A practical solution, adaptive-grid method utilizing a tension and torsion spring analogy is proposed for multidimensional fluid flow problems. The tension spring, which connects adjacent grid points to each other, controls grid spacings. The torsion spring, which is attached to each grid node, controls inclinations of coordinate lines and grid skewness. A marching procedure was used that results in a simple tridiagonal system of equations at each coordinate line to determine grid-point distribution. Multidirectional adaptation is achieved by successive applications of one-dimensional adaptation. Examples of applications for axisymmetric afterbody flow fields and two dimensional transonic airfoil flow fields are shown

Deiwert, G. S.

Nakahashi, K.

English

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NASA Technical Memorandum 85989 
NASA_TM-8598919840020676 
A Practical Adaptive-Grid Method 
for Complex Fluid-Flow Problems 
Kazuhiro Nakahashi and George S. Deiwert 
June 1984 
NI\S/\ 
National Aeronautics and 
Space Administration 
LIBRARY COpy 
J:A~GLEY RESEARC' .... CJ:NTER 
LlIORARY, NASA 
HM1PT"N, VIRGINIA 
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https://ntrs.nasa.gov/search.jsp?R=19840020676 2020-03-20T23:03:14+00:00Z
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NASA Technical Memorandum 85989 
A Practical Adaptive-Grid Method 
for Complex Fluid-Flow Problems 
Kazuhlro Nakahashi, 
George S. Delwert, Ames Research Center, Moffett Field, California 
NI\S/\ 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field, California 94035 
I 
A Practical Adaptive-Grid Method 
for Complex Fluid-Flow Problems 
KAZUHIRO N AKAHASHI AND GEORGE S. DEIWERT 
NASA Ames Research Center 
Moffett Field, California 94035, U.S.A. 
INTRODUCTION 
Adaptive-grid methods are an important subject of study in computational fluid dynamics because of their 
potential for improving the efficiency and accuracy of numerical methods. Various papers on grid adaptation 
have been presented in recent years, but few methods have been applied in practical applications. 
In this paper, a practical solution-adaptive-grid method utilizing a tension and torsion spring analogy is 
proposed for multidimensional fluid· flow problems. The tension spring, which connects adjacent grid points 
to each other, controls grid spacings so that clustering is realized in regions containing shock waves and 
shear layers. The torsion spring, which is attached to each grid node, controls inclinations of coordinate lines 
and prevents excessive grid skewness. A marching procedure is used that results in a simple tridiagonal 
system of equations at each coordinate line to determine grid-point distribution. Multidirectional grid 
adaptation to the flow solution is achieved by successive applications of one-diret.tional adaptation. Examples 
of applications for axisymmetric afterbody flow fields and two-dimensional transonic airfoil flow fields are 
shown. 
GRID ADAPTATION IN ONE COORDINATE DIRECTION 
For simplicity of illustration, consider grid adaptation to a flow field in which grid points are free to move 
along each q·coordinate line whose configuration is fixed. Let a grid point A in figure 1 be connected to its 
adjacent points, B and C, by tension springs whose spring constants are K'J-I and K'J' To distribute grid 
points along the q, -coordinate line in proportion to the gradient of selected flow properties, the relationship 
between the spring constant K and the gradient of the dependent variable I 
(1) 
is used, where C1 is a constant and 8s,j is the arc length calculated from point (i,l) along the "s-coordinate. 
Using equation (1), the distribution of grid points along the .,s-coordinate line, namely, new values of 8'J' 
is determined by 
(2) 
The idea of a spring analogy represented in equation (1) was introduced by Gnoffo(ref. 1). In his model, 
the distribution of points along the q, -coordina.te is determined only by the gradients of flow properties 
along that coordina.te line and is not affected by the distribution on adjacent ,,-coordinates, .,,-1 and "s+t. 
