VOL. 18, NO. 2, FEBRUARY 1980

ARTICLE NO. 78-10R

AIAA JOURNAL 159

Implicit Finite-Difference Simulations of Three-DimensionalCompressible Flow

Thomas H. Pulliam* and Joseph L. Steger*NASA Ames Research Center, Moffett Field, Calif.

An implicit finite-difference procedure for unsteady three-dimensional flow capable of handling arbitrarygeometry through the use of general coordinate transformations is described. Viscous effects are optionallyincorporated with a "thin-layer" approximation of the Navier-Stokes equations. An implicit approximatefactorization technique is employed so that the small grid sizes required for spatial accuracy and viscousresolution do not impose stringent stability limitations. Results obtained from the program include transonicinviscid or viscous solutions about simple body configurations. Comparisons with existing theories and ex-periments are made. Numerical accuracy and the effect of three-dimensional coordinate singularities are alsodiscussed.

I. Introduction

A COMBINATION of general curvilinear coordinatetransformations, well-ordered grids, and an implicit

algorithm is used here to construct a versatile three-dimensional program for unsteady or steady inviscid andviscous compressible flow. While we generally look to thefuture for computers with more speed and storage, thecomputers of today can calculate meaningful numericalsolutions about relatively simple three-dimensional bodyshapes. The need to progress to three dimensions on currentcomputers is twofold: I) to demonstrate to the user com-munity that complex geometries and body motions will notpresent any great difficulty to routine solution given com-puters one or two orders of magnitude more powerful thanwhat is available today, and 2) to provide simple, three-dimensional flowfield solutions to test current and futureturbulence models.

The three-dimensional program described hereindemonstrates one technique that can be applied to solveflowfields about simple (present application) and complicated(future applications) aerodynamic shapes. General trans-formations1'4 valid for any body configuration or grid systemare used. An implicit approximate factorization method5"8 ischosen chiefly for its unconditional linear stability property,as well as for its compatibility with the vectorized computerprocessors9 that seem to be the trend of the future. Theoverall algorithm constructed here is both simple and flexible;its extension to other problems and computers should bereasonably straightforward.

In Sec. II the equations, generalized transformations, andboundary conditions are presented, along with a "thin-layer"viscous approximation. The numerical algorithm is describedin Sec. Ill, while the geometry and grid systems are discussedin Sec. IV. Results and their discussion appear in Sec. V.

Presented as Paper 78-10 at the AIAA 16th Aerospace SciencesMeeting, Huntsville, Ala., Jan. 16-18, 1978; submitted June 5, 1978;revision received Aug. 6, 1979. Copyright © American Institute ofAeronautics and Astronautics, Inc., 1978. All rights reserved.Reprints of this article may be ordered from AIAA SpecialPublications, 1290 Avenue of the Americas, New York, N.Y. 10019.Order by Article No. at top of page. Member price $2.00 each,nonmember, $3.00. Remittance must accompany order.

Index categories: Computational Methods; Viscous Nonboundary-Layer Flows.

*Research Scientist; presently with Flow Simulations Inc., Sun-nyvale, Calif. Member AIAA.

II. Transformed EquationsTo enhance numerical accuracy and efficiency, coordinate

mappings of the governing equations are employed whichbring all body surfaces onto coordinate surfaces, and clustergrid points in flowfield regions where the dependent variablesare expected to undergo rapid changes of gradient. In thetransformed plane, uniform discretization formulas and well-ordered interior grid point solution algorithms can be used.The equations can also be written in strong conservation lawform for shock-capturing purposes and to avoid possibleweak instability from source terms. These considerations ledto the general transformed equations with the additionalsimplifications that are discussed in this section. Related workusing comparable transforms in flowfield applications can befound in Refs. 10-15.

A. Equations in Nondimensional FormIf inertial Cartesian velocity components are retained as

dependent variables, the three-dimensional, unsteady Navier-Stokes equations can be transformed to the arbitrary cur-vilinear space £,r/,f,T shown in Fig. la, while retaining strongconservation law form (see Refs. 1-4). The resulting trans-formed equations are not much more complicated than theoriginal Cartesian set, and can be written in nondimensionalform as

(i)where

ppu

pv

pw

e

pU

F=J

pV

pwV+Kizp

lK-7/ , /7

PW

(2)

160 T. H. PULLIAM AND J. L. STEGER AIAA JOURNAL

*- — *-

= £(x,y,z,t)= 77(x,y,z,t)- f(x.y,z.t)= t

^^

\

\

^-____

^

_ __

a) PHYSICAL DOMAIN COMPUTATIONAL DOMAIN

(INERTIAL)

f = CONSTANTBODYSURFACE

b)Fig. 1 Transformations and body coordinate.

and

W, ) +2fJLVv

(3y =

(3z =

~1dxeI + urxx + vrxy + wrxz

~IdyeI + UTvx + vry

~ 1 dzej -f wr.^ + VTZV

-0.5(u2 + v2 + w2) (5)

Here it is understood that the Cartesian derivatives are to beexpanded in £,i?,f space via chain-rule relations such as

and

V—TI(

(3)

where U, K, and W are contravariant velocities writtenwithout metric normalization (see, for instance, Ref. 16).(Note that other suggestive forms of the equations can bewritten; for example,

where (U — XT), etc. are relative Cartesian velocity com-ponents.)

