VOL. 24, NO. 9, SEPTEMBER 1986 AIAA JOURNAL 1453

Comparison of Finite Volume Flux Vector Splittingsfor the Euler Equations

W. Kyle Anderson* and James L. Thomas*NASA Langley Research Center, Hampton, Virginia

andBram Van Leer*

Delft University of Technology, Delft, the Netherlands

A flux-splitting method in generalized coordinates has been developed and applied to quasi-one-dimensionaltransonic flow in a nozzle and two-dimensional subsonic, transonic, and supersonic flow over airfoils. Com-putational results using the Steger-Warming and Van Leer flux splittings are compared. Discussed are several ad-vantages of a MUSCL-type approach (differencing followed by flux splitting) over a standard flux differencingapproach (flux splitting followed by differencing). With an approximately factored implicit scheme, spectralradii of 0.978-0.930 for a series of airfoil computations are obtained, generally decreasing as a larger portion ofthe flow becomes supersonic. The Van Leer splitting leads to higher convergence rates and a sharper representa-tion of shocks, with at most two (but more often, one) zones in the shock transition. The second-order accurateone-sided-difference model is extended to a third-order upwind-biased model with a small additional computa-tional effort. The results for both the second- and third-order schemes agree closely in overall features to a widelyused central difference scheme, although the shocks are resolved more accurately with the flux splitting approach.

Nomenclaturea = speed of sounde = total energy per unit volumeF,G = fluxes of mass, momentum, and energyp = pressureQ = conserved variablest = timeu, v = Cartesian velocitiesx,y = Cartesian coordinates7 = ratio of specific heats, = 1.4p = densitySuperscriptsn = time level = positive and negative flux contributions; also forward

and backward spatial differencing or extrapolation

Introduction

I N the solution of hyperbolic equations such as the time-dependent Euler equations, the theory of characteristics iscrucial in determining the directions of the signal propaga-tion. The information gained from the characteristic theory,for example, has been very useful in the development ofboundary conditions that are both physically correct andmathematically well posed. Recently, much interest has beengenerated in the development of computational methods forthe Euler equations that model the underlying physics, asdictated by characteristic theory, at each grid point. Thesemethods can be classified, in general, as upwind methodsand have the advantage of being naturally dissipative.Separate spatial dissipation terms, such as those generally re-quired in a central difference method to overcome oscilla-tions or instabilities arising in regions of strongly varyinggradients, need not be added.

Received Dec. 20, 1984; presented as Paper 85-0122 at the AIAA23rd Aerospace Sciences Meeting, Reno, NV, Jan. 14-17, 1985; revi-sion submitted Oct. 10, 1985. Copyright American Institute ofAeronautics and Astronautics, Inc., 1986. All rights reserved.

* Research Scientist. Member AIAA.

Many of these methods are applied to the nonconservativeform of the equations and consequently require the use ofshock-fitting techniques to obtain the correct location andstrength of shocks in transonic flows. Use of the conserva-tion law form allows the shock waves to be captured as weaksolutions to the governing equations and circumvents the dif-ficulty in applying shock-fitting techniques to arbitraryflows. Both flux difference and flux vector splitting methodscan be applied to the conservation law form; only flux vectorsplitting or, in brief, flux splitting, is considered in the pres-ent work.

Steger and Warming,1 making use of similarity transfor-mations and the homogeneity property of the Euler equa-tions, split the flux vector into forward and backward con-tributions by splitting the eigenvalues of the Jacobian matrix

1.5l.O.5

Massflux

-.5-1.0-1.51.51.0.5

Steger-Warming splitting

Massflux

Van Leer splitting

-.5-1.0-1.5

-1.50 -1.00 -.50 0 .50Mach number

1.00 1.50

Fig. 1 Variation of mass flux with Mach number for Steger-Warming and Van Leer splittings.

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1454 ANDERSON, THOMAS, AND VAN LEER AIAA JOURNAL

of the full flux into non-negative and nonpositive groups.The split flux contributions are then spatially differenced ac-cording to one-sided upwind discretizations. The Steger-Warming splitting has been widely used, with varyingdegrees of success, and has also been implemented ingeneralized coordinates.2 Its main shortcoming is that theforward and backward fluxes are not differentiable when aneigenvalue changes sign, such as occurs at sonic points; thisleads to the occurrence of small glitches or oscillations inthese regions. Van Leer3 introduced an alternate flux split-ting with continuously differentiable flux contributions thatlead to smoother solutions at sonic points. In addition, thesplitting is designed so that the shock structures can berealized with no more than two interior zones.

The use of this alternate splitting method in computingtwo-dimensional flow about airfoils in a general coordinatesystem has been very limited. The implementation of VanLeer's flux splitting in generalized coordinates and a com-parison of the results with those obtained with the Steger-Warming splitting are the main purposes of this inves-tigation.

