is the finite vol-ume V in E4 in Eq. (2) or the control volume CV .

2.3 Compact UpdatingFor any explicit time-marching scheme, flow data at theneighboring nodes at the previous time step are requiredto update the flow data at the current node to the presenttime level. In a finite difference method, a differencescheme is utilized to do the updating. In most finite vol-ume schemes, the computational cell (e.g., 1234-5678 inFig. 1) is the control volume (CV ). First, the fluxes onthe (hyper) surfaces S(V ) (e.g., 1265 in Fig. 1) needto be updated. Conventionally, flow data is extrapo-lated from several neighboring nodes to the center ofthe hyper-surface. In the CE/SE algorithm, a compactupdating is employed [17, 18]. The compact updatingachieves high resolution by using a cell stencil consist-ing only the immediate neighboring cells. For exam-ple, the CV is expanded from the hyper-hexahedral cell

1234− 5678 to a hyper-polyhedron of 24 faces (Fig. 1),with all the six neighboring cell centers (A, B, C...) in-cluded as vertices. When applying Eq. (2) to this hyper-polyhedron in E4, the flux associated with the face 1265is now replaced by the fluxes associated with the fourtriangles ∆A12, ∆A26, ∆A65 and ∆A51 for the newCV . The advantage of this procedure is that flow dataat the cell center A is known and since A lies in each ofthese triangles, no extrapolation through the interior ofthe CV is needed when evaluating the fluxes associatedwith these triangles. The updating is thus conservative.In the CE/SE scheme, in order to achieve higher orderaccuracy, flow data are extrapolated by Taylor expan-sion from A to the triangle centers along the triangle sur-face plane. Consequently, not only the conservative flowvariables U but their spatial derivatives also are consid-ered as the unknowns (totally 20 scalar unknowns) Notethat U t is obtained by evaluating Eq. (1). The hyper-24-face-polyhedron is the control volume where Eq. (2)is applied for conservative updating. Any of these 24surfaces (segments of solution elements or SEs) is as-sociated with one of the six neighboring average cen-ters (ACs,e.g. A, B, C, ...), where the solution U andits gradients are already given at the previous time stepn. Then both the inviscid and viscous fluxes on these24 faces (hyperplane surfaces in E4 space) can be calcu-lated by evaluating the flux functions F , G and H at theface geometrical centers through Taylor expansion. Con-sequently, the unknown U

n+1 at the new time level n+1is evaluated at the geometrical center of the polyhedronfrom the divergence theorem (2) in E4 space. Note thatthe ‘geometrical center’ or centroid of the polyhedron isin general different from its average center (AC). Afterevaluating the new gradients (see next subsection) Ux,Uy, U z , it takes only one more step of Taylor expansionto extrapolate U

n+1 to the average center (AC).The present compact updating avoids the uncertainty

of dimensional-splitting and extrapolation, and henceyields better accuracy. In addition, no Riemann solver isneeded at these hyper-surfaces. Incorporation of the gra-dients of U further enhances the accuracy of the schemeand makes it possible to use a compact cell stencil.

After all the Un+1 at the polyhedron centroids are up-

dated, the next step is to calculate their correspondingspatial gradients.

2.4 Evaluation of Spatial GradientsAs in the standard CE/SE procedure [17], unknown spa-tial derivatives/gradients Ux, Uy, and U z are evaluatedusing the weighted average technique or an extended vanAlbada limiter [19]. The six neighboring polyhedral cen-ters A, B, C, D, E, F around a hexahedron cell (Fig. 2)form an octahedron. The vertices of each of the trian-gular face of the octahedron and the polyhedral center O

3NASA/TM—2003-212314

A

B

C

D

E

F

O

Figure 2: An octahedron formed by the six neighboringpolyhedron centroids A, B, C, D, E, F , with the currentpolyhedron cell centroid O in the middle. With O alwaysas a vertex, eight tetrahedra are formed and hence eightsets of gradients exist.