The viscous flux terms are given by

0

rtxTxx+rlyTxy+rlz'rxz

(4)

GV=J

The Cartesian velocity components u, v, w are non-dimensionalized with respect to a^ (the freestream speed ofsound), density p is referenced to p^; and total energy e top^a2^ . Pressure is defined as

p= (y-l)[e-0.5p(u2 + v2 + w 2 ) ] (6)

and throughout 7 is the ratio of specific heats. Also, K is thecoefficient of thermal conductivity, /* is the dynamic viscosity,while X from the Stokes' hypothesis is - 2/V. The Reynoldsnumber is Re and the Prandtl number is Pr.

Finally, the metric terms are obtained from chain-ruleexpansion of x^, y^ etc., and solved for £v , £ v , etc., to give

=-XTrlx-yTrly-ZTriz

(7)

and

B. Thin-Layer ApproximationIn high Reynolds number flows, one usually has only

enough grid points to resolve viscous terms in a thin layer nearrigid boundaries. Typically, grid lines are clustered near abody and resolution along the body is similar to what isneeded in inviscid flow. Even though the full Navier-Stokesequations may be programmed, viscous derivatives along thebody are not resolved in general unless the streamwise andcircumferential grid spacings are sufficiently small, in manycases of 0(Re 'l/2) based on the effective viscosity coefficient.Consequently, a thin-layer approximation is used; all viscousderivatives in the £ and r? direction (along the body) areneglected, while terms in fare retained and the body surface ismapped onto f = const (see Figs. Ib and 2). Equation (1) thus

FEBRUARY 1980

simplifies to

where

IMPLICIT THREE-DIMENSIONAL COMPRESSIBLE FLOW

(8)

(9)

In the thin-layer model, a boundary-layer-like coordinate isadopted and the viscous terms that are dropped in boundary-layer theory are eliminated. It should be stressed that the thin-layer approximation is valid only for high Reynolds numberflows and that very large turbulent viscosity coefficients couldconceivably invalidate the model.

As noted earlier, the thin-layer model requires a boundary-layer-type coordinate system. To use this model near theintersection of a vertical and horizontal wall, for example,would require treating both walls as a single continuoussurface which is mapped into the same f= const plane. Ifthese conditions are violated, the neglected viscous termsshould and can be added to the present numerical procedure(Refs.6, 8,22, and 23).

The concept of a thin-layer viscous model has been em-ployed by others, notably, Cheng,17 Rubin and Lin,18 Lubardand Helliwell,19 McDonald and Briley,20 and Davis,21 withinthe framework of the parabolized Navier-Stokes equations orthe thin shock-layer approximations. Because these aresometimes space-marching procedures, we avoid this ter-minology here; that is, "parabolized Navier-Stokes." Inspace-marching methods, additional restrictions must beplaced on the streamwise (i.e., £) convection terms and theirboundary conditions, whereas, such restrictions are not madehere. Consequently, unlike the space-marching procedures,the present treatment of the unsteady equations, Eq. (8), isvalid for streamwise as well as crossflow separation and doesnot require additional approximation in subsonic flowregions.

C. Surface Boundary ConditionsThe tangency condition along the surface $(x,y,z,t) = const

for inviscid flow is that W= 0 and is used in

161

(11)where n is the normal direction to the body surface. Forviscous flows, the same relation is used with U— K=0. All ofthe preceding boundary conditions are valid for steady orunsteady body motion.

III. Numerical MethodThe finite-difference scheme is the implicit approximate

factorization algorithm used in the delta form described byBeam and Warming.7'8 An implicit method was chosen toavoid restrictive stability conditions which occur when smallgrid spacing is used. Highly refined grids are needed to obtainspatial accuracy and resolution of large gradients such asoccur in calculating viscous effects. Small grid sizes may alsooccur because nonoptimal mappings overly concentratepoints in a given region. Implicit methods are useful inavoiding stiffness in problems in which the solution is forced.In such cases, time steps that are large compared to thosedemanded by an explicit stability limit can often be takenwithout degradation of accuracy.

