added streamfunction/vorticity example

git-svn-id: https://svn.mcs.anl.gov/repos/nek5/examples@1004 24248562-e74d-0410-ab4b-d4d80c744c73

eddy_psi_omega/README

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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

eddy_psi_omega/SIZE

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+C     Dimension file to be included
+C
+C     HCUBE array dimensions
+C
+      parameter (ldim=2)
+      parameter (lx1=16,ly1=lx1,lz1=1,lelt=300,lelv=lelt)
+      parameter (lxd=3*lx1/2,lyd=lxd,lzd=1)
+      parameter (lelx=20,lely=20,lelz=1)
+c
+      parameter (lzl=3 + 2*(ldim-3))
+c
+      parameter (lx2=lx1-2)
+      parameter (ly2=ly1-2)
+      parameter (lz2=lz1  )
+      parameter (lx3=lx1)
+      parameter (ly3=ly1)
+      parameter (lz3=lz1)
+c
+c     parameter (lpelv=lelv,lpelt=lelt,lpert=3)  ! perturbation
+c     parameter (lpx1=lx1,lpy1=ly1,lpz1=lz1)     ! array sizes
+c     parameter (lpx2=lx2,lpy2=ly2,lpz2=lz2)
+c
+      parameter (lpelv=1,lpelt=1,lpert=1)        ! perturbation
+      parameter (lpx1=1,lpy1=1,lpz1=1)           ! array sizes
+      parameter (lpx2=1,lpy2=1,lpz2=1)
+c
+c
+c     parameter (lbelv=lelv,lbelt=lelt)          ! MHD
+c     parameter (lbx1=lx1,lby1=ly1,lbz1=lz1)     ! array sizes
+c     parameter (lbx2=lx2,lby2=ly2,lbz2=lz2)
+c
+      parameter (lbelv=1,lbelt=1)                ! MHD
+      parameter (lbx1=1,lby1=1,lbz1=1)           ! array sizes
+      parameter (lbx2=1,lby2=1,lbz2=1)
+c
+C LX1M=LX1 when there are moving meshes; =1 otherwise
+      parameter (lx1m=1,ly1m=1,lz1m=1)
+      parameter (ldimt= 2)                       ! 2 passive scalars + T
+      parameter (ldimt1=ldimt+1)
+      parameter (ldimt3=ldimt+3)
+      parameter (lp = 512)
+      parameter (lelg = 4100)
+c
+c     Note:  In the new code, LELGEC should be about sqrt(LELG)
+c
+      PARAMETER (LELGEC = 1)
+      PARAMETER (LXYZ2  = 1)
+      PARAMETER (LXZ21  = 1)
+c
+      PARAMETER (LMAXV=LX1*LY1*LZ1*LELV)
+      PARAMETER (LMAXT=LX1*LY1*LZ1*LELT)
+      PARAMETER (LMAXP=LX2*LY2*LZ2*LELV)
+      PARAMETER (LXZ=LX1*LZ1)
+      PARAMETER (LORDER=3)
+      PARAMETER (MAXOBJ=4,MAXMBR=LELT*6,lhis=100)
+C
+C     Common Block Dimensions
+C
+      PARAMETER (LCTMP0 =2*LX1*LY1*LZ1*LELT)
+      PARAMETER (LCTMP1 =4*LX1*LY1*LZ1*LELT)
+C
+C     The parameter LVEC controls whether an additional 42 field arrays
+C     are required for Steady State Solutions.  If you are not using
+C     Steady State, it is recommended that LVEC=1.
+C
+      PARAMETER (LVEC=1)
+C
+C     Uzawa projection array dimensions
+C
+      parameter (mxprev = 20)
+      parameter (lgmres = 30)
+C
+C     Split projection array dimensions
+C
+      parameter(lmvec = 1)
+      parameter(lsvec = 1)
+      parameter(lstore=lmvec*lsvec)
+c
+c     NONCONFORMING STUFF
+c
+      parameter (maxmor = lelt)
+C
+C     Array dimensions
+C
+      COMMON/DIMN/NELV,NELT,NX1,NY1,NZ1,NX2,NY2,NZ2
+     $,NX3,NY3,NZ3,NDIM,NFIELD,NPERT,NID
+     $,NXD,NYD,NZD
+
+c automatically added by makenek
+      parameter(lxo   = lx1) ! max output grid size (lxo>=lx1)
+
+c automatically added by makenek
+      parameter(lpart = 1  ) ! max number of particles
+
+c automatically added by makenek
+      integer ax1,ay1,az1,ax2,ay2,az2
+      parameter (ax1=lx1,ay1=ly1,az1=lz1,ax2=lx2,ay2=ly2,az2=lz2) ! running averages

eddy_psi_omega/psi_omega.box

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+base.rea
+2                      spatial dimension
+1                      number of fields
+#
+#    comments  Lee Ho Thesis, p. 76, p.128 
+#
+#
+#========================================================
+#
+Box
+-16  -16                       nelx,nely,nelz for Box
+0   1.  1.                     x0 x1 ratio 
+0   1.  1.                     y0 y1 ratio 
+P  ,P  ,P  ,P  ,              bc's  ! NOTE sigma given in usrdat2