associated with the current hexahedral cell form a tetra-hedron (e.g. ABEO in Fig. 2). The U

n+1 are alreadycalculated at all the vertices. For each tetrahedron, a setof U

n+1x , Un+1

y , and Un+1z can be directly calculated by

solving a linear equation system. Totally, there are eightdifferent tetrahedra, and so also eight sets of the spatialderivative data. The final values of the spatial derivativesare obtained by applying the extended van Albada limiter(weighted average) to these eight sets of data. Let U x

(i),Uy

(i), U z(i), i = 1, 2, ..., 8 be respectively the gradients

from the ith set data. Let

τi = [(Ux(i))2 + (Uy

(i))2 + (U z(i))2]α,

where α is any real index number. Let ti = τ1−1; if

τi = 0, ti = 0 and that

t = t1 + t2 + t3 + ... + t8,

then, with the van Albada limiter,

Ux =ΣUx

(i)tit

, Uy =ΣUy

(i)tit

, U z =ΣU z

(i)tit

.

As is well-known, the van Albada limiter is less dif-fusive. In practice, it was found that if the power in-dex number α is chosen to be slightly negative, say,α = −0.2, the numerical dissipation resulting from av-eraging could be further reduced.

Once the new gradients are evaluated, Un+1 at the ACof the hexahedron is obtained by Taylor expansion fromthe centroid of the polyhedron. One marching step is thuscompleted.

mesh details at the center

Figure 3: 2-D quadrilateral mesh in a cross section in a3-D hexahedral grid

3 Hexahedral Grid Generation and ParallelComputation

In order to substantially reduce the required numberof cells and to increase numerical stability in the compu-tation, a hexahedral grid is adopted for the 3-D CE/SENavier-Stokes (N-S) solver. For the simple geometry ofa circular jet, with the x axis assumed in the stream di-rection, the hexahedral grid can be generated as follows:

1. a 2-D quadrilateral unstructured mesh is generatedin a circular domain on the y − z plane (Fig. 3);

2. the 2-D mesh is translated step by step in the x di-rection to form the 3-D hexahedral mesh. At loca-tions that are occupied by the jet nozzle body, nocells are generated.

In the present 3-D screech noise computation, employ-ing a hexahedral grid instead of a tetrahedral one helps to

4NASA/TM—2003-212314

Figure 4: Schematic diagram of parallel computation.

reduce the number of cells from tens of millions to a fewmillions. Still, the number of computational cells is verylarge. Due to the large number of computational cells(about 3.67 millions in this case), parallel computationwith multiple processors becomes necessary in view ofcomputation turn-around time and memory size.

The parallelization procedure is similar to the one de-scribed in [15] and is sketched in Fig. 4:

1. the unstructured hexahedral grid is generated for theentire computational domain;

2. the domain is decomposed into subdomains accord-ing to the assigned number of processes (usuallyone-to-one with CPU’s, here 60 are used), using theMETIS code. METIS is an efficient mesh partition-ing code and is freely available from the Universityof Minnesota [20]—for example, Fig. 5 illustratesa typical partition of the computational domain forthe current circular jet noise problem;

3. the N-S flow solver is modified to use MPI librarycalls and applied to each subdomain to conductcomputations, with neighboring domains exchang-ing pertinent results.

MPI, or message passing interface, is an interprocessorcommunication protocol standard. The software libraryis prepared by the Argonne National Laboratory [21].

4 The 3-D Jet Screech Noise ProblemThe circular jet in 3-D space is sketched in Fig. 6. The

flow at the nozzle exit is choked, i.e., the nozzle exit

jetnozzle

Figure 5: A typical partition of the computational do-main by METIS, with different shading or color indicat-ing the subdomains.

Mach number, Me, is unity, and the ambient air is sta-tionary. The case of jet Mach number Mj = 1.42 isconsidered. These conditions correspond to the experi-mental conditions of Panda [5, 6]. In these experiments,it was shown that for Mj = 1.42, the jet noise field ex-hibits truly 3-D phenomena, e.g., the flapping ‘B’ mode.Hence, a 3-D Navier-Stokes solver described above is re-quired.

