Three-Dimensional Vortex Methods Derived from the Navier-Stokes Equations.

Abstract

Vortex methods have been used for many fluid engineering problems to simulate flow fields, in particular high Reynolds number flow fields. These methods have some advantages ; (1)are free of grid generation, (2)use a simple algorithm, (3)are inherently free of numerical viscosity, and (4)exhibit self-adaptation for strong shear flow. In three-dimensional flows, a few numerical schemes have been used in actual calculations, but their theoretical foundation is not sufficient in a sense that numerical schemes are not explicitly derived from the Navier-Stokes equations. The present paper aims to give the theoretical foundation of the vortex methods used for three-dimensional flows. In this paper, a basic integral equation concerning the vorticity is derived from the three-dimensional Navier-Stokes equations and typical algorithms of vortex methods, the vorton method, vortex element method and vortex stick method, are then derived from this integral equation. Furthermore, the vorticity field given by the integral equation is shown to be solenoidal, provided the initial one is solenoidal. Finally, the velocity field is analytically obtained for Gaussian distribution of vorticity, which is the typical vorticity distribution in vortex methods, and a new type of Gaussian vorton, which is solenoidal, is proposed and the velocity field induced by this vorton is obtained.