Direct Replacement of Arbitrary Grid-Overlap- ping by ...

NASA Technical ICOMP-94-07

Memorandum

106601

Direct Replacement of Arbitrary ping by Non-Structured Grid

Kai-Hsiung Kao Institute for Computational Lewis Research Center Cleveland,

Mechanics

Grid-Overlap-

in propulsion

Ohio

and Meng-Sing Liou -Lewis Research Cente.; Cleveland, Ohio

(NASA-TM-106601) REPLACEMENT

OF

DIRECT ARBITRARY

GRID-OVERLAPPING GRID 17 p

(NASA.

BY Lewis

N94-33120

NON-STRUCTURED Research

Center)

Unclas

G3/64

May 1994

National Aeronautics and Space Administration

0009999

Direct

Replacement of Arbitrary by Non-Structured

Grid-Overlapping Grid

by Kai-Hsiung Kao Institute for Computational Mechanics in Propulsion NASA Lewis Research Center, Cleveland, OH 44135 and Meng-Sing Liou Fluid Mechanics Division

Internal NASA

Lewis

Research

Center,

Cleveland,

OH 44135

Abstract A new tared

approach

regions

posite

of embedded

overlapping

regions

which

Delatmay DRAGON

can accommodate region

so that

generates

of embedded

code based

the compressible

grids.

the

Navier-Stokes

that

(1) eliminating

ping accelerates

convergence Numerical

shock

waves

mesh

finite volume,

has been schemes.

the

are demonstrated.

steady

present

and unsteady

into

In addition, wraps

over

approach,

difficulties

a the

termed

of the Chimera

stencils, (2) making grid into three dimensions is scheme

to include

state problems,

based on a Courant-Friedrichs-Leury tests on representative

com-

is subdivided

which

high resolution

further developed For steady

mesh

some

su'uc-

uses the Chimera

confgurafion.

triangular

It is designed

overlapped

of the flowfield

grid for complex

advantages:

equations

arbitrarily

methodology

domain

non-structured

on a time accurate,

grid and the non-structured

with strong

The present

the physical

easily-generated

technique

to replace

as the orphan points and/or bad quality of interpolation in a fully conservative way, and (3) implementation

straightforward. A computer

limit.

mesh

grids is presented. system

grid, has three important

scheme, such communication

stability

mesh

triangtdation

interconnecting

overset

that uses non-slructured

(CFL)

f6r solving

both the Chimera the local

number

supersonic

near

time stepthe local

inviscid

flows

I. Introduction During

the last decade

and applied

complex

geometry

while

in various

methods

tures is generally tured grids

classed

to be used

structured

structured

into the composite

without

excessive

versatile

geometry

grid catagory.

or inefficient

geometry

Overset

to

efficient

approaches

grids

use of grid density.

used to compute inviscid and high Reynolds flow about complex onstrated for unsteady three dimensional viscous flow problem#

to adapt

to use more

both pure-strain complex

there

imposes

grid schemes.

and easier

are considered However,

to resolve

structured

distortion

grid concepts,

in which

and unstructured

to be more

grids

been developed

have

various

fields

grid schemes

memory.

uses overset

systems

Among flow

grid methods

less computer

their strengths and weaknesses. The Chimera grid scheme I which

grid

CFD problems.

are considered

composite

and require

unstructured

for computing

- composite

grid

algorithms

and

approaches

conditions

unstructured

numerical have

two mainstream

boundary

In general,

structured

for the computations

are currently complex

both

or flow feaallow

struc-

It has

been

configurations, 2"3and even demin which one body moves with

respect to another. Overset grids have also been used as a solution adaptation found that the Chimera approach has more versatility than current unstructured

procedure, schemes.

e_ It is

Up to date, the Chimera method is an outgrowth of trying to generalize a powerful solution approach to more complex situations. However, it is also recognized that there are weakness which must be removed if Chimera is to remain competitive with unstructured grids. There ate two main

criticisms

complexity many

leveled

against

the current

implementations

is perhaps

as difficult

of the intereormectivity

times resulting

in orphan points

and/or

of the Chimera

as dealing

with

bad quality of interpolation

method:

