Flux-Limited Schemes for the Compressible Navier-Stokes ...aero-comlab.stanford.edu/Papers/AIAA-12422-882.pdf · unstructured meshes. This recent development has shed new light on

AIAA JOURNALVol. 33, No. 2, February 1995

Flux-Limited Schemes for the CompressibleNavier-Stokes Equations

S. Tatsumi,* L. Martinelli^ and A. Jameson*Princeton University, Princeton, New Jersey 08544

Several high-resolution schemes are formulated with the goal of improving the accuracy of solutions to thefull compressible Navier-Stokes equations. Calculations of laminar boundary layers at subsonic, transonic, andsupersonic speeds are carried out to validate the proposed schemes. It is concluded that these schemes, which wereoriginally tailored for nonoscillatory shock capturing, yield accurate solutions for viscous flows. The results of thisstudy suggest that the formulation of the limiting process is more important than the choice of a particular fluxsplitting technique in determining the accuracy of computed viscous flows. Symmetric limited positive and upstreamlimited positive schemes hold the promise of improving the accuracy of the results, especially on coarser grids.

I. Introduction

T HE calculation of compressible flows at transonic, supersonic,and hypersonic Mach numbers requires the implementation of

nonoscillatory discrete schemes which combine high accuracy withhigh resolution of shock waves and contact discontinuities. Theseschemes must also be formulated in such a way that they facilitatethe treatment of complex geometric shapes. In the past decade nu-merous schemes have been developed to meet these requirements inconjunction with the solution of the Euler equations.l More recently,the application of such schemes to the Navier-Stokes equations hasproduced algorithms which have progressively gained acceptance asanalysis tools in the aerospace industry. There remains, however, aneed to understand and improve Navier-Stokes schemes beyond thecurrent state of the art. The most compelling reason for this rests onthe fact that shock capturing requires the construction of schemeswhich are numerically dissipative, a requirement which could af-fect the global accuracy of the solution of the physical viscousproblem.

In a recent paper2 Jameson has shown that a theory of nonoscil-latory schemes can be developed for scalar conservation laws basedupon the local extremum diminishing (LED) principle that maximashould not increase and minima should not decrease. Moreover, al-though it is equivalent to the total variation diminishing principle(TVD) for one-dimensional problems, the LED principle can be ap-plied naturally to multidimensional problems on both structured andunstructured meshes. This recent development has shed new light onthe principles underlying the construction of both high-resolutionswitched and flux-limited dissipation schemes. In particular, it al-lowed the new formulation of two families of flux-limited schemesdenominated, symmetric limited positive (SLIP) and upstream lim-ited positive (USLIP), respectively. The present work merges severaldissipation schemes based on both the SLIP and USLIP constructionwith a well-developed cell-centered, finite-volume formulation forsolving the two-dimensional Navier-Stokes equations.3 The aim isto analyze and validate these new discretizations for the solution ofviscous flow problems.

In Sec. II the design principles of nonoscillatory discrete ap-proximations to a scalar convection equation are reviewed together

Presented as Paper 94-0647 at the AIAA 32nd Aerospace Sciences Meet-ing, Reno, NV, Jan. 10-13, 1994; received Feb. 1, 1994; revision receivedJune 7, 1994; accepted for publication June 13, 1994. Copyright © 1994by the American Institute of Aeronautics and Astronautics, Inc. All rightsreserved.

* Visiting Research Staff, Department of Mechanical and Aerospace En-gineering. Member AIAA.

"''Assistant Professor, Department of Mechanical and Aerospace Engineer-ing. Member AIAA.

* Professor, Department of Mechanical and Aerospace Engineering. Fel-low AIAA.

with the three alternative flux splitting techniques which have beenused to extend the construction to a system of conservation laws.Section III discusses some of the issues arising in the actual imple-mentation of the proposed schemes. It also reviews the multigridtime-stepping algorithm used to compute steady-state solutions. InSec. IV the results of computations for a laminar boundary layerat subsonic, transonic, and supersonic speeds are compared withtheoretical solutions.

II. Design Principles of Nonoscillatory SchemesConsider the one-dimensional scalar conservation law

(1)

which may be approximated at the grid node j in conservation formby the semidiscrete scheme

dt (2)

Here hj+\ is the numerical flux between cell j and 7 + 1. and A;cis the mesh interval. It is well known that the least diffusive first-order scheme which satisfies the LED property is the one obtainedby approximating the flux as

where dj+\ = \\aj+\ \&vj+i = ctj+\_(vj+i — Vj) is a first-orderdiffusive flux, computed at the cell interface j + ^, as the firstdifference of v scaled by the absolute value of the approximationof the wave speed a(v) = df/dv. This is exactly equivalent to afirst-order upwind scheme.

