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nse3d.f90
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nse3d.f90
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PROGRAM main
! This program numerically solves the 3D incompressible Navier-Stokes
! on a Cubic Domain [0,2pi]x[0,2pi]x[0,2pi] using pseudo-spectral methods and
! Implicit Midpoint rule timestepping. The numerical solution is compared to
! an exact solution reported by Shapiro
!
! Analytical Solution:
! u(x,y,z,t)=-0.25*(cos(x)sin(y)sin(z)+sin(x)cos(y)cos(z))exp(-t/Re)
! v(x,y,z,t)= 0.25*(sin(x)cos(y)sin(z)-cos(x)sin(y)cos(z))exp(-t/Re)
! w(x,y,z,t)= 0.5*cos(x)cos(y)sin(z)exp(-t/Re)
! gfortran nse3d.f90 -lfftw3 -lm
IMPLICIT NONE
INTEGER(kind=4), PARAMETER :: Nx=128, Ny=128, Nz=128
INTEGER(kind=4), PARAMETER :: Lx=1, Ly=1, Lz=1, Nt=20
REAL(kind=8), PARAMETER :: dt=0.2d0/Nt, Re=1.0d0, tol=0.1d0**10, theta=0.0d0
REAL(kind=8), PARAMETER :: pi=3.14159265358979323846264338327950288419716939937510d0
REAL(kind=8), PARAMETER :: ReInv=1.0d0/REAL(Re,kind(0d0)), dtInv=1.0d0/REAL(dt,kind(0d0))
REAL(kind=8) :: chg, factor, s, e, scalemodes=1.0d0/REAL(Nx*Ny*Nz,kind(0d0))
REAL(kind=8), DIMENSION(:), ALLOCATABLE :: x, y, z, time
COMPLEX(kind=8), DIMENSION(:), ALLOCATABLE :: kx, ky, kz
COMPLEX(kind=8), DIMENSION(:,:,:), ALLOCATABLE :: u, v, w, ux, uy, uz, vx, vy, vz, wx, wy, wz, uold, uxold, uyold, &
uzold, vold, vxold, vyold, vzold, wold, wxold, wyold, wzold,&
utemp, vtemp, wtemp, temp_r
COMPLEX(kind=8), DIMENSION(:,:,:), ALLOCATABLE :: uhat, vhat, what, rhsuhatfix, rhsvhatfix, rhswhatfix, nonlinuhat,&
nonlinvhat, nonlinwhat, phat,temp_c
REAL(kind=8), DIMENSION(:,:,:), ALLOCATABLE :: realtemp
INTEGER(kind=4), PARAMETER :: FFTW_IN_PLACE = 8, FFTW_MEASURE = 0, FFTW_EXHAUSTIVE = 8,&
FFTW_PATIENT = 32, FFTW_ESTIMATE = 64, FFTW_FORWARD = -1, FFTW_BACKWARD=1
INTEGER(kind=8) :: planfxyz, planbxyz
INTEGER(kind=4) :: i, j, k, n, t, allocatestatus, ind, ierr
PRINT *,'Grid:',Nx,'X',Ny,'Y',Nz,'Z'
PRINT *,'dt:',dt
ALLOCATE(x(1:Nx),y(1:Ny),z(1:Nz),time(1:Nt+1),u(1:Nx,1:Ny,1:Nz),&
v(1:Nx,1:Ny,1:Nz), w(1:Nx,1:Ny,1:Nz), ux(1:Nx,1:Ny,1:Nz),&
uy(1:Nx,1:Ny,1:Nz), uz(1:Nx,1:Ny,1:Nz), vx(1:Nx,1:Ny,1:Nz),&
vy(1:Nx,1:Ny,1:Nz), vz(1:Nx,1:Ny,1:Nz), wx(1:Nx,1:Ny,1:Nz),&
wy(1:Nx,1:Ny,1:Nz), wz(1:Nx,1:Ny,1:Nz), uold(1:Nx,1:Ny,1:Nz),&
uxold(1:Nx,1:Ny,1:Nz), uyold(1:Nx,1:Ny,1:Nz), uzold(1:Nx,1:Ny,1:Nz),&
vold(1:Nx,1:Ny,1:Nz), vxold(1:Nx,1:Ny,1:Nz), vyold(1:Nx,1:Ny,1:Nz),&
vzold(1:Nx,1:Ny,1:Nz), wold(1:Nx,1:Ny,1:Nz), wxold(1:Nx,1:Ny,1:Nz),&
wyold(1:Nx,1:Ny,1:Nz), wzold(1:Nx,1:Ny,1:Nz), utemp(1:Nx,1:Ny,1:Nz),&
vtemp(1:Nx,1:Ny,1:Nz), wtemp(1:Nx,1:Ny,1:Nz), temp_r(1:Nx,1:Ny,1:Nz),&
kx(1:Nx),ky(1:Ny),kz(1:Nz),uhat(1:Nx,1:Ny,1:Nz), vhat(1:Nx,1:Ny,1:Nz),&
what(1:Nx,1:Ny,1:Nz), rhsuhatfix(1:Nx,1:Ny,1:Nz),&
rhsvhatfix(1:Nx,1:Ny,1:Nz), rhswhatfix(1:Nx,1:Ny,1:Nz),&
nonlinuhat(1:Nx,1:Ny,1:Nz), nonlinvhat(1:Nx,1:Ny,1:Nz),&
nonlinwhat(1:Nx,1:Ny,1:Nz), phat(1:Nx,1:Ny,1:Nz),temp_c(1:Nx,1:Ny,1:Nz),&
