Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods

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Abstract
A class of nonlinear hydrodynamic problems is studied. Physical problems such as shear flow, flow with a sharp interface separating two fluids of different density and flow in a porous medium all belong to this class. Owing to the density difference across the interface, vorticity is generated along it by the interaction between the gravitational pressure gradient and the density gradient, and the motion consists of essentially two processes: the creation of a vortex sheet and the subsequent mutual induction of different portions of this sheet. Two numerical methods are investigated. One is based upon the well-known Green's function method, which is a Lagrangian method using the Biot-Savart law, while the other is the vortex-in-cell (VIC) method, which is a Lagrangian-Eulerian method. Both methods treat the interface as sharp and represent it by a distribution of point vortices. The VIC method applies the FFT (fast Fourier transform) to solve the stream-function/vorticity equation on an Eulerian grid, and computational efficiency is further improved by using the reality properties of the physical variables. Four specific problems are investigated numerically in this paper. They are: the Rayleigh-Taylor instability, the Saffman-Taylor instability, transport of aircraft trailing vortices in a wind shear, and the gravity current. All four problems are solved using the VIC method and the results agree well with results obtained by previous investigators. The first two problems, the Rayleigh-Taylor instability and the Saffman-Taylor instability, are also solved by the Green's function method. Comparisons of results obtained by the two methods show good agreement, but, owing to its computational economy, the VIC method is concluded to be the better method for treating the class of hydrodynamic problems considered here.

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