(1) the

an unstructured

stencils

grid,

left in the com-

putational domain, and (2) nonconservative interpolations to update interface boundaries are often used in practical cases. The fact that interlX)lation is generally used to connect grids implies that conservation is not strictly enforced. For some practical applications, such as the turbine-blade cooling passage flow in which heat transfer is involved and the supersonic shock waves travel in time, the interpolation errors become overall-significant are not fully conserved. It is remarked that while dures because idea

of Benek

boundaries has not schemes

weights ficult

and

ensures

simulations

robustness,

Calspan/AEDC of delta

space-time

condition a main

shuttle

use interpolants

conservation

yet been rigourously for overset grids have

conservation However,

of their

the space

drawback

for irregular

over

interface

grid cells.

and not straightforward. A fairly obvious way to ensure

used simple have

the global

where fluxes proce-

an unpublished

this quantity

field, but the penalty

on interface

of this approach

addressed). Many conservative interface recently, Wang and Yang 9 transformed the

treatment

conservation

methods

Implementation

interface

interpolation

implemented

Interpolating

grids to one of enforcing

for those

polygonal

simulations quantities.

examined (or clearly been devised. 7"8 More

for overlapped

have

inlet design if numerical

is in determination

into three

conservation

would

for patched

dimensions

of the area

is obviously

be to introduce

zones. dif-

an unstructured

flow solver in the vicinity of the interface boundaries. Hybrid schemes which incorporate the best featuresof both structuredand unstructured have already appeared t_n and will likelybecome more important in treating flow about complex geometries. sented which uses non-structuredmesh to replace the overset This

approach

Chimera

grid

not only eliminates scheme.

The

the above

extension

of the

criticisms present

2

In this paper a new approach is preregion for Chimera embedded grids.

but also method

preseves

to three

the advantages

dimensions

of the

is obviously

straightforward.

II. Structured Chimera

Grid

Scheme

The Chimera method ment

Usually,

boundaries

required

decomposition.

to provide

the flexibility

the means

Schemes

a major

which

provides

grid so as to resolve

are overset

on top of the major

grids

way. However,

of matching

a common

the solutions

of subdomains,

across

the Chimera

a conceptually

grid is generated

on the major

in any special

in the selection

technique

For instance,

the minor

to join

Grid

Grid System

grids are then overset

configuration. mesh

- Structured

scheme is a grid embedding

for domain and minor

and Unstructured

interesting grid

cle-

of the

requiring

region

interfaces.

also allows

simple

body

features

without

or overlap

boundary

scheme

about a main

the

is always To increase

an implementation

to remove regions of a mesh containing an embedded grid from that mesh. That is, an embedded mesh introduces a "hole" into the mesh in which it is embedded. Typically, a hole is defined through

a creation

boundary

which

consists

of a surface

or a group

of surfaces.

The purpose

of a

hole creation boundary is to identify points that are within this boundary. A mesh point is considered to be inside a hole creation boundary ff it is inside all surfaces that define the boundary. Figure

1 illustrates

blanked

the

connections

between

creation

boundary.

by a prescribed

interpolation For instance,

of boundary the scheme

composite

overlapping

Communication

grids,

between

with

overlaid

hole grids

points

being

is achieved

by

values from the mesh or meshes in which the boundaries are contained. employs nonconservative trilinear interpolation that as some simple

experiments have mera grid system

shown is superior to Taylor series expansion. A practical application of the Chifor complete Space Shuttle/Solid Rocket Booster geometry is shown in Fig. 2.

Details

the grid embedding

regarding

Figure

1. Interconnections

technique

of minor

can be found

and major

(Ref. 12)

3

in Ref.

grids for Chimera

12.

overset

grid scheme.

A flow

solver

must

be modified

to account

for the use of multiple

meshes

and the "hole"

in

the grids. These hole points must be blanked or excluded from the flow field solution. The main change in the flow algorithm itself is the treatment of the hole boundaries. The hole information from the Chimera grid package is stored in an array, IBLANK, which is defined for each point on each grid as

IBLANK

In the flow value

solver,

before

each

=

element

the solutions

{ 11 0,

in the correcter

The blanked

solutions

tine or logic

tests are required

DRAGON The

Grid

- Hybrid

application

two embedded generated

are updated

2. Overlapped

grids,

in step

(1)

step is multiplied

by the appropriate

IBLANK

axe updated:

Un+ 1 = un+

Figure

ifapointisnotblanked; if a point is blanked.

