NASA Technical Memorandum 86776 

NASA-TM-8677619850025803 

Use of Hyperbolic Grid Generation' 

Scheme in Simulating Supersonic 

Viscous Flow About Three ... 

Dimensional Winged 

Configurations 

Yehia M. Rizk, Joseph L. Steger and Denny S. Chaussee 

July 1985 

NI\S/\ 

National Aeronautics and 

Space Administration 

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11111111111111111111111111111111 111111111 1111 

NF00039 
NASA Technical Memorandum 86776 

lJse of Hyperbolic Grid Generation 

Scheme in Simulating Supersonic 

Viscous Flow About Three-

[)imensional Winged 

(~onfigurations 

Yehia M. Rizk, Informatics General Corporation, Palo Alto, California 

Joseph L. Steger 

Denny S. Chausse, Ames Research Center, Moffett Field, California 

JUly 1985 

NJ\S/\ 

National Aeronautics and 

Space Administration 

Ames Research Center 

Moffett Field, California 94035 
USE OF A HYPERBOLIC GRID GENERATION SCHEME 

IN SIMULATING SUPERSONIC VISCOUS FLOW ABOUT 

THREE-DIMENSIONAL WINGED CONFIGURATIONS 

YEHIA M. RIZK 

INFORMATICS GENERAL CORP., PALO ALTO, CALIFORNIA, 94303. 

JOSEPH L. STEGER AND DENNY S. CHAUSSEE 

NASA AMES RESEARCH CENTER, MOFFETT FIELD, CALIFORNIA, 94035. 

ABSTRACT 

The objectives of this paper is 1 )to describe an efficient numerical mesh generation 

technique capable of creating single grids around complex winged configurations, 

2)use the above grid generation scheme to simulate the flow field around a realistic three dimensional configuration by solving the Reynolds averaged Navier-Stokes 

equations using a time dependent implicit procedure. l'umerical results are presented for the flow around the complete configuration of the shuttle orbiter and a 

canard projectile at supersonic Mach numbers and zero angle of attack. 

INTRODUCTION 

The present paper describes a numerical mesh generation t.echnique (Ref. i 1 j) and 

an implicit finite-difference procedure (Refs. [2,3]) for simulating viscous flow about 

low- aspect-ratio wing body configurations using a single grid strategy. Meshes are 

generated for complex three-dimensional winged configurations using equations of 

mesh line orthogonality and cell volume which form a hyperbolic partial differential 

equation set. The grid generation equations are readily solved in the computational 

domain by marching outward from the body surface. In supersonic flow, the threedimensional Reynolds-averaged Navier Stokes equations are efficiently solved by 

segmenting the computational domain into multiple regions([3]).This is possible 

because the outflow boundary of each segment can be located in supersonic areas 

to avoid upstream influence between two contiguous regions. 

The numerical procedures have been used to compute the viscous high Reynolds 

number flow around a complete configuration of the shuttle orbiter (including the 

canopy. wing. OMS pods and vertical tail) and a canard projectile at supersonic 

Mach numbers and zero angle of attack. In the following sections, we briefly describe 

the grid generation procedure. the flow simulation procedure, and the computed 

results. 

NUMERICAL GRID GENERATION PROCEDURE 

Although, there are various grid generation techniques available, (see e.g. [4]), 

the hyperbolic grid generation method (11],[5]) was selected in the present applications because it has several desirable features. With the hyperbolic grid generation 
method, the user is able to specify an arbitrary surface point distribution, and the 

numerically generated grid is orthogonal or is as close to being orthogonal as possible. The hyperbolic grid generation procedure is also efficient in terms of computer 

time and memory. 