The basic algorithm is first- or second-order accurate intime and is noniterative. The equations are factored (spatiallysplit), which reduces the solution process to three one-dimensional problems at a given time level. Central-differenceoperators are employed and the algorithm produces blocktridiagonal systems for each space coordinate. The stabilityand accuracy of the numerical algorithm are described indetail by Warming and Beam.22 Linear analysis of thenumerical scheme shows that it is unconditionally stable,although in actual practice, the nonlinear equations aresubject to a time-step limitation. The limitation, though, isusually much less stringent than what is found for con-ventional explicit schemes. The numerical scheme can be first-or second-order accurate in time (for the steady-state casespresented here, first-order accuracy is employed), withsecond- or fourth-order accuracy in space.

A. Approximate Factorizations and LinearizationsThe finite-difference algorithm applied to Eq. (8) results in

the following approximate factorization

-qn}

(12)

V-r,, (10)

to obtain u,v, and w. For viscous flow, U= V=Q are enforcedin Eq. (10) as well.

A relation for pressure along the body surface is obtainedfrom a normal momentum relation (found by combining thethree transformed momentum equations)

where the <5's are central-difference operators, and A and Vare forward and backward-difference operators, e.g.,A f# = <7(£ , r / , f+Af) -<7(£,Tj , f ) . Indices denoting spatiallocation are suppressed for convenience and h — tcorresponds to Euler implicit first-order and h = At/2 totrapezoidal second-order time accuracy.

The Jacobian matrices A\ Bn, and Cn are obtained inthe time linearization of En, Fn, and G" and can be written as

162 T. H. PULLIAM AND J. L. STEGER AIAA JOURNAL

A,B, or C =

K 0

K2<t>2-vd

K3<t>2-w0

d[2<j>2->y(e/p)]

K0+6-KI(y-2)u

K}v-K2(y-l)u

KjW-K3(y-l)u

-(y-l)ud]

K2u-(y-l)K1v

K0+6-K2(y-2)v

K2w-K3(y-l)v

[K2[y(e/p)-<t>2]

-(y-l)vd]

K3u-(y-l)Klw Kj(y-l)

K3v-(y-l)K2w K2(y-l)

K0 + 6-K3(y-2)w K3(y-l)

[K3[y(e/p)-<i>2] K0+y6

-(y-l)wO]

(13)

where

and, for example, to obtain A,

= o.5 (y -1) (u2 + v2 + w2)

The viscous vector Sn + 1 is linearized by Taylor series as in Ref. 13, producing the coefficient matrix

0

m21

m3l

m41

m51

0 0

^52 m53

0

m

0

0

0

0

(14)

with

m2I=c

m,, = (.

-u/p)

-u/p)

-u/p)

-u2 Ip}

-v2/p)

- e/p2 )

m= -m4 -

-v/p) +a

-u/p) + a

-v/p) + a

—2uv/p)

-w/p)

-w/p)

—2uw/p)

) /p]

m53 = -ml -

(15)

B. Explicit and Implicit Numerical DissipationFourth-order dissipation terms such as eEJ~! (V^A^ )2Jq

in Eq. (12) are added explicitly to the equations to damp high-frequency growth and thus serve to control nonlinear in-stability. Linear stability analysis shows that the coefficient e£must be less than 1/24. The addition of the implicit second-difference terms operating on (qn + 1-qn) with coefficiente7 extends the linear stability bound of the fourth-order terms.Although linear stability theory based on periodic boundaryconditions shows that the stability-bound eE is again limited,unpublished analysis by Jean-Antoine Desideri of Iowa State

University suggests that unconditional stability can be ob-tained for fixed boundary conditions if e7 is sufficiently large,and numerical experiments confirm this. The presentprocedure is to set eE = At and e/ = 2e£, which results in aconsistent definition of e£; as the time step is increased, theamount of artificial dissipation added relative to the spatialderivatives of convection and diffusion remains constant. Theimplicit smoothing term adds an error Q(AtAx2qxx().

C. Second- and Fourth-Order Accuracy of Convection TermsGenerally, we have used second-order, central-difference

operators for convection and diffusion derivatives. In threedimensions we are forced into somewhat coarse grids due tolimitations of computer storage. For improved accuracy, it isdesirable to use fourth-order accurate convective dif-ferencing, especially in high Reynolds number viscous flow inwhich it can be difficult to keep the convection truncationerror from exceeding the magnitude of the viscous termsthemselves. One possibility is to use fourth-order Fade dif-ferencing for convection terms and second-order Fade dif-ferencing for diffusion terms.23 If the time variation of thesolution is small, however, the same effect is obtained byremaining first-order accurate in time and by using five-point,fourth-order accurate central differencing for the right-hand-side convection terms in Eq. (12). Because fourth-orderdifferences are applied only as an explicit operation, littleadditional work is necessary. One can show that for Eulerimplicit time differencing, the altered algorithm remainsunconditionally stable; it is, however, unconditionally un-stable for trapezoidal differencing.