Flux SplittingCartesian Coordinates

The two-dimensional Euler equations expressed in Carte-sian coordinates and conservation form are

where

pu

pv

e

F=

8Q 8F dGTL + -^ + -^dt dx by

pu

puu+p

puv

u(e+p)

(1)

pv

puv

pvv+p

v(e+p)

(2)

with the equation of state given by

f p(u2 + v2)~\P=

SEPTEMBER 1986 FLUX VECTOR SPLITTINGS FOR THE EULER EQUATIONS 1455

that follows, the superscript ( ) indicates variables in thegeneralized coordinates, while an overbar ( ) indicatesvariables in a locally Cartesian system. If no superscript isused, Cartesian coordinates are assumed. The strong conser-vation form of the Euler equations in generalized coor-dinates is

dQ dF dGdt d drj (U)

where

(12)The Jacobian of the transformation, 7, physically cor-responds to the inverse of the cell volume. The vectors (grad) / J and (grad 77) // correspond to directed areas of cell in-terfaces in the and 77 directions, respectively.^ For the purpose of determining a generalized splitting for

F, only the derivatives in the and t directions are con-sidered, while the 77 derivatives are treated as source terms.Conversely, when splitting G, the derivatives are treated assource terms. For determining the splitting of F, Eq. (11) istransformed by a local rotation matrix T given by

1 0 0 0

0 cos0 sin0 0

0 -sin0 cos0 0

0 0 0 1

= x / lg rad l

(13)

where

Multiplication of Eq. (11) with the matrix T then yields

whereP

pu

pv

e

(14)

(15)

F=TF=Igrad 1

J

pu

piiu+p

puv

J,e+p)u _,

(16)

The rotated velocity component u is the velocity normal to aline of constant representing the scaled contravariantvelocity component and v is normal to u,

w =(*"+ ,*>)/lgrad Iv= (-,M+ *!>)/Igrad f I

The transformed flux F is of the same functional form as theCartesian flux vector and thus can be split according to any

splitting developed for Cartesian coordinates. Therefore, theequations given in the previous section for both the Steger-Warming and the Van Leer splittings can be used to split theflux vector F after replacing the Cartesian velocity com-ponents u and v by the rotated velocity components u and v.Applying the rotation T to Eq. (11) simply allows us to splitthe flux vector in a one-dimensional fashion, along a coor-dinate axis perpendicular to the cell interface. After splittingF, the appropriate splitting for F is determined by applyingthe inverse transformation matrix T~l to Eq. (14), leading to

with

F+ = T-{ F- =

(17)

(18)Note that the inverse transformation restores the originalform of the equations, i.e., no additional source terms ariseand the form of G is unaffected. This allows a splitting of Gsimilar to the splitting of F shown above. Carried out withthe Steger- Warming splitting, the above procedure recoversidentically the generalized splitting given in Ref. 1.

Spatial DifferencingIn numerical solutions to the flux-split Euler equations,

the spatial derivatives of F+ and F~ , for example, can beapproximated with backward and forward differenceoperators, respectively, denoted herein as d~ and dx. Thestandard first- and second-order backward differences on auniformly spaced grid are given by

These formulas can be combined by writing dx as the sum ofa first-order term and a correction for second-order accuracy

12Ajc

(19)The order of accuracy of the approximation is governed bythe value of the switch 0~: second-order accuracy for0~ = 1, first order for ~ =0. Spatial variation of " allowsconservative switching between the second- and first-orderformulas, which is generally required to eliminate oscilla-tions in regions where the solution is discontinuous, such asshock waves. The switching function

1456 ANDERSON, THOMAS, AND VAN LEER AIAA JOURNAL

where the notation F (Q*) denotes F evaluated at gT,and

Qr+ 1/2 =: Q. + (f) r( Q. Qf_l)/2 (21)

Q++ 1/2 = Q.+ , - 0++ j (* = !, respectively; flux limiting is implemented through aspatial variation of * similar to that associated with Eq.(19).

The computational results presented in the next section in-dicate that MUSCL-type differencing [Eq. (20)] is clearlysuperior to standard flux differencing [Eq. (19)]. The ad-vantages of the MUSCL approach accrue for severalreasons. The fluxes are split according to the local cell inter-face Mach number, whereas in the flux differencing ap-proach the splitting depends on the Mach numbers in variouscell centers. Also, as noted in Ref. 6, the split fluxes aregenerally less differentiable than the conserved variableswhen transitioning through sonic and stagnation points. Thisis particularly true for the Steger-Warming fluxes that,unlike the Van Leer fluxes, are not even once differentiable.Additionally, the MUSCL approach is advantageous in ex-tensions to two and three dimensions in curvilinear coor-dinates, as an unambiguous choice for the metric factors inthe normal fluxes can be made.