Grid Generation Equations 

Body-fitted curvilinear coordinates of ~ =~ (x, y, z), 'r/ = 'r/ (x, y, z), and ~ = ~ 

(x, y, z), where ( is the outward direction, are sought to simplify the application 

of finite-difference viscous flow field simulation procedures. Following Ref. [1], two 

orthogonality relations and a cell volume constraint are used as grid generation 

equations to provide x, y and z in terms of e, 'r/ and (. The resulting equations are: 

XeX~ + YeY\ + zezi = 0 

x 17 x\ + YI7Yi + zl7 z\ = 0 

xeY.,.,z\ + xiYe z.,., + x.,.,Yize - xey\zl7 - x 17 Ye z\ - x\Y.,.,ze = ~v 

or with rdefined as (x,y,z)t 

re· - r- 0 \= 

r.,.,. ri = 0 

1

8 (x,y,z) II = J- 1 = ~V 18(e,'r/,~) 

(la) 

(1b) 

(lc) 

(2a) 

(2b) 

(2c) 

The first two equations represent orthogonality relations between ~ and ~ lines and 

between 'r/ and ( lines, whereas the last equation is the cell volume or the finite 

Jacobian constraint. 

Numerical Solution Algorithm 

The nonlinear syst.em of grid generation equations given by Eqs. (1) are linearized 

about a known stat.e ro, and are solved using a noniterative implicit finite-difference 

scheme. An unconditionally stable implicit scheme is chosen so that the marching 

step size in <; can be arbitrarily selected based only on considerations of accurately 

generating the grid. An iterative solution of the nonlinear grid generation equations 

is avoided by expanding the equations about the previous marching step. As a 

consequence, the resulting equations are solved with the nearby known state 0 

taken from the previous ~ step. 

Let ~e = ~'f/ = ~~ = 1 such that ~ = J - 1, 'r/ = k - 1, and ~ = 1- 1. Central 

spatial differencing of Equations (1) in ~ and 'r/ with two-point backward implicit 

differencing in ~ leads to 

(3) 

where I is the identity matrix, and Az, BI and CI are 3 x 3 coefficient matrices that 

are formed from the local linearization of Eqn. (2). The vector 91+1 contains the 

user specified cell volumes 

2 
The central difference operators used in t and T} are given by 

~ _ ~"+l - G-l 

uerJ" = , . 2 

whereas the backward differencing used in ~ is represented as 

To reduce the inversion cost, the difference equations are multiplied through by 

CI

- 1 and are approximately factored as 

(4) 

so that rl+ 1 is obtained by solving sequences of one-dimensional-like block tridiagonal systems 

(1 + CI

- 1 Bib." )g' 1+ 1 = CI

- 1 gl+ 1 

(1 + CI-IAI6E)'~7~rl+l = g'l+l 

(Sa) 

(5b) 

(Sc) 

In practice numerical dissipation terms are added in the € and 1] directions and 

for body shapes that have concavity, super implicitness is built into the algorithm 

following Ref. 161. 

Initial and Boundary Conditions 

The body surface is mapped to the bottom of the computational cube and the 

grid is obtained by marching outward from the body surface. The surface grid 

can be arbitrarly specifed by the user. For relatively simple body shapes, the 

user can generate an orthogonal grid along the body surface. This process assures 

that the resulting grid will be nearly orthogonal in all directions. For complicated 

configurations, however, the surface grid is generated along cross sections of constant 

axial distance. Depending on the body shape, the € and T) lines can be far from 

being orthogonal. However, this does not introduce any inconsistency because the 

orthogonality between the € and 1] is not enforced in the grid generation procedure. 

Boundary conditions in the € and T} directions at each marching step are needed 

to create the grid. Periodicity is always used in the T} (circumferential) direction. A 

stable boundary condition procedure is needed at ~ = 0 and € = €max for creating 

"0-0" meshes around closed bodies{Fig. (la)) and "C-O" or "H-O" meshes around 

3 
open bodies(Fig. (lb, 1c)) As shown in Ref. [1], the eigenvalues of either Gil A or 

CZ

- l B are of the form (a,O,-a) where a is real. Consequently, two stable boundary 

procedures have been used: 1) specify one combination of the dependent variables 

and use one sided differencing for the two remaining governing equations at the 

boundary, or 2)specify two combinations of the dependent variables and use one 

sided differencing for a third governing equation at the boundary. The first approach 

is used at a boundary surface (e.g. ~ = ~max in Figs. (lb, IC)) whereas the second 

approach is used at an axis (e.g. ~ = ° in Figs. (la, lb)). 