Higher order nonoscillatory schemes can be generally derived byintroducing antidiffusive terms in a controlled manner, for example,by an appropriate switch or by making use of flux limiters.4 Bothtechniques are reviewed in the following paragraphs.

High-Resolution Switched SchemesHigh-resolution switched schemes require the introduction of

antidiffusive terms. By subtracting neighboring differences one mayproduce a third-order diffusive flux

This scheme generates substantial oscillations in the vicinity ofshock waves which can be eliminated by switching locally to thefirst-order scheme. The switch introduced by Jameson et al.4 hasproved effective for this purpose and has recently been improved by

252

TATSUMI, MARTINELLI, AND JAMESON COMPRESSIBLE NAVIER-STOKES EQUATIONS 253

8.0 10.0=y*SQRT(Rex)/x

4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

l-a(3) : Tangential velocity profilesCUSP-Switch

4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

l-a(2) : Transverse velocity profilesScalar-Switch

1.0

0.8

.5 0.4D

A"*--AiA"*"*

/- 0ADO

64x16128x32256x64512x128

2.0 4.0 . 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

l-a(4) : Transverse velocity profilesCUSP-Switch

0.4

0.2

0.0

0A

D

0

64x16128x32256x64512x128

1.0

J2.0 4.0 6.0 8.0 10.0

ETA=y*SQRT( Rex) /xl-a(5) : Tangential velocity profiles

Characteristic-Switch

0.4

0.2

Blasius Solution64x16128x32256x64512x128

2.0 4.0 6.0_L_ J

10.0ETA=y*SQRT(Rex)/x

l-a(6) : Transverse velocity profilesCharacteristic-Switch

Fig. la Grid refinement study on boundary-layer velocity profiles: Mach = 0.15, Re = 1 x 10s, and x/L = 0.8; switched schemes.

______Table 1 Matrix of the schemes considered_________Scalar splitting Characteristic splitting CUSP

Switched x x xSLIP x x xUSLIP x x x

where if S is the maximum of Q in the chosen neighborhood, then

,«25)|fly.+ i|

= max

Swanson and Turkel.5 The improved switch is taken as the maxi-mum, in some neighborhood of j, of

A i > - , i — Av,7+2 J

(4)

where

The value of € is typically 1/2, and P0 is a threshold to makesure that the denominator cannot be zero. The diffusive flux is nowcalculated as

Usually (*! = 1/2, ^ = 1/4 to scale the diffusion to the levelcorresponding to upwinding, whereas a2 and /32 must be chosen toswitch the diffusion from third order to first order fast enough neara shock wave.

Symmetric Limited Positive SchemeFlux limiters offer an alternative avenue for devising high resolu-

tion nonoscillatory schemes, and their use dates back to the work ofBoris and Book.6 A particularly simple way to introduce limiters,proposed by Jameson7 in 1984, is to use flux limited dissipation. Inthis scheme, the third-order diffusion defined by Eq. (3) is modifiedby the insertion of limiters which produce an equivalent three-pointscheme with positive coefficients. Jameson has recently reformu-lated this scheme as follows.2'8

Let L(w, v) be a limited average of u and v with the followingproperties:

PI) L(u,v) = L(v,u).P2) L(ceu, civ) = ocL(u, v).

254 TATSUMI, MARTINELLI, AND JAMESON: COMPRESSIBLE NAVIER-STOKES EQUATIONS

1.2

1.0

oA

ao

64x16128x32256x64512x128

6.0 8.0 10.0ETA=y*SQRT( Rex) /x

l-b(l) : Tangential velocity profilesScalar-SLIP

6.0 8.0 10.0ETA=y*SQRT(Rex)/x

l-b(2) : Transverse velocity profilesScalar-SLIP



-b(3) : Tangential velocity profilesCUSP-SLIP

1.0

0.8

c5 0.6\3

0.4

0.2

—t ..... Rlocinc C/~i1n t-i,~vn

0.00.0

I2.0 4.0 6.0 8.0 10.0

ETA=y*SQRT(Rex)/xl-b(5) : Tangential velocity profiles

Characteristic-SLIP

§00*crc3

1.0

0.8

0.6

0.4

0.2


2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

l-b(4) : Transverse velocity profilesCUSP-SLIP

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

l-b(6) : Transverse velocity profilesCharacteristic-SLIP

Fig. Ib Grid refinement study on boundary-layer velocity profiles: Mach = 0.15, Re = 1 x 10s, and x/L = 0.8; SLIP schemes.