Computational results comparing the effects of spatial dif-ferencing approximations for a one-dimensional nozzle withtransonic flow are presented below. The Mach number varia-tion along the converging-diverging nozzle test problem usedin Ref. 4 is shown in Fig. 2. Figure 2a shows results usingsecond-order split-flux differencing [Eq. (19)] with theSteger-Warming flux splitting; Fig. 2b shows the correspond-ing result with the Van Leer flux splitting. As expected, thesmoother variation of the split fluxes with the Van Leersplitting has eliminated the glitch evident at the sonic pointwith the Steger-Warming splitting. A noticeable drawback toboth solutions is that overshoots in the Mach number are ob-tained just upstream of the shock. Although not shown, thedensity and pressure variations display similar overshootsand undershoots in the shock region. Figure 2c shows the ef-fect of using Eq. (7) with e = 0.04 rather than e = 0.0 as theeigenvalue switch in the Steger-Warming fluxes. As can beseen, the glitch at the sonic point has been successfullyeliminated, while the shock region is virtually unaffected.Although not shown , the use of a flux limiter resulted in theelimination of the overshoots in the shock region, while notdegrading the accuracy of the solution in the smooth regionof the flow.

Mach .5number

.4 .6 .8 1.0Distance along nozzle

Fig. 2 Mach number distribution along a one-dimensional nozzle us-ing split-flux differencing: a) Steger-Warming, e = 0; b) Van Leer; c)Steger-Warming, e = 0.04.

Second-order results obtained with MUSCL-type differ-encing [Eq. (20)] are shown in Fig. 3a for the Steger-Warming splitting with e = 0 and in Fig. 3b for the Van Leersplitting. As can be seen, in either solution the oscillations inthe shock region are absent, although no flux limiting hasbeen employed. This is an unexpected benefit of MUSCL-type differencing, when incorporated in a fully one-sidedscheme. The state quantity that jumps the most across theshock, i.e., the Riemann invariant associated with thecharacteristic speed that changes sign, is not differencedacross the shock (this is easily checked on the basis ofBurger's equation). The other quantities are, but carry muchsmaller jumps if the shock is not too strong. The result is amonotone profile for weak shocks.

The Van Leer splitting leads to a sharper representation ofthe shock than the Steger-Warming splitting. Note, however,that the Steger-Warming fluxes, although used with zerosmoothing parameter, no longer leads to glitches in the solu-tion as it passes through the sonic points. Several other tran-sonic cases were investigated and, in each one, results similarto those shown above were obtained.

Solution AlgorithmAfter flux splitting, the Euler equations in generalized

coordinates are given by

dQ dF+ dF-dt d d

- = 0 (23)

The method of solution is the approximately factored Beam-Warming implicit algorithm7 given in delta form by

dF~

(24)

where AQ = Qn + l - Qn . The spatial derivatives are approx-imated by the MUSCL-type differencing described previouslyand are implemented in what is commonly referred to as afinite volume formulation. The split-flux differencing in the direction, for example, can be written as

d F f i = [ F * (Qr+Kj,Mi+viJ)-F* (QT-wM^j)} (25)

The term M represents geometric terms involved in thetransformation to generalized coordinates, evaluated at thecell interface locations where the flux values are needed. Thevalues of Q^.y2j are determined as in the one-dimensional

1.2

Machnumber

.5

q

oooo000"00

oO00000

B-x

>0

oO000t>l

oO0l ^fi-X,^

^

.2 .4 .6 .8Distance along nozzle

1.0

Fig. 3 Mach number distribution along a one-dimensional nozzle us-ing MUSCL-type differencing: a) Steger-Warming; b) Van Leer.

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SEPTEMBER 1986 FLUX VECTOR SPLITTINGS FOR THE EULER EQUATIONS 1457

case

(26)Qr+

Since fluxes and geometric terms are evaluated only at cellinterfaces, the above MUSCL-type finite volume formulationleads to a vanishing right-hand side of Eq. (24) for free-stream conditions, in contrast to the standard flux-differencing formulation.2

First-order upwind differencing is used on the left-handside of Eq. (24), yielding a block-tridiagonal structure for theimplicit equations. Note that the true Jacobians arising fromthe time linearization of F and G are used in the resultspresented below. Appproximations to these matrices that aresimpler to form have been used previously, but lead to anexplicit-type stability limit for A/1.8 Since upwind spatial dif-ferencing is used, explicit artificial dissipation is not requiredand hence not used.

The boundary conditions for the airfoil solutionspresented subsequently are applied explicitly. On the body,the normal velocity is set to zero and the pressure is deter-mined from the normal momentum equation. The tangentialvelocity on the surface, needed to compute the normalpressure derivative, is determined by extrapolation from theinterior. The density on the surface, likewise needed, isdetermined by forcing the normal entropy derivative to bezero.

