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viscous.f90
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! *****************************************************************************
! * *
! * ECHO-QGP *
! * *
! * Version: 1.5.0-alpha *
! * *
! * Copyright (C) 2015 The ECHO-QGP team *
! * *
! * File: viscous.f90 *
! * *
! * License: GPL version 2.0 (Please, read the file LICENSE.TXT) *
! * *
! * This program is free software; you can redistribute it and/or *
! * modify it under the terms of the GNU General Public License *
! * as published by the Free Software Foundation; either version 2 *
! * of the License, or (at your option) any later version. *
! * *
! * This program is distributed in the hope that it will be useful, *
! * but WITHOUT ANY WARRANTY; without even the implied warranty of *
! * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
! * GNU General Public License for more details. *
! * *
! * You should have received a copy of the GNU General Public License *
! * along with this program; if not, write to the Free Software *
! * Foundation, Inc., 51 Franklin Street, Fifth Floor, *
! * Boston, MA 02110-1301, USA. *
! * *
! * Authors: Gabriele Inghirami (inghirami@fias.uni-frankfurt.de) *
! * *
! * Contributors: Vinod Chandra (vchandra@iitgn.ac.in) *
! * *
! * Acknowledgments: Luca Del Zanna (delzanna@unifi.it) *
! * *
! *****************************************************************************
module viscous
use parallel, only: ipe
use common, only: nv,krh,kvx,kvy,kvz,kpr,kpibu,kpizz,kpixz,kpixy,kpiyz,kpixx,kpiyy,g_cov,pitt,timeold,pitx,pity,pitz
use common, only: t, coordinates,nx,ny,nz, timeinterval, hbar,viscosity,g_cov0,init_type,g_cov_3_old,bulkvis
use common, only: dtx, dty, dtz, dxx, dxy, dxz, dyx, dyy, dyz, dzx, dzy, dzz, deriv, dtt, dxt, dyt, dzt, dtet
use common, only: ix1,ix2,iy1,iy2,iz1,iz2,run_crashed,derivatives_out,eta_over_s,tau_pi_coeff
use common, only: GUBSER_VISCOUS, SHEAR_VISCOUS_1D, GLAUBER_GEOMETRIC, TABULATED_INIT, MINKOWSKI, BJORKEN, GLAUBER_MONTECARLO
implicit none
integer, parameter :: nsigma=6 !number of sigma components needed to compute the viscosity tensor
integer, parameter :: sxz=5, sxy=4, syz=6, sxx=1, syy=2, szz=3 !stt=7, stx=8, sty=9, stz=10
real(8) :: tau_pi, eta_vis, zeta_vis, tau_pi_big
real(8), dimension(1:nsigma) :: sigma
contains
! ***********************************************************************************
! surbroutine dts. derive vs time simple
subroutine dts(quantity_array, quantity_array_old)
use eos
implicit none
integer :: ix, iy, iz
real(8), allocatable, dimension(:,:,:,:) :: quantity_array, quantity_array_old
real(8) :: gamma2,gamma1
real(8) :: temperature_old, temperature_cur
real(8) :: energy_bb, entropy_bb
integer :: errcode !AAA DDD this error code should be passed as a return value to the caller subroutine
if(timeinterval .eq. 0) then
deriv(:,:,:,dtt:dzt)=0.
return
end if
do iz=iz1,iz2
do iy=iy1,iy2
do ix=ix1,ix2
gamma1=1./sqrt(1.-g_cov(1)*quantity_array_old(ix,iy,iz,kvx)**2.-g_cov(2)*quantity_array_old(ix,iy,iz,kvy)**2.-&
&g_cov_3_old*quantity_array_old(ix,iy,iz,kvz)**2.)
gamma2=1./sqrt(1.-g_cov(1)*quantity_array(ix,iy,iz,kvx)**2.-g_cov(2)*quantity_array(ix,iy,iz,kvy)**2.-&
&g_cov(3)*quantity_array(ix,iy,iz,kvz)**2.)