P3) L(n, u) = u.P4) L(u, v) = 0 if u and v have opposite signs.

Note that properties P1-P3 are natural properties of an average,whereas P4 is needed for the construction of an LED scheme.

Then, one defines the diffusive flux for a scalar conservationlaw as

schemes. Corresponding to the original SLIP scheme defined byEq. (5), a USLIP scheme is obtained by setting

<0

(5)

This construction will be referred to as the Symmetric Limited Pos-itive (SLIP) scheme.

The requirement P4 on L(w, u) is the key for constructing a LEDscheme. In fact, if Auy.+ 3 and Av,._i have opposite signs, thenthere is an extremum at either j or j + 1. In the case of an odd-evenmode, however, they have the same sign, which is opposite to thatof Avy.+i, so that they reinforce the damping in the same way thata simple central fourth-difference formula would. At the crest of ashock, if the upstream flow is constant, then Auy._i =0, and thusAi>y+3 is prevented from canceling any part of AvT+ i because it islimited by Auy ._i .

Upstream Limited Positive SchemeBy adding the antidiffusive correction purely from the upstream

side one may derive a family of Upstream Limited Positive (USLIP)

Flux LimitersA variety of limiters may be defined which meet the requirements

(P1-P4). In particular, by defining

5(ii, v) = l/2{sign(w) + sign(v)}

so that

11 if u > 0 and v > 00 if u and i; have opposite sign

— 1 if u < 0 and v < 0

one may easily implement any of the three well-known limiters:minmod, Van Leer, or superbee, or construct alternative limitersstarting from the more general formulas presented in Ref. 2. In thepresent study we use a simpler limiter (a mean) which limits the

TATSUMI, MARTINELLI, AND JAMESON: COMPRESSIBLE NAVIER-STOKES EQUATIONS 255


l-c(l) : Tangential velocity profilesScalar-USLIP

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

l-c(2) : Transverse velocity profilesScalar-USLIP

1.2

l.O

b 0.6

0ADO

64x16128x32256x64512x128

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

l-c(3) : Tangential velocity profilesCUSP-USLIP


2.0 4.0 6.0 8.0 100ETA=y*SQRT(Rex)/x

l-c(5) : Tangential velocity profilesCharacteristic-USLIP

8.0 10.0=y*SQRT(Rex)/x

o.o 2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

l-c(6) : Transverse velocity profilesCharacteristic-USLIP

Fig. Ic Grid refinement study on boundary-layer velocity profiles: Mach = 0.15, Re = 1 X 10s, and x/L = 0.8; USLIP schemes.

arithmetic mean by some multiple of the smaller of \ u \ or \ v \ . It maybe cast in the following form:

L(u, v) = S(u, v) min[(\u , a\u\, a\v\]

In the present study the parameter a was fixed to be equal to 1 withthe exception of the calculations with the convective upwind andsplit pressure (CUSP) splitting where we set a = 2.

Extension to Systems of Conservation LawsThe crucial step for extending the construction of nonoscillatory

schemes to a system of conservation laws is the generalization ofthe concept of upwinding and the derivation of stable first-orderdiffusive schemes. Once this has been achieved, either the high-resolution switched, the SLIP, or the USLIP approach can be usedto construct higher order schemes. For the sake of clarity we willconsider only the one-dimensional system of equations

9w(6)

For the equations of gas dynamics the solution and flux vectors are

puH

where p is the density, u the velocity, E the total energy, p the

pressure, and H the stagnation enthalpy. If y is the ratio of specificheats and c the speed of sound, then

2 = (yp/P)- (u2/2)(p/p) = (c2/y -

In a steady flow H is constant. This remains true for the discretescheme only if the diffusion is constructed so that it is compatiblewith this condition.

The three different forms of flux splitting considered in the presentstudy are reviewed next.

