Application of adaptive quadrature to axi-symmetric vortex sheet motion

Studies of the formation of fine structures on free surfaces in liquids, such as curvature singularities or interface pinching, demand that the motion of the interface must be computed very accurately. Boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one. On the other hand, the boundary integrals are singular, and their accurate evaluation can prove quite challenging. In two-dimensional motion, the interface is just a curve, When this curve is closed or periodic, the singularity in the integrand may be removed and the trapezoidal rule may be applied with spectral accuracy. Unfortunately, the nature of the singularity in the integrand for three-dimensional motion is much more difficult to treat. In this paper, we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi-symmetric flow. The technique is based on a vector-potential formulation which offers some computational advantages over other methods based on the Biot-Savart integral, Direct numerical computations show that our technique is much more accurate and efficient than existing techniques. We apply our technique to study the evolution of an initially spherical vortex sheet. We present evidence of the formation of a 3/2 power singularity in the curvature of the vortex sheet. (C) 1998 Academic Press.