The boundary conditions in the far field with subsonicfreestream conditions are determined through a characteristicanalysis normal to the boundary, including a point vortexrepresentation for the induced velocities in the far field,similar to that described in Ref. 9. Along inflow boundariesfor supersonic freestream conditions, quantities are ex-trapolated from the exterior; along outflow boundaries,quantities are extrapolated from the interior.

Baseline Computational ResultsComputational results are shown in Figs. 4 and 5 for a

baseline series of test cases. Comparisons are made betweenthe Steger-Warming (e = 0) and the Van Leer flux splittingsusing MUSCL-type differencing with = l (second-orderspatial accuracy, no limiting). All of the results shown inFigs. 4 and 5 are for C-type grids with 161 grid points in thedirection tangential to the airfoil and 41 grid points normalto the surface and wake. The grids extend approximately 12chord lengths from the airfoil in all directions. Furthermore,each case was run using a variable time step throughout thegrid based on a local CFL number. All cases ran 1000 itera-tions with a local CFL number of 30 specified everywhere(unless otherwise noted), starting from freestream conditions.

The first case considered is subcritical flow around anNACA 0012 airfoil at a freestream Mach number ofMa, =0.63 and an angle of attack a = 2 deg. The distribu-tions of pressure coefficients cp are shown in Fig. 4a for theVan Leer and Steger-Warming flux splittings. Both solutionsare nearly identical with a lift coefficient c( of 0.333 and0.332, respectively, in close agreement with the value of0.335 obtained by Lock10 from an accurate solution basedon the potential equation. The history of residual and liftobtained with Van Leer's flux splitting is shown in Fig. 5.The residual is defined as the L2 norm of the right-hand sideof Eq. (24) evaluated for the continuity equation only. Theconvergence rate indicates a spectral radius of approximately0.978. The lift coefficient converges to its asymptotic valuevery rapidly, within 150-200 iterations. With the Steger-Warming flux splitting, the convergence was somewhat lowerthan that shown in Fig. 5 because of a reduction in theallowable CFL number.

The next case considered is supercritical flow over aNACA 0012 airfoil at M^O.80 and a =1.25 deg. Underthese conditions, a shock appears on both the upper andlower surfaces of the airfoil; the Mach number ahead of the

-2.0-1.6-1.2

-.8CP -.4

0.4

.8

1.2

Van Leer spl i t t ing- -0.0002; C7 = 0.333

Steger-Warming splittingc^ = 0.0006; c, = 0.332d L

Upper surfaceLower surface

\

.2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0

a) NACA 0012, Mx =0.63, a = 2 deg.

-2.0

1.2

Van Leer splittingc. = 0.0230; C7 = 0.370

\

0 .2 .4 .6 .8 1.0x/c

Steger-Warming splittingc = 0.0237; c, - 0.370

Upper surfaceLower surface

0 .2 .4 .6 .8 1.0x/c

b) NACA 0012, Mx =0.80, a = 1.25 deg.

Van Leer splitting-2.0

-1.6

-1.2

-.8

-.4

0

.4

.8

1.2

~ r = 0.0409; C7 = 1.097d L

------ - v" ,/"""""-.',""-,

f i i i i i0 .2 .4 .6 .8 1.0

Steger-Warming splittingc. = 0.0409; c, = 1.094

Upper surfaceLower surface

0 .2 .4 .6 .8 1.0x /c x / c

c) RAE 2822, M^ =0.75, a = 3.00 deg.

Van Leer splittingc. = 0.0965; c, - 0

Steger-Warming splittingc, = 0.0965; c, = 0

Upper surfaceLower surface

0 .2 .4 .6 .8 1.0 1.2 1.4 0 .2 .4 .6 .8 1.0 1.2 1.4

d) NACA 0012, M^ =1.2, a = 0.0 deg.Fig. 4 Pressure distribution comparison between the Steger-Warming and Van Leer flux splittings (161 X 41 C-mesh).

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1458 ANDERSON, THOMAS, AND VAN LEER AIAA JOURNAL

upper shock is approximately 1.4. The residual and lifthistory is shown in Fig. 5 for the Van Leer splitting; thespectral radius of 0.976 for the transonic case is slightly bet-ter than for the subsonic case. The convergence rate with theSteger-Warming flux splitting was almost identical to thatshown for the Van Leer splitting. A comparison between thepressure distributions of the Van Leer and Steger-Warmingsplittings in Fig. 4b shows that the Van Leer splitting leadsto a sharper representation of both the upper and the lowershock, but neither solution exhibits undershoots or over-shoots. This is the same qualitative behavior as seen in theone-dimensional test cases. The lift and drag coefficients, cfand cd9 obtained with the Van Leer splitting are 0.370 and0.0230, respectively; with the Steger-Warming splitting, thelift is the same and the drag differs by only 3%.

