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eddy_psi_omega
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c c This case demonstrates how to use the Nek5000 convection-diffusion c solver as a 2D Navier-Stoke solver using the streamfunction-vorticity c (psi-omega) formulation. c c Basically, vorticity is advected and the streamfunction is derived by c solving a Poisson equation. (The Poisson equation is not solved c efficiently here, this is just a demonstration of some of the capabilities c that users can develop with the userchk routine.) c c Details (below) of the particular example are from the "eddy" example. c c c This case monitors the error for an exact 2D solution c to the Navier-Stokes equations based on the paper of Walsh [1], c with an additional translational velocity (u0,v0). c c The computational domain is [0,2pi]^2 with doubly-periodic c boundary conditions. c c Walsh's solution consists of an array of vortices determined c as a linear combinations of eigenfunctions of having form: c c cos(pi m x)cos(pi n y), cos(pi m x)sin(pi n y) c sin(pi m x)cos(pi n y), sin(pi m x)sin(pi n y) c c and c c cos(pi k x)cos(pi l y), cos(pi k x)sin(pi l y) c sin(pi k x)cos(pi l y), sin(pi k x)sin(pi l y) c c While there are constraints on admissible (m,n),(k,l) c pairings, Walsh shows that there is a large class of c possible pairings that give rise to very complex vortex c patterns. c c Walsh's solution applies either to unsteady Stokes or c unsteady Navier-Stokes. The solution is a non-translating c decaying array of vortices that decays at the rate c c exp ( -4 pi^2 (m^2+n^2) visc time ), c c with (m^2+n^2) = (k^2+l^2). A nearly stationary state may c be obtained by taking the viscosity to be extremely small, c so the effective decay is negligible. This limit, however, c leads to an unstable state, thus diminsishing the value of c Walsh's solution as a high-Reynolds number test case. c c It is possible to extend Walsh's solution to a stable convectively- c dominated case by simulating an array of vortices that translate c at arbitrary speed by adding a constant to the initial velocity field. c This approach provides a good test for convection-diffusion dynamics. c In the current file set the translational velocity is specified as: c c U0 =[u0,v0] := [param(96),param(97)] ( in the .rea file ) c c c The approach can also be extended to incompressible MHD with non-unit c magnetic Prandtl number Pm. c c [1] Owen Walsh, "Eddy Solutions of the Navier-Stokes Equations," c in The Navier-Stokes Equations II - Theory and Numerical Methods, c Proceedings, Oberwolfach 1991, J.G. Heywood, K. Masuda, c R. Rautmann, S.A. Solonnikov, Eds., Springer-Verlag, pp. 306--309 c (1992). c