Flux Splitting by Characteristic DecompositionOn introduction of Roe's matrix9 A .+i, which is a mean value

Jacobian matrix exactly satisfying the condition

a splitting according to characteristic fields is obtained by decom-posing A;.+ i as

where the columns of T are the eigenvectors of Aj+\, and A is a di-agonal matrix containing the eigenvalues. Hence, the resulting splitcan be cast as

Af^ = rA±r~lAw^


4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

2-a(l) : Tangential velocity profilesSwitched schemes


2-a(2) : Transverse velocity profilesSwitched schemes

oA

D

ScalarCUSPCharacteristic

3.0 2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

2-b(l) : Tangential velocity profilesSLIP schemes


2-b(2) : Transverse velocity profilesSLIP schemes

1.2

1.0 naqc(BBM-B-g n; a --a—

1.2

1.0

Blasius SolutionScalarCUSPCharacteristic

l

Blasius SolutionScalarCUSPCharacteristic

D.O 2.0 4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

2-c(l) : Tangential velocity profilesUSLIP schemes

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

2-c(2) : Transverse velocity profilesUSLIP schemes

Fig. 2 Boundary-layer velocity profiles: Mach = 0.15, Re = I X 105, and x/L = 0.8.

Now the first-order diffusive flux vector which corresponds to a pureupwind scheme is

This particular splitting gives rise to the scalar diffusive flux

where

and |A| is the diagonal matrix containing the absolute values ofthe eigenvalues. This technique will be referred to as characteristicsplitting, and it has the advantage that it allows a discrete shockstructure with a single interior point. Moreover, it yields more accu-rate viscous solutions on coarser grids,3'10"12 because the eigenvalueq corresponding to the shear and entropy waves approaches zero atthe wall.

The implementation of the SLIP construction [Eq. (5)] isnow carried out on the differences of the characteristic variables

Scalar SplittingIt has been widely demonstrated3"5 that simple stable schemes

can be produced by considering

(f/+i - f7-)± = l(fj+i - fj) ± aj+\ (w7-+i - w/)

and will be referred to as scalar splitting.

Convective Upwind and Split Pressure SchemeThe eigenvalues of the Jacobian matrix A = df/dw are w, u -f- c,

andw— c.Ifw > 0 and the flow is locally supersonic (M = u/c > 1),all of the eigenvalues are positive, and simple upwinding is thus anatural choice for diffusion in supersonic flow. In general, however,it is convenient to consider the convective and pressure fluxes

fc = M l = W W C ,

pH

separately. Full upwinding of both fc and fp is incompatible withstability in subsonic flow, since pressure waves with the speed u — cwould be traveling backward, and the discrete scheme would nothave a proper zone of dependence. Since the eigenvalues of dfc/dware w, u and yu, whereas those of dfp/dw are 0, 0, and — (y — l)w,a split with

leads to a stable scheme13 in which downwind differencing is usedfor the pressure. However, such a scheme does not reflect the true


Table 2 Percentile errors with respect to Blasius solution, station x/L = 0.8

Discretization

Scalar-switched

Scalar-SLIP

Scalar-USLIP

CUSP-switched

CUSP-SLIP

CUSP-USLIP

Characteristic-Switched

Characteristic-SLIP

Characteristic-USLIP

Cell in boundary layer

816326481632648163264816326481632648163264816326481632648163264

% Error in C/

4.053.050.890.514.881.870.260.090.210.750.610.343.187.561.980.349.984.931.550.090.640.310.180.774.460.560.180.341.481.871.120.770.210.310.180.34

% Error in 5*

2.021.380.200.291.901.820.740.300.160.160.110.253.457.111.990.253.682.941.590.311.110.300.170.092.600.480.140.611.020.260.090.591.100.410.180.65

% Error in 0

1.400.460.370.131.210.970.160.213.531.150.480.710.106.161.320.246.443.321.090.182.580.980.430.600.660.680.500.891.660.290.300.962.310.770.370.95

zone of dependence in supersonic flow. Thus, we may seek a schemewhich is compatible with stability in subsonic flows and reduces tofull upwinding in the supersonic regions.

Full upwinding of the convective flux is achieved by

where M is the local Mach number attributed to the interval, whereasfull upwinding of the pressure is achieved by

dp j = sign(M)0

>Pj*0

By introducing blending functions f \ ( M ) and /2(M), with theasymptotic behavior f\(M) -> |M| and /2(M) -* sign(M) for|M| > 1, these equations can be written as

p.+

= MM}

The convective diffusion should remain positive when M = 0,whereas the pressure diffusion must be antisymmetric with respecttoM. A simple choice is to take f\ (M) = \M\ and /2(M) = sign(M)for \M\ > 1, and to introduce blending polynomials in M for\M\ < 1 which merge smoothly into the supersonic segments. Aquartic formula