Computations for the RAE 2822 airfoil at M^ =0.75 anda = 3 deg are shown in Figs. 4c and 5. The conditions corre-spond to a strong upper-surface shock with a Mach numberahead of the shock of about 1.5. The spectral radius for thiscase was 0.965; the residual was reduced to machine zero inless than 700 iterations, while the lift reached its final valuein 125-150 iterations. The rate of convergence with theSteger-Warming splitting was less than that with the VanLeer splitting, due to a reduction in the allowable CFLnumber. As indicated in the pressure distributions in Fig. 4c,the Van Leer splitting leads to a sharper representation ofthe shock, although both solutions yield almost identicalvalues for lift and drag. The lift and drag coefficients withthe Van Leer splitting are 1.097 and 0.0409, respectively,while corresponding values of 1.094 and 0.0409 are obtainedwith the Steger-Warming splitting.

The last test case considered was an NACA 0012 airfoil atnonlifting, supersonic conditions: M00 = 1.20 and a = Q deg.Although not shown, the residual was reduced to machinezero in less than 300 iterations with the Van Leer splitting,corresponding to a spectral radius of 0.93; with the Steger-Warming splitting approximately 800 iterations were re-quired because it was necessary to reduce the local CFLnumber to 10 in order to achieve convergence. The pressuredistributions obtained with the two splittings are shown inFig. 4d. The lift and drag coefficients for both splittings arezero and 0.0965, respectively. Both methods capture theoblique shock at the trailing edge very sharply. In this case, thestream wise flux differencing is identical in the majority of thecells, namely, in those cells with supersonic flow.

Extension to Third-Order Spatial AccuracyThe spatial differencing resulting from Eqs. (25-27) and

applied above corresponds to a fully one-sided approxima-tion of dF/d% and dG^/drj. By admitting an upwind-biasedapproximation, the accuracy can be increased to the thirdorder. A general upwind-biased MUSCL-type scheme can be

RAE 2822; 1^= 0.75; a = 3 NACA 0012; M = 0.63; a = 2 NACA 0012; 1^= 0.80; a = 1.25

1.61.41.2

Log,residua

42

0-2

+

-6-8

-10

-

-

-

^

-\\-\\

" \^N- V_^

1 1 1-12

0 1000Time steps

(

12000

Liftcoefficient *

obtained by defining

\U (28)

l / + u (29)where

(A + ) u = Qi + ltj - Qij (30)

( A _ ) ; ; = O / / -Q; i , (31)x t>J *-/t>J *-rl *-sJ v '

The fully one-sided approximation described earlier is ob-tained by inserting K= -1 in the above equations. The valueK=l/3 leads to a third-order upwind-biased approximation,while K = +1 yields the second-order central differencescheme. All upwind-biased approximations use the samenumber of cells for the residual computation as the fullyone-sided approximation and may be implemented with onlya slight increase in computational effort. It must be said thatthe third-order scheme is strictly third-order accurate only inone-dimensional calculations. To obtain a third-orderscheme in two dimensions, it is not sufficient to compute theflux across a cell face on the basis of an averaged state. Thedifference between the averaged flux and the flux of theaveraged state is a term of the second order and vanishesonly for a linear system of conservation laws. Nevertheless, byswitching from k = -1 to Vs, the accuracy of smooth solu-tions can be increased, as shown by Thomas and Walters11;in the present transonic solutions shown below, the dif-ferences are marginal.

In all computations with upwind-biased schemes, under-shoots and overshoots in the shock region are expected. Alimiter may be used to reduce the scheme to a fully one-sidedscheme of either first- or second-order accuracy in the shockregion, thus eliminating overshoots or undershoots. The firstlimiter chosen for this study is a so-called minimum-modulus(min-mod) limiter; it modifies the upwind-biased interpola-tions [Eqs. (28) and (29)] as follows:

(32)l / + u (33)

where

A+ =max[0,min(A + sgnA_,M_sgnA+ )]sgnA+ (34)A_ =max[0,min(A_sgnA+ ,M+sgnA_ )]sgnA_ (35)

b=(3-K)/(l-K) (36)The residual history using this limiter often enters into alimit cycle oscillation as shown in Fig. 6 although the final

0 1000 2000Time steps

Fig. 5 Residual and lift histories with Van Leer splitting (161 x 41C-mesh).

4

2

0

-2Log, _4

residual-6-8

-10-12

Non-differentiable 4limiter 2

^\ ~2x

Log, _4residual

-6-8

-10, , , i - 1 ?

Differentiatelimiter

-

\

A-

-

3 1000 2000 0 1000 2000Time steps Time steps

Fig. 6 Effect of limiting on residual history of NACA 0012,M^ =0.80, a = 1.25 deg (161x41 C-mesh).

