diff --git a/eddy_psi_omega/README b/eddy_psi_omega/README new file mode 100644 index 0000000..66759b4 --- /dev/null +++ b/eddy_psi_omega/README @@ -0,0 +1,66 @@ +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 diff --git a/eddy_psi_omega/SIZE b/eddy_psi_omega/SIZE new file mode 100644 index 0000000..e76dbba --- /dev/null +++ b/eddy_psi_omega/SIZE @@ -0,0 +1,97 @@ +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 diff --git a/eddy_psi_omega/psi_omega.box b/eddy_psi_omega/psi_omega.box new file mode 100644 index 0000000..ef65e4a --- /dev/null +++ b/eddy_psi_omega/psi_omega.box @@ -0,0 +1,14 @@ +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 diff --git a/eddy_psi_omega/psi_omega.map b/eddy_psi_omega/psi_omega.map new file mode 100644 index 0000000..e8ed124 --- /dev/null +++ b/eddy_psi_omega/psi_omega.map @@ -0,0 +1,257 @@ + 256 256 8 256 1024 256 0 + 218 226 225 205 173 + 219 225 233 173 175 + 222 233 232 175 174 + 223 232 229 174 180 + 202 229 234 180 164 + 203 234 238 164 166 + 206 238 237 166 165 + 207 237 228 165 207 + 250 228 230 207 194 + 251 230 231 194 196 + 254 231 235 196 195 + 255 235 236 195 201 + 234 236 239 201 185 + 235 239 248 185 187 + 238 248 247 187 186 + 239 247 226 186 205 + 216 205 173 204 178 + 217 173 175 178 176 + 220 175 174 176 177 + 221 174 180 177 179 + 200 180 164 179 169 + 201 164 166 169 167 + 204 166 165 167 168 + 205 165 207 168 203 + 248 207 194 203 199 + 249 194 196 199 197 + 252 196 195 197 198 + 253 195 201 198 200 + 232 201 185 200 190 + 233 185 187 190 188 + 236 187 186 188 189 + 237 186 205 189 204 + 210 204 178 206 170 + 211 178 176 170 172 + 214 176 177 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VISCOS + 0.000000E+00 BETAG + 0.000000E+00 GTHETA + 0.000000E+00 PGRADX + 0.000000E+00 FLOWRATE + 1.00000 RHOCP + -20.000 CONDUCT + 0.000000E+00 QVOL + 2.000000E+00 FINTIME + 0000.00 NSTEPS + -0.002 DT + 0.000000E+00 IOCOMM + 0.000000E+00 IOTIME + 000.0000E+00 IOSTEP + 0.000000E+00 EQTYPE + 0.000000E+00 AXIS + 0.250000E-01 GRID + -1.00000 INTYPE + 12.0000 NORDER + 1.000000E-10 DIVERGENCE + 1.000000E-09 HELMHOLTZ + 0.000000E+00 NPSCAL + 0.100000E-09 TOLREL + 0.100000E-09 TOLABS + 2.00000 COURANT + 3.00000 TORDER + 0.000000E+00 TMESH + 0.000000E+00 MAXCGHELM + 0.000000E+00 PoPRESSURE + 0.000000E+00 GAS CONS R + 0.000000E+00 Lo Ma GRAV + 100.000 MXGEOM + 100.000 MAXNEW + 0.100000E-03 + 0.100000E-03 XMAGNET + 0.100000E-01 NGRIDS + 0.000000E+00 NORDER2 + 0.000000E+00 NORDER3 + 0.000000E+00 NORDER4 + 0.000000E+00 NORDER5 + 0.000000E+00 SSFSLENG + 1.00000 VHScaleFac + 0.000000E+00 POISSONRA + 0.000000E+00 FSDTFAC + 0.000000E+00 IRSTART + 0.000000E+00 CNFAC + 0.000000E+00 PGEOM + 0.140000 MIXLFAC + 0.000000E+00 MAXCGFS + 0.000000E+00 + 0.000000E+00 HISTEP + 0.000000E+00 HISTIME + 0.000000E+00 RAMP_STEPS + 0.000000E+00 P55 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 new file each dump (1) +0 4.000000E+00 write form (0), unform(1), universal(2) +0 4.00000 read form (0), unform (1), universal(2) + 0.000000E+00 + 50000.0 