In the investigation, our attention is focussed on thenear field of the nozzle since this is the source regionof the noise. The inner diameter, D, of the jet nozzleis chosen as the length scale. The density, ρ0, speed ofsound, a0, and temperature T0 in the ambient flow aretaken as scales for the dependent flow variables.

In order to clearly display the upstream propagatingscreech waves, the computational domain was extended2D upstream of the nozzle exit. The full computationaldomain is a circular cylinder of 10D axial length and7.5D radius. At the nozzle exit, the inflow plane is re-cessed by two cells so as not to numerically restrict orinfluence the feed-back loop. A straight nozzle lip of0.25D in thickness is adopted in the computations. Notethat in the experimental setup of Panda [5, 6], the actualnozzle diameter D = 25.4 mm and the nozzle exit lip isbeveled.

The unstructured hexahedral grid currently used isgenerated as described in §2. The hexahedral cells arenon-uniform since good grid resolution is needed in thejet shear layer and in the screech feedback loop paths.Cell numbers in the x and radial directions are typically

5NASA/TM—2003-212314

L

2R

S

innerdiam. D

Figure 6: Geometry of the computational domain: theinner diameter of the nozzle is D; the nozzle extendsinto the computational domain by S = 2D, the domainlength L = 8D and the radius is R = 5.5D (spongezones are excluded)

400 and 160 respectively. In the azimuthal direction,there are 64 cells, corresponding to totally about 3.67million hexahedral cells. The last 6 cells in the stream-wise and radial directions have exponentially growingsize, respectively, and serve as sponge zones to substan-tially eliminate any small remaining numerical reflec-tion from the outflow and circumferential non-reflectingboundaries.

4.1 Initial ConditionsInitially, the flow of the entire domain is set at the am-bient flow conditions (using nondimensional variables),i.e.,

ρ0 = 1, p0 =1

γ,

u0 = 0, v0 = 0, w0 = 0.

4.2 Boundary ConditionsAt the inlet boundary, the conservative flow variables andtheir spatial derivatives are specified to be the same as 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, with Me = 1, are given by

ρe =γ(γ + 1)pe

2Tr,

pe =1

γ

[

2 + (γ − 1)M2j

γ + 1

]

γ

γ−1

,

ue =

(

2Tr

γ + 1

)1/2

, ve = 0, we = 0,

where Tr is the reservoir (plenum) temperature. We willalso follow the experimental cold-flow condition wherethe reservoir temperature equals the ambient one, i.e.,Tr = 1.

At the circumferential and outflow boundaries, theType II and Type I CE/SE non-reflecting boundary con-ditions as described in the next subsection are imposed,respectively. The no-slip boundary condition is appliedon all the nozzle walls.

4.3 Non-Reflecting Boundary ConditionsIn the CE/SE scheme, non-reflecting boundary condi-tions (NRBC) can be easily constructed based on plane-wave propagation theory for hyperbolic conservationlaws [22]. There are various implementations of thenon-reflecting boundary condition (NRBC) and in gen-eral they have proven to be well suited for aeroacousticproblems [14, 23]. The following 3-D NRBCs are em-ployed in this paper.

For a ghost grid node (j, n) lying at the outer radius ofthe domain the non-reflective boundary condition (TypeII) requires that

(Ux)nj = (Uy)n

j = (U z)nj = 0,

while Unj is kept fixed at the initial steady boundary

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

(Ux)nj = 0,

while Unj , (Uy)n

j and (U z)nj are now defined by simple

extrapolation from the nearest interior node j ′, i.e.,

Unj = U

nj′ ,

(Uy)nj = (Uy)n

j′ , (U z)nj = (U z)

nj′ .

As will be observed later, these NRBCs, when combinedwith the above buffer/sponge zones, are robust enoughto allow a near field computation without disturbing ordistorting the flow and acoustic fields.