IBLANK"

in the interpolation

to exclude

the blanked

grids for the Shuttle

Grid

routine. points

launch

With

from

vehicle

this approach,

no special

rou-

the flow field solution.

configuration.

(Ref. 3)

System

of the Chimera

scheme

and (2) execution

1. The PEGSUS

(2)

(U n+ 1 _ U n)

code

involves

two steps:

of a flow solver

is used to determine

4

(1) data communication

that uses the communication the interpolation

between information

coefficients

between

compositegrids. It

is now modified to persue the DRAGON grid technique. The approach of the grid is accomplished by using non-structured mesh in the vicinity of the interface to replace the arbitrarily overlapped structured regions. Descriptions regarding the

DRAGON boundaries

procedures to adopt the DRAGON grid method are summerized as follows: (1) The entire domain of a minor grid is used to form the hole creation (2) Grid points (3) Overset

inside

of the hole creation

grid meshes

the hole creation

boundary

are no longer as well

ical fluxes or primitive quantities (4) Both fringe points (hole treated

as blanked

values

are saved

points. points

Instead,

as the hole boundary

which

triangulation

adopt

(1) Boundary ric coordinates.

(s)

scheme

the triangular nodes

cells

points

are now forming

of unstructured grid generation is used for output of PEGSUS

to perform

in combination

of numerare no longer

and their IBLANK

the new

boundaries

structured

grid blocks, to cover

code

are reordered

I

i

grid generation.

grid scheme

according

m

I

iiI-.

u

m

m

replacement

(a) Chimera

files

a computa-

"U]-I

I

of arbitrary

grid method,

I I

grid-overlapping

(b) DRAGON

by non-structured

grid method

The steps

to their geomet-

iJ

m

3. Direct

output

are summarized.

E

Figure

for

will be addressed later in code which contains the

the unstruettm_

with Chimera

by the PEGSUS

|"i-"_-'l

by

by arbitrary embedded grids implies an irregular shape of to be represented by structured grids. The well-known

13 is applied

provided

points

points,

are deleted from the PEGSUS code. shaped cell is tile S_mpiest geometric shape

created difficult

is defined

In result, interpolations

as interior

of the unstructured region. process is not performed between

providing interpolation information It is known that the triangular

Delaunay

points.

grid.

gap region

code.

as the boundary

the unstructured grid generator. Details this section. A file named DRAGON

tional domain. A gap region domain and would be very

out in the major

a new generated

they are now represented

as well

boundary mesh coordinates (6) Since interpolation

are blanked

Instead,

are no longer performed. (Fig. 3(a)) boundary points) and interpolation boundary

to be 1 in the PEGSUS

(5) The fringe

boundary

exist.

boundary.

grid.

(2) Delaunaytriangulationmethodis thenperformedto connectthe boundarynodesbasedon the Bowyer algorithm.14 (3) Trianglesinterior to bodiesor exteriorto the domainaredeleted. (4) Interior nodesare insertedin the eircumcenterof triangleswhich are deemedto be too largebasedon the boundaryspacing and a clustering parameter (which governs the rate of change of triangle At

area). present,

This step completes no

further

improvement techniques, be used to achieve better domain

by using

Figure

With

an unslructured grid indices

inserting

or

deleting

triangulation

triangular

meshes

grid, individual as in a structured

method

methods

is shown

for multi-body

cells

grid. have

grid. Instead,

edges)

- 3 edges

IEDGECEIL(1:2,

edges)

- 2 neighboring

edges)

Mesh

refer

to their

a connectivity

Loellbaeh)

neighboring

matrix

cells

would

sim-

be used to

between structured and are summerized as fol-

- 2 nodes of each edge

cells)

IEDGETYPE(edges)

performed.

length sides, etc., earl about the multi-body

of a cell. This connectivity matrix usually conbased information. With the present approach,

