Merger and cancellation of strained vortices

Abstract

We study numerically the behaviour of and interaction between Burgers vortices – a known equilibrium solution to the Navier–Stokes equations which incorporates a balance between viscous diffusion and strain intensification of vorticity. A hybrid spectral/finite-difference method is employed to solve the Navier–Stokes equations in vorticity–stream function form for a unidirectional vorticity field on an infinite domain in the presence of a uniform three-dimensional strain field, one principal axis of which is parallel to the vorticity. Merging of two strained vortices is studied over a range of Reynolds numbers Re = Γ/2πν = 10–1280, and the results are used to calculate an energy spectrum for three-dimensional, homogeneous turbulence. The cancellation of two strained vortices with opposing circulation is investigated for Reynolds numbers Re = Γ0/2πν = 0.1–160 (τ0 is the circulation about one vortex), over a range of strain rates in the direction parallel to the line joining the vortex centres. A solution of the Navier–Stokes equations describing vorticity cancellation in the strain-induced collision of vortex layers (Kambe 1984) is used to estimate the asymptotic, timewise decay of circulation for each vortex. Good agreement with the present numerical results is obtained. Vortex core pressures calculated during the cancellation event are compared to a simple analytical model based on Moore & Saffman (1971).