Long-wavelength equations of motion for thin double vorticity layers

Abstract

We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thin-layer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The three-curve equations of contour dynamics that fully describe the double-layer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initial-value evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the single-layer equation of Moore (Stud. Appl. Math., vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the first-order equations to small-amplitude perturbations shows Kelvin–Helmholtz instability when the far-field fluid velocities on either side of the double layer are unequal. Equal velocities define a circulation-free double vorticity layer, for which solution of the initial-value problem using the Laplace transform reveals a double pole in transform space leading to linear algebraic growth in general, but there is a class of interesting initial conditions with no linear growth. This is shown to agree with the long-wavelength limit of the full linearized, three-curve stability equations.