Generalized Coordinate Forms of Governing Fluid Equations and Associated Geometrically Induced Errors

Abstract

The governing equations of fluid flow may be cast into various forms upon application of a generalized coordinate mapping. These forms are the nonconservation law form (NCLF) and strong, weak, and chain rule conservation law forms (SCLF, WCLF, CRCLF, respectively). This paper describes the geometrically induced errors resulting from the failure to satisfy a certain consistency condition for each of these four forms and also demonstrates the ability of the CRCLF to produce exactly the same numerical solution as the WCLF, provided a condition is met on where to evaluate the transformation metrics. It is also demonstrated that considerably fewer arithmetic operations are required to advance the solution from n to n +1 when the CRCLF is used in com- parison to both the SCLF and WCLF. HE effort devoted to the field of computational fluid dynamics is progressing at an ever-increasing rate. At certain points in time during this natural evolution of numerical methods, schemes, and ideas it is sometimes in- structive to pause to reflect upon past work in an attempt to gain the proper perspective. It is important that this reflection reaches all of the way to the fundamental rules and practices used to develop numerical techniques. Some of these practices are often taken for granted much too soon after their in- troduction into the literature. Only through a global view of this past effort can subtle, mutually experienced problem areas and possible causes be identified. The present work deals with the numerical solution to the transformed fluid flow equations (i.e., Euler equations, etc.) using the finite-difference approach. The transformed equations are obtained through application of a generalized mapping from physical coordinates to computational coordinates. At the time of application of the mapping to the original equations and prior to choosing the numerical in- tegration scheme, the analyst must decide in which form the equations should be written. The choices are the non- conservation law form (NCLF), strong conservation law form (SCLF), weak conservation law form (WCLF), and chain rule conservation law form (CRCLF). This decision is one of the fundamental practices alluded to in the previous paragraph. The present work illustrates how such a fundamental decision can strongly influence both the amount of analysis required to develop a consistent algorithm and the number of arithmetic operations required to execute the algorithm. In addition, it is shown that large solution errors and in some cases instabilities can result from failure to implement the results of such an analysis.