The vortex sheet model for a turbulent mixing layer

Abstract

The primary aim of this work is to study a temporal mixing layer using the model of instability-induced roll-up of a slightly perturbed vortex sheet in an inviscid and an incompressible fluid. A point vortex model evolves into a chaotic cloud of point vortices instead of smooth double-branched spirals. The vortex sheet model is derived from basic equations of vortex dynamics. The present model uses linear segments to interpolate the sheet. Computer simulation of the vortex sheet model is complicated as compared to point vortices. However, closed-form equations of motion do exist for vortex sheets. A prominent vortex core flanged by regular spiral windings and an irrotational fluid in between the layers is obtained. The sheet develops a highly peaked distribution of circulation density and curvature at a finite time. The problem of finite time singularity can be handled by a technique that invokes longitudinal circulation density diffusion along the sheet at singular points or by restriction of the smallest scales of motion. The accuracy of such simulations can be independently verified by using the laws of vortex dynamics and conserved physical quantities. We observe the growth of the two-dimensional shear layer with time and the merger of vortex-like structures. The dependence of the mixing layer on the initial conditions is studied and the mixing layer properties thus obtained are examined. It remains to be seen whether the vortex sheet model yields a truly turbulent mixing layer.