frequency of srf dump + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 p77 = Oscillation frequency (omega) + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 coarse grid weighting (default=10.0) + 0.000000E+00 + 0.000000E+00 + 0.000000E+00 +0 20.0000 Number of pressure sol'ns saved +0 5.00000 start saving pressure soln's after this +0 5.00000 start saving pressure soln's after this +0 16.0000E+00 p96 = u0 ! translational velocity for eddy.usr +0 5.00000E+00 p97 = v0 ! translational velocity for eddy.usr + 0.000000E+00 + 3.00000 p99 = 3 ==> dealias + 0.000000E+00 Turn off overlapp. Schw. for splitting + 0.000000E+00 No. of pdumps + 1.00000 Dump out divergence at each time step + .000000 weight of filter + 4 Lines of passive scalar data follows2 CONDUCT; 2RHOCP + 1.00000 1.00000 1.00000 1.00000 1.00000 + 1.00000 1.00000 1.00000 1.00000 + 1.00000 1.00000 1.00000 1.00000 1.00000 + 1.00000 1.00000 1.00000 1.00000 + 13 LOGICAL SWITCHES FOLLOW + F IFFLOW + T IFHEAT + T IFTRAN + F T F F F F F F F F F IFNAV & IFADVC (convection in P.S. fields) + F F T T T T T T T T T T IFTMSH (IF mesh for this field is T mesh) + F IFAXIS + F IFSTRS + F IFSPLIT + F IFMGRID + F IFMODEL + F IFKEPS + F IFMVBD + F IFCHAR + 8.00000 8.00000 -0.500000 -4.00000 XFAC,YFAC,XZERO,YZERO + **MESH DATA** 1st line is X of corner 1,2,3,4. 2nd line is Y. + 256 2 256 NEL,NDIM,NELV 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168 [ 1l] GROUP 0 + 4.3750000E-01 5.0000000E-01 5.0000000E-01 4.3750000E-01 + 6.2500000E-01 6.2500000E-01 6.8750000E-01 6.8750000E-01 + ELEMENT 169 [ 1m] GROUP 0 + 5.0000000E-01 5.6250000E-01 5.6250000E-01 5.0000000E-01 + 6.2500000E-01 6.2500000E-01 6.8750000E-01 6.8750000E-01 + ELEMENT 170 [ 1n] GROUP 0 + 5.6250000E-01 6.2500000E-01 6.2500000E-01 5.6250000E-01 + 6.2500000E-01 6.2500000E-01 6.8750000E-01 6.8750000E-01 + ELEMENT 171 [ 1o] GROUP 0 + 6.2500000E-01 6.8750000E-01 6.8750000E-01 6.2500000E-01 + 6.2500000E-01 6.2500000E-01 6.8750000E-01 6.8750000E-01 + ELEMENT 172 [ 1p] GROUP 0 + 6.8750000E-01 7.5000000E-01 7.5000000E-01 6.8750000E-01 + 6.2500000E-01 6.2500000E-01 6.8750000E-01 6.8750000E-01 + ELEMENT 173 [ 1q] GROUP 0 + 7.5000000E-01 8.1250000E-01 8.1250000E-01 7.5000000E-01 + 6.2500000E-01 6.2500000E-01 6.8750000E-01 6.8750000E-01 + ELEMENT 174 [ 1r] GROUP 0 + 8.1250000E-01 8.7500000E-01 8.7500000E-01 8.1250000E-01 + 6.2500000E-01 6.2500000E-01 6.8750000E-01 6.8750000E-01 + 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POINTS. Hcode, I,J,H,IEL + ***** OUTPUT FIELD SPECIFICATION ***** + 6 SPECIFICATIONS FOLLOW + T COORDINATES + T VELOCITY + T PRESSURE + T TEMPERATURE + F TEMPERATURE GRADIENT + 0 PASSIVE SCALARS + ***** OBJECT SPECIFICATION ***** + 0 Surface Objects + 0 Volume Objects + 0 Edge Objects + 0 Point Objects diff --git a/eddy_psi_omega/psi_omega.usr b/eddy_psi_omega/psi_omega.usr new file mode 100644 index 0000000..c28074c --- /dev/null +++ b/eddy_psi_omega/psi_omega.usr @@ -0,0 +1,334 @@ +c----------------------------------------------------------------------- + subroutine