Copyright ©1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1997A9715739, F33615-95-C-2538, AIAA Paper 97-0724

A 'grid-transparent' methodology for CFD

Oshin Peroomian, Sukumar Chakravarthy and Uriel C. _ Goldberg

AIAA, Aerospace Sciences Meeting & Exhibit, 35th, Reno, NV, Jan. 6-9, 1997

A new 'grid-transparent' methodology is introduced for the solution of the compressible Navier-Stokes equations. Thismethodology unifies the treatment of various types of grid topologies such as structured, unstructured, and hybrid multiblockgrids. The solver handles any mixture of tetrahedral, hexahedral, and triangular prism cells in 3D, any mixture ofquadrilateral and triangular cells in 2D, and line elements in 1D - thus the 'grid transparent\\" label. This methodology hasbeen implemented in a new CFD code (CFD++) using a finite volume formulation. The spatial discretization is achievedthrough a new multidimensional vertex-oriented TVD scheme along with a modified compact-storage Roe's Riemann solver.New concepts in polynomial limiting are introduced which enable efficient limiting using characteristic and primitivevariables. (Author)

Page 1

AIAA-97-0724-A "Grid-Transparent" Methodology for CFD

by

Oshin Peroomian , Sukumar Chakravarthy and Uriel C. GoldbergMetacomp Technologies, Inc.

Abstract

A new "grid-transparent" methodology is introducedfor the solution of the compressible Navier-Stokesequations. This methodology unifies the treatment ofvarious types of grid topologies such as structured,unstructured, and hybrid multi-block grids. The solverhandles any mixture of tetrahedral, hexahedral andtriangular prism cells in three dimensions, any mixtureof quadrilateral and triangular cells in two-dimensionsand line elements in one-dimension. Thus the "gridtransparent" label. This methodology has beenimplemented in a new CFD code (CFD++) using afinite volume formulation. The spatial discretization isachieved through a new multidimensional vertexoriented TVD scheme along with a modified compactstorage Roe's Riemann solver. New concepts inpolynomial limiting are introduced which enableefficient limiting using characteristic and primitivevariables.

IntroductionReal world engineering applications involve verycomplex geometries. Flow through a device which hascomplex machinery and moving parts, or flow over anobject with complicated geometry can present achallenge to many of the numerical algorithms used inthe current state of the art in CFD. Another challengeis to be able to model the many features of a flow fieldcorrectly. For example, it might be advantageous tohave a structured body fitted grid in the viscous layeraround an immersed object connected to an outerportion which is unstructured. Almost all CFDalgorithms currently in use will involve different typesof methodologies for the structured and unstructured

portions of the grid. In this paper a new "grid-transparent" methodology is introduced which isequally applicable to structured and unstructured gridtopologies.CFD techniques for the compressible Euler and Navier-Stokes equations have been the focus of manyresearchers. From the Total Variation Diminishing(TVD) schemes introduced by Harten [1], Osher andChakravarthy [2] [3] and Chakravarthy [4] to theEssentially Non-Oscillatory (ENO) schemes [5-10], thegoal has been to obtain robust and accurate solutionmethodologies which span the subsonic to hypersonicflow regimes. With the notable exception of [5] and[6], the formal extension of the methods presented inthe above references to an unstructured framework orunified-grid framework is not obvious. Manyconventional discretizations on triangles and tetrahedraare formally lower order accurate than nominallyassumed. In addition, many limiting techniquesintroduced in these papers do not have direct extensionsto tetrahedral and triangular elements.Numerous challenges exist in deriving numericalschemes which unify cell and grid topologies. Many ofthe concepts and algorithms derived for structured gridscannot be used in such a framework. Examples areapproximate factorization methods and uni-directionalinterpolation and limiting. A number of these issuesare addressed in the present paper by introducing a"grid-transparent" CFD methodology.In this methodology, the numerical solution of theReynolds-averaged Navier-Stokes equations is based onan integral form of the conservation laws that isdiscretized in space using a new second order multi-

Copyright © 1996 by Metacomp Technologies, Inc.Published by the American Institute of Aeronauticsand Astronautics, Inc. with permission.