deriv(ix,iy,iz,dtt)=(gamma2-gamma1)/(timeinterval)
deriv(ix,iy,iz,dxt:dzt)=(gamma2*quantity_array(ix,iy,iz,kvx:kvz)-gamma1*quantity_array_old(ix,iy,iz,kvx:kvz))/timeinterval
call get_derived_data(quantity_array_old(ix,iy,iz,krh), quantity_array_old(ix,iy,iz,kpr), energy_bb, temperature_old, &
&entropy_bb,errcode)
call get_derived_data(quantity_array(ix,iy,iz,krh), quantity_array(ix,iy,iz,kpr), energy_bb, temperature_cur, entropy_bb,&
&errcode)
if(derivatives_out) deriv(ix,iy,iz,dtet)=(temperature_cur-temperature_old)/timeinterval
end do
end do
end do
end subroutine dts
! ***********************************************************************************
subroutine viscous_initio(ix,iy,iz,nv,v,tauzero,errcode)
use eos
implicit none
integer,intent(in) :: ix,iy,iz, nv
real(8) :: dens, press, vx, vy, vz, tauzero
real(8) :: entropy_dens, energy_dens, temp
real(8), dimension(:,:,:,:), allocatable, intent(inout) :: v
integer :: errcode
errcode=0
dens=v(ix,iy,iz,krh)
vx=v(ix,iy,iz,kvx)
vy=v(ix,iy,iz,kvy)
vz=v(ix,iy,iz,kvz)
press=v(ix,iy,iz,kpr)
call get_derived_data(dens, press, energy_dens, temp, entropy_dens,errcode)
call get_viscous_parameters_0(eta_vis,zeta_vis,tau_pi, tau_pi_big, dens, press,errcode)
if(errcode .gt. 0) then
write(*,*) "An error occurred into the viscous_initio subroutine"
write(*,*) "Error code:", errcode
write(*,*) "Position on the grid: ix=",ix,"iy=",iy,"iz=",iz
run_crashed=.true.
return
end if
v(ix,iy,iz,kpibu:kpizz)=0.
if((init_type .eq. GLAUBER_GEOMETRIC) .or. (init_type .eq. TABULATED_INIT) .or. (init_type .eq. GLAUBER_MONTECARLO)) then
v(ix,iy,iz,kpizz)=-g_cov(3)*4.*eta_vis/(3.*tauzero)
v(ix,iy,iz,kpixx)=2.*eta_vis/(3.*tauzero)
v(ix,iy,iz,kpiyy)=2.*eta_vis/(3.*tauzero)
v(ix,iy,iz,kpibu)=-zeta_vis/tauzero
end if
end subroutine viscous_initio
! ***********************************************************************************
subroutine get_viscous_parameters_0(eta_vis,zeta_vis,tau_pi, tau_pi_big, dens, press,errcode)
use eos
implicit none
real(8) :: entropy_dens, energy_dens, temp,cs2
real(8), intent(in) :: dens, press
real(8), intent(out) :: eta_vis, zeta_vis, tau_pi, tau_pi_big
integer, intent(out) :: errcode
errcode=0
call get_derived_data(dens, press, energy_dens, temp, entropy_dens,errcode)
call eos_sound(dens,energy_dens,press,cs2,errcode)
eta_vis=hbar*eta_over_s*entropy_dens
if(init_type .eq. SHEAR_VISCOUS_1D) eta_vis=eta_over_s !for 1D viscous shear flow init_type
if(bulkvis) then
zeta_vis=2.*eta_vis*(1./3.-cs2)
else
zeta_vis=0.
end if
tau_pi=tau_pi_coeff*eta_vis/(temp*entropy_dens)
if(init_type .eq. GUBSER_VISCOUS) tau_pi=5.*eta_vis/(temp*entropy_dens)
tau_pi_big=tau_pi
end subroutine get_viscous_parameters_0
! ***********************************************************************************
subroutine get_viscous_parameters(eta_vis,zeta_vis,tau_pi, tau_pi_big, dens, press,errcode)
use eos
implicit none
real(8) :: entropy_dens, energy_dens, temp, cs2
real(8), intent(in) :: dens, press
real(8), intent(out) :: eta_vis, zeta_vis, tau_pi, tau_pi_big
integer, intent(out) :: errcode
call get_derived_data(dens, press, energy_dens, temp, entropy_dens,errcode)
call eos_sound(dens,energy_dens,press,cs2,errcode)
eta_vis=hbar*eta_over_s*entropy_dens
if(init_type .eq. SHEAR_VISCOUS_1D) eta_vis=eta_over_s ! for 1D viscous shear flow init_type
if(bulkvis) then
zeta_vis=2.*eta_vis*(1./3.-cs2)
else
zeta_vis=0.