Since, the grid is obtained by marching outward from the body, an exact location 

for the outer boundary can not be prescribed. Nevertheless, an approximate location 

for the outer boundary can be specified through proper selection of the cell volumes 

used in Eqn. (lc). 

FLOW SIMULATION PROCEDURE 

The main objective of the present work is to demonstrate an efficient numerical 

procedure for computing three-dimensional viscous flow over winged configurations 

(e.g., the shuttle orbiter) at supersonic free-stream Mach numbers. Although the 

flow is predominantly supersonic, some sections of the geometry generate local regions of embedded subsonic flow and/or axial separation. In such regions. the ftow 

equations are elliptic, precluding the use of the more efficient space-marching techniques such as the parabolized ~avier-St,okes (PNS) codes (e.g. Ref. 7). Instead. 

a time-dependent continuation procedure is used to find the steady-state solution 

of the three-dimensional conservation-law form of the Reynolds-averaged ~avierStokes equations with the thin layer approximation. 

The numerical procedure employs the vectorized Beam-Warming, approximate 

factorization, finite-difference scheme as presented in Refs. [2,3J• The algorithm is 

implicit. non-iterative and uses second or fourth-order accurate central-differencing 

in the spatial directions. A generalized coordinate transformation is employed to 

map the three- dimensional region between the body and the outer boundary into a 

cubical computational domain. The grid points are clustered in the direction normal 

to the body surface in such a manner as to adequately resolve both the viscous and 

the inviscid regions. The algebraic model of Ref. [8] is used for turbulent computations. The bow shock can either be fitted or captured while internal discontinuities 

are always captured. Calculations are performed using bilateral symmetry with the 

computational domain extending from the windvv'ard to the leeward side. 

In order to facilitate the solution of the Navier-Stokes equations around complex 

bodies, a segmentation procedure in which the ftow field is divided into multiple 

regions is used. The boundary between two adjacent regions is carefully located in 

areas where the outerinviscid flow is supersonic and the boundary layer is attached. 

This ensures the absence of any upstream influence between different regions and 

hence eliminates the otherwise costly interfacing between neighboring regions. Segmentation has been implemented in the current application to reduce the size of 

4 
the time dependent computational domain. Moreover, segmentation allows the use 

of the more efficient space marching procedure in any region not containing an 

embedded subsonic or axially separated flow. 

RESULTS 

The current numerical technique has been used to simulate the flow field around 

the entire configuration of the shuttle orbiter and a simple projectile with a canard. 

Shuttle Orbiter 

The supersonic viscous flow about the shuttle orbiter has been computed using 

the four computational segments or regions shown in Fig. (2). The bow shock 

segment, Region I, is treated as a typical blunt body problem with supersonic 

outflow and is solved using a time-relaxation code. Region II is centered about 

the canopy and' is also solved with the time dependent code because the canopy 

shock induces a small region of subsonic flow. Region III is purely supersonic and 

could be solved by space marching, but for ease of application the more robust 

time dependent code was again utilized. Finally, Region IV containing the wing, 

the OMS pods, and vertical tail has embedded regions of subsonic flow as well as 

regions of axial separation and must be solved using the time dependent code. In 

Region IV it is particularly important that the generated mesh be body normal, 

both in the cross-sectional and axial planes in order to resolve the viscous flow over 

different components of the orbiter which are inclined at relatively steep angles to 

the fuselage. 

The hyperbolic grid generation procedure was used to generate the grids in Regions II, III and IV, whereas an algebraic procedure was used for the simple nose 

geometry. Figure (3) indicates the 25 x 123 X 29 grid generated for Region IV in 

which there are 25 points in the axial direction, 123 points in the circumferential 

direction, and 29 points in the normal direction. Projections of either 'I} = constant 

or onto ( = consiant grid planes onto either y = constant or x = constant planes 

are used to illustrate the grid. Figure (3a), for example, shows a segment of the 

grid near the beginning of the vertical tail in the 'f) constant leeward plane of 

symmetry. Figures (3b) to (3d) show various ~ = constant grid planes projected 

into the y - z plane. It is seen that the grid lines are normal to the body surface. 