ICELLEDGE(1:3,

IEDGEFLUX(grids,

been

(by Dr. James

matrices are also formed to define the diseretized geometry grids. The connectivity matrices used in the present approach

lowing: .......... IEDGENODE(I:2,

(Fig. 3(b))

in Fig. 4.

domain.

can no longer

identify the relative position of the neighborhood tains the cell based information as well as edge connectivity unstructured

of the initial

such as grid smoothing, aspect ratio control, equal grid quality. An example of a coarse grid system

the Delaunay

4. Delaunay

ply by using

grid

the specification

of each triangular

cells of each edge

- edge

type (type of boundary

- edge

number

6

cen

that connects

condition) structured

and unstru-

IFLUXINDX(grids,

ctured grids -/-index of structured

edges)

with unstructured IFLUXINDY(grids,

edges)

Equations both sm_ctured

Euler equations

EqUations

and unstructured

can be expressed

the edge fluxes

mesh

-j-index of structured cell that shares with unstructured mesh

!11. Governing Governing Considering

cell that shares

and Solver

form in a volume

fluxes

Algorithm

grid systems, the time-dependent

in an integral

the edge

compressible

v with enclosing

surface

s as fol-

low

,

where

the conservative

variable

,

(3)

U is

(4)

u = [p,pa,pE]r

the inviscid

flux F T F =

and the equation

[p_, p_

+pl, pHi]

(5)

of state for ideal gas

p = (7-1)pc

(6)

Here e represents the internal energy. The vector quantities, expressed in terms dinates, are denoted with an overhead arrow, and the tensor with an overhead

of Cartesian coortilde or a dyadic

notationsuch as _la. Flux

Splitting Based on the finite

that the cell-centered

volume

conserved

method, variables

the governing are constant

equation within

is semi-diseretized a cell,

by assuming

and that the flux integral

at

ceil surface is also approximated by an average value of the numerical flux and the surface length. The numerical formulation uses a new class of flux spatting scheme. Accordingly, the name of the scheme

is coined

as Adveetion

full flux into convective

Upstream

and pressure

Splitting

fluxes,

Method

(AUSM))

s The

_11/2 and F_x/2 respectively,

scheme

first splits the

F1/2

scalar

variables

• -

(p, p_, pH)

cl/2 that is constructed in a similar state, i.e., "L" or "R", of the variables the following

= Cl/2dPL/R

1 L < _ < R. The interface pressure, appropriate domains of de_.ndence

at the cell interface negative data from The passive

= FCl/2 + Fel/2

+ P1/2

P1/2, simply comprises the positive and via characteristic speed decomposition.

is transported

fashion as P1/2" to be convected.

(7)

by a common

Then the As such,

convective

upwind idea the interface

velocity

is used to select the flux can be recast in

form:

(8)

where A1/2 ( ) = ( ) R - ( ) L" Here the first term on the fight hand side is clearly not a simple average of the "L" and"R" fluxes, but rather a weighted average via the convective velocity. Time

Integration An integration scheme assembles fluxes entering the cell interfaces and updates the conservafive variables at the cell center. The present method originates from the Taylor series expansion in time,

as was done in the Lax-Wendroff

racy is given

scheme.

A two-step

scheme

with second

order time

accu-

as:

predictor: Dun

u* =

(9)

correcfor."

U**

= U* +At_

U*

= 1 (u . + u'*)

0o)

2

The residual

_U

vanishes

the mmaerieald_tability no need

of defining

the transport

terms

as the solution

condition. a midpoint

in the general

Note

approaches

that both predictor

for the corrector curvi_ear

Local Time Stepping ................. For steady state cases, l_ time-stepp_g ing the solution

at each

cell in time

a steady

state. This

and corrector

step. This reduces

scheme

is res.tneted

steps are tdenueal,

the complexity

by with

of evaluating

coordinates.

accelerates

convergence

at a Courant-Friedricks-Leury

8

:....... to steady state by advanc(CFL)

number

near

the local

stability

limit.

