The General Interpolants Method (GIM) code which solves the multidimensional Navier-Stokes equations for arbitrary geometric domains is described. The geometry module in the GIM code generates two and three dimensional grids over specified flow regimes, establishes boundary condition information and computes finite difference analogs for use in the GIM code numerical solution module. The technique can be classified as an algebraic equation approach. The geometry package uses multivariate blending function interpolation of vector-values functions which define the shapes of the edges and surfaces bounding the flow domain. By employing blending functions which conform to the cardinality conditions the flow domain may be mapped onto a unit square (2-D) or unit cube (3-D), thus producing an intrinsic coordinate system for the region of interest. The intrinsic coordinate system facilitates grid spacing control to allow for optimum distribution of nodes in the flow domain

Anderson, P. G.

Spradley, L. W.

English

NASA Technical Reports Server

,N81 -14696 ':V~ 
- -
fINITE DIFFERENCE GRID GENERATION BY MULTIVARIATE 
BLENDING FUNCTIO~ INTERPOLATION* 
Peter G. Anderson and Lawrence W. Spradley 
Lockheed-Huntsville Research & Engineering Center 
Huntsville, Alahama 
ABSTRACT 
.., ..... " " 
The General Interpolants Method (GIM) code solves the multi-
dimensional Navier-Stokes equations for arbitrary geometric domains. 
The geometry module in the GIM code generates two- and three-
dimensional grids over specified flow regimes, establishes boundary 
condition information and computes finite difference analogs for use in 
the GIM code numerical solution module. The technique can be classified 
as an algebraic equation approach. 
The geometry package uses multivariate blending function interpola-
tion of vector-values functions which define the shapes of the edges and 
surfaces bounding the flow domain. By employing blending functions 
which conform to the cardinality conditions the flow domain may be mapped 
onto a unit square (2-D) or unit cube (3 -D), thus producing an intrinsic 
_- coordinate system for the region of interest. The intrinsic coordinate 
system facilitates grid spacing control to allow for optimum distribution 
of nodes in the flow domain. 
The GIM formulation is not a finite element method in the classical 
sense. Rather, finite difference methods are used exclusively but with the 
difference equations written in general curvilinear coordinates. Trans-
formations are used to locally transform the physical planes into regions 
of unit cubes. The mesh is generated on this unit cube and local metric-
like coefficients generated. Each region of the flow domain is likewise 
transformed and then blended via the finite element formulation to form 
the full flow domain. In order to treat "completely-arbitrary" geometric 
domains, different transformation functions can be employed in different 
regions. We then transform the blended domain to physical space and solve 
the Cartesian set of equations for the full region. The geometry part of the 
problem is thus treated much like a finite element technique while integration 
of the equations is done with finite difference analogs. 
*This work was supported, in payt, by NASA Langley 
Contracts NAS1-15341, 15783, and 15795. 
143 
https://ntrs.nasa.gov/search.jsp?R=19810006182 2020-03-21T14:27:20+00:00Z
144 
BU I LD I NG BLOCK CONCEPT 
The development is done in local curvilinear intrinsic coordinates based 
on the following concepts: 
• Analytical regions such as rectangles, spheres, cylinders, 
hexahedrals, etc., have intrinsic or natural coordinates. 
• Complex regions can be subdivided into a number of 
smaller regions which can be described by analytic 
functions. The degenerate caSe is to subdivide small 
enough to use very small straight-line segments. 
• Intrinsic curvilinear coordinate systems result in 
constant coordinate lines throughout a simply 
connected, bounded domain in Euclidean space. 
• The inter section of the lines of constant coordinates 