This can lead to excessive skewness of grid lines, especially when applied to complex flow fields, and this lack 
of control of grid inclination makes it difficult to extend the scheme to more than one family of coordinates. 
A force to control inclinations of e-coordinates in addition to that of grid spacings on .,-coordinates will 
correct this deficiency. This control force can be given by considering torsion springs attached to nodes 
along the .,,-1 line. The torsion spring enforces the inclination of line DA to that of a reference line. If the 
spring constant of the torsion spring is denoted by H, a mathematical statement of the force is 
(3) 
where (JDA is a inclination of line DA and ifJ the inclination of the reference line. The reference line can be 
chosen as an extension of F D to avoid kinks in the e-line at point D, as a line normal to the ",-coordinate 
to make the grid quasi-orthogonal, or as a streamline, and so forth. In practical calculations, a. combination 
of these reference lines is used. The torsion spring constant H can be prescribed for each coordinate line. 
A balance equation for the complete spring sylItem is 
(4) 
To facilitate solutions to equation (4), the third term is rewritten 
Hi-IJ(O,-I,J - ;,-1.1) => H'-I,J(B;J - i',J) (5) 
where i!Jl.. is arc length to the intersection of reference line DA' with the ",-coordinate as depicted in figure 
2. The H'-I,J term is set equal to H'-IJ divided by length of DA'. Finally, equation (4) reduces to the 
follOwing equation: 
(6) 
This is a tridiagonal system of equations for 8,.1 and can be readily solved. 
In this analysis only the torsion force on the upstream side (",-1) influences the distribution at ",. This 
permits simple marching schemes to be used, without any loss of generality, and contributes to the simplicity 
and robustness of the method. Conversely, if the influence from both sides (",-1 and ".+1) is considered 
simultaneously, the computational effort is increased considerably without any additional benefit. Note, 
too, that the downstream influence, (".+1), could be used instead of the upstream, without any additional 
complexity or loss of generality. 
GRm ADAPTATION IN MULTICOORDINATE DIRECTION 
A model using tension and torsion springs for a two-directional adaptation can be depicted as in figure 
3(80). Since each grid point is connected to its four adjacent points, the procedure for the grid movements of 
this model is more complicated and requires more computational effort. To minimize this complexity, a split 
model, which is a combination of one-directional adaptation, is used(fig. 3(b». Grid movement is achieved 
by successive applications of the one-directional adaptation method. This is analogous to ADI schemes for 
partial differential equations. It is not necessary to achieve convergence in this adaptation procedure and, 
in fact, one iteration is sufficient in practical applications of the scheme. 
The solution field is interpolated onto the newly adapted grid, using second-order, one-dimensional La-
grange interpolation after each one-directional adaptation. Before this interpolation, it is possible to add 
or delete grid points at the users discretion, thus enhancing the method without any loss of accuracy or in-
creased complexity. The successive application of one-directional adaptation also enhances the applicability 
of the method, in that unidirectional adaptation can be used in regions where gradients are large in only 
one direction. Extension of this scheme to three-dimensions is straightforward. 
RESULTS 
Shown in figures 4-8 are examples of applications of the method to axisymmetric, plume flow fields. 
These complex flow fields exhibit oblique shocks, barrel shocks, and slip surfaces, where grid points should 
be clustered for adequate resolution. The locations of these discontinuities are not known apriori, and a 
solution-adaptive grid is particularly useful in realizing an efficient and accurate simulation. 
The initial grid (fig. 4) was generated by an algebraic method and the flow field determined using a 
code(ref. 2) for the thin-layer Navier-Stokes equations. The free-stream Mach number is 2.01, the jet-exit 
Mach number is 2.5, and the static pressure ratio of the exhaust jet to the free-stream is 1. The density 
gradient was chosen as a reference variable for the grid clustering, and figure 5 shows a solution-adapted 
grid which clearly has clustered points to the oblique shock, the slip surface, and the barrel shock. Shown 
in figure 6 are computed density contours using the grid shown in figure 5. 
