AIAA 2004-4082Simulation of 3D Flows ofPropulsion Systems Using anEfficient Low Diffusion E-CUSPUpwind SchemeZongjun Hu and Gecheng ZhaDepartment of Mechanical and Aerospace Engineering,University of Miami, Coral Gables, FL 33124

40th AIAA/ASME/SAE/ASEEJoint Propulsion Conference and Exhibit

July 11-14, Fort Lauderdale, FloridaFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

Simulation of 3D Flows of Propulsion Systems Using anEfficient Low Diffusion E-CUSP Upwind Scheme

Zongjun Hu∗ and Gecheng Zha†

Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33124

AbstractA newly suggested E-CUSP upwind scheme is employed

for the first time to calculate 3D flows of propulsion sys-tems. The computed cases include a transonic nozzle withcircular-to-rectangular cross section, a transonic duct withshock wave/turbulent boundary layer interaction, and a sub-sonic 3D compressor cascade. The computed results agreewell with the experiments. The new scheme is proved to beaccurate, efficient and robust for the 3D calculations of theflows in this paper.

IntroductionThe aircraft engine propulsion systems usually work in

transonic regime, where the resolution of shock waves andboundary layers are very important. The 3D flow field cal-culation is usually CPU intensive. Hence, an accurate andefficient Riemann solver to resolve the shock waves andboundary layer is necessary.

The well known Roe scheme1 is accurate to resolve theshock waves and boundary layer. However, the Roe schemeneeds matrix operation for its numerical dissipation, whichis fairly CPU intensive. Recently, there are many efforts todevelop efficient Riemann solvers using scalar dissipationinstead of the matrix dissipation. For the scalar dissipationRiemann solver schemes, there are generally two types: H-CUSP schemes and E-CUSP schemes.2–4 The abbreviationCUSP stands for “convective upwind and split pressure”named by Jameson.2–4 The H-CUSP schemes have the totalenthalpy from the energy equation in their convective vec-tors, while the E-CUSP schemes use the total energy in theconvective vectors. The Liou’s AUSM family schemes, VanLeer-Hänel scheme,5 and Edwards’s LDFSS schemes6, 7 be-long to the H-CUSP group.

The H-CUSP schemes may have the advantages to betterconserve the total enthalpy for steady state flows. However,from the characteristic theory point of view, the H-CUSPschemes are not fully consistent with the disturbance propa-gation directions, which may affect the stability and robust-ness of the schemes.8 A H-CUSP scheme may have moreinconsistency when it is extended to moving grid system.It will leave a pressure term multiplied by the grid velocityin the energy flux that is not contained in the total enthalpy

∗Phd Student†Associate Professor, Director of CFD lab

Copyright c© 2004 by Zongjun Hu and Gecheng Zha. Published by the Ameri-can Institute of Aeronautics and Astronautics, Inc. with permission.

and the term will be treated as a part of the pressure term.From characteristics point of view, it is not obvious how totreat this term in a consistent manner.9

Recently, Zha and Hu suggested an efficient E-CUSPscheme (named as Zha CUSP) which is consistent with thecharacteristic directions.8 The scheme has low diffusionand is able to capture crisp shock profiles and exact contactdiscontinuities. The scheme is shown to be accurate, robustand efficient. In addition, it is straightforward to extend theZha CUSP scheme to moving grid system.9

The original E-CUSP scheme of Zha and Hu is furthermodified to remove the temperature oscillations occurringoccasionally near walls, in particular when the mesh isskewed.10 Zha modified the scheme by replacing the pres-sure in the dissipation of energy equation with the totalenthalpy.10 The modified scheme also yields more precisewall surface temperature at coarse grid.10 The modifiedscheme is named Zha CUSP2 scheme. Neither the origi-nal Zha CUSP scheme nor the Zha CUSP2 scheme has everbeen applied to 3D flow field calculations.

This paper is to extend and apply the Zha CUSP2 schemeto 3D calculation of internal flows of propulsion systems.The scheme is demonstrated to be accurate, efficient, androbust for the 3D flows calculated in this paper.

