On the ‘δ-equations’ for vortex sheet evolution

Abstract

We use a set of equations, sometimes referred to as the ‘δ-equations’, to approximate the two-dimensional inviscid motion of an initially circular vortex sheet released from rest in a cross-flow. We present numerical solutions of these equations for the case with δ2 = 0 (for which the equations are exact) and for δ2 > 0. For small values of the smoothing parameter δ a spectral filter must be used to eliminate spurious instabilities due to round-off error. Two singularities appear simultaneously in the vortex sheet when δ2 = 0 at a critical time tc After tc the solutions do not converge as the computational mesh is refined. With δ2 > 0, converged solutions were found for all values of δ2 when t < tc, and for all but the two smallest values of δ2 used when t > tc. Our results show that when δ2 > 0 the vortex sheet deforms into two doubly branched spirals some time after tc The limiting solution as δ→0 clearly exists and equals the δ = 0 solution when t < tc. For t > tc, the limiting solution appears to exist if only the converged solutions are used, but it is unclear what relation this limiting solution has to any δ2 = 0 solution for these times.