AIAA JOURNALVol. 33, No. 7, July 1995

Supersonic Flux-Split Procedure forSecond Moments of Turbulence

Foluso Ladeinde*State University of New York at Stony Brook, Stony Brook, New York 11794-2300

A finite volume procedure which combines flux-difference splitting and flux-vector splitting is presented in thispaper. The diffusive fluxes are treated explicitly to preserve the upwinding structure of the split Euler fluxes. Thisprocedure is extended in this paper from its essentially laminar or eddy viscosity form to include the equationsfor the six components of the Reynolds stress tensor and an additional equation for the solenoidal dissipation. Themodels used for the unclosed terms are presented as are the extensions of the numerical procedures to cover theturbulence equations. To validate the proposed procedures, a compressible turbulent flow over a flat plate at Machnumber MOO = 2.87 and Reynolds number per unit length Re/mof6.5X 107 was calculated and compared withexperimental measurements. Also simulated was the case when the foregoing boundary-layer flow was made topass over a ramp. A strong adverse pressure gradient case in which boundary-layer thickness-to-curvature ratio(8/Rc) is 0.1 was considered. In this case, the supersonic turbulent boundary layer experiences the combinedeffects of an adverse pressure gradient, bulk compression, and concave streamline curvature. For the various tests,the results obtained from our calculations are in agreement with the experimental measurements.

I. Introduction

IT seems that the computational fluid dynamics (CFD) approachto generate engineering design data is slowly gaining the ac-

ceptance of designers of aerodynamic systems. The potential of thisapproach to revolutionize aircraft analysis is exemplified by impres-sive calculations such as those of vortex asymmetry in a high-angle-of-attack flow about a body in supersonic and subsonic flow1 andthe complex flows over ogive-cylinder impinging shock-expansionflows.2 Unfortunately, many aerodynamic systems of interest in-volve turbulent or transitional flows and, because of our inadequateknowledge of these phenomena, the realization of the potential ofCFD is severely hampered. Moreover, for realistic systems, turbu-lence investigation approaches such as direct numerical simulation(DNS) are quite inappropriate, because of the high-computationalcost. Also, the large eddy simulation (LES) method has not maturedto a level where it can be used to generate engineering design datafor realistic problems. Thus, turbulence modeling appears to be themost feasible approach to obtain design data.

Turbulence models tend to be hierarchical. Exemplifying the zero-equation types is the Baldwin and Lomax model3; an approachthat has received a lot of attention from the aerodynamic com-munity, but one whose applicability is limited in complex, sepa-rated flow situations. The problem with this approach is the uni-versal near-wall length formulation. One-equation models, whichare the next in the hierarchy, also have limitations that limit theirapplicability in complex engineering systems. Two-equation mod-els, of which k-€ and k-co are the most popular are, in prin-ciple, more general and have been used, although with limitedsuccess, in predicting three-dimensional flows, including shock-wave/boundary-layer interactions. However, two-equation methodshave an inherent weakness which stems from the assumption thatthe Reynolds stress tensor is aligned with the mean strain tensor.With this kind of assumption one does not expect to accuratelymodel normal stresses and, consequently, two-equation models givepoor predictions of turbulence production and pressure strain whenused for flows where the normal stresses are important such asnonparallel flows.4

Received July 19, 1994; revision received Dec. 20, 1994; accepted forpublication Dec. 27, 1994. Copyright © 1995 by Foluso Ladeinde. Pub-lished by the American Institute of Aeronautics and Astronautics, Inc., withpermission.

* Assistant Professor, Department of Mechanical Engineering. MemberAIAA.

Although improvements are constantly being suggested,2 the ba-sic problem with the two-equation approach remains unresolved.Thus, inspite of the higher cost, it seems that second-momentclosures might be needed to solve many of the complicated en-gineering flow problems (see Lee et al.4 and the references therein).We are also interested in second-moment calculations because theyprovide more data for fundamental studies, compared to the lowerorder closures. An example of such data is the structural parameter(defined later in this paper), which is important in studies of coher-ent structures in supersonic turbulent boundary layers.5 Because thenormal stresses are involved, we do not expect current two-equationmodels to give reasonable calculation of this quantity.4

Until recently, turbulence modeling efforts were concentrated onincompressible flows. But attempts to apply variable density exten-sions of these incompressible models to compressible flows havenot been successful,6 implying the need for explicit compressibilityterms in the models. The failure of two-equation models in predict-ing the von Karman constant K in compressible flows has beendiscussed,7 and an expression for the dependence of K on turbu-lent Mach number Mt has been suggested.7 Also, it is now believedthat, in second-moment closures, pressure dilatation pujj == p9terms (0 is fluctuating dilatation) must be added explicitly, for anychance to simulate realistic high-speed aerodynamic systems.8"11

Further, for zero pressure gradient turbulent boundary layer (TBL),and perhaps for some other flows, the density gradient contributionto pO is indispensable for ensuring proper van Driest scaling offluctuations and mean velocity in the inertial sublayer. In the pres-ence of wall heat transfer, proper modeling of p9 might be neededto obtain the correct dependence of skin friction Cf on the walltemperature.11

Other explicit effects of_compressibility includes the dilatationaldissipation 6S = (4/3)vO2 (Refs. 6 and 12) which, studies haveshown, must be included to obtain a growth rate (vs convectiveMach number Mc) for compressible shear layers that agrees withexperiments such as those of Papamoschu and Roshko.13 Other roleshave been attributed to 6S in a turbulent boundary layer.11 Underpre-diction of flow separation and overprediction of heat transfer ratenear reattachment regions have been observed when the k-€ or k-a)model is used in complex supersonic and hypersonic flows.2 Con-trol of turbulence length scale was necessary to obtain reasonableresults for these cases.