Thus,

the expression

for the local time step at each grid point

At

where V is the cell volume,

= CFL

a is the local

. V/

(_, rl, _),

(IU1 + a) S

speed

of sound,

(11)

and S is the projected

area in _, 1"!,and

directions. Data

Communication For

Through

multi-block

according

grid system,

to different

Grid

Interface

the flow field

body geometry

is usually

decomposed

or flow features. However,

into

separated

regions

the success of the overlaid grid

or patched grid techniques relies on correct communication among grid Mocks through boundary interfaces. Since the interface treatment methods are not necessarily satisfying any form of a conservative

constraint,

may result

in instability

of overlaid grids. In the eun'ent center

the

scheme.

solutions

on overlaid

or oscillations,

work,

both

The rectangles

grids

especially

the structured and triangles

are often

when

unstructured

based

on the

conditions

a shock

and unstructured are the control

mulation. Figure 5 illustrates the space discretizations unstructured meshes. In the finite volume formulation at the cell interface

mismatched

flow

passes

solvers

volumes

through were

other.

This

boundaries

based

on a cell

used in the finite volume

for-

of the volume cells for both structured and (Eqn. 3), numerical fluxes will be evaluated

of neighboring

grid, the edge flux ,fk, of the interface

wave

with each

cells

(L and R cells

will be evaluated

using

in Fig.

the stuctured

5). For

cell value

as the right and the unstructured cell value as the left. On the other hand, for structured grid, interface fluxes which have been evaluated in the unstructured process are applied directly

the in

computation

has

of the cell volume

residuals,

i.e. J_-lr_j = f,: It is noted

been performed at any interface over the entire multi-block DRAGON grid approach automatically preserves both the local further

implementation

into three

dimensions

Unstructured Grid .. Control Volume

would

=. Structured/Unstmctured Interface

Figure

5. Control

volumes

for flux calculations.

domain. Obviously, the present and global conservation law, and

be straightforward.

Grid Control Volume

that no flux interpolation

IV. Tested Computational unsteady problems. problem.

Results

and Discussion

results are presented in this section for two-dimensional, inviscid, steady and Test eases include supersonic flow past a circular cylinder and the shock tube

Numerical

solutions

have

been

obtained

grid, (2) Chimera grid, and (3) DRAGON order-accurate version of the flux splitting

based

on the following

grid systems:

grid. Integration of the equations of motion scheme in both structured and unstructured

(1) single uses a firstregions.

Case 1. Supersonic Flow over a Circular Cylinder The first problem solved was that of a circular cylinder in a supersonic free stream (M.= with the associated bow shock. Three different mesh systems are employed. A single grid 6(a))

with

resolved domain

61 × 51 using

tion method major

beyond formed generates includes

Chimera

and the minor

laid meshes the

points

is first overset

and

being

at the grid

minor

grid

to provide scheme

grid (61 × 31) wraps

with hole points is used

used

grids.

blanked

interface

With

an analytic

while

around

in the major which

allows

the DRAGON

the

major

the cylinder

grid

solution. grid

meshes to fill in this region and 616 triangular cells.

(,,)

Figure

(b)

approach,

the

6. Computational

grid systems

field

is also

covers shows

entire

the over-

bi-linear

interpola-

to communicate

between

blanked

hole

is extended

and a new gap region is is then performed which

(Fig. 6(d)). The new generated

(¢)

6(b)

grid. Noneonservative fluid proporties

flow

(41 × 101)

body. Figure

the outer boundary of the minor grid, as shown in Fig. 6(c), between two embedded grids. Delaunay triangulation method triangular 390 nodes

The

2.5) (Fig.

unstructured

((r)

for supersonic flow over circular

cylinder.

(M. = 2.5) (a)singlegrid,(b)Chimera grid,(c)hole pointsblanked in DRAGON (d)DRAGON

grid

I0

grid,

mesh

The values

of the dependent

variables

at all the grid points

are set equal to their

free-stream

values initially. tion converged

The equations of motion with different grid systems are integrated until the soluto its steady-state value. Figure 7(a) shows the pressure contours obtained at con-

vergence

the single

with

Chimera

grid

showing

a comparable

shock As

wave

noted

grid

mesh.

and the DRAGON

in Figs.

in Fig.

shock

Numerical

grid

location

solutions

schemes,

as displayed

to that predicted

7(b) and 7(c) are seen to a slighter

7(b),

both mismatched

the vicinity of the grid interfaces. For the DRAGON grid approach,

countour both

are also

lines

structured

calculated

in Figs.

