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eddy_psi_omega

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