Aside from the fact that turbulence models on their own havenot always worked well, we are finding out from our studies that thechoice of numerical method might be a crucial factor for a successfulimplementation of a particular turbulence model. There are multi-

1185

1186 LADEINDE

tudes of numerical schemes, with varying degrees of efficiency, andthe choice among them is not an easy task. For example, optionsinclude the flux-difference split (FDS) schemes (such as those ofRoe,14 Harten and Lax,15 and Rusanov16) and the flux-vector split(FVS) schemes,17-18 as well as central-difference schemes.19'20 Ofthese, Roe's method has been shown to be the most accurate and theone that gives the sharpest shock resolution.21 For example, one-fifthto one-half as many points were required to resolve a shear layerusing Roe's fluxes in a first-order scheme, compared to using otherfluxes in second- or even third-order schemes. This would seem tobe quite important, especially in three-dimensional flows, where alarge payoff might be gained by the use of fewer points. However,all of the three schemes have defects. The basic Roe scheme couldproduce entropy-violating solutions,22 and it has a slower conver-gence compared to FVS.23 On the other hand, FVS tends to be toodiffusive,23 whereas the central difference schemes usually involvethe explicit addition of dissipation—with the attendant issue of "howmuch is enough at a location."

Upwinding is pursued in this paper. Our approach combines thetwo flux-splitting schemes to utilize the best of these two meth-ods; that is, faster convergence, in the case of FVS, whereas, forFDS, more accurate results and sharp resolution of shocks.23"25 Itis especially important to note that the hybrid scheme has a fasterconvergence than the fully FDS approach, for reasons explained inSimpson and Whitfield.23 This scheme will be used in the frame-work of a node-implicit, finite volume, lower-upper procedure. Wehave been motivated to pursue this hybrid procedure by the excel-lent performance that has been observed on a variety of complexaerodynamic flows, including Euler and Navier-Stokes equations,steady and truely transient calculations, and stationary and dynamicgrids.1'23"25 However, the experience with this approach has beenlimited to laminar or eddy viscosity flows. A contribution of thepresent paper is the extension of the method to include the equationsfor the six components of Reynolds stress tensor plus an additionalequation for turbulent dissipation. Models for the unclosed terms inthe turbulence equations are presented, as are the extensions of theflux-splitting procedure to handle the moment equations. Resultsshow accurate turbulence calculations when the numerical schemeis applied to zero pressure gradient turbulent boundary layer5 and toa turbulent boundary layer experiencing the combined effects of anadverse pressure gradient, bulk compression, and concave stream-line curvature.26'27

Finally, the paper by Lee et al.4 is relevant to the present one,although there are significant differences. For example, we use thefull three-dimensional Navier-Stokes equations, whereas Lee et al.4used the two-dimensional boundary-layer equations. Thus, we canpotentially address more complicated geometries, such as those in-volving viscous/inviscid interaction and large curvature. The nu-merical methods are also different: marching (parabolic) approachin Lee et al.4 but flux-splitting up winding approach in the presentcalculations.

The mean equations are presented in Sec. II. This is followed inSec. Ill by the turbulence closure models. The numerical procedureis contained in Sec. IV, results in Sec. V, and a summary of our workin Sec. VI.

II. Governing EquationsThe mean flow equations, following Reynolds- and Favre-

averaging procedures, can be written as

dq(1)

where q is the vector of conserved variables, or

Ppu\

q= (2)pu3

and G* is the tensor of convective terms, which is given byG jfc \r,l ^,2 /-r3l= O O Cz

li pu2 pu3

pu2u2 + p pu2ii3

pu2u3 pu3u3+p(3)

Gkv is the tensor of diffusive terms,

G^[Gi G2 G3]

0

0-21

0&i2 + Tn022 + ?22

0-32 + T32

GuS.2

013

0*33 + ^33

(4)

where

Gw5,i = + "30-13 - q\

- p»u'{

G 1,5,2 = U2a22

U3al3 ~ PE'U'l

- q2

Gv5,3 =

M2/flr22 "3^23

"2*32

In the preceding, an overbar indicates the mean relative toReynolds averaging, with a double prime for the fluctuation. Atilde and a single prime are the corresponding notations for Favre-averaging. Also, */ are the coordinate directions, t is time, p isdensity, w/ are the velocities in the three coordinatejiirections, andE is total energy, with the turbulent energy flux E'u'k defined as

E'u'k = CvT'u'k + ujUfkufj + (u'ju'ju'k/2)

Here, p is mean pressure. The equation of state is

p = (y - l)[pE - \puiUi - |p«X']

The mean viscous stress tensor a// is approximated as

_ / d w / diij\ 2_8uk0ij = Ml T— + -r— ~ T^T~^Vy dxj dxt J 3 dxk

and qk is the mean heat flux

Finally, r// is the Reynolds stress tensor, pu^u'-.