end if
if((temp .eq. 0) .or. (entropy_dens .eq. 0) ) then
tau_pi=0.
else
tau_pi=tau_pi_coeff*eta_vis/(temp*entropy_dens)
end if
if(init_type .eq. GUBSER_VISCOUS) tau_pi=5.*eta_vis/(temp*entropy_dens)
tau_pi_big=tau_pi
end subroutine get_viscous_parameters
! ***********************************************************************************
subroutine get_derived_pi_gcv3(p,pitt,pitx,pity,pitz,gcv3)
implicit none
real(8), intent(in), dimension(:) :: p
real(8), intent(inout) :: pitt,pitx,pity,pitz
real(8), intent(in) :: gcv3
!pitt=(g_cov(1)*p(kpixx)+g_cov(2)*p(kpiyy)+gcv3*p(kpizz))/(-g_cov0) !traceless condition
pitx=p(kpixx)*p(kvx)*g_cov(1)+p(kpixy)*p(kvy)*g_cov(2)+p(kpixz)*p(kvz)*gcv3
pity=p(kpixy)*p(kvx)*g_cov(1)+p(kpiyy)*p(kvy)*g_cov(2)+p(kpiyz)*p(kvz)*gcv3
pitz=p(kpixz)*p(kvx)*g_cov(1)+p(kpiyz)*p(kvy)*g_cov(2)+p(kpizz)*p(kvz)*gcv3
pitt=pitx*p(kvx)*g_cov(1)+pity*p(kvy)*g_cov(2)+pitz*p(kvz)*gcv3
end subroutine get_derived_pi_gcv3
! ***********************************************************************************
subroutine get_derived_pi_zz_gcv3(p,pitt,pitx,pity,pitz,pizz,gcv3)
implicit none
real(8), intent(in), dimension(:) :: p
real(8), intent(inout) :: pitt,pitx,pity,pitz,pizz
real(8), intent(in) :: gcv3
pitt=1./(1.-p(kvz)*p(kvz)*gcv3)*(p(kvx)*g_cov(1)*p(kpixy)*p(kvy)*g_cov(2)+p(kvx)*p(kvx)*g_cov(1)*g_cov(1)*p(kpixx)+&
&p(kvy)*g_cov(1)*p(kpiyz)*p(kvz)*gcv3+2.*p(kvx)*g_cov(1)*p(kpixz)*p(kvz)*gcv3+p(kvy)*g_cov(1)*g_cov(1)*p(kpixy)*&
&p(kvx)+p(kvy)*p(kvy)*g_cov(1)*p(kpiyy)*g_cov(2)+p(kvz)*gcv3*p(kpiyz)*p(kvy)*g_cov(2)-p(kpixx)*g_cov(1)*p(kvz)*p(kvz)*&
&gcv3-p(kpiyy)*g_cov(2)*p(kvz)*p(kvz)*gcv3)
pitx=p(kpixx)*p(kvx)*g_cov(1)+p(kpixy)*p(kvy)*g_cov(2)+p(kpixz)*p(kvz)*gcv3
pity=p(kpixy)*p(kvx)*g_cov(1)+p(kpiyy)*p(kvy)*g_cov(2)+p(kpiyz)*p(kvz)*gcv3
pitz=1./(1.-p(kvz)*p(kvz)*gcv3)*(p(kpixz)*p(kvx)*g_cov(1)+p(kpixz)*p(kvx)*g_cov(1)*p(kvz)*p(kvz)*gcv3+p(kpiyz)*p(kvy)*&
&g_cov(2)+p(kvz)*p(kvx)*g_cov(1)*p(kpixy)*p(kvy)*g_cov(2)-p(kvz)*p(kpixx)*g_cov(1)-p(kvz)*p(kpiyy)*g_cov(2)+p(kvz)*p(kvx)*&
&p(kvx)*g_cov(1)*g_cov(1)*p(kpixx)+p(kvy)*g_cov(1)*p(kpiyz)*p(kvz)*p(kvz)*gcv3+p(kvz)*p(kvy)*g_cov(1)*g_cov(1)*&
&p(kpixy)*p(kvx)+p(kvz)*p(kvy)*p(kvy)*g_cov(1)*p(kpiyy)*g_cov(2))