5 Numerical ResultsIn this section, 3-D numerical results for the under-

expanded circular jet described above are presented andcompared to experimental results [5, 6]. Computationsare conducted for the jet Mach number Mj = 1.42. Atthis moderate jet Mach number, the dominant unsteadymotion in the experiments, see [5], is truly three dimen-sional. With an appropriate time step size, a large num-ber (510,000) of time steps are performed in order toachieve sufficient accuracy in the Fourier analysis of timeseries data. Note that no harmonic forcing is imposedin the numerical simulation. The initial impact of theboundary condition at the nozzle exit stimulates the jetshear layer and triggers the feedback loop that generatesthe (then) self-sustained screech waves.

6NASA/TM—2003-212314

Figure 7: Time-averaged experimental (top) and instan-taneous numerical (bottom) Schlieren pictures, showingshock cell structure; Mj = 1.42.

5.1 Shock-Cell StructureExperimental results for jets are often documented interms of Schlieren pictures. It is straightforward to con-struct Schlieren plots (density-gradient modulus) fromthe numerical results. For the case of Mj = 1.42,Fig. 7 shows the experimental [5, Fig. 4(a)] and nu-merical Schlieren pictures. Good agreement in shock-cell structure is shown. For example, the shock cellwidth (spacing) is about 1.28D in the streamwise direc-tion. Note that the experimental Schlieren plot is a time-averaged results, while the numerical one is a snapshot ata relatively early stage (time step 110,000). In the exper-imental Schlieren plot, it is observed that the first shockcell appears to be sharp and clear since the shear-layer in-stability wave is too weak at this location to significantlyaffect the shock cell. However, once the instability wavehas gained a sufficient amplitude through its streamwisegrowth, it interacts strongly with the shock cells. Thisis evident from the deformations from the second shockcell and onwards and the additional blurring due to time-averaging in top panel of Fig. 7.

5.2 Near-Field Radiating Screech WavesFigs. 8- 12, which represent a series of instantaneoussnapshots of pressure iso-surfaces in the flow field, illus-trate the generation and propagation of screech waves.Since no forcing is applied in the numerical simulation,these waves are a clear indication of a self-sustained os-cillation. For Mj = 1.42, the jet screech is in the flap-ping B mode, which is a truly three dimensional one. Thescreech waves not only propagate in the upstream direc-tion but also swirl and flap around the jet core.

Figure 8: Pressure iso-surfaces at time step 120,000; ax − z cross section slice is also shown.

After the computation has run for about 110,000 timesteps, truly three dimensional asymmetric wave patternsbegin to appear. Fig. 8 and Fig. 9 demonstrate the pres-sure iso-surfaces at time steps 120,000 and 130,000 re-spectively and from different angle. The iso-surfacesclearly take on nonaxisymmetric forms. However, attime steps of 200,000 and 320,000, the pressure iso-surfaces again turn out to look more axisymmetric, asdemonstrated in Fig. 10 and Fig. 11 respectively. Attime step 510,000, when the computation was stopped, ahighly asymmetric pattern can be observed, see Fig. 12.

5.3 Screech Frequency and Sound PressureLevel (SPL)

The numerical time history is recorded for the location(2.0,0.6,0) at the nozzle lip in the flow field and laterpost-processed to obtain spectral information using FastFourier Transform (FFT) techniques. The recording be-gins after an initial time period has elapsed (180, 000 ac-tual time steps) ensuring that any start-up transients haveleft the computational domain. Figure 13 displays theSPL for the case of Mj = 1.42. The SPL plot showsthe spikes at the fundamental ‘B’ mode frequency andits harmonics. The fundamental frequency spike corre-spond to screech frequency of 4500 Hz (SPL=122 dB),which agrees quite well with the experimental value of4350 Hz.