III. Turbulence Models

(5)

(6)

(7)

(8)

The terms -pu^u'j, 0"ku", p"u'k, T'u'k, and u'jUfjUk/2 in the meanequations are not known or expressible in terms of the solutionvariables. Thus, models have to be given to assign values to themor to express them in terms of the solution variables. Reynolds-and Favre-averaged transport equations are solved for the Reynoldsstress tensor, for which the equations are given by

(9)

LADEINDE 1187

where

Tijk = p»u'fik) - (*&»

In Eq. (9), from left to right, we have the time rate of change ofthe Reynolds stress at a fixed point, the net convection of Reynoldsstress by the mean flow to the fixed point, local production (P/7-) ofReynolds stress, local pressure strain (n/7-), net diffusive transport(Tijk,k) of Reynolds stress to the fixed point, local dissipation tensor,and local pressure dilatation. The last four terms are the "production"of Reynolds stress at the fixed point by Favre- (mass-) averagedvelocity. In these equations, the following terms are closed: thetime rate of change of the Reynolds stress, the convective term,and the production term. Whereas, the terms n,-7-, Tijktk, e/y, and thepressure dilation and the mass-averaged velocity terms are unclosed.

Some of the unclosed terms in these equations are modeled usingthe variable density extension of the incompressible models whereasexplicit compressible models are required for others. More detailsare now provided on the various models used.

A. Solenoidal Dissipation and Dissipation Rate EquationSarkar et al.28 and Zeman12 have suggested that, for large

Reynolds number compressible flows, dissipation e consists of twoparts,

€ = €.y + €d (10)

where ey represents the solenoidal dissipation and €d represents thecompressible dissipation,

(ID€x =

Here, d" = u"kk is the fluctuating dilatation, and CD" =represents the fluctuating vorticity. The term €ijk is the permutationtensor. Models for £j are based on the assumption that compressibledissipation is a function of the turbulent Mach number. Zeman'sapproach assumes the existence of shock-like structures embeddedwithin the energetic turbulent eddies, whereas Sarkar et al.28 uselow-Mach number asymptotics to obtain a scaling relation. WithSarkar et al.,28 ed = M2ex, where Mt — q/c1 is the turbulent Machnumber, c is the speed of sound = V/P/p, and q is a velocityscale characteristic of turbulence (defined later). Zeman's model ismore general as it incorporates intermittency and dependence on y,the ratio of specific heats. One form of Zeman's model is

where ed = 0 if Mt* < M,0, with M,0 = 0.2, aM = 0.66, cd =0.75, and M?* = ^[2/(y + 1)]M,. The results presented here wereobtained using the approach of Sarkar et al.28

The transport equation we have used for the solenoidal dissipationis based on a variable density extension of the incompressible modelin Hanjalic and Launder29

«3xk

€S . ~ d i _ 6= -C€l-pu \u'.— - -C€2p^-. k J 3xj k

—— I Cf — u', u', —- I3xk\ e, * I3xtj

(12)

where the constants are Ce = 0.18, C6i = 1.44, and C€2 = 1.9.Wall value for e, is (e,)u; = 2v(3k^/dy)2.

The model we have used for the dissipation rate tensor is30

jfx + f (1 - fg)k8ij]€t = (13)

where fs is a function of the turbulent Reynolds number Rt = &2/veand is given by /, = [10/(10 + Rt)]. This model contracts to 2e atthe wall, meaning that it has the correct asymptotic behavior at thewall.

B. Pressure StrainMany options exist for modeling the pressure-strain correlation

(Fl//) (Refs. 4, 3 1 , and 32), but the more popular model by Launder,Reece, and Rodi32 (LRR) is used. A model for the fast term byLee et al.4 might be more suitable as it incorporates the effect ofMach number. However, it has not been calibrated for many flowsin comparison to the LRR models. The more simple isotropizationof production (IP) model of LRR is used,

btj - C2[Pjj - (14)

slow term fist term

where by is the anisotropy tensor, given by

bij = ffij/q2) ~ (*y/3)

and q2 = u'kuk = 2k denotes the trace of the Reynolds stress tensor.Here, k is kinetic energy of turbulence. We include wall (echo)effects in the pressure strain by assuming the model in Launder andShima33

(16)where the slow wall correction term D^.1 is given by

and the fast wall correction term n^'2 is given by

(17)

(18)

The near- wall damping function / is normalized to unity in the fullyturbulent region of a turbulent boundary layer. The quantity yn is thenormal distance to the wall. The quantity / represents the turbulentlength scale of the flow which is estimated to be

Launder and Shima have shown the near- wall damping function /to be

/(//?„) = 0.41 (*! /€yn)

The constants C{ and C'2 are CJ = 0.5 and C'2 = 0.3.

C. Transport TermsThe transport term 7} ,̂* consists of three parts,

Tijk,k = Cijk,k + Eijk,k ~~ ^ijk,k

turbulent diffusion pressure diffusion viscous diffusion

(19)

The model for the triple correlation, (Cijkyk) = pu\u'.n!k, is

Ciik-k = - c '(20)

The model coefficient C, is 0.018. We will assume that pressurediffusion effects are included in the model for the triple correlation,as did Hanjalic and Launder.29 The various other models proposed

1188 LADEINDE

for pressure diffusion have not performed significantly better thanthe one used here.34

The viscous diffusion term is modeled according to Launder andShima,33 and is given by

(21)

D. Mass-Averaged Fluctuating Velocity _The Favre-averaging procedure introduces the quantity wj , which

does not appear in the incompressible form of the Reynolds transportequation. Gradient transport is assumed,

dppeap

(22)

In this equation qrs is the vector of the unknown stresses given by

qrs =

pu\u'2pu(u'3pu'2u'2pu'2u'3

is the tensor of convective terms given by

The model constant C^ = 0.09 is taken from Sarkar and Laksh-manan,6 whereas the turbulent Schmidt number ap = 0.1.

