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NASA CONTRACTOR NASA CR-2 7 REPORT C_4 ! Qg _J Z BOUNDARY-FITTED CURVILINEAR COORDINATE SYSTEMS FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS ON FIELDS CONTAINING ANY NUMBER OF ARBITRARY TWO-DIMENSIONAL BODIES Joe F. Thompson, Frank C. Thames, and C. Wayne Mastin Prepared by MISSISSIPPI STATE UNIVERSITY Mississippi State, Miss. 39762 ]or Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY _1 _Z- __ _ _.: - • ZZ m- _--_Z " _---- u_-Z-- _- 1 ..... _H :j ---Z_ ._-Z z . 1. Z r 1. Report No. 2. Government Acc_sion No. 3. R_ipient's Catal_ No. NASA CR-2729 • 4. Title and Subtitle 5, Repo_ Date Boundary-Fitted Curvilinear Coordinate Systems For Solution of July 1977 Partial Differential Equations on Fields Containing Any Number of 6. Performing Organization Code Arbitrary Two-Dimensional Bodies 7. Author(s) 8. Performing Organization Report No. Joe F. Thompson, Frank C. Thames, and C. Wayne Mastin 10. Work Unit No. 9. Performing Organization Name and Addre_ 11. Contract or Grant No. Mississippi State University Mississippi State, Mississippi 39762 Grant NGR 25-001-055 13, Type of Report and Period Covered 12. Sponsoring Agency Name and Address Contractor Report National Aeronautics & Space Administration 14. Sponsoring Agency Code Washington, DC 2n546 3860 15, _pplementary Notes Ruby Davis of the Theoretical Aerodynamics Branch of Langley Research Center converted the UNIVAC 1106 code generated at Mississippi State University to the CDC 6000 Series code at Langley Research Center. Percy J. Bobbitt was the Langley Teclmical Monitor. Final report. 16. Abstract A method for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected two-dimensional region containing any number of arbitrarily shaped bodies is presented. No restrictions are placed on the shape of the boundaries, which may even be time-dependent, and the approach is not restricted in principle to two dimensions. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regardless of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system in the physical plane. A number of examples of coordinate systems and application thereof to the solution of partial differential equations are given. The FORTRAN computer program and instructions for use are included. 17. Key Words (Sugg_ted by Authoris)) 18. Distribution Statement Coordinate transformation Unclassified - Unlimited Multielement airfoils _.- Subject Category 02 Navier-Stokes 19. S_urity Clauif. (of this report} 20. Security Classif, (of this _) 21. No. of Pages 22. Price" Unclassified Unclassified 253 $9.00 Q For saleby the National Technical Information Service, Springfield, Virginia 22161 TABLE OF CONTENTS Page Ie Introduction ........................... 1 II. Mathematical Development ..................... Preliminaries .......................... 7 Doubly-Connected Region ...................... i0 Multiply-Connected Region ..................... 13 Simply-Connected Region ...................... 15 Coordinate System Control ..................... 15 Time-Dependent Coordinate Systems ........ ......... 19 III. Numerical Solution ........................ 21 Difference Equations ....................... 21 Multiple-Body Fields ....................... 23 Multiple-Body Segment Arrangements ................ 24 Initial Guess .......................... 26 Convergence Acceleration ..................... 30 Optimum Acceleration Parameters .................. 33 IV. Examples of Coordinate Systems .................. 36 Coordinate System Control ..................... 36 Various Body Shapes ........................ 37 V. Examples of Application to Partial Differential Equations ..... 38 Potential Flow .......................... 38 Viscous Flow ........................... 42 Velocity-Pressure Formulation ................ 44 Pressure Distribution and Force Coefficients ......... 45 Loaded Plate ........................... 46 _I. Instructions for Use - Coordinate Program ............ 47 Coordinate System ......................... 47 Dimensions ........................ 48 Input Parameters ....................... 48 Field Size .................... 48 Plotting ........................... 48 Initial Guess ........................ 49 Body Contours ............ ............ 49 Re-entrant Boundaries .................... 50 Acceleration Parameters ................... 51 Coordinate System Control .................. 51 Convergence of Very Large Fields ............... 54 Listing of Program TOMCAT ..................... 56 iii TABLE OF CONTENTS, CONTINUED Page Sample Cases ............. •............. 93 i. Simply-Connected Region ................ 94 Input ....................... 94 Output ...................... 96 Plot ....................... i01 2. Single-Body Field ................... 102 Input ....................... 102 Output ...................... 103 Plot ....................... 109 3. Double-Body Field ................... ii0 Input ....................... ii0 Output ...................... 112 Plot ....................... 122 VII. Instructions for Use - Factors Program ............. 123 Scale Factors .......................... 123 Listing of Program FATCAT .................... 124 Sample Case: Single-Body Field ................. 140 Input ........................... 140 Output ........................... 141 Vlll. References ........................... 147 IX. Tables ............................. 150 X. Figures ............................. 170 XI. Appendices ........................... A-I iv BOUNDARY-FITTED CURVILINEAR COORDINATE SYSTEMS FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS ON FIELDS CONTAINING ANY NUMBER OF ARBITRARY TWO-DIMENSIONAL BODIES By Joe F. Thompson,t Frank C. Thames,* and C. Wayne Mastin § Mississippi State University Mississippi State, Mississippi 39762 Research Sponsored by NASA Langley Research Center, Grant NGR 25-001-055 SUMMARY A method for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected, two-dimensional region containing any number of arbitrarily shaped bodies is presented. No restrictions are placed on the shape of the boundaries, which may even be time-dependent, and the approach is not restricted in principle to two dimensions. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regard- less of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system in the physical plane. A number of examples of coordinate systems and application thereof to the solution of partial differential equations are given. The FORTRAN computer program and instructions for use are included. I. INTRODUCTION There arises in all fields concerned with the numerical solution of partial differential equations the need for accurate numerical representa- tion of boundary conditions. Such representation is best accomplished when the boundary is such that it is coincident with some coordinate line, for then the boundary can be made to pass through the points of a finite dif- ference grid constructed on the coordinate lines; hence the choice of cylindrical coordinates for circular boundaries, elliptic coordinates for tprofessor of Aerophysics and Aerospace Engineering, Ph.D. *Engineering Specialist, Ph.D., Present Affiliation: LTV Aerospace Corporation, Dallas, Texas. §Associate Professor of Mathematics, Ph.D. elliptical boundaries, etc. Finite difference expressions at, and adjacent to, the boundary may then be applied using only grid points on the inter- sections of coordinate lines, without the need for any interpolation between points of the grid. The avoidance of interpolation is particularly important for boundaries with strong curvature or slope discontinuities, both of which are common in physical applications. Likewise, interpolation between grid points not coincident with _he boundaries is particularly inaccurate with differential systems that produce large gradients in the vicinity of the boundaries, and the character of the solution may be significantly altered in such cases. In most partial differential systems the boundary conditions are the dominant influence on the character of the solution, and the use of grid points not coincident with the boundaries thus places the most inaccurate difference representation in precisely the region of greatest sensitivity. The genera- tion of a curvilinear coordinate system with coordinate lines coincident with all boundaries (herein called a "boundary-fitted coordinate system" for pur- poses of identification) is thus an essential part of a general numerical solu- tion of a partial differential system. A general method of generating boundary-fitted coordinate systems is to let the curvilinear coordinates be solutions of an elliptic partial differen- tial system in the physical plane, with Dirichlet boundary conditions on all boundaries. One coordinate is specified to be constant on each of the bound- aries, and a monotonic variation of the other coordinate around each boundary is specified.

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