dimensional vertex-oriented TVD (Total VariationDiminishing) scheme coupled with a suitable pointwiseimplicit or explicit time integration. A new efficientmethod is used for limiting the polynomial coefficientsso that characteristic and primitive variable limiting isachieved with only a very small added overhead. Thecode is written so that it can efficiently handle 1-D, 2-Das well as 3-D calculations. The numerical frameworkincorporates a "grid-transparent" approach whichallows the solver to handle structured, unstructured,and hybrid grids seamlessly. The code can also handlemulti-block versions of the above grids. Itsmethodology is labeled "grid-transparent" because it isequally applicable to hexahedral, triangular prism, andtetraliedral elements in 3-D, quadrilaterals andtriangles in 2-D and line elements in 1-D. The finitevolume framework adopted for the integralconservation laws updates cell averages in timeinsuring correct signal propagation via a modifiedRoe's Riemann solver which reduces the storagerequired by the code immensely. In order to handleturbulent flow problems, pointwise turbulence modelsare employed which do not require information aboutdistance to wall or wall orientation. This feature is alsoadvantageous in multi-CPU implementations.

In what follows, the numerical framework summarizedabove will be discussed in some detail. Numericalexamples will be given to justify the various aspects ofthe formulation.

Governing Equations and Numerical Scheme

The conservation form of the governing differentialequations currently supported by CFD++ can be writtenas:

<?(/,-/,) <?('', - A,)

where the vector q represents the dependentconservation variables; / g, and h represent the fluxesin the three spatial directions; and S represents thesource terms. The subscript / and v stand for theinviscid and viscous terms, respectively. In theReynolds-averaged Navier-Stokes equations thedependent quantities and the inviscid fluxes can bewritten as follows:

' e ~

PpitfV

pwpa;

/ ,=

' («+/0»pu

pu +/:pttvfntw

pita.

g, =

f(e + p)v

P>pvtt

pvl +f

pv\vpna,

k =

' ' (e -f p)wpw

pwupwv

pvt* +p

pwa\

were e is the total energy; p is the density; u,v and w arethe x, y and z-direction velocities, respectively; p is thepressure; and o;'s represent color tracers or turbulencetransport quantities such as turbulence kinetic energyand "undamped" eddy viscosity in the pointwiseturbulence models. The infrastructure for inclusion ofmultiple species exists in the code. The first five rowsrepresent the standard Euler equations with the firstbeing the energy equation followed by the continuityand three momentum equations. The equation of statethat couples pressure to density and temperature is theperfect gas one (p=pRT) which is written in terms ofthe conservation variables in the following way:

P = (r- D(« - ~~2p

were y is the ratio of specific heats. The viscous termsare defined as follows:

/,=

f ft \K — - fu r + VT -t-wrat

0

ftr,pD ——

$& XpD -" ~"

da.

V jA^H0Ta

T»

r

oD3">P%

where T is the temperature, K is the coefficient ofthermal conductivity, D is the coefficient of diffusivity,and the viscous stresses r/, are defined as:

du 2 (du do

do 2 ( du

dw 2 ( du

8u dody dxda fftv

dv dw

where the Stokes theorem for gases is assumed to holdtme, relating the second coefficient of viscosity, K, tothe dynamic viscosity, p., via. K=2/3Lx. The temperatureis related to the conservation quantities via the equationof state:

ppR 2/>J R

where R is the gas constant. The source terms arewritten as:

( o "io

where gx, gy and g2 are body forces and can be activatedif necessary, and Q\ are the source terms such asproduction and dissipation of turbulence.

Finite Volume Framework

The equations are discretized using a finite volumeimplementation. Assume we have a computational cellwith a volume, V. The differential form of theequations is integrated over this volume in thefollowing way:

where / g, and h are now the full fluxes (viscous andinviscid taken together). The spatial terms arerewritten in the following way:

where

Next Gauss' and Leibnitz's rules are applied. Theformer allows the volume integral to be written as asurface integral and the latter allows the time derivativeto be taken outside the integral with the addition of aproper term within the integral. Thus the equation isnow written as follows.