produce nodal points evenly spaced in the domain. 
• Intrinsic curvilinear coordinate systems can be pro-
duced by a univalent mapping of a unit cube onto the 
simply connected bounded domain. 
Thus, if a transformation can be found which will map a unit cube uni-
valently onto a general analytical domain, then any complex region can be 
piecewise transformed and blended using general interpolants. 
Consider the general hexahedral configuration shown. The local intrinsic 
coordinates are 111,112,113 with origin at point PI' The shape of the geometry 
is defined by 
• Eight corner points, P. 
1 
• Twelve edge functions, E. 
1 
• Six surface functions, S. 
1 
This shape is then fully described if the edges and surfaces can be specified 
as continuous analytic vector functions S. (x, y, z), E. (x, y, z), 
1 1 
BUILDING BLOCK CONCEPT 
P.4..--____ -. P 3 
a. Point Designations 
E 3 
b. Edgl! Designations 
c. Surface Designations 
145 
146 
GENERAL INTERPOLANT FUNCTION 
Based on the work of Gordon and Hall we have developed a general 
relationship between physical Cartesian space and local curvilinear intrinsic 
coordinates. This relation is given by the general trilinear interpolant shown 
on the adjac ent figure. 
In this equation, ~ vector is the Cartesian coordinates 
and Si' Ei are the vector functions defining the surfaces and edges, respectively, 
and Pi are the (x,y, z) coordinates of the corner points. Edge and surface func-
tions that are currently included in the GIM code are the following: 
• EDGES (Combinations of up to Five Types) 
Linear Segment 
Circular Arc 
Conic (Elliptical, Parabolic, Hyperbolic) 
Helical Arc 
Sinu soldal Segment 
• SURF ACES (Bounded by Above Edges) 
Flat Plate 
Cylindrical Surface 
Edge of Revolution 
This library of available functions is simply called upon piecewise via input 
to the computer code. 
With this transformation, any point in local coordinates 1']1' 1']2' 1']3 can 
be related to global Cartesian coordinates x, y, z. Likewise any derivatives 
of functions in local coordinates can be related to that derivative in physical 
space. 
GENERAL INTERPOLANT FUNCT ION 
-' -l -" 
- (1 -7) 2.) 11 3 E 9 - , I 2. (1 - Ii 3) E 3 - 11 Z I') 3 E 1 } 
"-.-.,----
--' -" 
+ (1-1/1) III (1-1/3) P4 + (1-1/\) I)l 1/ 3 P s 
-" -" 
+ IiI (I-Ill) (1-1J 3) P z + IiI (I-liZ) 1')3 P 6 
...... 
+ II} I1 Z (1-1/3) P 3 + III liZ 1)3 P7 
---
-,,' 
147 
148 
INTERNAL FLOW GRID 
(Axisymmetric Rocket Nozzle) 
The grid shown was used to compute the flow in a model of the Space 
Shuttle engine us ing the GIM code. The mesh is stretched in the radial direc-
tion to cluster points near the wall and stretched axially to place points near 
the throat of the nozzle. Only a portion of the complete grid is shown for 
clarity and illustration. The grid shows the general format used by the GIM 
code for internal, two-dimensional flows in non-rectangular shapes. 
EXTERNAL FLOW G RID 
(Two-Dimensional BI unt Body Flow) 
This figure shows a polar-like grid used for computing external flow 
over a blunt body. The body surface is treated invlscidly, and thus does 
nOt require an extremely tight mesh. The outer boundary is the freestream 
flow. The grid illustrates the GIM code technique for two-dimensional ex-
ternal flows using a polar-like coordinate system. 
149 
150 
EXTERNAL FLOW G RID 
(Non-Orthogonal Cu rvilinear Coordinates) 
The nodal network for the external flow over an ogive cylinder illustrates 
the capability of the GIM code geometry package to stretch the nodal distribu-
tion. The grid is very compact in the leading edge region and greatly expanded 
in the far field areas, The axial points follow the body surface and could gen-
erally be called ., body-oriented coordinates" in the nomenclature of the litera-
ture. The radial grid lines are not necessarily normal to the lateral lines or 
to the body surface. The GIM code, through its tlnodal-analog" concept can 
operate on this general non-orthogonal curvilinear grid. 
OIUGF:.:\L r / .~~ IS 
.O~ r'\=<)l'( (:';_':k\·~.:":"'··-[ 
1/ 
"""-" 
THREE-DIMENSIONAL GRID 
(Simple Rectilinear Coordinates) 