The pressure contours in a cross section containing the wing, OMS pods and 

vertical tail are shown in Fig. (4). A side view of the simulated oil flow (i.e. 

limiting streamlines) at the body surface are shown in Fig. (5). It is seen that there 

is an area of axially separated flow upstream of the vertical tail and the OMS pods. 

The location of the separation line coincides with a line along which the limiting 

streamlines coalesce while the reattachment line coincides with a line at which the 

limiting streamlines diverge. 

The computations were performed on the CRAY-XMP-12 with one central processor and two millions words of memory. The code is vectorized and the metrics 

are not stored but are computed when needed. The computer time required per 

5 
grid point is 0.00012 sec per iteration. 

Canard Projectile 

The numerical procedure was also used to compute the flowfield around a projectile with a canard at a Mach number of 4 and zero angle of attack. The geometry 

(Fig. (6)) consists of an 8 degree blunted cone followed by a cylinder. A canard 

protrudes from the cylinder at a plane 90 degrees from the pitch plane of symmetry. 

The hyperbolic grid generation scheme was used t.o create the t.hree- dimensional 

mesh required for the viscous computations. Various views of the grid are depicted 

in Fig. (7). It is seen that the grid is body normal in both the streamwise and 

meridional directions. It is also seen that a clustering is employed in the outward 

direction in order to resolve the viscous effects near the body surface. In addition, 

a simple streamwise clustering is introduced to adequately resolve the flow features 

in the wake of the canard. 

In generat.ing the grid, the cell volumes were specified such that the outer boundary is nearly conical with a half cone angle of approximately 20 degrees. This assures 

that the bow shock is located well inside the outer boundary because the shock angle 

corresponding to an 8 degree cone at Mach number 4 is about 18 degrees. 

The above mesh was then partitioned into three longitudinal segments. The 

first segment contains the spherical nose. The second segment contains the conical 

section and the front portion of the canard. The third segment contains the back 

end of the canard and the cylindrical portion. An axisymmetric blunt body code 

was used to compute the flow field in the first segment. The flow field in the second 

and third segments were obtained using the time relaxation procedure of Ref. :3'. 

Because the body is at zero angle of attack, the computational domain was confined 

to the region between 4;= 0 and 4;= 90 degrees. The results of the calculations in 

the second segment revealed the absence of any embedded subsonic flow bubbles 

or axially separated flow regions. This means that a PNS code could have been 

utilized in this region. 

The results of the computations in the third segment are shown in Figs. (8-12). 

The Mach contours in the qJ= 90 degrees symmetry plane (canard plane) are shown 

in Fig. (8). It is seen that there is a relatively large subsonic bubble residing in 

the canard wake. The velocity vectors in the canard plane are shown in Fig. (9). 

The regions of axial flow separation and reattachment are clearly visible from the 

corresponding limiting streamlines depicted in Fig. (10). The pressure con toms and 

velocity vectors in a streamwise stations corresponding to the canard full extent and 

in the cylindrical portion are shown in Figs. (11, 12) and (13, 14) respectively. The 

above computations were obtained using 88125 grid points on the CRAY-XMP12. The computer time per time step is approximately 13 seconds. A total of 800 

iterations were required for convergence. 

SUMMARY 

The present paper described a numerical mesh generation technique to be used 

6 
with an implicit finite-difference method for simulating viscous supersonic flow 

about low-aspect-ratio wing body configurations using a single grid strategy. The 

computational domain was segmented into multiple regions, with borders located in 

supersonic areas to avoid the otherwise costly interfacing procedure between adjacent segments. The numerical procedure has been successfully applied to calculate 

thE~ turbulent flow around the shuttle orbiter and a canard projectile at supersonic 

free-stream Mach number. 

REFERENCES 

l)Steger, J.L., and Rizk, Y.M.,"Generation of Three Dimensional Body Fitted 

Coordinates Using Hyperbolic Partial Differential Equations." NASA TM 86753, 

June, 1985 (also AFWAL-TR 83-3129, Feb., 1984) 

2) Pulliam, T.H., and Steger, J.L, "On Implicit Finite-Difference Simulations of 

Three Dimensional Flow," AIAA Paper No. 78-10, Huntsville, Ala., Jan., 1978. 