,q}dA =~ (qV}

where q is the cell average of the dependent variables,n is the outward-pointing unit normal and nt isnormal movement of the control volume. Thesequantities are defined as the following:

n, = -xn, -yny - znt

Spatial Discretization using Vertex-oriented Multi-Dimensional TVP Polynomials

The integral term, $A (F • n + ntq)dA in the equations isevaluated by a midpoint rule formula:

where i denotes the centroidal location of each face.This yields a second order approximation to the integralof the fluxes over the cell faces. In our formulation thecell averages of the dependent variables (conservationvariables) are updated; therefore in order to findpointwise values at the cell-face centroids, polynomialsare constructed from the cell-average values. If thesevalues are used as approximations to the cell-facecentroidal values, then the scheme becomes a first orderone. For a second order approximation a linear multi-dimensional polynomial can be constructed in thefollowing way:B(x,y,z) = b, +6 ,O- x^ + b^y- yc) + b3(z-zc)

where (xayc,Zc) is the centroid of the cell for which thepolynomial is being constructed. The approach takenin CFD++ is to evaluate a polynomial for each vertex ofa given cell, i.e. a vertex-oriented polynomial. Forexample, a tetrahedral element has four vertices andthus four polynomial constructions. For the elementsconsidered in CFD++, each vertex is part of three facesin 3-D. The above polynomial has four coefficientswhich need to be solved for; therefore, for each vertexthe three neighboring cells (of the three faces) are usedalong with the cell itself to come up with thepolynomial coefficients. Thus a linear fit of the fourdata points is achieved. For vertices where any of thefaces are boundaries, a polynomial is not initiallyconstructed. At boundaries, after an extrapolation isdone to compute boundary conditions, these can then beincluded in polynomial evaluations. Thus at boundariestwo modes arc used, i.e. including or excluding theboundary points in the evaluation of the polynomials.CFD++ has the ability to detect geometrically"unsuitable" cell neighborhoods for which a leastsquares polynomial is constructed at each vertex.

It is well known that polynomial evaluations canintroduce new extrema in a given data field. In order to

avoid this phenomenon, which can cause oscillationsnear discontinuities, limiters are introduced which limitthe slopes by comparing them with their counterpartsfrom other vertex polynomials within that cell. Variouslimiting recipes have been suggested in the literaturebut they are mostly based on uni-directionalinterpolation. Harten and Chakravarthy [5] wereamoung the first to introduce the multi-dimensionalinterpolation based on cell averages. The limitersintroduced in that work grouped the polynomialcoefficients (Lj norms) in hierarchical fashion, i.e. firstderivatives together, second derivatives together and soon, and compared them to those of the neighboringcells. Peroomian and Chakravarthy [11] showed thatalthough that type of limiting was adequate for bluntbody calculations, it could fail in cases where differenttypes of phenomena were giving rise to the gradients inthe various spatial directions. They showed that insuch cases only a coefficient-by-coefficient approachcan give adequate results. In order to unify thetreatment of the grid and cell topologies, it is necessaryto develop a limiting algorithm that applies equally toall cell types supported by CFD++. In CFD++, insteadof limiting the polynomial coefficients directly, aslightly different approach is taken. For each cell-facea quantity Aq is calculated which represents thedifference between the cell centroidal and face valuescomputed by the average polynomial for that face. Thischange from the centroidal value is compared with asimilar quantity derived from the other polynomial sets(polynomials from vertices not included in theparticular face). This comparison is done usingcontinuous or minmod limiters which are explainedbelow. It essentially compares the in-face and out-of-face contributions to the average of the quantity V • Fwhere F=(q-qc)(dxfJ+dyJJ+dzfaxk). One notes thatthis limiting process is different from the traditionalTVD limiting found in Chakravarthy [4]. In thesetraditional methods, the limiting process is carriedalong a line that joins the centroids of the two cellssharing a common face. This roughly corresponds tofinding a value at the midpoint of the line joining thesecentroids, which can be far off the cell-face centroidallocation in high aspect ratio viscous flow grids. InCFD++ this is avoided by finding the Aq's at the cell-face centroidal locations.

The scheme can be summarized as follows:1. Compute all vertex polynomials.

2. Compute in-face and out-of-face Aq 's for each face.

3. For each face compare the in-face and out-of-facequantities using the following limiters:

(Minmod limiter)

(Continuous limiter)

where cmp is known as the compression parameter andtakes a value between one and two, and e is a verysmall number which is used to avoid division by zero inthe numerical implementation of the continuous limiter.Both limiters set the value of the slope to zero when theslopes being compared have opposite signs, this beingthe basic difference between TVD and ENOinterpolations; however, when the two slopes have thesame sign, the limiting applied is different in the twocases.4. Once the TVD Aq's are calculated for each cell face,a weighted average of these quantities is as follows.