5.4 Performance of the Parallel ComputationThe computation was mainly carried out on a Linux Pen-tium III PC cluster (NASA Glenn Research Center CW-7cluster) using 60 processes, running on 20 CPUs. Nor-mally, one would want a one-to-one relationship between

7NASA/TM—2003-212314

Figure 9: Pressure iso-surfaces at time step 130,000.

isobars200k steps

Figure 10: Pressure iso-surfaces at time step 200,000.

iso−surface levels: .7132, .71355, .7137

Figure 11: Pressure iso-surfaces at time step 320,000; ax − z cross section slice is also shown.

the number of processes and processors. This less thanoptimal situation lead to about 15 % increase in the wall-clock time and was due to the computation having beenstarted on a different Linux PC cluster using 60 CPUs.In the optimal situation, it takes about 8 seconds wall-clock time to march one step on the CW-7 cluster. Theparallel computation was also tested on the SGI (Sili-con Graphics ) Origin 3800 workstation cluster with 64and 128 processors at NASA Ames Research Center.From 64 to 128 processors, the clock time reduces lin-early with the increasing number of processors, but thereduction will begin to level off if more processors areused. As a result of the explicit time-marching in thescheme, when running with 64 processors, the numberof MFLOPS (megaflops) per processor remains between170 and 183, exceeding 20 % (160) of the theoreticalpeak MFLOPS. This performance is considered excel-lent by code-performance specialists at NASA Ames.

6 Concluding RemarksIn this paper, a 3-D CE/SE N-S solver using an un-

structured hexahedral grid is briefly described and testedin a 3-D circular jet screech problem. The use of a hexa-hedral grid rather than a tetrahedral one enhances the nu-merical stability of the scheme and significantly reducesboth memory size and CPU time, making the CE/SEscheme a viable tool for near-field CAA simulation.

For the test case of Mj = 1.42, the jet screech fre-quency of the dominant nonaxisymmetric ‘B’ mode andthe shock cell structure agree well with the experimen-tal data [5, 6]. Perhaps due to the relatively coarse gridused, the computed SPL is somewhat lower than the ex-perimental one. Further tests with refined grids are beingcarried out and will be reported in the future.

8NASA/TM—2003-212314

Figure 12: Pressure iso-surfaces at time step 510,000viewed from two different angles; x − z cross sectionslices are also shown.

SP

L (d

B)

frequency (Hz)

CE/SE simulation: M_j = 1.49 -- simplified nozzle geometry, 3.7M cells

(J. Panda experiment: B-mode 4350 Hz 161 dB)

V|

4500 Hz

2nd harmonic

3rd harmonic

0. 5000. 10000. 15000. 20000. 60.

100.

140.

180.

Figure 13: SPL at nozzle wall.

References[1] 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 Aeroa-coustics,” AIAA Paper 84-2275 (1984).

[3] Tam, C. K. W., “Jet Noise Generated by LargeScale Coherent Motion,” NASA RP-1258, pp. 311-390 (1991).

[4] Raman G., “Advances in Understanding Super-sonic Jet Screech: Review and Perspective,” Prog.Aerosp. Sci. vol. 34, pp. 45-106 (1998).

[5] Panda, J., “Shock Oscillation in UnderexpandedScreeching Jets,” J. Fluid Mech., vol. 363, pp. 173-198 (1998).

[6] Panda, J., “An Experimental Investigation ofScreech Noise Generation,” J. Fluid Mech., vol.378, pp. 71-96 (1999).

[7] Panda, J., Raman, G. and Zaman, K. B. M. Q., “Un-derexpanded Screeching Jets from Circular, Rect-angular and Elliptic Nozzles”, AIAA paper 97-1623 (1997).

[8] Ponton, M. K., Seiner, J. M. and Brown, M. C., “Near Field Pressure Fluctuations in the Exit Planeof a Choked Axisymmetric Nozzle”, NASA TM113137 (1997).

[9] Shen, H. and Tam, C. K. W., “Numerical Simula-tion of the Generation of Axisymmetric Mode JetScreech Tones,” AIAA Paper 98-0283 (1998).

[10] Shen, H. and Tam, C. K. W., “Three-DimensionalNumerical Simulation of the Jet Screech Phe-nomenon” AIAA Paper 2001-0820 (2001).

[11] Imamoglu, B. and Balakumar, P. “Computation ofShock Induced Noise in Imperfectly Expanded Su-personic jets”, AIAA Paper 2002-2527 (2002).