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SEPTEMBER 1986 FLUX VECTOR SPLITTINGS FOR THE EULER EQUATIONS 1459

1.2 L

Second orderFully one-sided ( K = -1)cd = 0.0231; cl = 0.363

Third orderUpwind-biased (K = 1/3)c = 0.0234; c = 0.363

Upper surfaceLower surface

0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0x/c x/c

-2.0-1.6-1.2-.8

CP -'40

.4

1.2

Second orderUpwind-biased (K = 0)

c. = 0.0235; c. = 0.362 - c. = 0.0230; c, = 0.362

0 .2 .4 .6 .8 1.0x/c

Central difference(K = 1; FL052)

Upper surfaceLower surface

0 .2 .4 .6 .8 1.0x/c

Fig. 7 Comparison of second and third-order accurate pressuredistributions for NACA 0012 airfoil, 71^=0.80, a = 1.25 deg(161 x 41 O-mesh).

lift value is reached in approximately the same number ofiterations as with the second-order scheme. The limit cycle isbelieved to be associated with the discontinuous derivative ofthe limiter in certain regions.

A second limiter, acting in a continuously differentiablemanner,6 was chosen in order to overcome the convergenceproblems encountered above. This limiter is implemented byrewriting Eqs. (28) and (29) as

(37)

where

+ A _ + e

(38)

(39)

and is a small number (e=10~6) preventing division byzero in regions of null gradients. The residual using thislimiter as shown in Fig. 6 decreases continuously, althoughnot as fast as for the fully one-sided scheme. The pressuredistribution obtained with the upwind-biased schemes agreeclosely with the fully one-sided scheme with no oscillations.

Finally, Fig. 7 shows a comparison of the present resultsfor the one-sided (K= - 1) and upwind-biased (both K = 0 andK= I /S ) schemes, with results from the central differencescheme developed by Jameson et al.12 for the NACA 0012airfoil at supercritical conditions. All results are obtained onthe same mesh, which is of the O-type, with 160 cells aroundthe airfoil and 40 cells in the direction normal to the airfoil.The three solutions yield similar pressure distributions overmost of the airfoil, as well as similar values of lift and drag.The upwind approximations, however, give much sharpershock profiles than the central difference scheme, which par-ticularly smears the lower surface shock. Further ex-

periments with the central difference scheme indicate that thelower-surface shock can be resolved if much finer meshes areused.

ConclusionsAn investigation has been conducted to compare the flux

splitting method of Steger-Warming with that of Van Leer,including applications to quasi-one-dimensional transonicflow in a nozzle and two-dimensional subsonic, transonic,and supersonic flow over airfoils in generalized coordinates.The studies in one dimension indicate a definite advantage ofthe MUSCL approach (differencing followed by flux split-ting) over the standard flux differencing approach (flux split-ting followed by differencing). With the Steger-Warmingsplitting, for example, the approach allows a smooth transi-tion from subsonic to supersonic flow without the need forextra smoothing. In addition, the fully one-sided MUSCL-type approach yields no over- or undershoots in the shockregion when either the Van Leer or the Steger-Warming split-ting is used. The Van Leer splitting leads to a sharperrepresentation of shocks, with at most two zones (butgenerally one zone) in the shock transition.

A baseline series of computations of flow over two-dimensional airfoils ranging from subcritical lifting flow(Mo, =0.63) to supersonic nonlifting flow (M00 = 1.2) hasbeen made. With an approximately factored implicit schemeand a fully one-sided second-order approximation to the fluxderivatives incorporating the Van Leer splitting, the L2 normof the residual was reduced to machine zero in less than 1000iterations for all cases on a 161x41 C--mesh. The spectralradius decreases from 0.978 for a subsonic case to 0.930 fora supersonic case. In all of the cases studied, the spectralradius decreases as a higher percentage of the flowfieldbecomes supersonic. With the Steger-Warming splittings, therate of convergence is less, in general, because of a decreasein the allowable CFL number.

The second-order one-sided scheme is easily generalized tosecond- and third-order upwind-biased schemes, at little ex-tra computational effort. These upwind-biased schemes,however, require a limiting of higher-order terms to avoidoscillations in the shock region. Two limiters have beentried, one of min-mod type, which is activated suddenly inregions of rapidly changing gradients, the other a smoothlimiter with a continuous action. The mid-mod limiter causeda limit cycle in the residual on fine grids, whereas with thesmooth limiter the solution continued to converge, althoughwith a somewhat higher spectral radius than for the one-sided scheme. The pressure distributions for the one-sidedsecond-order scheme as well as the upwind-biased third-order scheme both show sharp shocks and, in overallfeatures, agree closely to results obtained with a widely usedcentral-difference code. A weak shock arising on the lowerairfoil surface in supercritical flow was resolved more ac-curately with the flux splitting approach than with the cen-tral difference approach.