(°ul of face)

The quantity tj> is a weighting parameter which is takenfrom uni-directional concepts. A simple average of allthe vertex polynomials (non-TVD) of a face is used forthe evaluation of the viscous fluxes. The derivatives ofthe temperature and three velocity components in thesefluxes are constructed from polynomial representationsof the conservation variables. In the above discussion afull three-dimensional case was explained. Obvioussimplifications exist for the two-and one-dimensionalcases and are coded in CFD++ to achievecomputational efficiency.

Up to this point we have considered polynomiallimiting based on conservation variables. Severalresearchers [11-13] have shown that for certainproblems the choice of variables to be limited can makea big difference in the values of some of the variableswithin the flowfield. Since the code updates cellaverages, and at a given time step only the cell-averagesof the dependent variables are available, severaldifficulties arise if limiting of primitive variable orcharacteristic variables is to be achieved. Ifconventional means were followed, then up to secondorder the cell average values of the conservationvariables would be equal to the cell centroidal values.One could decode the primitive variables from them,invoke the polynomial interpolation for these variablesand then limit the slopes of the primitive variablepolynomials. If the conservation variables weredesired, the process would be similar; however, due tothe fact that direction normals are involved in decoding

the characteristic variables in multi-dimensionalproblems, different sets of geometrical quantities (suchas normals) must be used for each face. This procedureapplies only to second order, and for higher orders onemust actually find first (he polynomials for theconservation variables, decode them at the centroid andthen recompute them for, say, the primitive variables.In order to avoid these types of complications, thefollowing procedure is devised to limit the polynomialcoefficients of the conservation variables.Primitive Variables—Let the matrix, M, be theJacobian matrix dQldq where Q is the vectorrepresenting the primitive variables and q theconservation variables. Then, the "primitive-variable"limiting is done by multiplying each one of theconservation variable Aq's by this matrix and thenlimiting the resulting values. Finally these limitedvalues are multiplied by M"1 in order to get back theAq's for the conservation variables.

Characteristic Variables—Let A,B and C represent theflux Jacobian matrices in the three spatial direction.For each face, the conservation variable Aq's aremultiplied by the left eigenvectors of the matrix Anx +Bny + Cnz. The limiting is applied to these values andthen multiplied by the right eigenvectors of the abovematrix to recover the limited conservation variableAq's. Alternatively, if one was limiting polynomialcoefficients, then each coefficient (which correspond tothe derivatives) would be multiplied by the lefteigenvectors of the corresponding Jacobian matrix. Forexample, the coefficient bi would be multiplied by theleft eigenvectors of the Jacobian matrix A.

These two methods are quite efficient and add only asmall overhead to the code. For example, the CPU timeper time step on a 175 MHZ Indigo 2 for a 2-Dcompressible shear layer calculation with second orderRungc-Kutta integration (full results will be shownlater) on 40800 grid cells (240x170) is the following:

Limiting Method

Characteristic

Primitive

Conservative

CPU time/step

4 1.8 sec

37.2 sec

33.6 sec

As seen in 2-D, this type of characteristic variablelimiting can be achieved with less than 25% addedoverhead and primitive variable limiting with nearly12% overhead. The characteristic variable limiting wastested on the shear layer problem for which Peroomian

and Charkravarthy [11] have shown that simpleconservation variable limiting and interpolation wouldgive rise to oscillations in pressures and eventually tonumerically negative pressures. The pressure in thecenter region of the shear layer at t=0 and after 10 timesteps is shown in Figures la and Ib under conservationand characteristic variable limiting. It is clear that thecharacteristic variable limiting greatly reduces theoscillations seen in the pressure when conservationvariable limiting is used.Once the limited polynomials are computed, the goal isto find the values of the fluxes at each integration point.For each such point, there are two possible values fromthe polynomial evaluations: a value from the left (fromthat cell's polynomial) and a value from the right (froma neighboring cell). For the viscous fluxes a simpleaverage of the two values is taken; however, for theinviscid fluxes a Riemann solver is invoked in order toinsure a correct signal propagation of the hyperbolic(inviscid) terms.

