Numerical Methods for the Navier-Stokes Equations

Abstract

In the early 1970’s Murman and Cole in a landmark paper [1] set the groundwork for the development of computational fluid dynamics for years to come. Their paper demonstrated the use of type dependent finite difference approximations, chosen according to the characteristic speeds of the flow, and Gauss-Seidel line relaxation to solve the transonic small disturbance equation. Following their success, Jameson [2] introduced a “rotated” difference scheme and extended the Murman-Cole procedure to solve the full potential equation. These early contributions were responsible for the rapid growth and widespread application of computational fluid dynamics to aerodynamic design in the 1970’s. Last year Chakravarthy [3] applied flux-split type-dependent difference approximations and Gauss-Seidel line relaxation to solve the Euler equations. And this year Napolitano and Walters [4] and this author [5] used these procedures to solve the Navier-Stokes equations. On the one hand it’s amazing that the same key features of the Murman-Cole scheme can be applied to the more complete sets of governing equations, and on the other it’s amazing that it took a decade and a half to realize it. This paper outlines the use of these features for solving the Navier-Stokes equations and presents some computed results demonstrating high numerical efficiency.