Governing EquationsThe governing equations are the 3D Reynolds averaged

Navier-Stokes equations in conservation law form and ingeneralized coordinates, and is given as,

∂ Q

∂ t+ ∂ E

∂ ξ+ ∂ F

∂ η+ ∂ G

∂ ζ

= 1Re

(

∂ R∂ ξ

+ ∂ S∂ η

+ ∂ T∂ ζ

)

(1)

where Q is the conservative variable vector, E, F , G andR, S, T are the inviscid and viscous flux vectors in ξ, η, ζdirections respectively.

To save space, only the contents of Q, and fluxes in ξdirection, E and R are given below.

Q =1

J

ρρuρvρwρe

(2)

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082

E =

ρUρuU + lxpρvU + lypρwU + lzp(ρe + p) U

(3)

R =

0lxτxx + lyτxy + lzτxz

lxτxy + lyτyy + lzτyz

lxτxz + lyτyz + lzτzz

lxβx + lyβy + lzβz

(4)

where, J is the transformation Jacobian, ρ is the density, eis the total energy per unit mass, u, v, w are the velocitycomponents in x, y, z direction. The static pressure p isdetermined as,

p = (γ − 1)

[

ρe − 1

2ρ

(

u2 + v2 + w2)

]

(5)

U is the normal contravariant velocity in generalized ξdirection,

U = V · l = ulx + vly + wlz (6)

The vector l is the control volume interface area vectorpointing in the direction normal to the interface with itsmagnitude equal to the interface area. When ∆ξ = ∆η =∆ζ = 1, l is expressed as the following,

l = lxi + lyj + lzk =1

J(ξxi + ξyj + ξzk) (7)

Shear stress components, τxx, τxy , τxz , τyy, τyz , τzz aredefined as,

τij = −2

3(µ + µt)

∂ uk

∂ xk

δij

+(µ + µt)

(

∂ ui

∂ xj

+∂ uj

∂ xi

)

(8)

where, µ is the molecular viscosity, µt is the turbulent vis-cosity.

βx, βy , βz are defined as,

βi = ujτij − qi (9)

where, qx, qy , qzare the heat fluxes in x, y, z direction.

q̄i = −[

µ̄

(γ − 1) Pr+

µ̄t

(γ − 1) Prt

]

∂ a2

∂ xi

(10)

Where, Pr and Prt are the molecular and turbulencePrandtl number taken the value of 0.72 and 0.9. The tur-bulence viscosity is calculated using Baldwin-Lomax tur-bulence model.11 a is the speed of sound.

The control volume method is used to discretize the gov-erning equations on mesh cell (i, j, k) as:

∂∂ t

∫

dVi,j,k

Qdξdηdζ +(

Ei+ 1

2

− Ei− 1

2

)

+(

F j+ 1

2

− F j− 1

2

)

+(

Gk+ 1

2

− Gk− 1

2

)

= 1Re

[(

Ri+ 1

2

− Ri− 1

2

)

+(

Sj+ 1

2

− Sj− 1

2

)

+(

T k+ 1

2

− T k− 1

2

)]

(11)

where i± 12

, j ± 12

and k ± 12

represent the control volumeleft and right interface locations in ξ, η and ζ directions.

The E-CUSP Scheme (Zha CUSP2)Take ξ direction as example, the original Zha CUSP

scheme splits the inviscid flux E on interface i± 12

into con-vective vector Ec and wave vector Ep to represent the ve-locity and pressure wave characteristics.8 The total energyis included in the convective vector. In subsonic regime, theconvective vector Ec is treated in an upwind manner, andthe pressure vector Ep is averaged with the weight of theeigenvalues U ± C from both the upwind and the down-wind directions. C is the speed of sound multiplied by theinterface area:

C = a√

l2x + l2y + l2z (12)

where a =√

γRT is the speed of soundThe interface flux E 1

2

is evaluated as the following.