uservp (ix,iy,iz,eg) + include 'SIZE' + include 'TOTAL' + include 'NEKUSE' + integer e,eg,f +C + udiff =0. + utrans=0. + return + end +c----------------------------------------------------------------------- + subroutine userf (ix,iy,iz,eg) + include 'SIZE' + include 'TOTAL' + include 'NEKUSE' + integer e,eg,f +C + ffx = 0.0 + ffy = 0.0 + ffz = 0.0 + return + end +c----------------------------------------------------------------------- + subroutine userq (ix,iy,iz,eg) + include 'SIZE' + include 'TOTAL' + include 'NEKUSE' + integer e,eg,f +C + qvol = 0.0 + source = 0.0 + return + end +c----------------------------------------------------------------------- + subroutine userchk + include 'SIZE' + include 'TOTAL' +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), + + ifield = 2 ! for outpost + + ifto = .true. + + n = nx1*ny1*nz1*nelv + +c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +c Compute streamfunction from vorticity + + call col3 (rhs,bm1,t,n) ! B*omega + 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 + call dudxyz(vy,psi,rxm1,sxm1,txm1,jacm1,1,1) ! quick to code... + call chsign(vx,n) + + call dsavg(vx) ! make velocity continuous + call dsavg(vy) + + u0 = param(96) + v0 = param(97) + call cadd (vx,u0,n) ! add base translation compoents + call cadd (vy,v0,n) +c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + + + +! ERROR ANALYSIS + + + visc = param(2) + u0 = param(96) + v0 = param(97) + call exact (ue,ve,xm1,ym1,n,time,visc,u0,v0) + if (istep.eq.0) call outpost(ue,ve,vx,pr,t,' ') + + call sub3 (ud,ue,vx,n) + call sub3 (vd,ve,vy,n) + + umx = glamax(vx,n) + vmx = glamax(vy,n) + uex = glamax(ue,n) + vex = glamax(ve,n) + udx = glamax(ud,n) + vdx = glamax(vd,n) + call copy (pr,ud,n) + + if (nid.eq.0) write(6,11) istep,time,udx,umx,uex,u0,' X err' +c if (nid.eq.0) write(6,11) istep,time,vdx,vmx,vex,v0,' Y err' + 11 format(i5,1p5e14.6,a7) + + + if (istep.le.5) then ! Reset velocity to eliminate + call copy (vx,ue,n) ! start-up contributions to + call copy (vy,ve,n) ! temporal-accuracy behavior. + + call dudxyz(h1,vx,rym1,sym1,tym1,jacm1,1,2) + call dudxyz(h2,vy,rxm1,sxm1,txm1,jacm1,1,1) + call sub3 (t,h1,h2,n) + call dsavg (t) + + endif + + vmx = glmax(vx,n) + vyx = glmax(vy,n) + tmx = glmax(t,n) + if (nid.eq.0) write(6,11) istep,time,tmx,vmx,vyx,v0,' Tmax ' + + return + end +c----------------------------------------------------------------------- + subroutine userbc (ix,iy,iz,iside,eg) + include 'SIZE' + include 'TOTAL' + include 'NEKUSE' + integer e,eg,f + ux=0.0 + uy=0.0 + uz=0.0 + temp=0.0 + return + end +c----------------------------------------------------------------------- + subroutine useric (ix,iy,iz,eg) + include 'SIZE' + include 'TOTAL' + include 'NEKUSE' + integer e,eg,f + + common /exacu/ ue(lx1,ly1,lz1,lelt),ve(lx1,ly1,lz1,lelt) + common /exacd/ ud(lx1,ly1,lz1,lelt),vd(lx1,ly1,lz1,lelt) + + integer icalld + save icalld + data icalld /0/ + + n = nx1*ny1*nz1*nelv + if (icalld.eq.0) then + icalld = icalld + 1 + time = 0. + u0 = param(96) + v0 = param(97) + call exact (ue,ve,xm1,ym1,n,time,visc,u0,v0) + endif + + e = gllel(eg) + ux=ue(ix,iy,iz,e) + uy=ve(ix,iy,iz,e) + uz=0.0 + temp=0 + + return + end +c----------------------------------------------------------------------- + subroutine usrdat + include 'SIZE' + include 'TOTAL' + integer e + + one = 1. + twopi = 8.