realtemp(1:Nx,1:Ny,1:Nz), stat=AllocateStatus)
IF (AllocateStatus .ne. 0) STOP
CALL dfftw_plan_dft_3d_(planfxyz,Nx,Ny,Nz,temp_r(1:Nx,1:Ny,1:Nz),&
temp_c(1:Nx,1:Ny,1:Nz),FFTW_FORWARD,FFTW_MEASURE)
CALL dfftw_plan_dft_3d_(planbxyz,Nx,Ny,Nz,temp_c(1:Nx,1:Ny,1:Nz),&
temp_r(1:Nx,1:Ny,1:Nz),FFTW_BACKWARD,FFTW_MEASURE)
CALL cpu_time(s)
DO i=1,Nx/2+1
kx(i)= cmplx(0.0d0,1.0d0)*REAL(i-1,kind(0d0))/Lx
END DO
kx(1+Nx/2)=0.0d0
DO i = 1,Nx/2 -1
kx(i+1+Nx/2)=-kx(1-i+Nx/2)
END DO
ind=1
DO i=-Nx/2,Nx/2-1
x(ind)=2.0d0*pi*REAL(i,kind(0d0))*Lx/REAL(Nx,kind(0d0))
ind=ind+1
END DO
DO j=1,Ny/2+1
ky(j)= cmplx(0.0d0,1.0d0)*REAL(j-1,kind(0d0))/Ly
END DO
ky(1+Ny/2)=0.0d0
DO j = 1,Ny/2 -1
ky(j+1+Ny/2)=-ky(1-j+Ny/2)
END DO
ind=1
DO j=-Ny/2,Ny/2-1
y(ind)=2.0d0*pi*REAL(j,kind(0d0))*Ly/REAL(Ny,kind(0d0))
ind=ind+1
END DO
DO k=1,Nz/2+1
kz(k)= cmplx(0.0d0,1.0d0)*REAL(k-1,kind(0d0))/Lz
END DO
kz(1+Nz/2)=0.0d0
DO k = 1,Nz/2 -1
kz(k+1+Nz/2)=-kz(1-k+Nz/2)
END DO
ind=1
DO k=-Nz/2,Nz/2-1
z(ind)=2.0d0*pi*REAL(k,kind(0d0))*Lz/REAL(Nz,kind(0d0))
ind=ind+1
END DO
time(1)=0.0d0
factor=sqrt(3.0d0)
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
u(i,j,k)=-0.5*( factor*cos(x(i))*sin(y(j))*sin(z(k))&
+sin(x(i))*cos(y(j))*cos(z(k)) )*exp(-(factor**2)*time(1)/Re)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
v(i,j,k)=0.5*( factor*sin(x(i))*cos(y(j))*sin(z(k))&
-cos(x(i))*sin(y(j))*cos(z(k)) )*exp(-(factor**2)*time(1)/Re)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
w(i,j,k)=cos(x(i))*cos(y(j))*sin(z(k))*exp(-(factor**2)*time(1)/Re)
END DO
END DO
END DO
CALL dfftw_execute_dft_(planfxyz,u(1:Nx,1:Ny,1:Nz),uhat(1:Nx,1:Ny,1:Nz))
CALL dfftw_execute_dft_(planfxyz,v(1:Nx,1:Ny,1:Nz),vhat(1:Nx,1:Ny,1:Nz))
CALL dfftw_execute_dft_(planfxyz,w(1:Nx,1:Ny,1:Nz),what(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=uhat(i,j,k)*kx(i)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),ux(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=uhat(i,j,k)*ky(j)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),uy(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=uhat(i,j,k)*kz(k)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),uz(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=vhat(i,j,k)*kx(i)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),vx(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=vhat(i,j,k)*ky(j)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),vy(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=vhat(i,j,k)*kz(k)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),vz(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=what(i,j,k)*kx(i)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),wx(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=what(i,j,k)*ky(j)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),wy(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=what(i,j,k)*kz(k)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),wz(1:Nx,1:Ny,1:Nz))