1. In subsonic region, |U | 12

< C 1

2

,

E 1

2

= 12

[

(ρU) 1

2

(qcL + qc

R) − |ρU | 12

(qcR − qc

L)]

+ 12

0P+p lxP+p lyP+p lz

12p

[

U + C 1

2

]

L

+

0P−p lxP−p lyP−p lz

12p

[

U − C 1

2

]

R

(13)

where,

qc =

1uvwe

(14)

The subscripts L and R represent the left and righthand sides of the interface. The interface speed ofsound C 1

2

is computed as,

C 1

2

=1

2(CL + CR) (15)

where CL and CR are the speed of sound determinedfrom the left and the right side of the interface.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082

The mass flux on the interface is introduced as the fol-lowing,

(ρU) 1

2

=(

ρLU+L + ρRU−

R

)

(16)

where,

U+L = C 1

2

{

ML+|ML|2

+αL

[

14

(ML + 1)2 −

(

ML+|ML|2

)]}

(17)

U−R = C 1

2

{

MR−|MR|2

+αR

[

− 14

(MR − 1)2 −

(

MR−|MR|2

)]}

(18)

αL =2 (p/ρ)L

(p/ρ)L + (p/ρ)R

(19)

αR =2 (p/ρ)R

(p/ρ)L + (p/ρ)R

(20)

ML =UL

C 1

2

(21)

MR =UR

C 1

2

(22)

The coefficient P is defined as,

P+L =

1

4(ML + 1)

2(2 − ML)

+αML

(

M2L − 1

)2(23)

P−R =

1

4(MR − 1)

2(2 + MR)

−αMR

(

M2R − 1

)2(24)

where

α =3

16

2. In supersonic region, |U | 12

> C 1

2

, E 1

2

is simply com-puted using upwind variables.

When U 1

2

> C 1

2

,

E 1

2

= EL (25)

When U 1

2

< −C 1

2

,

E 1

2

= ER (26)

The above formulations are for the original Zha CUSPscheme, which can capture sharp shock profiles and exactcontact surface with low diffusion.8 However, the schemeis found to have temperature oscillations near the wall whenthe grid is skewed. The Zha CUSP scheme is hence modi-fied to the following Zha CUSP2 scheme:10

The total enthalpy instead of the static pressure is used tocompute the numerical dissipation coefficients αL and αR

for the energy equation,

αL =2 (H/ρ)L

(H/ρ)L + (H/ρ)R

(27)

αR =2 (H/ρ)R

(H/ρ)L + (H/ρ)R

(28)

The total enthalpy is calculated as,

H = e +p

ρ(29)

It needs to emphasize that, when computing the fluxes ofcontinuity and momentum equations, the formulations ofthe αL and αR given in Eq. (19) and Eq. (20) must be used.Eq.(27) and (28) are only for the energy equation.

The temperature oscillations are removed by usingeq.(27) and (28) and the wall temperature is more preciselypredicted by this modified scheme if a coarse grid is used.10

This modified scheme is used for the computations in thispaper.

Turbulence ModelThe turbulent viscosity µt is computed based on

Baldwin-Lomax two-layer algebraic model.11 At innerlayer,

µti = ρl2 |ω| (30)

where

l = ky

[

1 − exp

(

− y+

A+

)]

(31)

ω is the local vorticity, y and y+ are the dimensional anddimensionless distance to the wall.

At the outer layer,

µto = KCcpρFwakeFkleb (32)

Fwake = min(

ymaxFmax, Cwakeymaxu2diff/Fmax

)

Fkleb =

[

1 + 5.5

(

Ckleby

ymax

)6]−1

In the above formulations, k, A+, Cwake, Ckleb, Ccp andK are constants.

The quantities udiff , Fmax and ymax are determined bythe velocity profile following a line normal to the wall.Fmax and ymax are the maximum value and the correspond-ing distance of function Fy,

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082

Fy = y |ω|[

1 − exp

(

− y+

A+

)]

(33)

udiff =(√

u2 + v2 + w2)

max−

(√u2 + v2 + w2

)

min(34)

In the 3D computation of this paper, the wall is locatedat the 4 side of the computation domain. The value of µt

is simply computed according to the closest wall surface.Fmax and ymax is searched from the wall to the center ofthe passage. In the wake region, the exponential part is setto zero in Eq. (31) and Eq. (33). The second part of Eq. (34)is set to be zero outside of the wake region.