[12] Loh, C. Y., Hultgren, L. S., and Jorgenson, P. C.E., “Near Field Screech Noise Computation for anUnderexpanded jet by the CE/SE Method,” AIAAPaper 2001-2252 (2001).

[13] Loh, C. Y. and Hultgren, L. S., “Computing JetScreech - A Complex Aeroacoustic Feedback Sys-tem”, presented at the 2nd Int’l Conference on CFD(ICCFD2), also NASA/TM-2002-211807 (2002).

[14] Jorgenson, P. C. E. and Loh, C. Y., “Computing Ax-isymmetric Jet Screech Tones Using UnstructuredGrids”, AIAA Paper 2002-3889 (2002).

9NASA/TM—2003-212314

[15] Loh, C. Y., Himansu, A. Wang, X.-Y. and Jorgen-son, P. C. E., “Computation of an Underexpanded3-D Rectangular jet by the CE/SE Method”, AIAAPaper 2001-0986 (2001).

[16] Zhang, Z.-C., Yu, S. T. and Chang, S.-C., “A Space-Time Conservation Element and Solution Elementmethod for Solving the Two and Three Dimen-sional Unsteady Euler Equations Using Quadrilat-eral and Hexahedral Meshes”, J. Comp. Phys. vol.175, pp. 168-199 (2002).

[17] Chang, S.-C., Wang, X.-Y. and Chow, C.-Y., “TheSpace-Time Conservation Element and SolutionElement Method—A New High Resolution andGenuinely Multidimensional Paradigm for SolvingConservation Laws,” J. Comp. Phys. vol. 159, pp.89-136 (1999).

[18] Wang, X.-Y. and Chang S.-C., “ A 2-D Non-splitting Unstructured Triangular Mesh EulerSolver Based on the Space-Time Conservation Ele-ment and Solution Element Method” C.F.D. J. vol.8, pp. 309-325 (1999).

[19] van Albada, G. D., van Leer, B. and Roberts, W. W.,“A Comparative Study of Computational Methodsin Cosmic Gas Dynamics,” Astronomy and Astro-phys., vol. 108, pp. 76-84 (1982).

[20] Karypis, G. and Kumar, V. “Multilevel k-way Par-titioning Scheme for Irregular Graphs”, Univ. ofMinnesota Dept. of Comp. Sc./Army HPC Re-search Center Tech. Report 95-064 (1995).

[21] website: http://www-unix.mcs.anl.gov/mpi

[22] Loh, C. Y., “On a Nonreflecting Boundary Condi-tion for Hyperbolic Conservation Laws,” AIAA Pa-per 2003-3975 (2003).

[23] Loh, C. Y., Hultgren, L. S. and Chang S.-C.,“Computing Waves in Compressible Flow Usingthe Space-Time Conservation Element Solution El-ement Method,” AIAA J., Vol. 39, pp. 794-801(2001).

10NASA/TM—2003-212314

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16

A 3-D CE/SE Navier-Stokes Solver With Unstructured Hexahedral Gridfor Computation of Near Field Jet Screech Noise

Ching Y. Loh, Ananda Himansu, and Lennart S. Hultgren

CE/SE method; Unstructured hexahedral grid; Jet screech noise

Unclassified -UnlimitedSubject Categories: 02, 71, and 61 Distribution: Nonstandard

Prepared for the Nineth Aeroacoustics Conference and Exhibit cosponsored by the American Institute of Aeronauticsand Astronautics and the Confederation of European Aerospace Societies, Hilton Head, South Carolina, May 12–14,2003. Ching Y. Loh and Ananda Himansu, Taitech, Inc., Beaver Creek, Ohio 45430; Lennart S. Hultgren, NASA GlennResearch Center. Responsible person, Ching Y. Loh, organization code 5880, 216–433–3981.

A 3-D space-time CE/SE Navier-Stokes solver using an unstructured hexahedral grid is described and applied to acircular jet screech noise computation. The present numerical results for an underexpanded jet, corresponding to a fullyexpanded Mach number of 1.42, capture the dominant and nonaxisymmetric 'B' screech mode and are generally ingood agreement with existing experiments.