ReferencesSteger, J. L. and Warming, R. F., "Flux Vector Splitting of the

Inviscid Gasdynamic Equations with Application to Finite Dif-ference Methods," Journal of Computational Physics, Vol. 40, No.2, April 1981, pp. 263-293.

2Buning, P. G. and Steger, J. L., "Solution of the Two-Dimensional Euler Equations with Generalized Coordinate Transfor-mation Using Flux Vector Splitting," AIAA Paper 82-0971, 1982.

3Van Leer, B., "Flux-Vector Splitting for the Euler Equations,"ICASE Rept. 82-30, Sept. 1982 (also Lecture Notes in Physics, Vol.170, 1982, pp. 507-512).

4Steger, J. L., "Preliminary Study of Relaxation Methods for theInviscid Conservative Gasdynamics Equations Using Flux Splitting,"NASA CR-3415, 1981.

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1460 ANDERSON, THOMAS, AND VAN LEER AIAA JOURNAL

5Van Leer, B., "Towards the Ultimate Conservative DifferenceScheme V: A Second-Order Sequel to Gudonov's Method," Journalof Computational Physics, Vol. 32, 1979, pp. 101-136.

6Mulder, W. A. and Van Leer, B., "Implicit Upwind Methods forthe Euler Equations," AIAA Paper 83-1930, July 1983.

7Beam, R. and Warming, R. F., "An Implicit Finite-DifferenceAlgorithm for Hyperbolic Systems in Conservation-Law-Form,"Journal of Computational Physics, Vol. 22, Sept. 1976, pp. 87-110.

8Jespersen, D. C. and Pulliam, T. H., "Flux Vector-Splitting andApproximate Newton Methods," AIAA Paper 83-1899, July 1983.

9Thomas, J. L. and Salas, M. D., "Far-Field Boundary Condi-

tions for Transonic Lifting Solutions to the Euler Equations,"AIAA Paper 85-0020, Jan. 1985.

10Lock, R. C., "Test Cases for Numerical Methods in Two-Dimensional Transonic Flows," AGARD Rept. 575, 1970.

11 Thomas, J. L. and Walters, R. W., "Upwind Relaxation

Algorithms for the Navier-Stokes Equations," AIAA Paper 85-1501,July 1985.

12Jameson, A., Schmidt, W., and Turkel, E., "Numerical Solu-tions of the Euler Equations by Finite Volume Methods UsingRunge Kutta Time-Stepping Schemes," AIAA Paper 81-1259, June1981.

From the AIAA Progress in Astronautics and Aeronautics Series.

FUNDAMENTALS OFSOLID-PROPELLANT COMBUSTION - v. 90

Edited by Kenneth K. Kuo, The Pennsylvania State Universityand

Martin Summerfield, Princeton Combustion Research Laboratories, Inc.

In this volume distinguished researchers treat the diverse technical disciplines of solid-propellant combustionin fifteen chapters. Each chapter presents a survey of previous work, detailed theoretical formulations and ex-perimental methods, and experimental and theoretical results, and then interprets technological gaps andresearch directions. The chapters cover rocket propellants and combustion characteristics; chemistry ignitionand combustion of ammonium perchlorate-based propellants; thermal behavior of RDX and HMX;chemistry of nitrate ester and nitramine propellants; solid-propellant ignition theories and experiments; flamespreading and overall ignition transient; steady-state burning of homogeneous propellants and steady-stateburning of composite propellants under zero cross-flow situations; experimental observations of combustioninstability; theoretical analysis of combustion instability and smokeless propellants.

For years to come, this authoritative and compendious work will be an indispensable tool for combustionscientists, chemists, and chemical engineers concerned with modern propellants, as well as for appliedphysicists. Its thorough coverage provides necessary background for advanced students.

Published in 1984, 891 pp., 6x9illus. (some color plates), $60 Mem., $85 List; ISBN0-915928-84-1

TO ORDER WRITE; Publications Order Dept., AIAA, 1633 Broadway, New York, N.Y. 10019

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233. Ge-Cheng Zha, Doyle KnightComputation of 3D asymmetric crossing shock wave/turbulent boundary layer interactionusing a full Reynolds stress equation turbulence model . [Citation] [PDF] [PDF Plus]

234. Sang-Hyeon Lee, In-Seuck Jeung, Soogab LeeApplication of shock-enhanced mixing to circular cross-sectionalcombustor . [Citation] [PDF] [PDF Plus]

235. Francesco Grasso, Carlo MeolaEuler and Navier-Stokes equations for compressible flows 159-282. [CrossRef]236. Ramesh Pankajakshan, W. Roger BrileyParallel solution of viscous incompressible flow on multi-block structured grids

using MPI 601-608. [CrossRef]237. C. P. van Dam, K. Nikfetrat, K. Wong, P. M. H. W. Vijgen. 1995. Drag prediction at subsonic and transonic speeds

using Euler methods. Journal of Aircraft 32:4, 839-845. [Citation] [PDF] [PDF Plus]238. Grant Palmer, Ethiraj Venkatapathy. 1995. Comparison of nonequilibrium solution algorithms applied to chemically

stiff hypersonic flows. AIAA Journal 33:7, 1211-1219. [Citation] [PDF] [PDF Plus]239. W Briley, Shyam Neerarambam, David WhitfieldImplicit lower-upper/approximate-factorization algorithms for viscous

incompressible flows . [Citation] [PDF] [PDF Plus]240. David Bishop, Ralph NoackAn implicit flow solver with upwind differencing for three-dimensional hybrid grids .