Shown in figure 7 is an adapted grid for a high-jet-pressure case (pressure ratio of 6), and density contours 
are shown in figure 8. The same initial grid (fig. 4) used in the previous case was used here. 
The grids were adapted to the flow-field solutions periodically (typically two or three times) during the 
course of reaching a steady-state solution. In this way, the time required for adaptation is a negligible 
fraction of the total time required for flow-field solution. 
Examples of the application to transonic flow fields past a NACA0012 airfoil are shown in figures 9-12. 
The free-stream Mach number is 0.8 and the angle of attack is 1.250 • Two-dimensional Eule~ equations 
were solved, using the code ARC2D(refs. 3,4). Figure 9 shows an initial O-grid generated by an algebraic 
method. Grid-point distributions along e-coordinate lines, which are parallel to the airfoil surface, were 
adapted to the density gradients in the flow-field solution. Shown in figure 10 is an adapted grid showing 
appropriate clustering of grid points to the shocks. Computed Mach contours obtained using the adapted 
grid are shown in figure 11. Figure 12 shows a comparison of pressure coefficients obtained using nonadapted 
and adapted grids. This figure shows that shock waves are crisply resolved using the adapted grid. 
An example for supersonic flow is shown in figures 13-15. The free· stream Mach number is 1.2 and the 
angle of attack is 1.250 • In this case, both e and 'I-coordinate lines were adapted to the density gradients. 
: 
, 
" 
SUMMARY 
The principal features of the proposed adaptive-grid method are as follows: 
1. It is a simple concept consisting of a tension and torsion spring analogy. The combination of these springs 
produces a suitable adaptive grid without excessive grid skewness. 
2. A marching-type of calculation procedure minimizes the computation time. 
3. The split-solution procedure for multidirectional adaptation is simple and practical. 
4.. The method can be applied independently to selected parts of the entire grid. 
S. The controlla.bility of grid inclinations with the torsion spring makes it possible to generate a nearly 
orthogonal adaptive grid. Also, the inclinations of grid lines near boundary can be specified arbitrarily. 
These features make the proposed adaptive-grid method practical and robust, and enhance its applica-
bility. The method can be applied to two- and three-dimensional fluid·flow problems without any difficulty 
and requires little computational time and effort. 
REFERENCES 
1. Gnoffo, P. A.: A Finite-Volume, Adaptive Grid Algorithm Applied to Planetary Entry Flowfields, AIAA 
J., vo1.21, no.9, 1983, pp. 1249-1254. 
2. Deiwert, G. S.; Andrews, A. E.; and Nabhashi, K.: Theoretical Analysis of Aircraft Afterbody Flow, 
AIAA Paper 84-1524, 1984. 
3. Pulliam, T. H.; Jespersen, D. C.; and Childs, R. E.: An Enhanced Version of an Implicit Code for the 
Euler Equations, AIAA Paper 83-0344, 1983. 
4.. Pulliam, T. H.: Euler and Thin Layer Navier-Stokes Codes: ARC2D, ARC3D, Notes for Computational 
Fluid Dynamics User's Workshop, The University of Tennessee Space Institute, Tullahoma, Tenn., Mar. 
1984. 
1) 1-1 
c 
, J+1 
c 
~~ __ ~ ______ ~~ __ --~~ 'J 
--.... ----_~~ ____ t_P' <"J-1 
Fig.1 Schematic of adaptive-grid algorithm 
with tension and torsion spring analogy. 
(a) Two-dimensional model. 
Fig.2 Notations for torsion spring analogy. 
+ 
(b) Split model. 
Fig.3 Multidirectional adaptation. 
20 
y y 
10 1 111 
o ,t=t:=t=t:H1:1iE 
:"'20 -10 0 10 20 
oL-----~~~~~~~ 
8 9 10 11 
x x 
(a) Complete configuration. (b) Base region detail. 