Time Marching MethodThe linearized governing equation, Eq. (11), is solved

implicitly using the line Gauss-Seidel iteration method. Theiteration is swept line by line in the vertical η directionwithin each time step. The updated variables are used im-mediately in the previous neighboring line during the sweepas the Gauss-Seidel iteration requires. Two alternating di-rection sweeps are used in each time step with one sweepfrom inlet to outlet, and the other from outlet to inlet. Firstorder Euler scheme is used to discretize the temporal term.First order scheme discretization in space is also used forthe implicit left hand side operator to enhance the diagonaldominance.

The discretized implicit equations is given as the follow-ing:

I∆Qi,j,k + A+∆Qi+1,j,k + A∆Qi,j,k

+ A−∆Qi−1,j,k + B+∆Qi,j+1,k

+ B∆Qi,j,k + B−∆Qi,j−1,k

+ C+∆Qi,j,k+1 + C∆Qi,j,k

+ C−∆Qi,j,k−1

= RHS (35)

where,∆Q = Qn+1 − Qn

RHS =∆t

∆V

{

1

Re

[(

Ri+ 1

2

− Ri− 1

2

)

+(

Sj+ 1

2

− Sj− 1

2

)

+(

T k+ 1

2

− T k− 1

2

)]

−[(

Ei+ 1

2

− Ei− 1

2

)

+(

F j+ 1

2

− F j− 1

2

)

+(

Gk+ 1

2

− Gk− 1

2

)]}n

(36)

The superscripts n and n + 1 denote two sequential timesteps. A, B, C, A+, B+, C+, A−, B−, C−are derivedJacobian coefficient matrices. I is the identity matrix of

order 5. The van Leer scheme12 is used to construct theJacobians for its diagonal dominance nature.

The equation system is rewritten into the following formfor η-direction line Gauss-Seidel iteration.

B−∆U i,j−1,k + B∆U i,j,k + B+∆U i,j+1,k

= RHS∗ (37)

whereB = I + A + B + C (38)

RHS∗ = RHS − A+∆U i+1,j,k − A−∆U i−1,j,k

−C+∆U i,j,k+1 − C−∆U i,j,k−1 (39)

Results and DiscussionThe 3D Zha CUSP2 scheme is applied to subsonic and

transonic 3D internal flow cases for validation. The casesinclude: 1) a transonic nozzle with circular-to-rectangularcross section; 2) a transonic channel flow with shockwave/turbulent boundary layer interaction; and 3) a sub-sonic compressor cascade. In the following computa-tions, third order accuracy is used for the inviscid fluxeswith MUSCL type differencing13 and the Minmod lim-iter. Central-differencing is used for computing the viscousfluxes. The meshes are clustered in regions close to thewall, and the y+ on the first inner mesh point is kept lessthan 3 on wall boundaries for all cases. Local time steppingis used to speed up the convergence.

Circular-to-rectangular Nozzle

A transonic nozzle with circular-to-rectangular cross sec-tion tested at NASA14 is calculated. The transition ductconnects the axisymmetric engine to the nonaxisymmetricnozzle through a smooth progression of geometrically sim-ilar cross sections. There is no swirl flow at the inlet. TheReynolds number is Re=1.522 × 105/in. Due to its sym-metric structure, only a quarter of the nozzle geometry iscomputed as shown in Fig. 1. Two symmetric boundariesare located at the bottom and back sides. The wall is divideinto two parts to generate the H-type mesh. For clarity, ev-ery two other grid line is omitted in each direction in theplot. The baseline mesh size is 100×50×50 and is highlystretched near the walls. No shock wave exists in the flowfield. The total pressure, total temperature and flow anglesare fixed at the inlet as the boundary conditions. Becauseof the supersonic flow at the outlet, zero-gradient boundarycondition is used at the nozzle exit. No slip and adiabaticwall conditions are used for the walls. The optimum CFLnumber used is 200.