7(b)

with the single extent

simply

grid

based and

small

in fact,

gaps

those

interrupting

lines seem

the contour

center values ate displayed at their cell vertex locations

to have

oscillations

and unstructured

lines are observed

in the structured ....

(a)

grid region

slope

along while

results

across

unstructured

grid density.

7. Pressure P,,_=6.18, (a) single

contours

package on an the intersection

boundaries grid values

(c)

for Supersonic

flow over a circular

cylinder.

AP=0.28) grid, (b) Chimera

grid, and (c) DRAGON

11

grid.

in

can be displayed

\

Figure

captured

are observed

the interfaces.

the interface

(b)

The

due to a coarser

and pressure

a continuous

7(c) respectively,

system.

on the same plot using the Flow Analysis Software Toolkit ui (FAST) visualization IRIS workstation. As plotted in Fig. 7(c), the countour lines smoothly across boundaries;

on both the

(P,,,i,,=0.71,

Note

that

as the cell are ploted

Figure 8 compares the pressure distributions along the centerline of the flow field. It is shown that, for this test problem, both Chimera and DRAGON grid methods perform very accurate prediction for shock location and strength.

8.0

Stagnation

Point

Bow Shock 6.0

4.0

Single Grid O Chtmem Gdd • DRAGON Grid

2.0

I

OO.

1.0

I

m

I

I

2.0

1.5

I

2.5

i

I

3.0

x (center line)

Figure

8.Pressure

distributions

along the ocnterline of the flow field. (M.

= 2.5)

Case 2. Shock Tube Problem It is of great concern that inaccurate shock speed and strength may be captured if the numerical scheme is not fully conserved. An attempt is to test the time-dependent, plane moving shock problem on the embedded grid system. The shock wave is moving into a quiescent region in a constant-area channel with a designed shock speed M s = 4. Like the first test case, the computational domain is composed using the single grid, the Chimera grid as well as the DRAGON grid schemes. As shown in Fig. 9(a), the single grid system containing a grid dimension of g01 x 21 is used to provide an analytic solution for shock location. Figure 9(b) illustrates the Chimera grid system which includes a major grid (upstream region) of size 41 x 21, and a minor grid (downstream region) of dimension 781 x 21. The grid lines between two overlaid grids are embedded in any arbitrary orientation. Upon the replacement of grid-overlapping by non-structured grid, the DRAGON grid for the moving shock problem is shown in Fig. 9(c). Note that all three grid systems apply equivalent grid densities in the overall regions, except for that near the interface boundaries. It is designed that the initial shock position is marked in the upstream region. (Fig. 9) The initial conditions for the flow field are assumed to have the left and fight values as PL = 4.57

|

PR = 1

uL = 3.125 PL vL == 013.21

__

uR = 0 PR vR == 00.71

12

(a)

i

|j

_tl|li|*llamlm|lanlJHl|llul

ll||ll

ii|il|llnnlp

I|llm

nlml|llnn

t Initial Shock Position Figure

9. Computational grid systems (a) single grid, Co) Chimera

for moving shock problem. (M s = 4) grid, and (c) DRAGON grid.

Interface Boundaries

1

(' Figure

10. Pressure

I I ! I

contours

for moving

shock problem.

(500 iterations,

At=0.1)

14

i9

_"" "-

" - --"" " 5""

" - "---"-- " _

0

_

4

-1

S;_Id

WDAA_ !

180

I

!

190

I

m

200

I

210

n

Qd_ I

220

m

230

x-d/_nce

Figure

H. Comparison

of pressure

distributions

13

along the centefline

of the channel.

Figure 10 displays the pressure contours with various grid systems at a downstream location after the shock wave passing through the interface boundaries. It is hardly to identify any difference on the contour lines between three grid systems. However, a critical comparison of the pressure distributions

along

scheme

a faster

approach

predicts

accurately

in a fully conservative

the centerline moving

captures

of the channel, shock

the shock

in the speed

as plotted

tube.

which

As

in Fig.

a result,

ensures

11, shows the

that the Chimera

present

the numerical

DRAGON

solution

grid

is preserved

nature.