D. Differencing the Metrics to Maintain the FreestreamThe metric terms £x, rjx, etc., are formed from x,y,z data

using second-order, central-difference approximations of x%,*n, etc. in Eq. (7). In three dimensions, a freestream error canbe introduced, unless, for example, this differencing is donewith special weighted averages (see Ref. 24). However,

FEBRUARY 1980 IMPLICIT THREE-DIMENSIONAL COMPRESSIBLE FLOW 163

calculations have been made on smoothly varying gridswithout attempting to maintain freestream; compared withresults in which freestream is maintained using the averagingprocedure, there was no discernible difference in the solutionon the body.

Perfect maintenance of the freestream can also be achievedby simply subtracting the freestream fluxes from thegoverning equations; that is,

(16)

where the error in the viscous term is neglected since it will besmall for high Reynolds number flow. This procedure isadopted for all subsequent calculations.

E. Implementing Boundary ConditionsUnknown values of q on the boundaries are updated ex-

plicitly. Thus, A<7 is set to zero at the boundaries leading toa first-order error in time. One intuitively expects implicitboundary conditions to be more stable than explicit ones.With our treatment of boundary conditions, however, this hasnot been our experience, at least for the various implicitboundary schemes that have been successfully implemented sofar in two dimensions. In any event, explicit treatment of theboundaries leads to a far more simple and flexible scheme,where boundary conditions become a modular element thatcan be put in or pulled out of a computer program withoutdisturbing the implicit algorithm.

For inviscid flows, values of p, U, and V along the bodysurface are found by linear extrapolation from above, while inviscous cases p is extrapolated and U= K=0. In either case,W= 0 and values of u, v, and w are obtained from Eq. (10).

Surface pressure is obtained by integrating Eq. (11). In Eq.(11) the right-hand side is known from the previous ex-trapolation process, and the basic approximate factorizationalgorithm is applied along the body using backward dif-ferencing in f and central differences in £ and r/. Scalartridiagonals are thus inverted in the £ and ry directions. At thefar field boundaries, freestream values are specified except onthe downstream boundary. There, a simple first-order ex-trapolation is used for M^>\, while extrapolation and thecondition that pressure is fixed at/?^ is used for Mx < 1 .

We remark that a more pleasing boundary treatment wouldresult if the governing equation, Eq. (8), could be differenceddirectly on the boundary surface using inward, one-sided,spatial-difference operators. However, the use of one-sidedspatial differencing is usually a destabilizing process fordiffusion terms, as well as for convection terms whoseJacobian matrices have real eigenvalues of the wrong sign.Consequently, in the present boundary condition treatment,the simple time-independent-like relations of tangency, ex-trapolation, and, for pressure, Eq. (11), were imposed andapplied as described earlier.

IV. Grid SystemsThe flow equations have been written in a generalized

curvilinear coordinate system that can handle arbitrarysurface geometry, including steady and unsteady bodymotions. The only restriction is due to the thin-layer ap-proximation where it is required that the thin-layer coordinatebe f, where f equals a constant taken as the body surface.Simple body motions such as plunge, rotation, and trans-lation can be handled with linear transformations. Morecomplicated motions and distortions of the grid and body canbe introduced as necessary. In our current three-dimensionalapplications, only steady problems have been solved in whichthe body and grid system remain stationary.

Because of limited storage capacity on current computers,only simple bodies and geometries have been used. Figure 2shows the two-grid systems that are used — a warped cyiin-

A

rU'

^^^

D'C. '^f<fi~^-Ay^"'f;

A

.STING/BODY//7^Jv yJ-""11

CROSS SECTIONAL PLANE

CROSS SECTION PLANE

b)Fig. 2 Coordinate systems: a) warped cylindrical coordinate; b)warped spherical coordinate.

drical and a warped spherical coordinate. In each case, thegrid unfolds into a rectangular computational domain withthe body projected onto the lower face. Note that bothsystems have a coordinate singularity (S in Fig. 2) which isunavoidable in mapping a closed three-dimensional bodysurface. The singular line is equivalent to either the r = 0 lineof a cylinder or the polar singularity of a sphere; and, while itslocation is arbitrary, it is placed here at the nose of the bodies.

The warped cylindrical coordinate (Fig. 2a) has thedisadvantage that, as grid lines are clustered near the body foraccuracy and viscous resolution, grid points also clustertoward the singular axis. Points are thus wasted in upstreamregions where changes in the flow gradients are small;whereas in the warped spherical system (Fig. 2b), grid linesclustered to the body do not accumulate toward the singularpole. Furthermore, radial grid lines diverge away from thesingularity, providing a more efficient distribution of pointsin the far field.

While a limiting form of the equations would be needed atthe singular line, the choice of cylindrical coordinates ininviscid flow or spherical coordinates in inviscid or thin-layerflow allows us to avoid the singularity altogether in the finite-difference formulation. This is because, in either case, E andG are identically zero on the singular line, even for a time-deforming grid. (Note that £ X / J = (y^z^-y^) and that r/derivatives of x,y,z are zero because x,y,z do not change as werotate around the axis. Likewise ,£ y /J , £ Z / J , ZX/J, etc., arezero; consequently, from Eq. (2), E and G are zero.)

