... N81-14697 .., .~, A ~

Introducti on:

Component-Adaptive Grid Embedding

E. H. Atta Lockheed-Georgia Company

One of the major problems related to transonic flow prediction

about realistic aircraft configuration is the generation of a suitable

grid which encompasses such configurations. In general, each aircraft

component (wing, fuselage, nacelle) requires a grid system that is usually incompatible with the grid systems of the other components; thus, the implementation of finite-difference methods for such

geometrically-complex configurations ;s a difficult task.

In this presentation a new approach is developed to treat such a problem. The basic idea is to generate different grid systems, each suited for a particular component. Thus, the flow field domain is divided into overlapping subdomains of different topology. These

grid systems are then interfaced with each other in such a way that stability, convergence speed and accuracy are maintained.

157

https://ntrs.nasa.gov/search.jsp?R=19810006183 2020-05-06T20:54:18+00:00Z

Mode 1 :

To evaluate the feasibility of the present approach a two-dimensional model is considered (figure 1). The model consists of a single airfoil embedded in rectangular boundaries, representing an airfoil in a wind tunnel or in free air. The flow field domain is divided into two overlapping subdomains, each covering only a part of the whole field. The

inner subdomain employs a surface-fitted curvilinear grid generated by an

elliptic grid-generator (ref. 1). while the outer subdomain employs a

cartesian grid. The overlap region between the two subdomains is bounded by the outer boundary of the curvilinear grid and the inner boundary of the cartesian grid.

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Figure 1.- Composite grid for an airfoil.

158

I ~ .. ~J ......... "

~roach:

Figure 2 shows the two subdomains (A,B) of the flow field; each has a grid adapted to suit its geometry. The flow in both subdomains is governed by the transonic full-potential equation. While a Neumann-type boundary condition is used at the inner bound­

ary of subdomain B (overlap inner boundary), a Dirichlet-type boundary condition is used at the outer boundary of subdomain A (overlap outer boundary). These boundary conditions are updated

during the solution process. The implicit approximate factoriza-,

tion scheme is used in both grid systems. The code of ref. 1 ;s modified to fit into the present scheme.

The solution process ;s performed in cycles, starting by solving for the flow field in subdomain A, then switching after a number of iterations to solve for the flow field in subdomain B. During each cycle the overlap boundary conditions are updated by

using a two dimensional second order Lagrangian interpolation scheme. This process is then repeated until convergence ;s achieved ;n both su bdoma i ns.

Figure 2.- Grid topology for the different subdomains.

1 S9

160

Comparison with a homogeneous grid:

The results of the present method are compared with the results obtained from using one homogeneous grid for the entire flow field (ref. 1). In all the test cases considered, a standard grid with (31 x 147) points and a circular outer boundary located 6 chord­

lengths away from the airfoil are used. (See figure 3.)

Figure 3.- Uniform grid for an airfoil (ref. 1).

~- Computed Results:

Results of the present method are compared with the results obtained from the code of ref. 1. Two sets of parameters affecting

the performance of the numerical scheme are listed in tables I and

II. Figures 4 and 5 display the pressure-coefficient distributions for a NASA-0012 airfoil resulting from the flow field solutions.

The results are in good agreement for both subcritical and super­

critical cases; savings in computing time are achieved by reducing

the size of the flow field covered by the curvilinear grid

(subdomain A).

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-.4

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Figure 4.- Comparison of pressure coefficient for NACA-0012. (M = 0.75, a = 0.)

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161

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Figure 5.- Comparison of pressure coefficient for NACA-DD12. (Moo = D.8, a = D.)

! U

Flow Field Topology:

The extent of the overlap region between the different grids and

the relative size of each subdomain are the main factors affecting

the accuracy and convergence speed of the present scheme. Figure 6

shows the flow field topology for several test cases. In these cases the overlap extent and subdomain sizes are varied to determine their

optimum va1ues that will minimize the computing effort. while maintain­

ing a reasonab1e accuracy.

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Figure 6.- Flow-field topology with different grid-overlap.

163

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,-,J Overlap arrangement:

Test cases with different grids-arrangement are compared to determine the optimum choice for the extent of the overlap region. A work factor w [number of iterations for convergence x number of grid points (curvilinear grid)] is taken as a measure of the computing effort. Numerical results show that increasing the ex­tent of the overlap region decreases the number of iterations for convergence; however, this also increases the computing effort (figure 7). To minimize the computing time the Cartesian grid

should overlap 15-25% of the curvilinear grid, and the inner

boundary of the Cartesian grid should not be located less than

0.25 chord-length away from the airfoil.

w x 10 5

7 L

"1 Y

.25

R

Figure 7.- Effect of overlap parameters on work factor w. (NACA-0012, Moo = 0.8, a = 0.)