3) Rizk, Y.M., and Ben-shmuel, S.,"Computation of the Viscous Flow Around the 

Shuttle Orbiter at Low Supersonic Speeds." , AIAA Paper 85-0168, Reno, Nevada, 

Jan., 1985. 

4)Thompson, J.F.,"Numerical Grid Generation", Proceedings of a Symposium 

on the Numerical Generation of Curvilinear Coordinate Systems and Their Use in 

the Numerical Solution of Partial Differential Equations.1982, Nashville, Tennesee. 

5)Steger, J.L., and Chaussee, D.S.," Generation of Body Fitted Coordinate Using 

Hyperbolic Partial Differential Equations," SIAM Journal Sci. Stat. Comput., Vol. 

1, No.4, Dec. 1980, pp. 431-437 

6)Kinsey, D.W., and Barth, T.J. "Description of a Hyperbolic Grid Generation 

Procedure for Arbitrary Two-Dimensional Bodies", AFWAL Technical Memorandum No.84-191-FIMM (July, 1984) 

'l) Schiff, L.B., and Steger, J.L., " Numerical Simulation of Steady Supersonic 

Viscous Flow," AIAA Journal, Vol.18, Dec. 1980., pp 1421-1430. 

8) Baldwin, B.S., and Lomax, H.,"Thin Layer Approximation and Algebriac 

Model for Separated Turbulent Flows," AIAA Paper No. 78-257, Huntsville, Ala, 

1978. 

7 
AXIS 

~ = 0 

0-0 MESH (SPHERICAL LIKE) 

BOUNDARY SURFACE 

~ = ~ max 

AXIS 

~ = ~ max 

--AXIS 

~=O 

C-O MESH (SEMI-SPH_ERICAL) 

Fig. 1 Mesh topologies around closed and open 

bodies. 

8 
BOUNDARY SURFACE 

~ = ~ max / 

BOUNDARY SURFACE 

~=O 

~ 

H-O MESH (CYLINDRICAL LIKE) 

Fig. 1 continued. Fig. 2 Flow regions around the shuttle orbiter. 

650 300 

250 

550 200 

150 

450 100 

Z 400 Z 50 

350 0 

300 -50 

250 -100 

200 -150 ----- "----- --

150 -200 

700 800 900 1000 1100 1200 0 100 200 300 400 500 

X Y 

Fig. 3a Grid in the leeward plane of symmetry. Fig. 3b Projection of the grid on an axis normal 

plane containing the strake. 

9 
300 

250 

200 

150 

100 

Z 50 

0 

-50 

-100 

-150 

-200 

0 100 200 300 400 500 

Y 

Fig. 3c Projection of the grid on an axis normal 

plane containing the wing. 

300 

250 

200 

150 

100 

Z 50 

0 

-50 

= -100 ---

-150 

-200 

0 100 200 300 400 500 

Y 

Fig. 4 Pressure contours in a cross section near 

the end of region IV. 

10 

300 

250 

200 

150 

100 

Z 50 

0 

-~O 

-100 

-150 

-200 

0 100 200 300 400 500 

Y 

Fig. 3d Project ion of the grid on an axis normal 

plane containing the wing, OMS pods and vertical 

tail. 

300 

250 

200 ~-:-::~ 

150 --~..:"----

.--~-... ~ 

100 '-~. 

Z 50 

0 

-50 ' . . ~ ",e-

-100 

-150 

-200 

700 800 900 1000 1100 1200 

.x 

Fig. 5 Side view of the oil flow (limiting streamlines) in region IV. 
16,-----------------------------

12 

y 8 

4 

2 4 6 8 10 12 14 16 

X 

Fig. 6 Flow regions around a projectile with a Fig. 7a Grid in the canard plane of symmetry. 

canard. 

16 2.00 

14 1.75 

12 1.50 

10 1.25 - .. 

Z 8 Z 1.00 

6 .75 

4 .50 

2 .25 

0 

16 12 8 4 0 .50 1.00 1.50 2.00 

X Y 

Fig. 7b Grid in the leeward plane of symmetry. Fig. 7c Grid in a cross section containing the 

canard. 