Results will be shown for both coordinate systems. Simpleshear grids are used about either ogive cylinders or bluntedbodies. Exponential clustering is used in the radial-likedirection to keep grid points reasonably clustered near thebody. For more complex shapes automatic grid generationprocedures (see, e.g., Refs. 25-28) can be used which will alsoproduce smoothly varying grids.

V. ResultsSolutions have been obtained about isolated body con-

figurations, either ogive sting or a hemisphere cylinder, atvarious Mach numbers and angles of attack. A typical inviscid

164 T. H. PULLIAM AND J. L. STEGER AIAA JOURNAL

.04

.08

LEEWARD

-7STING

P/Pcx

TTV

Moo= 1.2, a =2°ELLIPTICITY = 2.0VA NUMERICAL SOLUTION• EXPERIMENT, REF. 29

WINDWARD

Fig. 3 Parabolic arc body with elliptic cross section at M^ = 1.2 anda = 2 deg.

EXPERIMENT, REF. 322nd ORDER \ NUMERICAL COARSE GRID4th ORDER /

- 2ndORDER ) NUMERICAL FINE GRID4th ORDER /

Moo= 1.4 a = 10° INVISCID

I I I

2.0

P/Poo

1.0

2 4 6 8 10X/R

Fig. 5 Hemisphere cylinder, M^ = 1.4, a = 10 deg.

Moo= 1.2,a = 19°Re = 222,500V NUMERICAL SOLUTION, LAMINAR FLOW• EXPERIMENT, REF. 32

= 0° LEEWARD

V

S : SEPARATION POINTR : REATTACHMENT POINT

P/Pcx

Moo = 0.95, a = 0°• EXPERIMENTAL DATA, REF. 30V THREE-DIMENSIONAL INVISCID SOLUTIONO AXISYMMETRIC INVISCID SOLUTION, REF. 31

Fig. 4 Transonic flow about hemisphere cylinder, a = Q deg,A/ =0.95.

2.0

V7

-V

P/Pcx

1.0

!>= 180° WINDWARD

m v*v v • v *v

_L _L0 1 2 3 4 5 6 7

X/R

Fig. 6 Hemisphere cylinder at Mx =1.2 and a = 19 deg.

cylindrical grid consists of 48 points along the body axis—20points radially, and 12 points in the circumferential direction.A restriction of bilateral symmetry is imposed solely to reducethe computational domain and is not a limitation of thealgorithm. The spherical grid usually consists of 30 pointsalong the axis, 30 points in the radial direction, and 12-18circumferential points with bilateral symmetry imposed. Allviscous results were obtained on the hemisphere cylinder atmoderate angles of attack so that crossflow asymmetries do

not invalidate the bilateral symmetry assumption. For theviscous grids, exponential stretching are used to cluster pointsnear the body.

For freestream Mach numbers close to one, an inviscidsteady-state solution is typically obtained in approximately 1h of CPU time on a Control Data Corporation 7600, while 2-3h of CPU time are needed for viscous cases. Note that whileimplicit methods usually imply that large time steps can betaken, it does not automatically follow that fast convergence

FEBRUARY 1980 IMPLICIT THREE-DIMENSIONAL COMPRESSIBLE FLOW 165

to a steady state is achieved. Techniques to improve thealgorithm as a viable relaxation procedure are currently beinginvestigated.

A. Inviscid Flow about Ogive CylinderInviscid flow results for an elliptical cross-sectional body at

MOO = 1.2 and a. = 4 deg are shown in Fig. 3 and are comparedwith experimental data of McDevitt and Taylor.29 Thenumerical and experimental results agree quite well, exceptfor numerical oscillations at the nose that result from theshock-capturing process. Additional inviscid flow results overpointed nose elliptical and circular cross-sectional bodies aregiven in Ref. 24, which show similar good agreement withexperiment.

B. Hemisphere-Cylinder Results

Verification of Accuracy in Inviscid FlowSwitching now to the hemisphere-cylinder cases, a zero

angle of attack inviscid solution at Mx =0.95 is presented inFig. 4. Comparisons are made with the experimental results ofHsieh30 and a numerical calculation by Chaussee using histwo-dimensional inviscid axisymmetric code.31 Bothnumerical solutions use second-order accurate differencingand are in excellent agreement. Agreement with the ex-periment is also quite good.

A comparison between solutions using second- and fourth-order spatial differences for the convection terms is shown inFig. 5 for the leeward side of the hemisphere cylinder atMOO = 1.4 and ct= 10 deg. The numerical solutions are alsocompared with the experimental results of Hsieh.32 Thesecond-order coarse grid results are inaccurate. Althoughrefining the grid improves the results, they are still somewhatunsatisfactory; better results are obtained when fourth-orderdifferences are used on the coarse grid, and excellentagreement is achieved for fourth-order differences on the finegrid.

