The hyperbolic scheme is used to efficiently generate smoothly varying grids with good step size control near the body. Although only two dimensional applications are presented, the basic concepts are shown to extend to three dimensions

Sorenson, R. L.

Steger, J. L.

English

NASA Technical Reports Server

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~'f.N81-1472S 
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USE OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS 
TO GENERATE BODY FITTED COORDINATES 
-- * Joseph L. Steger 
Flow Simulations, Inc., Sunnyvale, CA 94086 
and 
Reese L. Sorenson 
NASA Ames Research Center, Moffett Field, CA 94035 
Abstract 
Interpreting previous work, hyperbolic grid 
generation procedures are formulated in the style 
of the elliptic partial differential equation 
schemes used to form body fitted meshes. For 
problems in which the outer boundary is not 
constrained, the hyperbolic scheme can be used 
to efficiently generate smoothly varying grids 
with good step size control near the body. 
Although only two dimensional applications are 
presented, the basic concepts are shown to 
extend to three dimensions. 
Now with Stanford University. 
Preceding page blank 
463 
https://ntrs.nasa.gov/search.jsp?R=19810006209 2020-03-21T14:25:39+00:00Z
464 
The task of generating the exterior mesh about 
an arbitrary closed body as indicated in this slide 
is undertaken. The locati'on of the outer botmdary 
is not specified; it only need be far removed from 
the inner boundary. Such a grid generation problem 
is encountered in external flow aerodynamics. 
We seek a grid composed of constant s and n 
lines as indicated in this slide, given initial x,y 
data along ~ at n = O. The grid generation 
equations, just as the flow field equat ions, are 
solved in the uniform transform plane. 
SKETCH OF PHYSICAL AND COMPUTATIONAL PLANE 
77max 
I:: = 0 x, y SPECIFIED I:: <,; <,;max 
Partial differential equations are sought to 
generate a smoothly varying mesh such that grid lines 
of the same family do not inter'ect or coalesce. Two 
systems of nonlinear hyperbolic partial differential 
equations have been considered for the given initial 
data sketched in the previous slide. As indicated 
in this slide, these systems each use the condition of 
orthogonality and a geometric constraint. 
HYPERBOLIC GRID GENERATION EQUATIONS 
ARC LENGTH-ORTHOGONALITY SCHEME 
2 2 2 2 2 
X ~ + y ~ + X77 + Y 77 = (~S) 
X~X77 + Y~Y77 = 0 
VOLUME-ORTHOGONALITY SCHEME 
X~Y77 - x77Y~ = V 
x~x77 + Y~Y77 = 0 
IN BOTH CASES ~~ = A77 = 1 
465 
466 
The previously described nonlinear partial 
differential equations must be shown to be properly 
posed for the given initial value data. As a first 
step. the equations are cast in a locally linearized 
form so that they can be analysed as a system of two 
first order partial differential equations. For the 
locally linearized form to be meaningful. the equations 
are expanded about a nearby known solution or state. 
LOCALLY LINEARIZED FORM 
X~X11 + Y~Y 77 = 0 
x~Y77 - x77Y~ = V 
EXPAND X AND y ABOUT KNOW STATE X, Y 
E.G. x~y 11 = (x + X-X) ~ (Y + Y-Y)11 
= X~Y11 + (x~ - x~)Y11 + (Y11 - Y11)x~ + 0(.62) 
- - - - (2) 
= Y11x~ + x~Y11- x~Yl1 + 0 Ll 
OBTAIN LOCALLY LINEARIZED FORM 
- - -(~~ ~X:) (~)~ + (:J~ 
Analysis of the locally linearized partial 
differential equations indicates that the equations 
can be marched in n provided that x~ + y~ 1 o. 
That is, the grid spacing in ~ cannot be of zero 
length. The fact that B-1A is a symmetric matrix 
ensures that it has real, distinct eigenvalues. 
This then means that the system is hyperbolic 
if n is used as the marching or time-like 
direction. 
HYPERBOLICITY 
LOCALLY LINEARIZED VOLUME-ORTHOGONALITY EQUATION 
FIND: 
a) B-1 EXISTS I F x~ + Y~ =1= 0 
b) B-1 A IS SYMMETRIC 
THEREFORE LINEARIZED EQUATIONS ARE HYPERBOLIC 
467 
468 
The grid generation equations can be solved 
using standard numerical techniques for first order 
systems of hyperbolic partial differential equations. 
In our case we have used a noniterative implicit 
finite difference procedure. An unconditionally 
stable implicit scheme was selected so that an 
arbitrary mesh step size can be specified in the 
marching direction. The same kind of numerical 
procedure is used to solve the flow field equations. 
NUMERICAL SOLUTION OF VOLUME-ORTHOGONALITY "-
EQUATIONS 
USES IMPLICIT FINITE DIFFERENCE SCHEME FOR 
x~YTl - xTlY~ = V 
x~xTl + Y~YTl = 0 
SCHEME IS: a) UNCONDITIONALLY STABLE 
b) NONITERATIVE 
c) SECOND ORDER IN ~, FIRST ORDER IN Tl 
d) REQUIRES A BLOCK TRIDIAGONAL INVERSION 
The volume orthogonality scheme requires that the 
user specify the volume (area in 2-D) of each mesh 
cell. The quality of the grid will, to a large extent, 
be determined by the user's cleverness in specifying 
