By Georges-Henri Cottet, Petros D. Koumoutsakos
Vortex tools have matured in recent times, providing a fascinating replacement to finite distinction and spectral equipment for top answer numerical ideas of the Navier Stokes equations. some time past 3 a long time, examine into the numerical research facets of vortex equipment has supplied an excellent mathematical history for knowing the accuracy and balance of the tactic. whilst vortex tools keep their beautiful actual personality, which was once the inducement for his or her creation. This booklet offers and analyzes vortex equipment as a device for the direct numerical simulation of impressible viscous flows. it is going to curiosity graduate scholars and researchers in numerical research and fluid mechanics and likewise function a great textbook for classes in fluid dynamics.
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Extra resources for Vortex Methods: Theory and Applications
Let us now come to the precise definition of the method we plan to analyze. From now on, we make constant use of the notations and results given in Appendix A. Let Q =]0, 1[2, r > 1, and let coo be a periodic function in Q. Let f be a C°° cutoff function of order r. We define the periodic convolution of a periodic function (or distribution) / defined on Q with a function g defined in R2 by g *p f = g * / , where / denotes the periodic extension of / in R 2. Then, if £ is a cutoff function, we set f£(x) = e~ 2 £(x/£), a)e0 = ££ *, o)0, K£ = K * &.
6: the case of a bounded flow in a simply connected domain and the case of flows in periodic boxes. In the first case, we assume that the normal component of the velocity is zero, the so-called no-through-flow boundary condition. In terms of the stream function \j/ such that u = V x f, this means that the tangential derivative of \jr vanishes at the boundary. Since the boundary has a single connected component, \j/ is a constant, which can be set to zero as ty is obviously determined up to an additive constant, at the boundary.
For periodic or unbounded geometries, it is thus infinite-order accurate in the sense that the distance in some appropriate distribution space between the exact vorticity and its particle approximation can be bounded by Chm for all m, provided the vorticity has derivatives of an order up to m bounded. For rectangular domains without periodicity assumptions, the midpoint rule is only second order, but higher-order initializations can be obtained if initial particle locations coincide with quadrature points associated with Gauss-type quadrature formulas.