The results in Fig. 5 represent extremes of the calculations.It has been our experience that adequate resolution can beobtained for simple body shapes and small angles of attack bymeans of second-order differencing. However, for morecomplicated bodies, fourth-order differencing will probablybe needed to obtain good accuracy. The addition of fourth-order accuracy (as previously described for steady-stateproblems) does not substantially increase the computationtime.

= Q° LEEWARD

HEMISPHERE-CYLINDERMoo= 1.2, a = 19°LAMINAR, Re = 222,500X = 7.8 NOSE RADIISP : PRIMARY SEPARATION

POINTSs : SECONDARY SEPARATION

POINT

= 180° WINDWARD

Fig. 7 Velocity vectors in crossflow plane, M^ = 1.2, a = 19 deg.

Viscous Separated Flow at High Incidence and M^ > 1A laminar viscous-flow calculation is presented in Fig. 6 for

a hemisphere cylinder at M^ = 1.2 and a =19 deg. Ex-perimental pressure profiles from Hsieh32 are compared withthe numerical calculations obtained using fourth-order ac-curate differencing. On the leeward side, streamwiseseparation occurs at the nose which was also reported byHsieh.33. Points of streamwise separation and reattachmentpredicted by the numerical calculation are denoted by S and Rin Fig. 6. The boundary layer remains attached on the wind-ward side and numerical results for the windward side are alsoshown in Fig. 6.

The thin-layer model is capable of predicting the leesidecrossflow separation which occurs at this high incidence aswell as the preceding predicted streamwise separation.Crossflow velocity vectors from the numerical calculation areshown in Fig. 1 at a downstream station of approximatelyeight nose radii. Notice the existence of a major recirculationregion on the leeside of the cylinder and also a secondaryseparation region which is seen more clearly in Fig. 8. Thecalculated crossflow separation lines are shown in Fig. 9 andare compared with the experimental data taken from oil flowpictures.33 The calculated primary separation line reproducesthe experimental data in the region between X/R = 4 andX/R = 8. Past this point, the combined effects of coarse-gridresolution and the use of the supersonic-outflow boundary

= 40° i.

HEMISPHERE CYLINDERM00 = 1.2,a=19°LAMINAR, Re = 222,500X = 7.8 NOSE RADIISs = SECONDARY SEPARATION POINT

£ = 70°

Fig. 8 Velocity vectors in crossflow plane, M^ — 1.2, a = 19 deg.

1 = 1.2, a = 19°VA NUMERICAL SOLUTION—— EXPERIMENT, REF. 33

100

80

O 60

<DC<

40

20

V

5 10X/R

Fig. 9 Separation angles.

15

166 T. H. PULLIAM AND J. L. STEGER AIAA JOURNAL

1.8<\

1.4

1

.6

9

0 = 0° LEEWARD M°° °;9) a = 5r Red = 425,000

V NUMERICAL~~ • EXPERIMENT, REF

V

v *? v* v

VV V

- £srVvv V

S T R

1, I i , i 1 <

32

f

i

1.8

1.4V_V

180° WINDWARD

V

S

A__0 1 2 3 4 5

X/R

Fig. 10 Transition calculation on hemisphere cylinder.

condition destroy the accuracy of the calculation. Thecomputed secondary separation line is also compared withexperimental data in Fig. 9.

Viscous Separated Flow at Moderate Incidence and M^ < 1Hsieh33 also carried out an experiment on a hemisphere

cylinder at 5 deg angle of attack in a subsonic freestream,M^ =0.9 and ReD = 425,000. Numerical laminar and fullyturbulent results for this case were presented previously,24 butwere not satisfactory. Leeside streamwise separation wasreported by Hsieh33 and although the laminar numericalresults correctly reproduce this feature, a possibly spuriousunsteady motion of the flow also occurred. In the numericalturbulence calculation, the results lacked streamwiseseparation altogether and closely resembled the inviscidresults.

Hsieh reports in a private communication that the flow isprobably laminar through separation and then transitions toturbulence. To simulate this flow, a transition-turbulencemodel has been implemented in the thin-layer equations.

To simulate transition, the Baldwin-Lomax34 algebraicturbulence model was modified so that the eddy viscositycoefficient is set equal to zero until a specified maximumvalue of eddy viscosity is reached. Downstream of this point,the usual turbulent eddy-viscosity coefficient of the Baldwin-Lomax model is used. For airfoils, Baldwin and Lomax34

suggest a characteristic constant for transition of 14 in at-tached boundary layers; whereas in separated regions, a valueof 500 is more appropriate.