E. Unclosed Terms in the Mean Energy EquationThe turbulent quantities that need to be closed in the mean energy

equation are

u'jU'jU'k/2, T'n1k

From the definition of the transport term in the Reyjiolds stresstransport equation, the quantities cr"ku", p"u"k, and u'jU'jUf

k/2 can becombined to yield

a/t«J - p"u"k - («X<4/2) = -(Tjjt/2)

Thus the modelsjn Sec. III.C are applicable.Concerning T'u'k, evolution equations can be written11 which in-

troduce three additional transport equations. For simplicity, we areusing a gradient transport6 as

(23)

where aT = 0.7 is the turbulence Prandtl number.

pU\U\U2 PU\U\U3

pu'2u'2ui

pu\uf3u2

pu'2u2u2

p€sui pesu2 p€su

Srsi and SrS2 are source vectors given by

n12 - £12 + nn13 - 613 + nn22 - ^22 + nn23 - e23 + nn33 - 633 + n

+ 0-12,2

+ ̂ 22,2

MIT" +wi(°'iox

U\ —— + u'2(aoy

wiT-ox

U2 ——ox_ _ — dp — _ _ _ — dp

i +̂ 32,2 + ̂ 33,3) -u'\~r- +"3(̂ 11,1 +̂ 12,2 + 0-13,3) -u'3 —oz ox

l + ̂ 22,2 + ̂ 23,3) -U'2^r~+ "2(̂ 21,1 + ̂ 22,2 + ̂ 23,3) ~ u'2 ——oy oy

1 + 0̂ 32,2 + ̂ 33,3) - U2—— + W3(a2l,l + ̂ 22,2 + ̂ 23,3) ~ "3 ——dz oy

—- (<T31,1 + ̂ 32,2 + ̂ 33,3) ~ "3-7-oz3—-oz0

F. Pressure Dilatation _The model used in our code for pO is taken fromZeman.11 How-

ever, for the calculations presented in this paper, p9 effects are notsignificant and are omitted in the governing equations given subse-quently. A more significant role of pO is expected at higher Machnumbers and/or larger heat fluxes than those considered in this paper.

With the foregoing, the final form of the turbulence equations canbe written as

(24)

and D* is the tensor of diffusive terms

-T

—T\i2 —Tm

—T222 —T22i— T232 —7233

—7-332 -7-333

A-/7.1 A-s7,3.

LADEINDE 1189

where

r\ r^ P / / / v' "--v , / / " '-•vA-.v7,3 = C 6—— U^——— + «2W3 ——— •

Initial conditions and boundary conditions will be discussed underspecific sample calculations.

IV. Numerical ProcedureA cell-centered finite volume formulation is used, where the vec-

tor of dependent variables q or qrs is assumed to "reside" at thecentroid of the cell. Following the finite volume procedures, we candiscretize Eq. (1) as

(25)

where A#" = qn+l - qn, F = &nk, Fv = Gkvn\ and S is the area

of a cell face, with an outward normal, nk.Beam and Warming35 linearization gives

(26)

In the FDS scheme, the exact Jacobian of the Roe matrix is ex-pensive to compute whereas approximated Jacobians have been ob-served to give poor convergence, compared to FVS.23 In the presentwork, the FVS scheme is used to evaluate the Jacobians in the im-plicit part (left-hand side) of the matrix equation (31). The Eulerflux Fn in the residual part (right-hand side) is computed using theFDS scheme.

For the FVS technique,

(F±)"+1 = (F±)n + (A±Aq)n (27)

where the plus and minus signs indicate evaluation based on thepositive and negative eigenvalues of matrix (A), respectively. Interms of the centroid values, surface quantities are approximated,as an example, by

(A- (28)

The diffusive fluxes do not contain information on wave propaga-tion. Therefore, the upwinding structure of the split (Euler) matriceswill be destroyed if the diffusive terms are treated implicitly. Thus,they are linearized as

(29)

and treated explicitly, to preserve the upwinding scheme. For high-Reynolds number flows, Gatlin36 has shown that this linearizationis stable for CFL numbers to as high as 15.

For FDS via Roe splitting,37 the flux at the surface is evaluated as

(30)

Here, Xk are the characteristic wave speeds of the Riemann problemand ak represent the projection of A# onto the eigenvectors of Roe'sapproximate Jacobian. Rk are the right eigenvectors of Roe's matrix.The fluxes F/+IJ,*, and Fjjjk and in Eq. (30) are evaluated usingthe conserved variables qi+ij-k and #,-,_/,*, respectively. FollowingFrink,38 the sum ̂ a* [A* \Rk is written in terms of five flux jumps,AF*, each associated with a distinct eigenvalue Xk of Roe's matrix.In our procedure, the mean flow equations are segregated from theturbulence equations. That is, at a time step we first solve the (5 x 5)matrix system for the mean flow variables at a node, for all nodes

in the mesh. We then follow with the solution of the (7 x 7) systemfor the turbulence equations before proceeding to the next time step.LU decomposition of the mean flow equations, using the two-passalgorithm gives the following.

1) Forward sweep:

(31)

where

(32)

and [•] indicates that, for example, the term (A+)jI_1JJtS/_!jM op-erates on (Agr")/_i )7?jt- The unknown in this equation is Aw, whichis obtained by solving a sparse block lower tridiagonal matrix withforward substitution. Because boundary conditions are treated ex-plicitly, Aw" is set to zero at the left and bottom boundaries of thedomain.