Fig. 2 shows the contour lines of Mach number on thebottom symmetric plane for Zha CUSP2 scheme. The flowaccelerates from subsonic at the inlet, reaches sonic at thenozzle throat, and becomes supersonic at the exit. The topwall and side wall static pressure distributions are comparedwith experimental results in Fig. 3 and Fig. 4 respectively.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082

Good agreement is obtained. The pressure distributions ofZha CUSP2 scheme and Roe scheme are indistinguishableon both the top and the side walls.

Because Zha CUSP2 scheme uses scalar dissipation andhence is more CPU efficient than the Roe scheme. On an In-tel Xeon 1.7G Hz processor, the CPU time used to computethe inviscid flux per step per node is 1.84×10−5s for theZha CUSP2 scheme and 2.9723×10−5s for the Roe scheme. The Zha CUSP2 scheme is about 40 percent more effi-cient.

Mesh refinement study is carried out by doubling meshdensity in ξ, η and ζ directions respectively. The com-putation results with mesh 200×50×50, 100×100×50 and100×50×100 are also plotted in Fig. 3 and Fig. 4. The samemesh height is kept on the first inner cell close to the wallboundary for the mesh refinement. The mesh refinementgives about the same results as the original baseline meshas shown in Fig. 3 and 4, which indicate that the solution isconverged based on the mesh refinement.

3D Compressor Cascade

The third case is to calculate a 3D subsonic compressorcascade tested at NASA GRC.16 The cascade has the chordlength of 8.89 cm, the stagger angle of 60◦ and the solid-ity of 1.52. In the experiment, the geometry incidence isset to 0◦. However, due to the side wall boundary layer ef-fect, the actual incidence angle is considered to be 1◦to1.5◦

higher. In the numerical simulation, the incidence angle isset as 1.5◦. The Reynolds number based the chord length is9.67×105. A mesh of 100×60×60 is used in the computa-tion (Fig. 5).

The simulation includes the top and bottom end walls,where the cascade airfoil shape is very different from theone in the midspan as shown in Fig. 6. The computationstarts from zero initial velocity field. The total pressure,total temperature and flow angles are fixed at the inlet.The static pressure is specified at the outlet. No slip andadiabatic wall boundary conditions are applied on bladesurfaces and top, bottom end walls. Periodic boundary con-ditions are applied upstream and downstream of the blade.

Fig. 7 shows the Mach number contours of Zha CUSP2scheme on the mid-span plane. Fig. 8 shows that the com-puted surface pressure (Cp = p−p∞

1

2ρ∞U2

∞

) distribution agrees

very well with the experiment. The result of the Zha CUSP2scheme is also virtually identical to that of the Roe scheme.

Transonic Channel Flow

The second case is the transonic channel flow with shockwave/turbulent boundary-layer interaction and is studiedexperimentally in.15 The test section of the transonic chan-nel has an entrance height of 100 mm and a width of 120mm. It is composed of a straight top wall, two straight sidewalls. A bump with varying shape in span-wise direction ismounted on the bottom wall. The boundary conditions areto fix the total pressure, total temperature and flow angles atthe inlet and static pressure at the outlet. No slip adiabatic

wall boundary conditions are used on the walls.In the present computation, the inlet Mach number is

about 0.5. The Reynolds number based on the entranceheight is 5×105. A mesh of 90×60×60 is used for com-putation. The mesh is mostly uniformly distributed in hor-izontal direction, but clustered in the bump region to betterresolve the shock wave. Fig. 9 shows the 3D mesh withevery two other grid line omitted for clarity. To resolve theturbulent boundary layer, the mesh is clustered near the fourwalls.