time(1)=0.0
n=0
DO n=1,Nt
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
uold(i,j,k)=u(i,j,k)
uxold(i,j,k)=ux(i,j,k)
uyold(i,j,k)=uy(i,j,k)
uzold(i,j,k)=uz(i,j,k)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
vold(i,j,k)=v(i,j,k)
vxold(i,j,k)=vx(i,j,k)
vyold(i,j,k)=vy(i,j,k)
vzold(i,j,k)=vz(i,j,k)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
wold(i,j,k)=w(i,j,k)
wxold(i,j,k)=wx(i,j,k)
wyold(i,j,k)=wy(i,j,k)
wzold(i,j,k)=wz(i,j,k)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
rhsuhatfix(i,j,k) = (dtInv+(0.5d0*ReInv)*&
(kx(i)*kx(i)+ky(j)*ky(j)+kz(k)*kz(k)))*uhat(i,j,k)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
rhsvhatfix(i,j,k) = (dtInv+(0.5d0*ReInv)*&
(kx(i)*kx(i)+ky(j)*ky(j)+kz(k)*kz(k)))*vhat(i,j,k)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
rhswhatfix(i,j,k) = (dtInv+(0.5d0*ReInv)*&
(kx(i)*kx(i)+ky(j)*ky(j)+kz(k)*kz(k)))*what(i,j,k)
END DO
END DO
END DO
chg=1
DO WHILE (chg > tol)
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_r(i,j,k)=0.25d0*((u(i,j,k)+uold(i,j,k))*(ux(i,j,k)+uxold(i,j,k))&
+(v(i,j,k)+vold(i,j,k))*(uy(i,j,k)+uyold(i,j,k))&
+(w(i,j,k)+wold(i,j,k))*(uz(i,j,k)+uzold(i,j,k)))
END DO
END DO
END DO
CALL dfftw_execute_dft_(planfxyz,temp_r(1:Nx,1:Ny,1:Nz),nonlinuhat(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_r(i,j,k)=0.25d0*((u(i,j,k)+uold(i,j,k))*(vx(i,j,k)+vxold(i,j,k))&
+(v(i,j,k)+vold(i,j,k))*(vy(i,j,k)+vyold(i,j,k))&
+(w(i,j,k)+wold(i,j,k))*(vz(i,j,k)+vzold(i,j,k)))
END DO
END DO
END DO
CALL dfftw_execute_dft_(planfxyz,temp_r(1:Nx,1:Ny,1:Nz),nonlinvhat(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_r(i,j,k)=0.25d0*((u(i,j,k)+uold(i,j,k))*(wx(i,j,k)+wxold(i,j,k))&
+(v(i,j,k)+vold(i,j,k))*(wy(i,j,k)+wyold(i,j,k))&
+(w(i,j,k)+wold(i,j,k))*(wz(i,j,k)+wzold(i,j,k)))
END DO
END DO
END DO
CALL dfftw_execute_dft_(planfxyz,temp_r(1:Nx,1:Ny,1:Nz),nonlinwhat(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
phat(i,j,k)=-1.0d0*( kx(i)*nonlinuhat(i,j,k)+ky(j)*nonlinvhat(i,j,k)&
+kz(k)*nonlinwhat(i,j,k))/(kx(i)*kx(i)+ky(j)*ky(j)+kz(k)*kz(k)+0.1d0**13)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
uhat(i,j,k)=(rhsuhatfix(i,j,k)-nonlinuhat(i,j,k)-kx(i)*phat(i,j,k))/&
(dtInv-(0.5d0*ReInv)*(kx(i)*kx(i)+ky(j)*ky(j)+kz(k)*kz(k)))
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
vhat(i,j,k)=(rhsvhatfix(i,j,k)-nonlinvhat(i,j,k)-ky(j)*phat(i,j,k))/&
(dtInv-(0.5d0*ReInv)*(kx(i)*kx(i)+ky(j)*ky(j)+kz(k)*kz(k)))
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
what(i,j,k)=(rhswhatfix(i,j,k)-nonlinwhat(i,j,k)-kz(k)*phat(i,j,k))/&
(dtInv-(0.5d0*ReInv)*(kx(i)*kx(i)+ky(j)*ky(j)+kz(k)*kz(k)))