11 
7.-----------,---------, 

6 

5 

4 

V 

3 

,~.,'--+-- 2 - ?O _' __ . 

1.6 . __ ~. 

I.S ._~ __ 

"I.,~ 

o~--~--~--~--~----------~ ·8 9 10 11 12 13 14 15 

X 

Fig. 8 Mach number contours in the canard symmetry plane. 

3.0 r---__ ~_-------___, 

2.0 

V 1.6 

1.0 

.5 

O+---~--~----~--~--~-~ 

9.0 9.5 10.0 10.5 11.0 11.5 12.0 

X 

Fig. 9 Velocity vectors in the c.anard plane of 

symmetry. 

Fig. 10 Limiting streamlines (simulated oil flow) 

" in the canard region. 

12 
2,,5 

2.0 

Z 1.5 

1.0 

o .5 1.0 1 .. 5 

Y 

2.0 2.5 3.0 

Fig. 11 Pressure cont.ours in a cross section corresponding to the full extent of the canard. 

:31.0 -- I.r~ 

- - - --- ! 02- __ 

I.n; 1,1'-' J.OO~ ____ _ 2.5 ------0.96 __ 

2.0 

Z 1.5 

1.0 

o .5 1.0 1.5 2.0 2.5 3.0 

y 

Fig. 13 Pressure contours In a cross section In 

the wake of the canard. 

3.0 

2.5 

2.0 

Z 1.5 

1.0 

.5 

o .5 1.0 

, , 

1.5 

Y 

2.0 2.5 3.0 

Fig. 12 Velocity vectors in a cross section corresponding to the full ext.ent of the canard. 

13 

3.0 11111111, 'I 

2.5 

2.0 1111111111"", 

Z 1.5 

o .5 

I I" 

'" 

""" '. '. 

1.0 

" 

'" '. 

" "1, 

". 

, 

1.5 

Y 

'. 

" 

, , 

, 

, 

" 

" 

2.0 2.5 3.0 

Fig. 14 Velocity vectors in a cross section In the 

wake of the canard. 
1. Report No. 2. Government Accession No. 3. Recipient's Catelog No. 

NASA TM- ~h77h 

4. Title and Subtitle 5. Report Date 

USE OF A HYPERBOLIC GRID GENERATION SCHEME IN July 1985 

SIMULATING SUPERSONIC VISCOUS FLOW ABOUT THREE- 6. Performing Organization Code 

DIMENSIONAL WINGED CONFI~nR'<\'TION 

7. Authorls) Yehia H. Rizk (Informatics General Corp., 8. Performing Organization Report No. 

Palo Alto, Calif.) , Joseph L. Steger, and 85344 

Dennis S Chaussee 10. Work Unit No. 

9. Performing Organization Name and Address 

Ames Research Center 11. Contract or Grant No. 

Moffett Field, CA 94035 

13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address Technical Memorandum 

National Aeronautics and Space Administration 14. Sponsoring Agency Code 

Washington, DC 20546 505-31-01 

15, Su pplementary Notes 

Point of contact: Yehia M. Rizk, Ames Research Center, MS N202A-14, 

Moffett Field, CA 94035 (415) 694-6329 or FTS 464-6329 

16. Abstract 

The present paper describes a numerical mesh generation technique to 

be used with an implicit finite difference method for simulating viscous 

supersonic flow about low-aspect-ratio wing body configurations using a 

single grid strategy. The computational domain is segmented into multiple 

regions, with borders located in supersonic areas to avoid the otherwise 

costly interfacing procedure between adjacent segments. The numerical 

procedure is applied to calculate the turbulent flow around the shuttle 

orbiter and a canard projectile at supersonic free stream Mach number. 

, 

17. Key Words ISuggested by Author!s)) 1 B. Distribution Statement 

Computational fluid dynamics Unlimited 

Grid generation 

Three-dimensional viscous flow Subject category: 02 

19. Security Classif. lof this report) 20. Security Classif. (of this page) 21. No. of Pages I 22. Price" 

Unclassified Unclassified 15 I A02 

"For sale by the National Technical Information Service, Springfield, Virginia 22161 
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