Some Algorithms for Unsteady Navier—Stokes Equations

Abstract

There are essentially four different formulations of the incompressible Navier—Stokes equations—the primitive-variable (velocity and pressure), streamfunction-vorticity, streamfunction and velocity-vorticity formulations. The primitive-variable formulation can be found in any text on fluid dynamics (e.g., Batchelor (1967)), as can the two-dimensional version of both stream-function formulations. The three-dimensional streamfunction equations are given by Murdock (1986). Both streamfunction formulations avoid the complications of dealing with the pressure, as does the straightforward velocityvorticity approach. Murdock (1977, 1986) and Vanel, Peyret and Bontoux (1985) have developed spectral algorithms based on the two-dimensional streamfunction-vorticity formulation. Murdock (1986) has extended this to three dimensions. Gottlieb and Orszag (1977) and Maday and Métivet (1986) discuss spectral methods for the pure streamfunction version in two-dimensional flows. The velocity-vorticity formulation (Dennis, Ingham and Cook (1979)) has not yet been employed in spectral calculations. The primitive-variable formulation has been the one most extensively employed in three-dimensional spectral calculations and spectral methods based on it will be the focus of this chapter.