Two counter-rotating diffusing vortices

Abstract

The problem of the diffusion of two counter-rotating vortices of equal strength is studied numerically and analytically. Asymptotic expansions are derived for the limiting behavior of the solution for small times, for small Reynolds numbers, and for large times. The results are used to more fully understand the drift and decay of the vortex system. Thus it is shown that different measures for the position of the vortex system used by previous authors may give significantly different values for the drift velocity of the vortices. The expansion for small Reynolds number shows that these differences remain even in the Stokes limit Re-40, in which the vorticity system becomes symmetric about the line connecting the vortex centers. But surprisingly, the large time expansion shows that for large times all drift velocities become identical. Moreover, this universal velocity is different from the average velocity in each half plane although it equals the velocity of the centers of vorticity of those planes. The small time expansion shows that increasing Reynolds number makes the vortices more symmetric. This tends to reduce the differences between the drift velocities. The small time expansion describes the numerical solution well as long as the vortices remain small compared to their spacing. The numerical results show that the Stokes solution describes various flow quantities fairly well for Reynolds numbers up to 600 based on the circulation; however, nonzero Reynolds number reduces the decay of the circulation of the vortices even on a diffusive time scale. C 1995 American Institute of Physics.

A New Diffusion Procedure for Vortex Methods

“…The first is the diffusion of a pair of counterrotating point vortices. Addi-tional physical discussions of the results below, including the long time behavior, can be found in [65]. Results of computations of flows about rotating and translating cylinders using our scheme can be found in [54][55][56].…”

“…Another advantage of the redistribution equations is that they tend to localize the errors in velocity that result We refer elsewhere for actual physical applications using the redistribution method; see [54][55][56]65]. from the numerical diffusion.…”

A fast resurrected core-spreading vortex method with no-slip boundary conditions

“…This requirement increases the computational effort and prevents schemes from being entirely grid-free. Instead of approximating the Laplace operator with integral operators, the vortex-redistribution method that Shankar and van Dommelen proposed [8,9] distributes fractions of circulation of each vortex to its neighboring vortices within a chosen distance; the fractions are computed on equating the Fourier transforms of 0021-9991/$ -see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2008.…”

Generation and decay of counter-rotating vortices downstream of yawed wind turbines in the atmospheric boundary layer

“…2011), such as viscous diffusion, strong external turbulence, cross-diffusion across the line of symmetry, and Crow instability breakup. When turbulence levels and shed vorticity strength are moderate, evidence from many of these earlier simulations points to the cross-diffusion mechanism (Cantwell & Rott 1988; Ohring & Lugt 1993; van Dommelen & Shankar 1995; Leweke, Dizès & Williamson 2016) playing a dominant role. In the following sections, we develop a model first to predict the generation and then the decay of the yawed wind turbine CVP.…”