Convection of Concentrated Vortices and Passive Scalars as Solitary Waves

Abstract

A new version of a computational method, Vorticity Confinement, is described. Vorticity Confinement has been shown to efficiently treat thin features in multi-dimensional incompressible fluid flow, such as vortices and streams of passive scalars, and to convect them over long distances with no spreading due to numerical errors. Outside the features, where the flow is irrotational or the scalar vanishes, the method automatically reduces to conventional discretized finite difference fluid dynamic equations. The features are treated as a type of weak solution and, within the features, a nonlinear difference equation, as opposed to finite difference equation, is solved that does not necessarily represent a Taylor expansion discretization of a simple partial differential equation (PDE). The approach is similar to artificial compression and shock capturing schemes, where conservation laws are satisfied across discontinuities. For the features, the result of this conservation is that integral quantities such as total amplitude and centroid motion are accurately computed. Basically, the features are treated as multi-dimensional nonlinear discrete solitary waves that live on the computational lattice. These obey a “confinement” relation that is a generalization to multiple dimensions of 1-D discontinuity capturing schemes. A major point is that the method involves a discretization of a rotationally invariant operator, rather than a composition of separate 1-D operators, as in conventional discontinuity capturing schemes. The main objective of this paper is to introduce a new formulation of Vorticity Confinement that, compared to the original formulation, is simpler, allows more detailed analysis, and exactly conserves momentum for vortical flow. First, a short critique of conventional methods for these problems is given. The basic new method is then described. Finally, analysis of the new method and initial results are presented.