Newton’s Method for the Navier-Stokes Equations

Summary

The work described in this paper is part of a larger effort whose objective is the development of efficient rapid convergence algorithms for finite difference equations which approximate the steady-state, compressible, Reynolds-averaged, Navier-Stokes equations. The overall program objective has been divided into three parts: (a) Investigate the effects of Jacobian-approximations on the convergence rate of Newton’s method applied to the NS equations. (b) Investigate the usefulness of ‘steady-state solvers’, i.e., those algorithms which are not time accurate, (c) Develop an efficient algorithm for solving finite-difference approximations to the steady-state compressible Reynolds-averaged NS equations.

Many of the current algoirthms used to find steady-state solutions for the Navier-Stokes equations are approximations to Newton’s method, however their convergence rates per iteration do not approach that of Newton’s method. A direct application of Newton’s method is expensive per iteration but provides a useful tool for obtaining objectives (a), (b), and (c). In this paper we discuss the development and application of a computer program which uses Newton’s method to solve the equations in two spatial dimensions.

The finite-difference equations are written in generalized curvilinear coordinates and strong conservation-law form and a turbulence model is included. We compute the flow field about a lifting airfoil for subsonic and transonic conditions. We investigate the requirements for an initial guess to insure convergence. We demonstrate the ability of Newton’s method to find temporally unstable solutions and show the necessity for auxiliary methods to evaluate the temporal stability of the steady-state solutions. Finally, we use Newton’s method to find nonunique solutions of the finite-difference equations, i.e., two different solutions for the same flow conditions.