Fig.4. Initial grid for afterbody flow field. 
y 
.511 
o_~-___ II 
x 
Fig.5 Adapted grid for afterbody How field: 
Moo = 2.01,MJ = 2.5: PJ/Poo = 1.0. 
2_1IIi1l 
Y 1 
105 125 
x 
Fig.7 Adapted grid for afterbody flow field: 
Moo = 2.01,MJ = 2 5 :PJ/Poo = 6.0. 
1.5 r-----,----,------r-----.--,-----, 
1.0 
Y 
O~-~~~~_L~_L~~~LLLL~_~ 
8.5 95 105 115 
x 
Fig.6 Computed density contours with adapted grid: 
Y 1 
Moo = 2.01,AfJ = 2.5. PJ/roo = 1.0. 
10.5 
x 
125 
Fig.8 Computed densIty contours with arlapted end: 
Moo = 2.01,MJ = 2.5: PJ/Poo = 6.0. 
15 
10 
,l 
5 
Y 0 
-5 
-10 
Fi!.9 Initial O-grid for NACAOO12 airfoll. 
0 
0:1 
0 
0 
-1.0 L-__ ..L.-__ ....L... __ .l....L. __ ----' 
-.5 o .5 1.0 1 5 
x 
Fig.l1 Computed Mach contours 
with adapted grid: Moo = 0.8, a = 1.250 • 
x x 
1.0 ••• __ TIl-Zl771 
.5 
-5 
-10~~~~~~~~~~~~ 
-.5 0 5 10 1.5 
x 
Fi!.10 Adapted grid: 
Moo = 0.8, a = 1.250 • 
-15 
-1 0 
-5 
Cp 0 
.5 
10 
1 5 0 4 
x 
8 
c* p 
1.2 
Fig.12 Cp for solutions with (dot) 
and without (line) adapted grid. 
15 
1 0 
5 
Y 0 
-5 
-1 0 
-15 
2 -1 0 
x 
2 
Fig.14 Adapted grid: FI&'.15 Computed Mach contours: 
Moo = 1.2, a = 1.250 • Moo = 1.2, a = 1.250 • 
1 Report No 2 Government Accession No - 3 Recipient's Catalog No 
NASA TM-85989 
4 Title and Subtitle 5 Report Date 
A PRACTICAL ADAPTIVE-GRID METHOD FOR COMPLEX June 198!. 
FLUID-FLOW PROBLEMS 6 Performing Organozatlon Code ATP 
7 Author(s) 8 Performing Organozatlon Report No 
Kazuhiro Nakahashi* and George S. Deiwert A-9803 
10 Work Unit No 
9 Performing Organization Name and Address T-6462 
*NRC Associate. 
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 S.50nsorlng Agency Code 
Washington, DC 20546 50 -31-01 
15 Supplementary Notes 
Point of Contact: Kazuhiro Nakahashi, Ames Research Center, MS 202A-1, 
Moffett Field, CA 94035 (415) 965-6667 or FTS 448-6667 
16 Abstract 
A practical solution-adaptive-grid method utilizing a tension and tor-
sion spring analogy is proposed for multidimensional fluid-flow problems. 
The tension spring, which connects adjacent grid points to each other, con-
trols grid spacings. The torsion spring, which is attached to each grid 
node, controls inclinations of coordinate lines and grid skewness. A march-
ing procedure is used that results in a simple tr~diagona1 system of equa-
tions at each coord~nate l~ne to determine grid-point distribution. Mu1ti-
direct~ona1 adaptation is achieved by successive applications of 
one-directional adaptation. Examples of applications for axisymmetric 
afterbody flow fields and two-dimensional transonic airfoil flow fields are 
shown. 
17 Key Words (Suggested by Author(s)) 18 Distribution Statement 
Adaptive grid Unlimited 
Numerical method 
Subject Category - 02 
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A practical adaptive-grid method for complex fluid-flow problems

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