Fig. 10 shows the computed shock wave structure (Machnumber contour) compared with the experiment at 3 span-wise planes. They are located at 60 mm, 75 mm and 90mm away from the back wall respectively. The plane at60mm is the central plane of the channel. The outlet staticpressure is adjusted to achieve the same shock location asthat in the experiment. The back pressure has the value ofpoutlet/Pt = 0.53. The Zha CUSP2 scheme and the Roescheme result in the shock wave structure very similar to theexperiment. The computed maximum Mach numbers usingRoe and Zha CUSP scheme are a little lower than that inthe experiment. However, the Zha CUSP2 scheme gives themaximum Mach number closer to the experiment than theRoe scheme. Both the schemes predict the boundary layerthicker than that measured in the experiment. This may bemainly due to the inadequacy of the turbulence modeling.

At Z =60 mm, the experiment shows no flow separa-tion. The Zha CUSP2 scheme also yields the attachedflow. The Roe scheme predicts a flow separation at thatlocation, which is different from the experiment. At loca-tion Z =75mm, both the Zha CUSP2 scheme and the Roescheme predict the flow separation similar to the experi-ment. However, at the location Z =90 mm which is closeto the side wall, the results computed by both the schemespredict larger separation zone than that of the experiment.

The mesh refinement study with the mesh size of 180 ×60 × 60, 90×90×60 and 90×60×90 gives very similar re-sults, which indicates that the solution is mesh independent.

ConclusionThe newly suggested E-CUSP upwind scheme with

scalar dissipation, Zha CUSP2 scheme, is applied for thefirst time to calculate the 3D flows for propulsion systems.

For the transonic nozzle with circular-to-rectangularcross section and the subsonic compressor cascade, thewall static pressure distributions computed by both the ZhaCUSP2 scheme and the Roe scheme are in good agreementwith the experiments. The CPU time to calculate the fluxusing the Zha CUSP2 scheme is about 40% more efficientthan that used by the Roe scheme.

For the transonic channel case, the shock wave structuregiven by Zha CUSP2 scheme agrees well with the experi-ment. The result of Zha CUSP2 scheme agrees better withthe experiment than the one predicted by the Roe scheme,which gives flow separation that does not exist in the ex-periment. The Zha CUSP2 scheme also predicts the peak

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082

Mach number closer to the experiment than that of the Roescheme.

The Zha CUSP2 scheme is shown to be accurate, efficientand robust for the 3D flows calculated in this paper.

AcknowledgmentThis work is partially supported by AFOSR Grant

F49620-03-1-0253 monitored by Dr. Fariba Fahroo.

References1P. Roe. Approximate riemann solvers, parameter vectors, and differ-

ence schemes. Journal of Compuational Physics, 43:357–72, 1981.2A. Jameson. Analysis and design of numerical schemes for gas

dynamics I: Artificial diffusion, upwind biasing, limiters and their effecton accuracy and multigrid convergence in transonic and hypersonic flow.AIAA Paper 93-3359, 7 1993.

3A. Jameson. Analysis and Design of Numerical Schemes for GasDynamics I: Artificial Diffusion, Upwind Biasing, Limiters and Their Ef-fect on Accuracy and Multigrid Convergence in Transonic and HypersonicFlow. Journal of Computational Fluid Dynamics, 4:171–218, 1995.

4A. Jameson. Analysis and Design of Numerical Schemes for GasDynamics II: Artificial Diffusion and Discrete Shock Structure. Journal ofComputational Fluid Dynamics, 5:1–38, 1995.

5D. Hänel, R. Schwane, and G. Seider. On the accuracy of upwindschemes for the solution of the navier-stokes equations. AIAA Paper 87-1105 CP, 1987.

6J. R. Edwards. A low-diffusion flux-splitting scheme for navier-stokes calculations. AIAA Paper 95-1703-CP, 6 1995.

7J. R. Edwards. A low-diffusion flux-splitting scheme for navier-stokes calculations. Computer & Fluids, 6:635–659, 1997.

8Gecheng Zha and Zongjun Hu. Calculation of transonic internalflows using an efficient high resolution upwind scheme. to be published inAIAA Journal, 2004. AIAA Paper 2004-1097.

