Chapter 9 - Simulation of viscous diffusion in discrete vortex modelling

Summary

Introduction

The ‘random walk’ model for simulation of viscous diffusion in discrete vortex clouds was first proposed by Chorin (1973) for application to high Reynolds number flows and has been widely used since. The principle involved is to subject all of the free vortex elements to small random displacements which produce a scatter equivalent to the diffusion of vorticity in the continuum which we are seeking to represent. Such flows are described by the Navier Stokes equations which may be expressed in the following vector form, highlighting the processes of convection and diffusion of the vorticity ω,

where q is the velocity vector and ∇² the Laplacian operator. The third term, applicable only in three-dimensional flows represents the concentration of vorticity due to vortex filament stretching. Otherwise in two-dimensional flows, with which we are concerned here, the vector Navier-Stokes equation reduces to

Normalised by means of length and velocity scales ℓ and W_∞ this may be written in the alternative dimensionless form

where the Reynolds number is defined by

For infinite Reynolds number (9.3) describes the convection of vorticity in in viscid flow, for which the technique of discrete vortex modelling was developed in Chapter 8. At the other end of the scale, for very low Reynolds number flow past an object of characteristic dimension ℓ, the viscous diffusion term on the right hand side (9.3) will predominate.