\M\ < 1

preserves continuity of f\ and d/i/dM at |M| = 1 if a2 = 3/2—2ao, #4 = a® — 1/2. Then a$ controls the diffusion at M = 0.For transonic flow calculations a good choice is aQ = 1/4, whereasfor very high-speed flows it may be increased to 1/2. Notice that a0controls also the amount of dissipation in the inner part of viscousregions. In principle one would prefer to decrease its value as muchas possible. In practice, however, it was found that the value of aQcannot not be decreased below 1/4 without impairing convergence.A suitable blending formula for the pressure diffusion is

\M\ < 1

The diffusion corresponding to the convective terms is identicalto the scalar diffusion of Jameson ef al.4 with a modification ofthe scaling, whereas the pressure term is the minimum modifica-tion needed to produce perfect upwinding in the supersonic zone.The scheme retains the property of the original scheme in that itis compatible with constant stagnation enthalpy in steady flow. Ifone derives the viscosity corresponding to the flux splitting recentlyproposed by Liou and Steffen,14 then it can be shown2 that theirscheme produces first-order diffusion with a similar general form,and the present scheme may thus be regarded as a construction ofartificial viscosity approximately equivalent to Liou-Steffen split-ting. The high-resolution scheme is obtained by applying either thelimited average or the switch to (dc l -f dp L).

III. ImplementationThe nine schemes which have been implemented and tested in the

present work can be summarized in Table 1. The actual implementa-tion of these schemes to the solution of two-dimensional problemsis straightforward: the flux splitting is applied separately in eachcoordinate direction.

For the scalar dissipation scheme, a blending formula for thespectral radii of the flux Jacobians has been used. Following Ref. 3


oADO

x/L=.20x/L=.40x/L=.61x/L=.82

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex)/x

3-a(l) : Tangential velocity profilesMach=0.15

oA

ao

x/L=.20x/L=.40x/L=.61x/ 1 .̂82


3-a(2) : Transverse velocity profilesMach=0.15

1.2

1.0

0.8 -<+-«| 0.6\3

0.4

0.2

0.0

/——— Bla

o x/LA x/L

sius Solution i=.20=.40

r 1

o x/L=.82

0.0 2.0 4.0 6.0 8.0 10.0«ETA=y*SQRT(Rex)/x

3-a(3) : Tangential velocity profilesMach=0.7

1.2

1.0

0.8cD 0.6\3

0.4

0.2

0.0

................. ....-.-.--...̂ SBBW.

..jr........... I.............

-ttr-OBo-ascr-jafttD— o -A- -o

Rlao x/LA x/La x/Lo x/L

sius Solution=.20=.40=.61=.82

j_2.0 4.0 6.0 8.0 10.0

ETA=y*SQRT( Rex) /\3-a(5) : Tangential velocity profiles

Mach=2.0

1.2

_ 1.0xo>I"Si 0.6

| 0.4

>

"•" 0.2

O.Q

Blasius Solutionx/L=.20x/L=.40x/L=.61x/L=.82

i i6.0 8.0 10.0ETA=y*SQRT(Rex)/x


1.2

l.O

0.6

0.4Blasius Solutionx/L=.20x/L=.40x/L=.61x/L=.82



Fig. 3a Boundary-layer velocity profiles: Mach = 0.15,0.7, and 2.0 (top to bottom), Re = I x 10s, 32 cells in the layer; scalar-switch.

the scaling of the dissipative flux in the 7th and 7th coordinatedirections is taken as follows:

A./ —

Notice that these formulas reduce to a pure directional scaling forote = 0, and to a fully isotropic scaling for a€ = I . This techniquehas been shown to be effective in reducing the numerical dissipa-tion on highly stretched grids while maintaining good convergencecharacteristics. For the calculation of the boundary layers presentedhere, an optimum value of ot€ was chosen for each grid density.However, the same value of a€ was maintained on a given grid forall of the Reynolds and Mach numbers. On finer grids, comprisingmore than 16 cells in the boundary layer, the value ae = 2/3 wasfound to give the best results.3 On coarser grids, however, we foundthat ot€ needed to be reduced to 1 /4.

Multigrid Time-Stepping Integration SchemeTime integration is carried out by making use of a five-stage

scheme which requires re-evaluation of the dissipative operators atalternate stages.3 This scheme couples the desirable feature of a wide

stability region along both the imaginary and the real axis with goodhigh-frequency damping. The efficiency of the scheme is enhancedby using implicit residual averaging with variable coefficients andan effective multigrid strategy which utilizes a W cycle.3

In the present study it was found that 100 multigrid cycles aregenerally sufficient to achieve convergence to a steady state withfive orders of magnitude reduction of the rms density residual.