2) Backward sweep:

= AW" (33)

In Eq. (33), a sparse upper tridiagonal matrix is solved using back-ward substitution. At the right and top boundaries of the domain,Aqn is set to zero for the same reason given earlier.

3) Update solution:

qn+l = qn + Aqn (34)

For the turbulence equations, the analogous equations toEqs. (31), (33), and (34) are

1) Forward sweep:

= (At/V)R"a

where

K.=

(35)

2) Backward sweep:

= AH-" (37)

1190 LADEINDE

3) Update solution:

0» = i

V. Results

(38)

The performance of the schemes presented in this paper was in-vestigated extensively by comparing the calculations with knownbenchmarks for laminar and turbulent flows. For laminar tests wecalculated the shock tube problem and supersonic laminar boundary-layer flow. The benchmark input conditions in Sod39 were used forthe shock tube tests, and the three coordinate directions were tested,one coordinate at a time. Excellent agreement was found in compari-son with the results obtained by Steger and Warming18 and by Roe,40

as the results lie on the same curves (Fig. 1). The present schemesseem to give accurate calculations compared with the various finitedifference schemes investigated in Sod.39 For example, whereasthe MacCormack and Lax-Wendroff schemes produce oscillationsbehind the shock and shock transition occupies about 10 cells, inthe present work the shock occupies roughly 3-5 cells with nooscillations.

We also compared our calculation of Carter's problem (Shakib41)for laminar supersonic flow over a flat plate at M^ = 3 with Carter'sown calculations and the finite element calculations by Shakib.41

The agreement was excellent for the quantities compared. The mo-tivation for the laminar tests of our code was not to validate theschemes used, since this has been done elsewhere,1'23"25 but to de-tect programming errors and build confidence in the procedures. Thecode was subsequently validated for second-moment calculations,as described subsequently.

The compressible turbulent flow over a flat plate was one of theproblems chosen to validate our code for turbulent calculations. Wecompare our results with those by Spina and Smits,6 who carried outa fundamental study of this problem, using experimental measure-ments. The Reynolds number per unit length, Re/m, is 6.5 x 107,and MOO = 2.87. Mean flow and turbulence data along the plate areavailable in Ref. 42.

The dynamic viscosity is assumed to depend only on temperature,and to obey the Sutherland law:

The constant S^ = 1.454 x 10~6 is based on the value of ^ at atemperature of 100 K.

The boundary conditions for this problem are as follows.Upstream:

(x = 0, y)

p = 0.8478 kg/m3

pu\ =479kg/m2s

pu2 = 0 kg/m2s

pE = 193,988.13 N/m2

pufr'j = 0 N/m2

p€s — 0 kg/ms3

COc<D

•

<Dv.Q.

>^

"oO LEGEND

• = velocityO = densityD =pressure

0.00 0.25 0.50x-location

0.75 1.00

Fig. 1 Shock tube test problem showing the distribution of velocity u, density p, and pressure//; symbols are the computed results, solid lines are thecorrect benchmark solutions in Sod.39

LADEINDE 1191

1.25

Fig. 2 Profile of mean Mach number for supersonic turbulent boundary layer flow with zero pressure gradient.

Plate:

(0 < jc < 0.5, y = 0)

pu\ = Okg/m2s

pu2 = 0 kg/m2s

= 0 N/m3

pu'.u'. = ON/m2

m2/s3

Finally, the condition of zero normal gradients of the con-served variables is specified at the freestream and downstreamboundaries.

As is often the case in near-wall turbulence calculations, nu-merical experiments were needed to determine the grid requiredto resolve the viscous sublayer, y+ — Avaii^r/AWi < 30 (wherethe friction velocity ur = Vawaii/Avail) and the cell aspect ratioSi+Ljtk/Sij+^tk for stable calculations. For the various turbulencecalculations, we observed very good convergence when 1) the cen-troid of the first cell along the plate is located such that y+ ~ 1, 2)approximately 25 cells are used below y+ = 50, and 3) cell aspectratio is as large as 2500. With the proper initial conditions (nextparagraph), the calculations are fairly insensitive to the grid used.The mesh used for this test consists of 171 grid points in the x di-rection and 71 exponentially varying grid points in the y direction.The centroid of the first cell along the plate is located at y+ ~ 0.35.

Concerning initial conditions, a generalization of the procedurereported for k-6 models by Gerolymos43 is used. Our procedureconsists of estimating initial mean velocity profile between y = 0and y = 8 by using the ^ power law, estimating initial mean density

from Crocco's theorem, and using equilibrium conditions in De-muren and Sarkar44 to determine initial stress levels. This seems towork very well. The initial external turbulence intensity is assumedto be 0.1%.

We compared the results at locations jc = 0.3 m and ;c = 0.5 malong the plate and find the differences to be small; results are pre-sented for jc = 0.5 m.

The behavior of the boundary layer in the mean is comparedin Figs. 2 and 3 for the profiles of the Mach number and massflux (pui/pooUit00). Agreement between the present work and ex-perimental results is evident. Next, the velocity profile is mappedinto the incompressible plane with the van Driest transformation45

or1

== sii ' - B

-f sin~

where'(TTO)] (40)

B = ̂ - + A2 -

The subscripts e and w refer to properties taken at the edge of theboundary layer and at the wall, respectively. Figure 4, which containsthe log-law plot at x = 0.5 m, shows a very good agreement withthe experimental results of Spina.

