The effect of viscosity upon an unsteady vortex

The effect of viscosity upon an unsteady vortex had been investigated ana lytically by M. Kurosaka (University of Tennessee Space Institute). He created the Circumferential Average Method and divided a vortex into two parts: a rotational core and several outer spiral turns in order to analyze the problem. In general, the solutions for circumferentially averaged vorticity ajid tangential velocity are in the form of integral representations. The integrand is products of a modified Bessel function, exponential function and others. He also briefly analyzed these solutions in some special cases corresponding to power-law growth.

The Gausian quadrature with Labatto integration is the numerical method used to calculate the circumferential average vorticity and taiigental velocity as the functions of radial positions. Provision of sub-intervals near the singularities of integrand is needed in order to control the error of numerical calculations. The polynomials are approximated for modified Bessel function. Some exact solutions from certain limit cases such as invicid limit and far field are used to check the numerical results.

The effect of viscosity on radial vorticity and tangential velocity is predicted to be substantial. Before the separation from the mother body, the viscosity removes the inviscid singularity at the center of vortex and smoothes out the inviscid discontinuity near the outer edge of the rotational core. After the separation, the viscosity causes transformation of the radial profiles into the Lamb-Oseen one.