IV. ResultsA laminar boundary layer developing over a flat plate at zero in-

cidence provides the test case for the various schemes. A low valueof the incoming flow Mach number (M^ = 0.15), well withinthe incompressible regime, is selected to make a comparison witha Blasius solution meaningful. Also, this flow condition facilitatesthe evaluation of the numerical schemes toward their limit of appli-cability as MOO -* 0.

The computational domain is a rectangle with the inflow bound-ary located two plate lengths upstream of the leading edge and thedownstream boundary located at the plate trailing edge. The upperboundary is located at a distance of four plate lengths. The meshpoints are clustered in the streamwise direction near the leadingedge in order to provide adequate resolution of the flow near thestagnation point. The finest grid contains a total of 512 cells placedin the streamwise direction with 384 of these cells placed alongthe plate. Within the boundary layer, the grid is equally spacedin the boundary-layer coordinate in the direction perpendicular tothe plate. This ensures a constant level of resolution for all of the


.0Aao

x/L=.20x/L=.40x/L=.61x/L=.82

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

Tangential velocity profilesMach=0.15

« 0-8

Q* O ABlasius Solutionx/L=.20x/L=.40x/L=.61x/L=.82

3-b(2)

.0 4.0 6.0 8.0 10.0ETA=y*SQRT(Rex)/x

: Transverse velocity profilesMach=0.15

1.2

1.0

0.8

c5 0.6\3

0.4

0.2

O.Q


2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

3-b(3) : Tangential velocity profilesMach=0.7


_n6.0 8.0 10.0ETA=y*SQRT( Rex) /x

3-b(4) : Transverse velocity profilesMach=0.7




3-b(5) : Tangential velocity profilesMach=2.0

3-b(6) : Transverse velocity profilesMach=2.0

Fig. 3b Boundary-layer velocity profiles: Mach = 0.15,0.7, and 2.0 (top to bottom), Re = 1 x 105,32 cells in the layer; scalar-USLIP.

boundary-layer profiles. It also ensures an identical resolution in theboundary layer independently of the Reynolds number. Outside ofthe boundary layer the grid is exponentially stretched toward thefar field. The finest grid used contains a total of 128 cells in thedirection normal to the plate with half of these cells in the boundarylayer. Three coarser grids containing 8,16, and 32 cells, respectively,within the boundary layer are obtained by elimination of alternatepoints and are used in the grid refinement study.

No-slip boundary conditions are used on the plate, and symmetryof the incoming flow is assumed upstream of the plate leading edge.Appropriate nonreflective boundary conditions, based on the solu-tion of the one-dimensional Riemann problem normal to the gridlines, are used at the three outer boundaries.

Calculations were performed at Reynolds numbers of 1,000,10,000, 100,000, and 500,000 based on the length of the plate.However, due to editorial constraints, only the computed resultsfor Re^ = 100, 000 are presented here.

A grid refinement study is presented first, to analyze the accuracyof the nine schemes. The computed results are found to obey thesimilarity law. Thus, in Fig. 1, we only show the comparison of thecomputed velocities on four computational grids at one streamwiselocation (x/L = 0.8). Results of the grid refinement study for thethree switched schemes are compared in Fig. la. It is seen that eventhe characteristic splitting can produce overshoots in the normalvelocity profiles on coarser grids. However, these overshoots arereduced, for all of the three flux splitting techniques considered inthis study, if one chooses the SLIP construction. This finding is

documented in Fig. Ib. The USLIP construction seems to give thebest results on coarser grids as it is illustrated in Fig. Ic. Noticethat all nine schemes produce solutions which converge to the the-oretical one as the grid is refined. A more quantitative analysis ofthe numerical errors is summarized in Table 2 where the percentileerrors in skin friction C/, displacement thickness 5*, and momen-tum thickness 6 are tabulated. Although the skin friction is obtaineddirectly from the computations, one needs to postprocess the resultsto obtain an estimate of the integral parameters. In this study weuse simple trapezoidal integration applied to the definitions givenby Schlichting.15 We verified that different integration techniqueswould produce results within 0.5% of those reported in the table.