[Citation] [PDF] [PDF Plus]241. David WhitfieldPerspective on applied CFD . [Citation] [PDF] [PDF Plus]242. Ichiro Nakamori, Yoshiaki NakamuraAn upwind scheme based on the flux-vector splitting . [Citation] [PDF] [PDF

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turbulent flows about appended submarine configurations . [Citation] [PDF] [PDF Plus]244. Lafayette Taylor, Abdollah Arabshahi, David WhitfieldUnsteady three-dimensional incompressible Navier-Stokes

computations for a prolate spheroid undergoing time-dependent maneuvers . [Citation] [PDF] [PDF Plus]245. H. Jiang, P.A. Forsyth. 1995. Robust linear and nonlinear strategies for solution of the transonic Euler equations.

Computers & Fluids 24:7, 753. [CrossRef]246. P. Durgapal, Grant Palmer. 1994. Strongly coupled radiative transfer and Joule heating in an arc heater electrode. Journal

of Thermophysics and Heat Transfer 8:4, 730-736. [Citation] [PDF] [PDF Plus]247. Peter M. Hartwich. 1994. Comparison of coordinate-invariant and coordinate-aligned upwinding for the Euler equations.

AIAA Journal 32:9, 1791-1799. [Citation] [PDF] [PDF Plus]

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248. Morgan WilliamsA spatially high-order unstructured grid based computational scheme for incompressible fluid-structure interaction . [Citation] [PDF] [PDF Plus]

249. Alexander Boschitsch, Todd QuackenbushPrediction of rotor aeroelastic response using a new coupling scheme .[Citation] [PDF] [PDF Plus]

250. Feng Liu, Xiaoqing ZhengA strongly-coupled time-marching method for solving the Navier-Stokes and k-omegaturbulence model equations with multigrid . [Citation] [PDF] [PDF Plus]

251. D. Drikakis, F. Durst. 1994. Investigation of flux formulae in transonic shock wave/turbulent boundary layer interaction.International Journal for Numerical Methods in Fluids 18:10.1002/fld.v18:4, 385-413. [CrossRef]

252. J.C. Mandal, S.M. Deshpande. 1994. Kinetic flux vector splitting for euler equations. Computers & Fluids 23, 447-478.[CrossRef]

253. Yang-Yao Niu, J. Scott, G. GregorekA modified flux-vector split algorithm for oscillating airfoil flow simulations .[Citation] [PDF] [PDF Plus]

254. Yasuhiro Wada, Meng-Sing LiouA flux splitting scheme with high-resolution and robustness for discontinuities .[Citation] [PDF] [PDF Plus]

255. K. Wang, Joe CaramComparison of Space Shuttle Orbiter wing leading edge heating rate calculations with experimentaland flight data . [Citation] [PDF] [PDF Plus]

256. Greg W. Burgreen, Oktay Baysal, Mohamed E. Eleshaky. 1994. Improving the efficiency of aerodynamic shapeoptimization. AIAA Journal 32:1, 69-76. [Citation] [PDF] [PDF Plus]

257. W.Kyle Anderson, Daryl L Bonhaus. 1994. An implicit upwind algorithm for computing turbulent flows on unstructuredgrids. Computers & Fluids 23, 1-21. [CrossRef]

258. GRANT PALMER, ETHIRAJ VENKATAPATHY. 1993. Effective treatment of the singular line boundary problemfor three-dimensional grids. AIAA Journal 31:10, 1757-1758. [Citation] [PDF] [PDF Plus]

259. G-C. Zha, E. Bilgen. 1993. Numerical solutions of Euler equations by using a new flux vector splitting scheme.International Journal for Numerical Methods in Fluids 17:10.1002/fld.v17:2, 115-144. [CrossRef]

260. GERARD WELCH, RODRICK CHIMATwo-dimensional CFD modeling of wave rotor flow dynamics . [Citation][PDF] [PDF Plus]

261. P. DURGAPAL, GRANT PALMERStrongly coupled radiative transfer and Joule heating in the cathodeof an archeater . [Citation] [PDF] [PDF Plus]

262. THOMAS ROBERTS, GARY WARRENAnalysis of implicit second-order upwind-biased stencils . [Citation] [PDF][PDF Plus]

263. J.