clean up psi-omega example, pff

git-svn-id: https://svn.mcs.anl.gov/repos/nek5/examples@1019 24248562-e74d-0410-ab4b-d4d80c744c73

eddy_psi_omega/psi_omega.usr

@@ -1,3 +1,70 @@
+c-----------------------------------------------------------------------
+c
+c     This case demonstrates how to use the Nek5000 convection-diffusion
+c     solver as a 2D Navier-Stokes 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
 c-----------------------------------------------------------------------
       subroutine uservp (ix,iy,iz,eg)
       include 'SIZE'
@@ -40,10 +107,10 @@ c
       common /exacu/ ue(lx1,ly1,lz1,lelt),ve(lx1,ly1,lz1,lelt)
       common /exacd/ ud(lx1,ly1,lz1,lelt),vd(lx1,ly1,lz1,lelt)
 
-      common /xtream/ psi(lx1*ly1*lz1*lelt),
-     $                rhs(lx1*ly1*lz1*lelt),
-     $                h1 (lx1*ly1*lz1*lelt),
-     $                h2 (lx1*ly1*lz1*lelt),
+      common /xtream/ psi(lx1*ly1*lz1*lelt)
+     $              , rhs(lx1*ly1*lz1*lelt)
+     $              , h1 (lx1*ly1*lz1*lelt)
+     $              , h2 (lx1*ly1*lz1*lelt)
 
       ifield = 2  ! for outpost
 
@@ -58,14 +125,9 @@ c     Compute streamfunction from vorticity
       call rone (h1 ,n)
       call rzero(h2 ,n)
 
-      eps = 1.e-4
-      call add2s2(rhs,binvm1,eps,n)
-
       ifield = 2
       tol    = param(22)
       imsh   = 2
-      write(6,12) istep,rhs(1),h1(1),h2(1),tmask(1,1,1,1,1),'RHS'
-   12 format(i5,1p4e14.6,1x,a3)
       call hmholtz('psi ',psi,rhs,h1,h2,tmask,tmult,imsh,tol,200,1)
 
       call dudxyz(vx,psi,rym1,sym1,tym1,jacm1,1,2) ! Inefficient, but
@@ -185,8 +247,8 @@ c-----------------------------------------------------------------------
       enddo 
       enddo 
 
-c     param(66) = 0
-c     param(67) = 0
+      param(66) = 0
+      param(67) = 0
 
       return
       end