The results presented in Fig. 2 reinforce the finding of previousstudies which have shown that 32 cells are generally sufficient toresolve adequately the viscous layer.3 Moreover, it is seen that theonly noticeable differences are in the transverse velocity compo-nent and that the results depend more on the construction of thelimiting process rather than on the particular form of flux split-ting. Notice that the USLIP construction performs slightly betterthan either the switched or the SLIP formulation, independently ofthe particular form of flux splitting. This observation provides themotivation for the last set of calculations which are aimed at inves-tigating the behavior of the USLIP schemes over a range of Machnumbers. The results presented in Fig. 3 are obtained for a Reynoldsnumber of 100,000 on a grid with 32 cells in the boundary layer.Figure 3a shows the velocity profiles computed with the standardscalar switched scheme3 for three Mach numbers: M = 0.15, 0.7,



I).0 2.0 4.0 6.0 8.0 10.0

ETA=y*SQKT(Rex)/x3-c(l) : Tangential velocity profiles

Mach=0.15

0.0 •«0.0

pln

o x/LA x/La x/Lo x/L

sius Solution=.20=.40=.61=.82

0

2.0 4.0

3-c(2) : TransverseMach=0.15

i . . . i6.0 8.0 10.0ETA=y *SQRT( Rex) /x

velocity profiles

1.0

0.8

0.6

0.4

0.2

j* ^

....../ j.............Rid

0 X/lA X/l

a x/Lo x/l

sius Solution=.20=.40.=.61.=.82

0.0

3-c(3)

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

: Tangential velocity profilesMach=0.7

K 0.6

§ 0.4\>~ 0.2

O.Oi

/

*~

^afta**"1 : i TO ^

: : :

o x/L=.20A x/L=.40n x/L=.61o x/L=.82

2.0 4.0 6.O 8.0 10.0ETA=y*SQRT(Rex)/x

3-c(4) : Transverse velocity profilesMach=0.7


I I6.0 8.0 10.0ETA=y*SQRT(Rex)/x

3-c(5) : Tangential velocity profilesMach=2.0

50.4 o

D

x/L=.20x/L=.40x/L=.61x/L=.82

2.0 4.0 6.0 8.0 10.0ETA=y*SQRT( Rex) /x

3-c(6) : Transverse velocity profilesMach=2.0

Fig. 3c Boundary-layer velocity profiles: Mach = 0.15,0.7, and 2.0 (top to bottom), Re = 1 X 105,32 cells in the layer; characteristic-USLIP.

and 2, respectively. The transonic as well as the supersonic resultsare scaled by using the Illingworth-Stewartson transformation15 andare again compared with the Blasius solution. The results at fourstreamwise locations are overplotted to verify the self-similarity ofthe computed flow. Figures 3b and 3c show similar behavior of thecomputed results with the scalar-USLIP, and characteristic-USLIPschemes. The results for the subsonic and transonic cases are in ex-cellent agreement with the theory, although some deviations fromthe theoretical velocity distributions are noticeable for the super-sonic one. This discrepancy is partially attributable to the numericalimplementation of the Illingworth-Stewartson transformation in thepostprocessing stage. In fact, grid refinement studies, not reportedhere for the sake of brevity, show that the computed solutions pre-sented are indeed grid independent, even for the supersonic case.

V. Concluding RemarksSeveral discretization schemes have been developed and applied

to the solution of the compressible Navier-Stokes equations. Theresults presented indicate that these schemes, which have beeno-riginally tailored for nonoscillatory shock capturing, yield accuratesolutions for viscous flows. In particular, it appears that the construc-tion of SLIP and USLIP schemes holds the promise of improvingthe accuracy, especially on coarser grids. This result reinforces theobservation reported by two of the authors,3 and could have a bene-ficial impact on three-dimensional applications since current avail-able computers do not allow yet the adequate resolution of complexthree-dimensional viscous flows.

The results obtained in the present study with the scalar-switchedscheme do not show a degradation in accuracy which is anywherenear as large as that reported by Allmaras.1 Moreover, our studytends to suggest that the cause for the overshoot observed withthe scalar scheme is somewhat different from the one reported inreference,10 where the poor performance of a scalar scheme forlaminar flow at high Reynolds number is attributed to the scalingof the artificial dissipation terms. Our results suggest that the flux-limiting process plays an important role, and that good accuracy canbe achieved even with the scalar dissipation in conjunction with theSLIP or the USLIP construction. It appears that the higher orderdifferences required for an antidiffusive construction may result inovershooting unless limiters are introduced.

Preliminary tests on both two-dimensional airfoils16 and three-dimensional wings, not reported here for the sake of brevity, confirmthat the findings of the present study extend to viscous flows withpressure gradients.