Compact Storage Riemann Solver

A Riemann solver is a routine which solves exactly orapproximately the Riemann problem (constant states tothe left and right of a discontinuity). The Riemannsolver in this code approximates the solution of theRiemann problem along the t=0 line and uses thosevalues of fluxes for the cell-face centroidal fluxes. TheRiemann flux, /(#"), at the t=0 line can be expressed asfollows:

/to* ) = /to*) -ATwhere A/+ is the change in fluxes across the rightrunning characteristics and shock waves (positiveeigenvalues) and A/" is the change in fluxes across theleft running characteristics and shock waves (negativeeigenvalues). These two equations can be combinedinto a single equation:

In Roe's scheme, all the waves are treated as lineardiscontinuous ones. The Euler equations are written inquasi-linear form and only the direction normal to thecell face is considered. For the Euler equations inmultiple dimensions, the Roe's equation being solvedis:

dt dn

where n is the normal direction to the cell face ai\AAn isa combination of the Jacobian matrices in the threespatial directions (A = $ldi, B = <%/<% and C^chlOi )and the motion of the grid normal to the cell face, n,,i.e.

An = n, + An x + Bny + Cn,Roe's scheme assumes that the coefficient (Jacobian)matrix, An, has constant values and that the solution tothe equation is governed by the linear constantcoefficient jump conditions. For linear equations, thetotal jump in fluxes across all waves can be written interms of the Jacobian matrix:

F(qR ) - F(qL ) = = (R AI) Aq

Roe found that a special density-weighted averagingbased on the left and right states would make the aboveequality hold for the nonlinear Euler equations. Thisdensity weighting is given below:

PRO, =

ItRoe —+ UL

where h is the enthalpy of the gas. The speed of soundbased on the Roe averages can be found using thefollowing equation:

+ V

For this linearized problem the jumps across the leftrunning and right running characteristics can bewritten as:

AT =]Tr';i;a,

where r' is the right eigenvalue vector corresponding tothe positive (/l+) or negative (/I") eigenvalues, and «, isthe product of the left eigenvector (/') and Aq. Theeigenvalues of the Euler equations with two additionaltransport equations are:

At=USJ 1=2-4,6,7

where Sf is cell face area and U =n,+ uns +vny

A modification is done to the eigenvalues at sonicrarefactions in order to add some dissipation to theproblem (where the actual eigenvalue might be veryclose to zero):

iff

This scheme is easy to implement in a structured gridframework since line-by-line methods can be used toreduce the storage needed for the polynomialcoefficients. However, in an unstructured gridframework, storage must be allocated for thepolynomial coefficients at each cell if both qL and qR areneeded simultaneously at a given face, or they have tobe recomputed, leading to increased computationaloverhead. We minimize both memory andcomputational overhead. For example, in a 1,000,000point 3-D calculation (5 equations) and linearpolynomials (3 coefficients bi, b2, b3) nearly 120 MB ofstorage is needed. A problem with extra species andturbulence equations can increase this storage to be inthe Gigabytes. In order to reduce this storage, Roe'sRiemann solver is modified so that a cell-by-cellapproach can be taken. As shown above, Roe's schemecan be written as:

/(»*) = 2 2

which can be written also as:

/V)-^A £•

We reiterate that here qL and qR are the second orderapproximate values of the dependent variables at thecell face centroid, and the left and right eigenvectors aswell as the positive and negative eigenvalues arecalculated using Roe's density-weighted average usingqL and qR. The following modification is made to thisscheme. If the calculations of the left and righteigenvectors, as well as the eigenvalues, are done usingcell average values, then a cell by cell approach can beadopted and the polynomial coefficients need not bestored. Therefore one can carry out the Riemann solvercalculation in two steps:

/,(**) =

TV ) = /,(<?*) +A (<?*)Note that the second term in the first two equationsabove can be combined using the absolute value of theeigenvalues. This modification to Roe's schemechanges only the net numerical diffusion added to thescheme and no longer has the propertyF(qK)-F(qL) = An(qR-qL) for non-constant leftand right states; however, in all the cases tested, the useof this Riemann solver did not affect the solution ascompared with the conventional scheme. One mustalso note that the scheme is still second order, since theflux and Aq calculations are being done using a secondorder approximation.

Once the solution to the Riemann problem is found, theinviscid flux at the integration node is known; thus,together with the viscous fluxes, the entire right handside is known and can be updated using an explicit or apointwise implicit time stepping scheme.