V. Conclusion

tured

In this paper we presented a new approach, termed the DRAGON mesh to replace the arbitrarily overlapped structured regions

designed

to further

conservative and unsteady interface

boundaries.

the DRAGON

enhance

the flexibility

grid communication problems indicated Future plans

for Chimera

overlapping

grid, that uses non-strucof embedded grids. It is

grid

technique

between embedded grids. Numerical results that the fluid proporties have been accm'ately include

grid are now ongoing

unstructured

grid improving

and to enforce

obtained for steady carried through the

and 3D implementation

of

and will soon be reported.

Acknownledgements The first author would

like to thank

Dr. James

Loellbach

for his valuable

discussions

and pro-

viding the main coding structures for Delaunay tria gulafion and the unstructured flow solver. The computational results for the test problems are obtained using the CRAY-YMP computer system at NASA Lewis Research Center and partly calculated on the C90 at NASA Ames Research Center.

References

IBenek,

J. A., Baning,

P. G. and Steger,

J. L., "A 3-D Chimera

Grid Embedding

Teelmique,"

AIAA

Paper 85-1523, 1985. _Stegcr, J. L. and Benek, J. A., "On the Use of Composite Grid Schemes in Computational Aerodynamics" Corny Methods Awl. Mech. and Eng., Vol. 64, Nos. 1-3, 1987. 'I_XSA A_ Sp'ace ShuttleFlo_v Simulation Group: P. G. Buning, L T. Chiu, F.. W..Martin Jr., R. LaMeakin, S. Obayashi, Y. M. Rizk, J. L. Steger and M. Yarrow, 'Flowfield S1mulauon or the Space Shuttle Vehicle in ascent," Proceedings of the Fourth Internation Conference on Supercom_MeakinUting, Santa Clara, CA, (1989). . , R. L. and Suhs, N., 'Unsteaay Aerodynamic Simulation of Multiple Bodies in Relative Motion," AIAA 9th Computational Fluid Dynamics Conference. 5Kao, IC H., I.,iou, M. S. and Chow, C. Y., "Grid Adaptation Using Chimera Composite Overl_oin_ Meshes," AIAA Paver 93-3389, Orlando, FL, July 1993. -'-" 6l_er, M. J. and Oliger, J., "Adaptive Mesh Refinements for Hyperbolic Partial Differential Equa_tions," J. Comp. Physics, Vol. 53, 1987. 7Merger, NL J., "On Conservation at Grid Interfaces," ICASE Report Research Center, Hampton, Vh'ginia, 1984. 8Moon, Y. J. and Liou, M.-S., "Conservative Treatment of Boundary

14

84-43, Interfaces

NASA

Langley

for Overlaid

Grids andMulti-level Grid Adaptations,"AIAA Paper$9-1980-CP, 9Wang, Z. J. and Yang,

H. Q., "A Unified

Conservative

1989. Zonal Interface

Treatment

for Arbi-

traril_toNPatched and Overlapped Grids," AIAA Paper 94-0320, January 1994. akahashi, IC, 'FDM-FEM Zonal Approach for Computations of Compressible Viscous Flows," 10th International Conference on Numerical Methods in Fluid Dynamics, Beijing, China, June, 1986. nSoetrisno, M., Imlay, S. T. and Roberts, D. W., "A Zonal Implicit Procedure for Hybrid Strucmre_-Unstrucmred Grids," IAA Paper 94-0645, January 1994. t2Suhs, N. E. and Tramel, R. W., "PEGSUS 4.0 User's Manual," AEDC-TR-91-8, Calspan Corpg. ration/AEDC Operations, November 1991. 13Hase, J. E., Anderson, D. A. and Patpia, I., "A Delaunay Triangulation Method and Euler Solver for Bodies in Relative Motion," AIAA-91-1590-CP, Hawaii, June 1991. _4Bowyer, A., "Computing Dirichlet Tess_llations," The Computer J., VoL 24, No. 2, pp. 162166, 1981. lSLiou, M. S., "On a New Class of Flux Splitting," 13th International Conference on Numericol Methods for Fluid Dynamics, Rome, Italy, July 1992. l°Walatka, 1:).P., Clucas, J., McCab¢, R. IC, Plessel, T. and Potter, R.," FAST User Guide," NASA Ames Research Center, RND-93-010, July 1993.