Since the SLIP and USLIP constructions can be carried outfor unstructured triangular (tetrahedral) meshes, providing multi-dimensional upwinding on arbitrary geometries, it is expected thatschemes of this class will also improve the resolution of complexviscous flows on unstructured triangular and tetrahedral grids.

AcknowledgmentsThe first author gratefully acknowledges Mitsubishi Heavy In-

dustries, Ltd. whose support has made his stay at Princeton Univer-sity possible. This work has benefited from the generous support of


the Advanced Research Projects Agency under Grant N00014-92-J-1796 and the Air Force Office of Scientific Research under GrantAFOSR-91-0391.

ReferencesJameson, A., "Successes and Challenges in Computational Aerodynam-

ics," AIAA Paper 87-1184, June 1987.2 Jameson, A., "Computational Algorithms for Aerodynamic Analysis and

Design," Technical Rept., INRIA 25th Anniversary Conference on ComputerScience and Control, Paris, France, Dec. 1992; Applied Numerical Methods(to be published).

3Martinelli, L., and Jameson, A., "Validation of a Multigrid Method forthe Reynolds Averaged Equations," AIAA Paper 88-0414, Jan. 1988.

4Jameson, A., Schmidt, W., and Turkel, E., "Numerical Solutions of theEuler Equations by Finite Volume Methods with Runge-Kutta Time SteppingSchemes," AIAA Paper 81-1259, Jan. 1981.

5Swanson, R. C., and Turkel, E., "On Central-Difference and UpwindSchemes," Journal of Computational Physics, Vol. 101, No. 2, 1992,pp. 297-306.

6Boris, J. P., and Book, D. L., "Flux Corrected Transport, 1 SHASTA, aFluid Transport Algorithm that Works," Journal of Computational Physics,Vol. 11, No. 1, 1973, pp. 38-69.

7Jameson, A., "A Non-Oscillatory Shock Capturing Scheme Using FluxLimited Dissipation," Mechanical and Aerospace Engineering Rept. 1653,Princeton Univ., Princeton, NJ, April 1984; also, Lectures in AppliedMathematics, edited by B. E. Engquist, S. Osher, and R. C. J. Sommerville,American Mathematical Society, Pt. 1, Vol. 22, 1984, pp. 345-370.

8Jameson, A., "Artificial Diffusion, Upwind Biasing, Limiters and theirEffect on Accuracy and Multigrid Convergence in Transonic and HypersonicFlow," AIAA Paper 93-3359, July 1993.

9Roe, P. L., "Approximate Riemann Solvers, Parameter Vectors, and Differ-ence Schemes," Journal of Computational Physics, Vol. 43, 1981, pp. 357-372.

10Allmaras, S., "Contamination of Laminar Boundary Layers by ArtificialDissipation in Navier-Stokes Solutions," Proceedings of the Conference onNumerical Methods in Fluid Dynamics, Reading, England, UK, 1992.

HSwanson, R. C., and Turkel, E., "Aspect of a High-Resolution Schemefor the Navier-Stokes Equations," AIAA Paper 93-3372, July 1993.

12Garriz, J. A., Vatsa, V. N., and Sanetrik, M. D., "Issues Involved inCoupling Navier-Stokes Mean-Flow and Linear Stability Codes," AIAAPaper 94-0304, Jan. 1994.

13Denton, J. D., "An Improved Time Marching Method for Turbomachin-ery Flow Calculations," Journal of Engineering for Gas Turbines and Power,Transaction of ASME (American Society Mechanical Engineers), Journalof Engineering for Power, Vol. 105, July 1983, pp. 514-524a.

14Liou, M.-S., and Steffen, C. J., "A New Flux Splitting Scheme," NASA-TM 104404, 1991; also, Journal of Computational Physics, Vol. 107, No. 1,July 1993, pp. 23-39.

^Schlichting, H., Boundary Layer Theory, 7th ed., McGraw-Hill, NewYork, 1979.

16Tatsumi, S., "Design, Implementation, and Validation of Flux LimitedSchemes for the Solution of the Compressible Navier-Stokes Equations,"Mechanical and Aerospace Engineering Rept. 2009, Princeton Univ., Prince-ton, NJ, Aug. 1994.

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Flux-Limited Schemes for the Compressible Navier-Stokes ...aero-comlab.stanford.edu/Papers/AIAA-12422-882.pdf · unstructured meshes. This recent development has shed new light on