165

166

Computed Results:

The use of nonoptimal parameters for grids arrangement (overlap extent, relative grid sizes) can produce inaccurate results and/or slow down convergence. The Peaky pressure coefficient distribution shown in figure 8 is corrected by increasing the extent of the overlap region described in Table III .

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-.2

-.4

• o

o homoaeneous 0rid

.0 composite grid

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Figure 8.- Comparison of pressure coefficient for NACA-0012. (Moo = 0.75, a = 0.)

Figures 9 and 10 display the pressure coefficient distributions

for two lifting cases for the parameters described in Tables IV and V. The evolution of circulation, and hence lift, is slowed down as

the solution process alternates between the different grids. This

is dealt with by decreasing the number of iterations performed in

each grid.

o homo~eneous 9rid

• • comoosite grid

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• -.4

-.6

Figure 9.- Comparison of pressure coefficient for NACA-0012. (Moo; 0.63, a = 2°.)

167

168

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0 homogeneous grid

• composite qrid

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• o

o

Figure 10.- Comparison of pressure coefficient for NACA-0012. (Moo = 0.75, a = 2°.)

Errors in sonic line position:

Should the shock wave extend into the overlap region, the interpolation process can produce errors in the shock location and strength. Comparisons of the results of the present method with those of a homogeneous grid shows that the maximum relative error did not exceed 1.5%. (See figure 11.)

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.1

o

y

.7

.1

o

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.4

homogeneous grid

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Figure 11.- Effect of interface location on sonic line position. (NACA-0012, Moo = 0.8, a = 0.)

169

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boundary

x I.

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Figure 11.- Concluded.

--' Conclusion:

A method for interfacing grid systems of different topology is developed. This offers a new approach to the problem of transonic flow prediction about mUltiple-component configurations. The method

is implemented in a 2-D domain containing two grid systems of differ­ent topology. The numerical scheme in the present method proved to

be stable and accurate. Savings in computer time and/or storage is

achieved by the proper choice of the overlap region between the differ­

ent grids.

Reference:

1. Holst, T. L., "Implicit Algorithm for the Conservative Transonic Full­Potential Equation Using an Arbitrary Mesh," AIAA J., Vol. 17, No. 10,

October 1979.

171

TABLE I

Code of Ref. 1 Present Method

TAIR Code

Curvilinear grid 31 x 147 15 x 147 21 x 147

Cartesian grid 30 x 30 30 x 30

X cpu time reduc ti on as cOr.1pared to TAIR 30% 10% Code

location of subdomain 6 chord- 6 chord-S ou ter bounda ry length 1 ength

location of subdomain 1 chord- 2 chord-B inner boundary 1 ength 1 ength

location of subdomain 1 c hord- 4 chord-A outer bounda ry length length

number of cycles for 9 10 convergence

TABLE II

Code of Ref. 1

TAIR Code Pre sent Met hod

Curvilinear grid 31 x 147 18 x 147 14 x 147

Ca rtesi an gri d 30 x 30 50 x 50

% cpu time reduction as compared to TAIR 20% 10% Code

location of subdomain 6 chord- 6 chord-B outer boundary 1 ength length

location of subdomain 1 chord- 1/4 chord-S inner boundary 1 ength 1 ength

location in subdomain 2 chord- 1 chord

A outer boundary lellgth length

number of cycles for 12 15

convergence

172

TABLE I II

Code of Ref. 1

TAIR Code Present ~lethod

Curvilinear grid 31 x 147 lOx 147 15 x 147

Cartes ian grid 30 x 30 40 x 40

location of subdomain 6 chord- 6 chord-B outer boundary length length

location of subdomain 1/4 chord- 1/4 chord-B inner boundary length length

location of subdomain 1.5 chord- 1 chord-A outer boundary length length

TABLE IV

Code of Ref. 1

lAIR Code Present Method

Curvilinear grid 31 x 147 15 x 147

Cartesian grid 30 x 40

% cpu time reduction as 39;; compared to lAIR Code

location of subdomain B 6 chord-length outer boundary

location of subdomain B inner boundary 1 chord-length

location of subdomain A 3 chord-length outer boundary

lift coefficient 0.334 0.337

number of cycles for convergence 16

173

TABLE V

Code of Ref. 1

lAIR Code Present Method

Curvilinear grid 31 x 147 21 x 147

Cartesian Grid 30 x 30

% cpu time reduction as compared to TAIR Code 2%

location of subdomain B outer boundary 6 chord -1 ength

location of subdomain B inner boundary 2 chord-length

location of subdomain A outer bounda ry 4 chord-length

lift coefficient 0.574 .584

number of cycles for 14 convergence

174