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Center for Nonlinear Studies |
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. It has also been shown to be effective in representing thin boundary layers on surfaces immersed in uniform Cartesian grids. We define these thin vortical = or scalar regions as features. Outside these 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 convecting features, the result of this conservation is that integral quantities such as total momentum and amplitude, and centroid motion are accurately computed. Effectively, 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 rotationally invariant limiter a single limiter that is a function of rotationally invariant variables. This is in contrast to conventional discontinuity capturing schemes which may involve a concatenation of separate 1-D limiters, each a function of variables along each axis. Results will be shown for convection of thin streams of passive scalars, thin convecting vortex filaments, treatment of small vortical scales in turbulent wakes, and flow over complex surfaces immersed in uniform grids.