15

REPORT

DOCUMENTATION

Form Approved OMBNo. 0704-0188

PAGE

Public reporting burden forthiscollection ofinformation isestimated toaverage 1 hourperresponse, including thetimeforreviewing instructions, searching existing datasources, gather ngandmaintaining thedataneededandcompleting andrevtew{ng thecollection ofinforrnatiom Se. ndc_'nmants regarding thisburden estimate oranyotheraspect ofthis conection ofinformation, including suggestions forreducing thisburden, toWashington Headquarters _ervices, Dtrectorate forInformation Operations andReports, 1215Jefferson Davis Highway, Suite1204,Arlington, VA 22202.4302, andtotheOffice ofManagement andBudget, Paperwork Reduction Project (0704-0188), Washington, IX: 20503, 1. AGENCYUSE ONLY(Leaveblank)

2. REPORTDATE May 1994

3. REPORTTYPE AND DATESCOVERED Technical Memorandum 5. FUNDINGNUMBERS

4._TITLEANDSUBTITLE Direct

Replacement

of Arbitrary

Grid-Overlapping

by Non-Structured

Grid WU-505-90-5K

6. AUTHOR(S) Kai-Hsiung

Kao and Meng-Sing

Liou 8. PERFORMINGORGANIZATION REPORTNUMBER

7. PERFORMINGORGANIZATIONNAME(S)AND ADDRESS(ES) National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135- 3191

E-8875

10. SPONSORING/MON_ORING AGENCYREPORTNUMBER

9: SPONSORING/MONITORINGAGENCYNAME(S)AND ADDRESS(ES) _,T National Aeronautics and Space Administration Washington, D.C. 20546-0001

NASATM-106601 ICOMP-94-7

11. SUPPLEMENTARYNOTES Meng-Sing Liou, NASA Lewis Research Center and Kai-Hsiung Kao, Institute for Computational Mechanics in Propulsion (work funded by NASA Cooperative Agreement NCC3-233). ICOMP Program Director, Louis A. Povinelli, organization code 2600, (216) 433-5818. 12b. DISTRIBUTIONCODE 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified Subject

- Unlimited

Category

64

13. ABSTRACT(Maximum200 words) A new approach

that uses non-structured

mesh to replace

the arbitrarily

overlapped

structured

regions

of embedded

grids is presented. The present methodology uses the Chimera composite overlapping mesh system so that the physical domain of the flowfield is subdivided into regions which can accommodate easily-generated grid for complex configuration. In addition, a Delaunay triangulation technique generates non-structured triangular mesh which wraps over the interconnecting region of embedded grids. It is designed that the present approach, termed DRAGON grid, has three important advantages: (1) eliminating some difficulties of the Chimera scheme, such as the orphan points and/or bad quality of interpolation stencils, (2) making grid communication in a fully conservative way, and (3) implementation into three dimensions is straightforward. A computer code based on a time accurate, finite volume, high resolution scheme for solving the compressible Navier-Stokes equations has been further developed to include both the Chimera overset grid and the non-structured mesh schemes. For steady state problems, the local time stepping accelerates convergence based on a Courant-Friedrichs-Leury (CFL) number near the local stability limit. Numerical tests on representative steady and unsteady supersonic inviscid flows with strong shock waves are demonstrated.

15. NUMBEROF PAGES 17 16. PRICECODE A03

14. SUBJECTTERMS Overset

grid; Hybrid

grid; Supersonic

17. SECURITYCLASSIFICATION OF REPORT Unclassified NSN 7540-01-280-5500

flow

18. SECURITYCLASSIFICATION OF THIS PAGE Unclassified

19. SECURITYCLASSIRCATION OF ABSTRACT Unclassified

20. LIMITATIONOF AB_HACT

Standard Form 298 (Rev. 2-89) Prescribed byANSIStd.Z39-18 298-102