Results

Several preliminary results are shown to demonstratethe method described above.

1. Ramp problem with combination of triangular andquadrilateral cells in a single grid. The flow isassumed to be inviscid, the inflow Mach number is2.0 and the ramp angle is 10 degrees. Figure 2a isa portion of the grid in the computational domainwhich clearly shows a mixture of triangular andquadrilateral cells. Figure 2b shows pressurecontours, and Figure 2c shows the pressurepressure along the line y=0.2. This simpleexample shows the grid transparent nature of themethodology as well as the robustness of the TVDinterpolation.

2. Laminar boundary layer problem. The initialviscous flow tests are being conducted on a flatplate at Mach 2.0 and Re = 1.85xl06/m. Thegeometry corresponds to that used in Chakravarthyet al. [14]. The computational grid is 0.31m in the(streamwise) x-direction and O.lm in the (normal)y-direction. The grid is evenly spaced in x (50points) and clustered in y (100 points) with aminimum step size of 0.0001m. Figure 3a showsthe the velocity profile at x=0.11 as compared withthat from White [15]. Figure 3b shows the self-similar behavior of the velocity profiles at several x

stations. The non-dimensional adiabatic walltemperature computed by CFD++ is 1.68 which isthe same as that given in Whites [15] (recoveryfactor=VPr, Pr=0.72)

3. Isothermal cold wall. A Mach 2 laminar boundarylayer was run in the same computational box as (2).The isothermal wall temperature was set to that ofthe freestream (Tw = Tm). The non-dimensionalheat transfer at the wall (kdT/dy)rf) at x=0.31 wascomputed by CFD++ as 0.002931 whichdimensionally becomes 3290.634 W/m2 . TheStanton Number normalized by the square-root ofthe Reynolds number is given as:

j.,0

The Reynolds number of the current run was1.85xl06 /m, The density 0.046347 kg/m3, thevelocity 684.71 mis, and the temperature 282.4 K.Substituting these into the above equation, thenormalized Stanton Number is computed as 0.394,very close to the value given in Figure 7-4 inWhite[15].

4. Turbulent flat plate flow problem. A one-equation pointwise turbulence model given inGoldberg [16] is used for the adiabatic Mach 2.0flat plate flow. Two computaional grids were usedfor the problem. The first was similar to thecomputational box used in (2) but with a minimumy spacing of 0.00001m and the second was atriangular grid derived from the first grid. Figure 4shows the velocity profiles predicted on the twogrids at x=0.11m. The two solutions are almostidentical, showing the unification power of CFD++even for viscous flow problems. Note that atx=0.11m, the local Reynolds number correspondsto transitional, rather than fully turbulent flow, asthe velocity profiles indicate.

5. 2-D supersonic Shear Layer. This is a case takenfrom Peroomian and Chakravarthyfll], It is asupersonic shear layer perturbed at the dominantfrequency using eigenfunction forcing at the leftedge of the computational domain. Figure 5ashows the centerline pressure. The measuredgrowth rate matches within 1% to that from linearanalysis. Figure 5b shows pressure contours in thecomputational domain, clearly showing theexpansion-compression structures which arecharacteristic of acoustic instabilities. Thisproblem was computed using characteristic

variable limiting and a second order Runge-Kuttaintegration to ensure an accurate temporalevolution.

Currently 3-D calculations are being conducted on aFast-Back car geometry simulated in [17], and a two-hole injector geometries simulated in [18].

Conclusions

This paper presented grid-transparent aspects of a new,general and versatile CFD capability. Theimplementation includes a unified computingframework and the software can run on the entirehierarchy of computers ranging from a LINUX-basedPC to massively parallel computers such as the IBMSP2, Intel Paragon, CRAY T3D, and SGI PowerChallenge Array with excellent scalability.

Acknowledgments

This work was partially funded by Air Force SBIR FC-33615-95-C-2538. The authors wish to thank Dr. BaluSekar of WL/POPS Wright Laboratories for hisvaluable guidance, and the Air Force for their supportthrough their SBIR program.

References

[1] Harten, A. (1983) "High Resolution Schemes forHyperbolic Conservation Laws", J. Comp. Phys., 49,357-393..[2] Osher S and Chakravarthy, S. (1984) "Very HighOrder Accurate TVD Schemes ApproximatingHyperbolic Conservation Laws", ICASE Report No. 84-44.