these volumes. To specify these volumes, we currently 
define a simple geometric shape (e.g. circle or straight 
line) which has exactly the same arc length as the body 
we wish to grid. An algebraically clustered grid is 
then created by the user for the simple geometric shape. 
The volumes of this simple grid, the control volume grid, 
are then used directly on a point by point basis in the 
hyperbolic grid generation equations. 
SELECTION OF VOLUMES 
SPECIFIED 
CONTROL 
VOLUME 
GRID 
a 
PHYSICAL 
GRID 
"/// .///////// .////////// b 
469 
470 
These slides show views of a viscous grid generated 
for a typical cross section of an aircraft fuselage. 
The control volumes were specified about a circle whose 
arc length matches that of the fuselage. Initially 
these control volumes are proportional to the grid-point 
arc-length spacing around the fuselage. Ultimately, 
however, control volumes that are uniformly spaced in 
the circumferential (i.e. e) direction are specified. 
Thus in far field a polar coordinate system is formed. 
y 
y 
VISCOUS GRID GENERATED ABOUT TYPICAL 
AIRCRAFT·FUSELAGE CROSS SECTION 
OVERVIEW 
5.Sr--------------, 
""" .. 
..  >~.':.'it~--
...... 
............ 
. S.S.';;-S---------------:', 
! 
.---~ 
I 
I 
+.-
x 
GRIO OETAlllIIEAR CORNER 
x 
This slide shows views of an inviscid grid 
generated about a highly cambered profile. The 
same type of control volume used previously is 
employed. A uniform grid spacing was specified 
in the direction away from the body as is clear 
from the view showing grid detail near the body. 
y 
y 
INVISCID GRID GENERATED ABOUT HIGHLY 
CAMBERED AIRFOIL 
OVERVIEW 
5,----------------------------, 
. ',' 
.' .' ~ ... .' .. ~: .. : . ",' '.', ",'. :'. " . " . " " ,: 
" '. ',',: :. "., 
....... ' . \ 
',. \"" .' 
-~5~---------------------------J 
x 
GRID DETAIL OF SLADE 
471 
472 
In this case a viscous grid is generated about 
the cambered profile. The normal grid spacing at 
the body is 0.01% of the chord length. Note that 
because volume is specified the grid spacing grows 
in the marching direction so as to prevent the 
circumferential spacing from vanishing. For a 
profile with twice the camber, however, this 
process breaks down and grid lines do coelesce. 
In these cases a more sophiticated means of 
specifying the volumes is needed. 
Y 
Y 
VISCOUS GRID GENERATED ABOUT HIGHLY 
CAMBERED AIRFOIL 
GRID DETAIL NEAR BODY 
x 
GRID DETAIL NEAR LEADING EDGE 
.5r-----~--~----~--~~--~,--~,------r-~ 
, \\- . ........-\~,/ ". - ~\'-
\ 
.-'. 
--/ /<..... /:, -/ 
""""'. \. 
, '. 
\. 
\ 
-.25 ~----....:'.'------->"""-__ .....::....--'--__ ..:=--.L ___ ~ 
-.50 
.25 
x 
473 
474 
These views show the hyperbolic grid generation 
scheme applied to generating a "C-grid" about a cambered 
airfoil. Here the control volume grid is generated 
about a straight line, that is, it is nothing more 
than a clustered rectangular grid. It is clear from 
the view at the trailing edge that some adjustments 
are needed to the current numerical treatment of 
discontinuous boundary data. 
GENERATION OF C-GRID ABOUT 
CAMBERED AIRFOIL 
OVERVIEW 
GRID DETAIL NEAR BODY 
GRID DETAIL AT TRAILING EDGE 
475 
HYPERBOLIC GRID GENERATION ADVANTAGES 
• SMOOTHLY VARYING GRID IS FOUND 
• GOOD USER CONTROL OF CLUSTERING NEAR BOUNDARY 
• FAST GRID GENERATION 
• ORTHOGONAL OR NEARLY ORTHOGONAL 
• AUTOMATICALLY TREATS COMPLEX SHAPES 
HYPERBOLIC GRID GENERATION DISADVANTAGES 
• OUTER BOUNDARY CANNOT BE SPECIFIED (UNLESS 
ITERATIVE SHOOTING METHOD DEVISED) 
• SCHEME TENDS TO PROPAGATE DISCONTINUOUS 
BOUNDARY DATA 
• POORLY SPECIFIED BOUNDARY DATA AND CONTROL 
VOLUMES CAN RESULT IN "SHOCK-WAVE" LIKE BREAKDOWN 
476 
The hyperbolic grid generation scheme can also be 
formulated in three di~nsions. With volume specified 
as one constraint, orthogonality can only be enforced 
in two of the coordinate directions. The three partial 
differential equations shown form a hyperbolic system 
for marching in~. Proof that the equations are 
hyperbolic was quite tedious, required considerable 
insight, and was carried out by Dennis Jespersen of 
Oregon State University. 
EXTENSION TO THREE DIMENSIONS -
VOLUME AND TWO ORTHOGONALITY 
~ ~ 
dr . dr = 0 
d17 d~ 
I 
a(X,y,Z) 1= V 
a(~,17,~) 
SYSTEM IS FOUND TO BE 
HYPERBOLIC 
477 
The three constraints of orthogonality do not 
form a hyperbolic system of partial differential 
equations. Neither are the equations of elliptic 
type. In fact, their classification and what if 
any type of boundary data makes them unique is 
unknown to the authors. 
EXTENSION TO THREE DIMENSIONS 
478 
THREE ORTHOGONALITY 
~ ~ 
dr . dr = 0 
d17 d~ 
SYSTEM CANNOT BE MARCHED 
AND IS NOT ELLIPTIC 
-
'"" 


Use of hyperbolic partial differential equations to generate body fitted coordinates

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