Numerical results (using fourth-order differencing) ob-tained with the transition model are shown in Fig. 10 andcompared with data from Hsieh.32 For the windward side, asmall area of streamwise flow separation occurs, as reportedin the experiment, and the results are in good agreement withthe experiment. On the leeward side, separation occurs as inthe experiment and a steady flow is obtained which agrees

quite well with the experimental pressures. Transition (T)occurs at approximately X/R = 1.5 on the leeside and does notoccur at all on the windward side of the body. Note that theslight recompression, which occurs after separation in Fig. 10(also shown in Fig. 6), is due to the rapid growth of theboundary layer after separation. It should be pointed out thatthese results are rather qualitative and no attempt has beenmade, at this time, to investigate the finer details of theflowfield.

VI. ConclusionsA general purpose, implicit, finite-difference computer

program has been developed to solve compressible unsteadyinviscid or thin-layer viscous three-dimensional flow. Appliedto current computers, the numerical procedure is capable ofproviding reasonably accurate flow simulation to simpleaerodynamic configurations. Solutions of such airplane-likecomponents can be interesting in their own right and can beuseful in testing turbulence models as well. In conjunctionwith automatic generation of highly warped spherical gridsbeing developed simultaneously with the flowfield solver,simulation of flow about complex aerodynamic con-figurations can progress when more powerful computersbecome available.

ReferenceslPeyret, R. and Viviand, H., "Computation of Viscous Com-

pressible Flows Based on the Navier-Stokes Equations," AGARD-AG-212, 1975.

2Lapidus, A., "A Detached Shock Calculation by Second-OrderFinite Differences," Journal of Computational Physics, Vol. 2, 1967,pp. 154-177.

3 Viviand, H., "Conservative Forms of Gas Dynamic Equations,"La Recherche Aerospatiale, No. 1, Jan.-Feb. 1974, pp. 65-68.

4Vinokur, M., "Conservation Equations of Gas Dynamics inCurvilinear Coordinate Systems," Journal of Computational Physics,Vol. 14, Feb. 1974, pp. 105-125.

5Lindemuth, I. and Killeen, J., "Alternating Direction ImplicitTechniques for Two-Dimensional MagnetohydrodynamicCalculations," Journal of Computational Physics, Vol. 13, 1973, pp.181-208.

6Briley, W. F. and McDonald, E., "An Implicit Numerical Methodfor the Multi-Dimensional Compressible Navier-Stokes Equations,"Rept. M911363-6, United Aircraft Research Laboratories, 1973.

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.

8Beam, R. and Warming, R. F., "An Implicit Factored Scheme forthe Compressible Navier-Stokes Equations," AIAA Paper 77-645,June 1977.

9 "Proceedings of NASA Workshop on Future ComputerRequirements for Computational Aerodynamics at the AmesResearch Center," Moffett Field, Calif., Oct. 4-6, NASA CP, 2032,19.

10MacCormack, R. W. and Paullay, A. J., "The Influence of theComputational Mesh on Accuracy for Initial Value Problems withDiscontinuous or Nonunique Solutions," Computers and Fluids, Vol.2, 1974, pp. 339-361.

nRizzi, A., "Transonic Solutions of the Euler Equations by theFinite Volume Method," in Proceedings of Symposium TranssonicumH, K. Oswatitsch and D. Rues, eds., Springer-Verlag, 1976, pp. 567-574.

12Deiwert, G. S., "Numerical Simulation 6f High ReynoldsNumber Transonic Flows," AIAA Journal, Vol. 13, Oct. 1975, pp.1354-1359.

13Steger, J. L., "Implicit Finite Difference Simulation of FlowAbout Arbitrary Geometries with Applications to Airfoils," AIAAPaper 77-665, 1977.

14Lerat, A. and Sides, J., "Numerical Calculation of UnsteadyTransonic Flows," Paper presented at AGARD Meeting of UnsteadyAirloads in Separated and Transonic Flow, Lisbon, April 1977.

15Kutler, P., Chakravarthy, S. R., and Lombard, C. K.,"Supersonic Flow Over Ablated Nosetips Using an Unsteady ImplicitNumerical Procedure," AIAA Paper 78-213, 1978.

16Korn, G. and Korn, T., Mathematical Handbook for Scientistsand Engineers, McGraw-Hill Book Co., Inc., New York, 1961.

FEBRUARY 1980 IMPLICIT THREE-DIMENSIONAL COMPRESSIBLE FLOW 167

17Cheng, H. K., Chen, S. Y., Mobley, R., and Huber, C. R., "TheViscous Hypersonic Slender-Body Problem: A Numerical ApproachBased on a System of Composite Equations," The Rand Corp., SantaMonica, Calif., RM 6193-PR, May 1970.

18Rubin, S. G. and Lin, T. C., "Numerical Methods for Two- andThree-Dimensional Viscous Flow Problems: Application toHypersonic Leading Edge Equations," PIBAL Rept. 71-8 (AFOSR-TR-71-0778), 1971.