In Fig. 5 we present the computed near-wall energy budget (againat x = 0.5 m), nondimensionalized by pw4/vwaii- The calculationsagree with those reported by Grasso and Falconi.46 Further, we cal-culated an approximate value ofJX3 across the boundary layer forthe structural parameters* = —u^u^/u^u^. This value is essentiallythe same as that reported by Jayaram et al.26 for the upstream (zeropressure gradient) portion of a Mach 2.87 turbulent boundary layerover a concave surface curvature (see next paragraph).

As a final demonstration of the potential usefulness of the pro-cedures presented in this paper we will discuss the calculation ofa supersonic turbulent boundary layer experiencing the combinedeffects of an adverse pressure gradient, bulk compression, and con-cave streamline curvature.26'27 These effects result from flows over

1192 LADEINDE

0.0 1.25

Fig. 3 Profile of mass flux pu\ /pui,oo for supersonic turbulent boundary layer flow with zero pressure gradient.

Wall LawSpinaCurrent Work

O.o 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Fig. 4 Profile of log-law velocity u+ after Van Driest transformation, for supersonic turbulent boundary layer flow with zero pressure gradient.

short regions of concave surface curvature. The upstream of theramp consists of a Mach 2.87 boundary layer with a thickness of8 = 0.025 m. The total pressure is 6.9 x 105 N/m2 and the Reynoldsnumber is 6.3 x 107/m. The strong adverse pressure gradient casein which boundary-layer thickness-to-curvature ratio (8/Rc) is 0.1is calculated.

The predictions are compared with experimental measurements inFigs. 6-9, where we show the wall pressure Pw/p^and skin frictionCf as functions of jc/5, and the normal stress pu^/p^u2^ and theshear stress —pu'v'/p^u1^ as functions of y/8 for x/8 & 0.5, 1,and 2. The predicted results for the various quantities comparedshow reasonable agreement with the experimental measurements.

The exception is the shear stress result for x/8 & 2, y/8 < 0.8,where a significant deviation can be observed. The precise causeof the disparity is not known, but it does not appear to be numer-ical. Finally, comparisons were drawn between the present workand those in Lee et al.,4 in the introductory section of this paper.We will like to point out that, relative to the experimental measure-ments, the pressure and skin friction results from our calculationsare better than those reported by Lee et al., although more stud-ies are needed for a conclusive statement. For reasons of space, thecomparison exercise is not presented; the interested reader can makethe assessment by comparing our results with the published work ofLee et al.

LADEINDE 1193

—— Production•— Transport----Dissipation

50 100

y~ 150 200 250

Fig. 5 Near-wall energy budget for supersonic turbulent boundary layer flow with zero pressure gradient.

2 OO

8 "°h

' 25

100

0-751——L-6-0 -4-0 -2-0 00 30 40 6-0 8-0 10-0

x/6Fig. 6 Normalized wall pressure as a function of x for supersonic tur-bulent flow over a ramp; symbols are experimental measurements, solidlines are computed results.

Fig. 8 Normalized shear stress as a function of x/6 andy/6 for super-sonic turbulent flow over a ramp; symbols are experimental measure-ments, solid lines are computed results.

Oio-<i—I

X

3-0

c-oLL

CURVED

0.0048

0.0024

-20 0-0 2-0 4-0 6;0 8-0 100

x/80.4 0.8 1.2

Fig. 7 Skin friction as a function of x for supersonic turbulent flowover a ramp; symbols are experimental measurements, solid lines arecomputed results.

Fig. 9 Normalized shear stress as a function of x/6 and y/6 for super-sonic turbulent flow over a ramp; symbols are experimental measure-ments, solid lines are computed results.

1194 LADEINDE

A. Computing PerformanceFor turbulence calculations involving the mean flow equations,

the six equations for the second moments, and an additional onefor dissipation, speed of computation is 0.00394 s/time step/cell onthe IBM RS/6000 Model 530. This is approximately four times theCPU time for supersonic laminar flow and nine times that for theshock tube problem.

VI. SummaryThe present work can be summarized as follows:1) A numerical procedure for the full three-dimensional Navier-

Stokes equations in which FVS and FDS are combined and theviscosity terms treated explicitly is presented in this paper. Theprocedure obviates the need for an explicit dissipation to controlnonlinear stability. It also exploits the sharp shock resolution ca-pability of approximate Riemman solvers as well as the relativelyfast convergence of flux vector splitting methods. Very accuratecalculations of laminar supersonic flows were observed, using thefull Navier-Stokes equations. This conclusion is based on previousresults1'23"25 and the test problems in this paper.

2) In this work we have extended the foregoing approach to in-clude the equations for the six components of the Reynolds stressand an additional equation for turbulence dissipation. Comparison ofour turbulent calculations with recent experimental measurementsof a Mach 2.87 supersonic turbulent boundary layer by Spina andSmits6 show a reasonable agreement for the quantities compared.We also calculated a supersonic turbulent boundary layer experi-encing the combined effects of an adverse pressure gradient, bulkcompression, and concave streamline curvature.26-27 The agreementwith the experimental measurements was also very reasonable. Theforegoing should establish the procedures presented in this paper asa viable one for supersonic turbulent flows.

AcknowledgmentsThis work was partially funded by NSF Grant CTS-9221630.