Supersonic flow in expanding ducts of arbitrary crOss section is a common 
occurrence in computational fluid dynamics. This figure illustrates a simple 
grid for a three-dimensional duct whose cross section varies sinusoidally with 
the axial coordinate. The 11 top" wall and the "front" wall have this sinusoidal 
variation while the "bottom" and "back" walls are flat plates. The grid shown 
was used to resolve the expanding-recompressing supersonic flow including the 
intersection of the two shock sheets. 
151 
152 
THREE-D IMENSIONAL GRID 
(Pipe Flow in a 90 deg ElbowTurn) 
There are numerous flow fields of interest which contain a sharp turn 
inside a smooth pipe. The GIM code has treated certain of these for applica-
tion to jet deflector nozzle flow in VTOL aircraft. The portion of a grid shown 
in the adjacent figure was used for this calculation. 
The 90 deg elbow demonstrates the capability to model three-dimensional 
non-Cartesian geometries. The internal nodes were emitted for clarity. The 
elbow grid was generated by employing edge-oI-revolution surfaces with circular 
arc segments as the edges being revolved. 
GRID FOR SPACE SHUTILE MAIN ENGINE 
(Hot Gas Manifold Geometry Model) 
The recent problems encountered with the Space Shuttle main engine 
tests have resulted in a GIM code analysis of the system. The I'hot gas mani-
fold" is a portion of this analysis for the high pressure turbopump system. 
The grid shown in the adjacent figure was used for this calculation. Only a 
small number of nodes are shown for clarity; the full model consists of approx-
imately 14,000 nodes. The extreme complexity of this geometry illustrates 
the necessity of using a GIM-like technique. Transforming this case to a 
square box computational domain is, of course, impossible. The results of 
the GIM code analysis agree qualitatively with flow tests that have been run 
on the hot gas manifold. 
Hot Gas Manifold Configuration 
153 
154 
Inner Wall 
Omitted 
GRID FOR SPACE SHUTTLE MAIN ENGINE 
(Hot Gas Manifold Geometry Model) 
Outer Wall 
Inner Duct 
SUMMARY 
• Finite difference grids can be generated for very general con-
figurations by using multivariate blending function interpolation. 
• The GIM code difference scheme operates on general non-
orthogonal curvilinear coordinate grids. 
• This scheme does not require a single transformation of the 
flow do~nain onto a square box. Thus, GlM routines can indeed 
treat arbitrary three-dimensional shapes. 
• Grids generated for both internal and external flows in two and 
three dimensions have shown the ver satility of the algebraic 
approach. 
• The GIM code integration module has successfully computed 
flows on these complex grids, including the Sp:ice Shuttle 
main engine turbopump system. 
• Plans for future application of the code include supersonic flow 
over missiles at angle of attack and three-dimensional, viscous, 
reacting flows in advanced aircraft engines. Plans for future 
grid generation work include schemes for time-varying networks 
which adapt themselves to the dynamics of the flow. 
. 155 
156 
BIBLIOGRAPHY 
Spradley, L. W., J.F. Stalnaker and A. W. Ratliff, IIComputation of Three-
Dimensional Viscous Flows with the Navier -Stokes Equations,1I AIAA Paper 
80-1348, July 1980. 
Spradley, L. W., and M. L. Pearson, "GIM Code User's Manual for the 
STAR-IOO Comp'.lter,f1 NASA-CR-3157, Langley Research Center, Hampton, 
Va., 1979. 
Spradley, L. W., P. G. Anderson and M. L. Pearson, 11 Comp'.ltation of Three-
Dimensional Nozzle-Exhaust Flows with the GIM Code,f1 NASA CR-3042, 
Langley Research Center, Hampton, Va., August 1978. 
Prozan, R.J., L.W. Spradley, P.G. Anderso:l:lnd M.L. Pearson, TIThe 
General Interpolants Method," AIAA Pap'2r 77-642, June 1977. 
Gordon, W.J., and C.A. Hall, "Construction:):;: Curvilinear Coordinate 
Systems and Applications to Mesh Generation,1I J. Numer. Math., Vol. I, 
1973, pp. 461-477. ---------


Finite difference grid generation by multivariate blending function interpolation

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