19Lubard, S. C. and Helliwell, W. S., "Calculation of the Flow ona Cone at High Angle of Attack," AIAA Journal, Vol. 12, July 1974,pp. 965-974.

20McDonald, H. and Briley, W. R., "Three-DimensionalSupersonic Flow of a Viscous or Inviscid Gas," Journal of Com-putational Physics, Vol. 19, 1975, pp. 150-178.

21Davis, R. T., "Numerical Solution of the Hypersonic ViscousShock-Layer Equations," AIAA Journal, Vol. 8, May 1970, pp. 843-851.

22Warming, R. F. and Beam, R., "On the Construction andApplication of Implicit Factored Schemes for Conservation Laws,"SIAM-AMS Proceedings of the Symposium on Computational FluidMechanics, Vol. 11, New York, 1977.

23Steger, J. L. and Kutler, P., "Implicit Finite-DifferenceProcedures for the Computation of Vortex Waves," AIAA Journal,Vol. 15, April 1977, pp.

24Pulliam, T. H. and Steger, J. L., "On Implicit Finite-DifferenceSimulations of Three-Dimensional Flow," AIAA Paper 78-10, 1978.

25Chu, W. H., "Development of a General Finite DifferenceApproximation for a General Domain," Journal of ComputationalPhysics, Vol. 8, 1971, pp. 392-408.

26Thompson, J. F., Thames, F. C., and Mastin, C. M..,"Automatic Numerical Generation of Body-Fitted CurvilinearCoordinate System for Fields Containing any Number of ArbitraryTwo-Dimensional Bodies," Journal of Computational Physics, Vol.15, 1974, pp. 299-319.

27Thames, F. C., Thompson, J. F., and Mastin, C. M.,"Numerical Solution of the Navier-Stokes Equations for ArbitraryTwo-Dimensional Airfoils," NASA SP-347, Pt. I, March 1975, pp.469-530.

28Ghia, U. and Ghia, K. N., "Numerical Generation of a System ofCurvilinear Coordinates for Turbine Cascade Flow Analysis,"University of Cincinnati Rept. No. AFL 75-4-17, 1975.

29McDevitt, J. B. and Taylor, R. A., "Force and PressureMeasurements at Transonic Speeds for Several Bodies HavingElliptical Cross Sections," NACA TN 4362, 1958.

30Hsieh, T., "Low Supersonic, Three-Dimensional Flow About aHemisphere-Cylinder," AIAA Paper 75-836, June 1975.

31Chaussee, D., "On the Transonic Flow Field SurroundingTangent Ogive Bodies with Emphasis on Nose Drag Calculation,"AIAA Paper 78-212, 1978.

32Hsieh, T., "An Investigation of Separated Flow About aHemisphere-Cylinder at 0- to 90-deg Incidence in the Mach NumberRange from 0.6 to 1.5," AEDC-TR-76-112, July 1976.

33Hsieh, T., "An Investigation of Separated Flows About aHemisphere-Cylinder at Incidence in the Mach Number Range from0.6 to 1.5," AIAA Paper 77-179, Jan. 1977.

34Baldwin, B. S. and Lomax, H., "Thin Layer Approximation andAlgebraic Model for Separated Turbulent Flows," AIAA Paper 78-257, 1978.

From the AIAA Progress in Astronautics and Aeronautics Series..

EXPERIMENTAL DIAGNOSTICSIN COMBUSTION OF SOLIDS—v. 63

Edited by Thomas L. Boggs, Naval Weapons Center, and Ben T. Zinn, Georgia Institute of Technology

The present volume was prepared as a sequel to Volume 53, Experimental Diagnostics in Gas Phase Combustion Systems,published in 1977. Its objective is similar to that of the gas phase combustion volume, namely, to assemble in one place a setof advanced expository treatments of the newest diagnostic methods tha t have emerged in recent years in experementalcombustion research in heterogenous systems and to analyze both the potentials and the shortcomings in ways tha t wouldsuggest directions for fu ture development. The emphasis in the first volume was on homogenous gas phase systems, usuallythe subject of idealized laboratory researches; the emphasis in the present volume is on heterogenous two- or more-phasesystems typical of those encountered in practical combustors.

As remarked in the 1977 volume, the part icular diagnostic methods selected for presentation were largely undeveloped adecade ago. However, these more powerful methods now make possible a deeper and much more detailed understanding ofthe complex processes in combustion than we had thought feasible at that t ime.

Like the previous one, this volume was planned as a means to disseminate the techniques hitherto known only tospecialists to the much broader community of reesearch scientists and development engineers in the combustion field. Webelieve that the articles and the selected references to the current l i t e ra ture contained in the articles wi l l prove useful andst imulat ing.

339pp., 6x9illus., including one four-color plate, $20.00 Mem., $35.00 List

TO ORDER WRITE: Pub l i ca t ions Dept., A I A A , 1290 Avenue of the Americas, New York, N .Y . 10019