The author will also like to express appreciation to D. L. Whitfieldand Linda Kidd, both of Mississippi State University, for makingavailable to him the extensive studies on this subject by Whitfieldand his students. The useful discussions the author had with RobertMelnik of Northrop-Grumman are very much appreciated.

References1 Vanden, K. J., and Belk, D. M., "Numerical Investigation of Subsonic

and Supersonic Axisymmetric Vortical Flows," AIAA Journal, Vol. 31, No. 8,1993, pp. 1377-1383.

2Huang, P. G., and Coakley, T. J., "Turbulence Modeling for ComplexHypersonic Flows," AIAA Paper 93-0200, Jan. 1993.

3Baldwin, B. S., and Lomax, H., "Thin-Layer Approximation and Al-gebraic Model for Separated Turbulent Flows," AIAA Paper 78-257, Jan.1978.

4Lee, J., Taulbee, D. B., and Holden, M. S., "Study of Turbulence onSupersonic Surfaces Using Reynolds Stress Model," AIAA Journal, Vol. 30,No. 7, 1992, pp. 739-746.

5Sarkar, S., and Lakshmanan, B., "Application of a Reynolds Stress Tur-bulence Model to the Compressible Shear Layer," AIAA Journal, Vol. 29,No. 5, 1991, pp. 743-749.

6Spina, E. F, and Smits, A. J., "Organized Structures in a Compressible,Turbulent Boundary Layer," Journal of Fluid Mechanics, Vol. 182, 1987,pp. 85-109.

7Wilcox, D. C, "Progress in Hypersonic Turbulent Modeling," AIAAPaper 91-1785, June 1991.

8Hortsman, C. C., "Prediction of Hypersonic Shock-Wave/TurbulentBoundary Layer Interaction Flows," AIAA Paper 87-1367, June 1987.

9Zeman, O., "On the Decay of Compressible Isotropic Turbulence," Cen-ter for Turbulence Research, CTR Manuscript 115, Stanford Univ., Stanford,CA, Aug. 1990.

10Zeman, O., and Blaisdell, G. A., "New Physics and Models for Com-pressible Turbulent Flows," Advances in Turbulence, Vol. 3, edited byA. V. Johanson and P. H. Alfredsson, Springer-Verlag, Berlin, pp. 445-454.

HZeman, O., "A New Model for Super/Hypersonic Turbulent BoundaryLayers," AIAA Paper 93-0897, Jan. 1993.

12Zeman, O., "Dilation Dissipation: The Concept and Application in Mod-eling, Compressible Mixing Layers," Physics of Fluids, Vol. A2, No. 2,1990,pp. 178-188.

13Papamoschu, D., and Roshko, A., Journal of Fluid Mechanics, Vol. 197,1988, p. 453.

14Roe, P. L., "The Use of the Riemann Problem in Finite-DifferenceSchemes," Lecture Notes in Physics, Vol. 141, 1980, pp. 354-359.

15Harten, A., and Lax, P. D., "A Random Choice Finite Difference Schemefor Hyperbolic Conservation Laws," SIAM Journal of Numerical Analysis,Vol. 18, 1981, pp. 289-315.

16Rusanov, V V, Zh. Vych. Matem. Mat. Fiz, Vol. 1, No. 2, 1961, pp.267-279.

17Van Leer, B., "Flux-Vector Splitting for the Euler Equations," EighthInternational Conference on Numerical Methods in Fluid Dynamics, LectureNotes in Physics, Vol. 170, 1982, pp. 507-512.

18Steger, J. L., and Warming, R. F, "Flux Vector Splitting of theInviscid Gasdynamic Equations with Application to Finite-DifferenceMethods," Journal of Computational Physics, Vol. 40, 1981, pp. 263-293.

19Jameson, A., Schmidt, W., and Turkel, E., "Numerical Solutions ofthe Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes," AIAA Paper 81-1259, June 1991.

20MacCormack, R. W., "The Effect of Viscosity in Hypervelocity ImpactCratering," AIAA Paper 69-354, May 1969.

21Van Leer, B., Thomas, J. L., Roe, P. L., and Newsome, R. W., "AComparison of Numerical Flux Formulas foir the Euler and Navier-StokesEquation," AIAA Paper 87-1104-CP, June 1987.

22Quirk, J. J., "A Contribution to the Great Riemann Solver Debate,"Institute of Computer Applications in Science and Engineering ICASE Rept.92-64, 1992.

23Simpson, L. B., and Whitfield, D. L., "Flux-Difference Split Algorithmfor Unsteady Thin-Layer Navier-Stokes Equations," AIAA Journal, Vol. 30,No. 4, 1992, pp. 914-922.

24 Whitfield, D. L., "Implicit Upwind Finite Volume Scheme for the Three-Dimensional Euler Equations," Mississippi State Univ., Rept. MSSU-EIRS-ASE-85-1, Mississippi State, MS, Sept. 1985.

25Whitfield, D. L., "Implicit Finite Volume High Resolution Wave-SplitScheme," Mississippi State Univ. Rept. MSSU-EIRS-ASE-88-2, MississippiState, MS, Feb. 1988.

26Jayaram, M., Taylor, M. W, and Smits, A., "The Response of a Com-pressible Turbulent Boundary Layer to Short Regions of Concave Sur-face Curvature," Journal of Fluid Mechanics, Vol. 175, 1987, pp. 343-362.

27Degani, D., and Smits, A. J., "Response of a Compressible, TurbulentBoundary Layer to a Short Region of Surface Curvature," AIAA Journal,Vol. 27, No. 1, 1989, pp. 23-28.