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=uhat(i,j,k)*kx(i)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),ux(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=uhat(i,j,k)*ky(j)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),uy(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=uhat(i,j,k)*kz(k)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),uz(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=vhat(i,j,k)*kx(i)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),vx(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=vhat(i,j,k)*ky(j)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),vy(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=vhat(i,j,k)*kz(k)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),vz(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=what(i,j,k)*kx(i)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),wx(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=what(i,j,k)*ky(j)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),wy(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
temp_c(i,j,k)=what(i,j,k)*kz(k)*scalemodes
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,temp_c(1:Nx,1:Ny,1:Nz),wz(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
utemp(i,j,k)=u(i,j,k)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
vtemp(i,j,k)=v(i,j,k)
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
wtemp(i,j,k)=w(i,j,k)
END DO
END DO
END DO
CALL dfftw_execute_dft_(planbxyz,uhat(1:Nx,1:Ny,1:Nz),u(1:Nx,1:Ny,1:Nz))
CALL dfftw_execute_dft_(planbxyz,vhat(1:Nx,1:Ny,1:Nz),v(1:Nx,1:Ny,1:Nz))
CALL dfftw_execute_dft_(planbxyz,what(1:Nx,1:Ny,1:Nz),w(1:Nx,1:Ny,1:Nz))
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
u(i,j,k)=u(i,j,k)*scalemodes
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
v(i,j,k)=v(i,j,k)*scalemodes
END DO
END DO
END DO
DO k=1,Nz
DO j=1,Ny
DO i=1,Nx
w(i,j,k)=w(i,j,k)*scalemodes
END DO
END DO
END DO
chg =maxval(abs(utemp-u))+maxval(abs(vtemp-v))+maxval(abs(wtemp-w))
!PRINT *,'chg:',chg
END DO
time(n+1)=n*dt
PRINT *,'time',n*dt
END DO
! DO k=1,Nz
! DO j=1,Ny
! DO i=1,Nx
! utemp(i,j,k)=u(i,j,k)-(-0.5*( factor*cos(x(i))*sin(y(j))*sin(z(k))&
! +sin(x(i))*cos(y(j))*cos(z(k)) )*exp(-(factor**2)*time(Nt+1)/Re))
! END DO
! END DO
! END DO
! DO k=1,Nz
! DO j=1,Ny
! DO i=1,Nx
! vtemp(i,j,k)=v(i,j,k)-(0.5*( factor*sin(x(i))*cos(y(j))*sin(z(k))&
! -cos(x(i))*sin(y(j))*cos(z(k)) )*exp(-(factor**2)*time(Nt+1)/Re))
! END DO
! END DO
! END DO
! DO k=1,Nz
! DO j=1,Ny
! DO i=1,Nx
! wtemp(i,j,k)=w(i,j,k)-(cos(x(i))*cos(y(j))*sin(z(k))*exp(-(factor**2)*time(Nt+1)/Re))
! END DO
! END DO
! END DO
! chg=maxval(abs(utemp))+maxval(abs(vtemp))+maxval(abs(wtemp))
! PRINT*,'The error at the final timestep is',chg
CALL cpu_time(e)
PRINT *, "Time took to execute: ", (e-s), "s"
CALL dfftw_destroy_plan_(planfxyz)
CALL dfftw_destroy_plan_(planbxyz)
DEALLOCATE(x,y,z,time,u,v,w,ux,uy,uz,vx,vy,vz,wx,wy,wz,uold,uxold,uyold,uzold,&
vold,vxold,vyold,vzold,wold,wxold,wyold,wzold,utemp,vtemp,wtemp,&
temp_r,kx,ky,kz,uhat,vhat,what,rhsuhatfix,rhsvhatfix,&
rhswhatfix,phat,nonlinuhat,nonlinvhat,nonlinwhat,temp_c,&
realtemp,stat=AllocateStatus)
IF (AllocateStatus .ne. 0) STOP
PRINT *,'Program execution complete'
END PROGRAM main