[3] Chakravarthy, S. and Osher S. (1985) "A NewClass of High Accuracy TVD Schemes for HyperbolicConservation Laws", AIAA Paper No. 85-0363.[4] Chakravarthy, S. (1986) "The Versatility andReliability of Euler Solvers Based on High-AccuracyTVD Formulations", AIAA Paper No. 86-0243.[5] Harten, A. and Chakravarthy, S. (1991) "Multi -Dimensional ENO Schemes for General Geometries",ICASE Report No. 91-76.

[6] Chakravarthy, S. (1994) Rockwell Science CenterReport No. SC71039TR.[7] Casper, J. (1991) "Finite-Volume Application ofHigh Order ENO Schemes to Two-DimensionalBoundary-Value Problems," AIAA Paper No. 91-0631.

[8] Shu, C.W. and Osher, S. (1989) "EfficientImplementation of Essentially Non-Oscillatory ShockCapturing Schemes II", J. Comp. Phys., 83, 32-78.[9] Atkins, H. L., (1991) "High-Order ENO Methodsfor the Unsteady Compressible Navier-StokesEquations", AIAA Paper No. 91-1557.[10] Suresh, A. "Centered Nonoscillatroy Schemes ofThird Order", (1995), submitted to publication in I.Comp. Phys.[11] Peroomian, O. and Chakravarthy S. (1996) "AFeasibility Study of a Cell-Averaged-Based Multi-Dimensional ENO Scheme for use in Supersonic ShearLayers", AIAA Paper No. 96-0524.[12] Chakravarthy, S. (1990) "Some Aspects ofEssentially Nonoscillatory (ENO) Formulations for theEuler Equations", NASA Contractor Report 4285.[13] Shu, C.W., Erlebacher, G., Zang, T.A., Whitaker,D., and Osher, S. (1991) "High-Order ENO SchemesApplied to Two- and Three-Dimensional CompressibleFlow", ICASE Report No. 91-38.[14] Chakravarthy, S.R., K-Y. Szema, U.C. Goldberg,J. J. Gorski and S. Osher "Application of a New Class ofHigh Accuracy TVD schemes to the Navier-StokesEquations", AIAA Paper No. 85-0165.[15] White, Frank M. Viscous Fluid Flow. McGraw-Hill Pub. Copmany, 1974.[16] Goldberg, U.C. "A Pointwise One-EquationTurbulence Model for Wall Bounded and Free ShearFlows", Turbulence. Heat and Mass Transfer. BegellHouse Inc. Pub., Jan. 1996.[17] Han T., Sumantran, V., Harris, C. Kuzmanov, T.,Huebler, M., Zak, T. "Flow-field Simulations of ThreeSimplified Vehicle Shapes and Comparisons withExperimental Measurements",[18] B. Sekar "Three Dimensional Computation ofParallel and Non-Parallel Injection in SupersonicFlow", AIAA Paper No. 95-0886.

• - P At lOtlma itep

o-oo D.QS •o.io -0.05 0.00 aos

Figure la. Pressure at middle of shear layer withcharacteristic variable limiting

Figure Ib. Pressure at middle of shear layerwith conservative variable limiting.

CJ15 <M4 8.4S 050 '0.5S . OJH 9.SS

Figure 2a. Detail of grid made up of triangles and quadrilaterals.

0.4 oex

Figure 2b. Pressure contours showing the oblique shock.

Figure 2c. Pressure as a function of x at y=0.2.

Boundaiy Layer prcfUos lot M. « 2.0 Fkjw

0.0 O.S

Figure 3a. Velocity profile at x=0.11 compared

with the exact solution from White [15].

- X. 0.08 (X -0.05) j• XsO.11 (x-0.10]

Figure 3b. Velocity profiles in similarity form.

- Triangular grid• Cartesian grid

Figure 4. Velocity profiles of turbulent flat plate flow using Cartesian and Triangular grids

0.13

0.11

0.09

0.07

0.05

0.03

0.0112 16

Figure 5a. Instantaneous centerline pressure (y=0) as a function of x.

10

Figure 5b. Pressure contours at t=56.0.

11

Copyright ©1997, American Institute of Aeronautics and Astronautics, Inc.