*atan(one) + + do e=1,nelv ! Rescale mesh to [0,2pi]^2 + do i=1,4 ! Assumes original domain in .rea file on [0,1] + xc(i,e) = twopi*xc(i,e) + yc(i,e) = twopi*yc(i,e) + enddo + enddo + + param(66) = 0 + param(67) = 0 + + return + end +c----------------------------------------------------------------------- + subroutine usrdat2 + return + end +c----------------------------------------------------------------------- + subroutine usrdat3 + return + end +c----------------------------------------------------------------------- + subroutine exact(uu,vv,xx,yy,n,time,visc,u0,v0) +c +c This routine creates initial conditions for an exact 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 +c The approach can also be extended to incompressible MHD with 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 2/23/02; 6/2/09; pff +c +c + include 'SIZE' + include 'INPUT' +c + real uu(n),vv(n),xx(n),yy(n) +c + real cpsi(2,5), a(2,5) + save cpsi , a + +c data a / .4,.45 , .4,.2 , -.2,-.1 , .2,.05, -.09,-.1 / ! See eddy.m +c data cpsi / 0,65 , 16,63 , 25,60 , 33,56 , 39,52 / ! See squares.f +c data cpsi / 0,85 , 13,84 , 36,77 , 40,75 , 51,68 / + + +c This data from Walsh's Figure 1 [1]: + + data a / -.2,-.2, .25,0., 0,0 , 0,0 , 0,0 / + data cpsi / 0, 5 , 3, 4 , 0,0 , 0,0 , 0,0 / + + one = 1. + pi = 4.*atan(one) + + aa = cpsi(2,1)**2 + arg = -visc*time*aa ! domain is [0:2pi] + e = exp(arg) +c +c ux = psi_y, uy = -psi_x +c + do i=1,n + x = xx(i) - u0*time + y = yy(i) - v0*time + + sx = sin(cpsi(2,1)*x) + cx = cos(cpsi(2,1)*x) + sy = sin(cpsi(2,1)*y) + cy = cos(cpsi(2,1)*y) + u = a(1,1)*cpsi(2,1)*cy + v = a(2,1)*cpsi(2,1)*sx + + do k=2,5 + s1x = sin(cpsi(1,k)*x) + c1x = cos(cpsi(1,k)*x) + s2x = sin(cpsi(2,k)*x) + c2x = cos(cpsi(2,k)*x) + + s1y = sin(cpsi(1,k)*y) + c1y = cos(cpsi(1,k)*y) + s2y = sin(cpsi(2,k)*y) + c2y = cos(cpsi(2,k)*y) + + c1 = cpsi(1,k) + c2 = cpsi(2,k) + + if (k.eq.2) u = u + a(1,k)*s1x*c2y*c2 + if (k.eq.2) v = v - a(1,k)*c1x*s2y*c1 + if (k.eq.2) u = u - a(2,k)*s2x*c1y*c1 + if (k.eq.2) v = v + a(2,k)*c2x*s1y*c2 + + if (k.eq.3) u = u - a(1,k)*s1x*c2y*c2 + if (k.eq.3) v = v + a(1,k)*c1x*s2y*c1 + if (k.eq.3) u = u - a(2,k)*c2x*c1y*c1 + if (k.eq.3) v = v - a(2,k)*s2x*s1y*c2 + + if (k.eq.4) u = u + a(1,k)*c1x*c2y*c2 + if (k.eq.4) v = v + a(1,k)*s1x*s2y*c1 + if (k.eq.4) u = u + a(2,k)*c2x*c1y*c1 + if (k.eq.4) v = v + a(2,k)*s2x*s1y*c2 + + if (k.eq.5) u = u - a(1,k)*s1x*c2y*c2 + if (k.eq.5) v = v + a(1,k)*c1x*s2y*c1 + if (k.eq.5) u = u - a(2,k)*s2x*c1y*c1 + if (k.eq.5) v = v + a(2,k)*c2x*s1y*c2 + enddo + uu(i) = u*e + u0 + vv(i) = v*e + v0 + enddo + + return + end +c-----------------------------------------------------------------------