9Xiangying Chen, Gecheng Zha, and Zongjun Hu. Numerical sim-ulation of flow induced vibration based on fully coupled fluid-structuralinterations. submitted to 34th AIAA Fluid Dynamics Conference, 2004.

10Gecheng Zha. A low diffusion efficient e-cusp upwind scheme fortransonic flows. AIAA Paper 2004-2707, Submitted to AIAA Journal, 34thAIAA Fluid Dynamics Conference, June 28 - July 1, 2004.

11B. S. Baldwin and H. Lomax. Thin layer approximation and alge-braic model for separated turbulent flows. AIAA Paper 78-257, 1978.

12B. van Leer. Flux-vector splitting for the Euler equations. LectureNote in Physics, 170, 1982.

13B. Van Leer. Towards the ultimate convservative difference scheme,III. Jouranl of Computational Physics, 23:263–75, 1977.

14James R. Burley II, Linda S. Bangert, and John R. Carlson. Static in-vestigation of circular-to-rectangular transition ducts for high-aspect-rationonaxisymmetric nozzles. NASA Technical Paper 2534, 1986.

15Jean M. Délery. Experimental investigation of turbulence propertiesin transonic shock/boundary-layer interaction. AIAA Journal, 21, 1983.

16Daniel H. Buffum, Vincent R. Capece, Aaron J. King, and Yehia M.EL-Aini. Oscillating cascade aerodynamics at large mean incidence. Tech-nical Memorandum 107247, NASA, 1996.

Fig. 1 The mesh of the nozzle with circular-to-rectangularcross section

0.215215

0.268999 0.295891

0.295891

1.50

603

0.75

3055

Fig. 2 Mach number contours of the nozzle with circular-to-rectangular cross section

x

p/p

t

-5 -4 -3 -2 -1 0 1 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ExperimentMesh 100x50x50Mesh 200x50x50Mesh 100x100x50Mesh 100x50x100

Fig. 3 Top wall surface pressure distributions of the nozzlewith circular-to-rectangular cross section

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082

x

p/p

t

-5 -4 -3 -2 -1 0 1 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ExperimentMesh 100x50x50Mesh 200x50x50Mesh 100x100x50Mesh 100x50x100

Fig. 4 Side wall surface pressure distributions of the nozzlewith circular-to-rectangular cross section

Fig. 5 3D cascade mesh

Fig. 6 Cascade Mesh on the bottom plane (left), midspan(middle), and top plane (right)

0.499883

0.458237

0.45

8237

0.604

0.604

0.604

0.520707

0.499883

0.54153

0.624823

0.43

7414

Fig. 7 Cascade Mach contours on the midspan plane

x/c

Cp

0.2 0.4 0.6 0.8

-0.5

0

0.5Roe 100x60x60

Zha CUSP2 200x60x60

Zha CUSP2 100x120x60

Zha CUSP2 100x60x120

Experiment

Zha CUSP2 100x60x60

suction side

pressure side

Fig. 8 Cascade midspan plane surface pressure coefficientdistributions

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082

Fig. 9 Transonic duct mesh

1.21

314

1.64684

1.48

42

0.887861

0.99

6287

1.10471

x

y

2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 Z = 60 mm

1.20

259

1.63602

1.47

349

0.877521

0.9858781.14841

x

y

2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 Z = 75 mm

1.18819

1.61843

1.51

087 0.865513

1.02685

1.13441

x

y

2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 Z = 90 mm

1.24167

1.61746

1.45

64

0.919559

1.02

693

1.1343

x

y

2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 Z = 60 mm

1.22

383

1.59489

1.4888

8 0.90

5784

1.01181.11782

x

y

2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 Z = 75 mm

1.22

249

1.59045

1.48532

0.907089

1.012221.11735

x

y

2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1 Z = 90 mm

(A) Zha CUSP 2 scheme (B) Roe scheme (C) Experiment

Fig. 10 Transonic duct Mach number contours

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2004-4082