Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review

Introduction R interest in the generation of general bodyoriented curvilinear coordinate systems," for the purpose of solving the complete Navier-Stokes system of equations for subsonic and supersonic flows, has given rise to many forms of presentations of the equations both in conservative and nonconservative formulations. Based on the available solutions of the gasdynamic equations (e.g., Ref. 5) the conservation-law form of the equations seems definitely preferable, particularly when shocks are present. Although the above statement cannot be repeated in a definitive sense for the case of viscous subsonic and supersonic flow prediction through the Navier-Stokes equations, nevertheless, it is expected that the conservation-law form may eventually be more acceptable for numerical purposes. The purpose of this paper is to derive the conservation-law form of the Navier-Stokes equations in general nonsteady coordinate systems in a simple and direct fashion. Previous work on this subject has been done by McVittie, Viviand, and Vinokur. It will be shown in this Note that the equations in the conservation-law form can be obtained simply by a little manipulation of some standard vector and tensor formulas.

Conservation form of the Navier-Stokes equations in general nonsteady coordinates

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.

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Grid generation for the solution of partial differential equations

A time-dependent, two-dimensional Navier-Stokes code employing the body-fitted coordinate technique has been developed for supersonic flows past blunt bodies of arbitrary shape. The computer program is based on the finite-difference approximation of the compressible Navier-Stokes equations transformed to nonorthogonal curvilinear coordinates with the contravariant components of the velocity vector as dependent variables. The bow shock ahead of the body is obtained as part of the solution, by 'shock capturing'. Numerical solutions of the complete equations are presented in detail for free-stream Mach number 4.6, Reynolds number 10,000, and an isothermal wall temperature of 556 K for a circular cylinder with the free-stream outer boundaries forming a hyperbola in the front and a circular arc in the back.

Numerical solution of the navier-stokes equations for arbitrary blunt bodies in supersonic flows

In this paper a noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves has been developed. The basic generating equation is the Gaussian equation for a Euclidean space under the constraint of orthogonality. The resulting equation has been solved analytically for the case of isothermic coordinates. A coordinate transformation then yields nonisothermic coordinates in which the outgoing coordinate can be distributed in any desired manner. The method has been applied in many cases and these test results demonstrate that the proposed method can be readily applied in a variety of problems.

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A noniterative method for the generation of orthogonal coordinates in doubly-connected regions

: This paper presents a study in depth on the numerically generated body-fitted coordinate systems All fundamental ideas have been developed from a very basic point of view which leads to a well structured method of coordinate system control in any region of interest It is also shown how the developed concepts are instrumental in obtaining the machine solutions of the field differential equations in general dependent variables on the generated coordinate systems (Author)

J F Thompson

Papers

Machine Solutions of Partial Differential Equations in the Numerically Generated Coordinate Systems.