28Sarkar, S., Erlebacher, G., Hussaini, M. Y., and Kreiss, H. O., "TheAnalysis and Modeling of Dilational Terms in Compressible Turbulence,"Journal of Fluid Mechanics, Vol. 227, 1991, pp. 473-493.

29Hanjalic, K., and Launder, B. E., "A Reynolds Stress Model of Turbu-lence and its Application to Thin Shear Flows," Journal of Fluid Mechanics,Vol. 52, 1972, pp. 609-638.

30Hanjalic, K., and Launder, B. E., "Contribution Towards a Reynolds-Stress Closure for Low-Reynolds-Number Turbulence," Journal of FluidMechanics, Vol. 74, 1976, pp. 593-610.

31Speziale, G., Sarkar, S., and Gatski, T. B., "Modelling the Pressure-Strain Correlation of Turbulence: An Invariant Dynamical Systems Ap-proach," Journal of Fluid Mechanics, Vol. 227, 1991, pp. 245-272.

32Launder, B. E., Reece, G. J., and Rodi, W., "Progress in the Develop-ment of a Reynolds-Stress Turbulence Closure," Journal of Fluid Mechanics,Vol. 68, 1975, pp. 537-566.

33Launder, B. E., and Shima, N., "Second-Moment Closure for the Near-Wall Sublayer: Development and Applications," AIAA Journal, Vol. 27,No. 10, 1989, pp. 1319-1324.

34Lee, J., "Reynolds Stress Model Predictions of Supersonic BoundaryLayer Over Compression Surfaces," Ph.D. Dissertation, State Univ. of NewYork, Buffalo, NY, Jan. 1991.

35Beam, R. M., and Warming, R. F, "An Implicit Finite Difference Al-gorithm for Hyperbolic Systems in Conservation Law Form," Journal ofComputational Physics, Vol. 22, 1976, pp. 87-110.

36Gatlin, B., "An Implicit, Upwind Method for Obtaining Symbiotic So-lutions to the Thin-Layer Navier-Stokes Equations," Ph.D. Dissertation,Mississippi State Univ., Mississippi State, MS, Aug. 1987.

37Roe, P. L., "Approximate Riemann Solvers, Parameter Vectors andDifference Schemes," Journal of Computational Physics, Vol. 43, 1981,pp.263-293.

38Frink, N. T., "Upwind Scheme for Solving the Euler Equations on Un-structured Meshes," AIAA Journal, Vol. 30, No. 1, 1992, pp. 70-77.

39Sod, G. A., "A Survey of Several Finite Difference Methods for Systemsof Nonlinear Hyperbolic Conservation Laws," Journal of ComputationalPhysics, Vol. 27, 1978, pp. 1-31.

40Roe, P. L., "Characteristic-Based Schemes for the Euler Equations,"Annual Review of Fluid Mechanics, Vol. 18, 1986, pp. 337-365.

LADEINDE 1195

41Shakib, R, "Finite-Element Analysis of the Compressible Euler andNavier-Stokes Equations," Ph.D. Dissertation, Stanford Univ., Stanford, CA,1988.

42Fernholtz, H. H., Finley, P. J., Dussauge, J. P., and Smits, A. J., "ASurvey of Measurements and Measuring Techniques in Rapidly DistortedCompressible Turbulent Boundary Layers," AGARD 315, 1989.

43Gerolymos, G. A., "Implicit Multi-Grid Solution of the CompressibleNavier-Stokes Equations Using k-€ Turbulence Closure," AIAA Journal,Vol. 28, No. 10, 1990, pp. 1707-1717.

44Demuren, A. O., and Sarkar, S., "Systematic Study of Reynolds StressClosure Models in the Computations of Plane Channel Flows," Institute forComputer Applications in Science and Engineering, ICASE Rept. 92-19,1992.

45 Van Driest, E. R., "Turbulent Boundary Layer in Compressible Fluids,"Journal of Aeronautical Science, Vol. 18, 1951, pp. 145-160.

46Grasso, F, and Falconi, D., "High-Speed Turbulence Modeling ofShock-Wave/Boundary Layer Interaction," AIAA Journal, Vol. 31, No. 7,1993. pp. 1199-1206.

Integrated Productand ProcessDevelopment(IPPD) forAerospaceSystemsLed by Dr. Daniel P. Schrage,Georgia TechSeptember 22-23, 1995Los Angeles, CAHeld in conjunction with the 1st AIAAAircraft Engineering, Technology, andOperations Congess

American Institute of Aeronautics and Astronautics

This course will use a generic Integrated Product and Process Development(IPPD) methodology to show you how new systems engineering and qualityengineering techniques and tools can be integrated into a design tradedecision support process using a computer integrated environment.WHO SHOULD ATTENDEngineers, managers, and academic faculty involved in the design,development, and manufacturing of aerospace systems.HOW YOU WILL BENEFIT FROM THIS COURSELearn how to make better product design trades at the system, sub-system/major component, and sub-component/Line Replaceable Unity (LRU)/partlevels.Learn how emerging information technologies will permit design by function,integrated product-process data and description models, and novel designconsiderations.Discover how a computer integrated environment can build on informationtechnology to allow parallel product and process design trades early in theproduct development process.

For more information contact AIAA Customer Service,Phone 202/646/7400 or 800/639-2422 or Fax 202/646-7508.e-mail custerv@aiaa.org