pizz=1./(gcv3*(1.-p(kvz)*p(kvz)*gcv3))*(p(kvx)*g_cov(1)*p(kpixy)*p(kvy)*g_cov(2)-p(kpixx)*g_cov(1)-p(kpiyy)*g_cov(2)+&
&p(kvx)*p(kvx)*g_cov(1)*g_cov(1)*p(kpixx)+p(kvy)*g_cov(1)*p(kpiyz)*p(kvz)*gcv3+2.*p(kvx)*g_cov(1)*p(kpixz)*p(kvz)*&
&gcv3+p(kvy)*g_cov(1)*g_cov(1)*p(kpixy)*p(kvx)+p(kvy)*p(kvy)*g_cov(1)*p(kpiyy)*g_cov(2)+p(kvz)*gcv3*p(kpiyz)*&
&p(kvy)*g_cov(2))
end subroutine get_derived_pi_zz_gcv3
! ***********************************************************************************
subroutine get_derived_pi(p,pitt,pitx,pity,pitz)
implicit none
real(8), intent(in), dimension(:) :: p
real(8), intent(inout) :: pitt,pitx,pity,pitz
!pitt=(g_cov(1)*p(kpixx)+g_cov(2)*p(kpiyy)+g_cov(3)*p(kpizz))/(-g_cov0) !traceless condition
pitx=p(kpixx)*p(kvx)*g_cov(1)+p(kpixy)*p(kvy)*g_cov(2)+p(kpixz)*p(kvz)*g_cov(3)
pity=p(kpixy)*p(kvx)*g_cov(1)+p(kpiyy)*p(kvy)*g_cov(2)+p(kpiyz)*p(kvz)*g_cov(3)
pitz=p(kpixz)*p(kvx)*g_cov(1)+p(kpiyz)*p(kvy)*g_cov(2)+p(kpizz)*p(kvz)*g_cov(3)
pitt=pitx*p(kvx)*g_cov(1)+pity*p(kvy)*g_cov(2)+pitz*p(kvz)*g_cov(3)
end subroutine get_derived_pi
! ***********************************************************************************
subroutine get_derived_pi_zz(p,pitt,pitx,pity,pitz,pizz)
implicit none
real(8), intent(in), dimension(:) :: p
real(8), intent(inout) :: pitt,pitx,pity,pitz,pizz
pitt=1./(1.-p(kvz)*p(kvz)*g_cov(3))*(p(kvx)*g_cov(1)*p(kpixy)*p(kvy)*g_cov(2)+p(kvx)*p(kvx)*g_cov(1)*g_cov(1)*p(kpixx)+&
&p(kvy)*g_cov(1)*p(kpiyz)*p(kvz)*g_cov(3)+2.*p(kvx)*g_cov(1)*p(kpixz)*p(kvz)*g_cov(3)+p(kvy)*g_cov(1)*g_cov(1)*p(kpixy)*&
&p(kvx)+p(kvy)*p(kvy)*g_cov(1)*p(kpiyy)*g_cov(2)+p(kvz)*g_cov(3)*p(kpiyz)*p(kvy)*g_cov(2)-p(kpixx)*g_cov(1)*p(kvz)*p(kvz)*&
&g_cov(3)-p(kpiyy)*g_cov(2)*p(kvz)*p(kvz)*g_cov(3))
pitx=p(kpixx)*p(kvx)*g_cov(1)+p(kpixy)*p(kvy)*g_cov(2)+p(kpixz)*p(kvz)*g_cov(3)
pity=p(kpixy)*p(kvx)*g_cov(1)+p(kpiyy)*p(kvy)*g_cov(2)+p(kpiyz)*p(kvz)*g_cov(3)
pitz=1./(1.-p(kvz)*p(kvz)*g_cov(3))*(p(kpixz)*p(kvx)*g_cov(1)+p(kpixz)*p(kvx)*g_cov(1)*p(kvz)*p(kvz)*g_cov(3)+p(kpiyz)*p(kvy)*&
&g_cov(2)+p(kvz)*p(kvx)*g_cov(1)*p(kpixy)*p(kvy)*g_cov(2)-p(kvz)*p(kpixx)*g_cov(1)-p(kvz)*p(kpiyy)*g_cov(2)+p(kvz)*p(kvx)*&
&p(kvx)*g_cov(1)*g_cov(1)*p(kpixx)+p(kvy)*g_cov(1)*p(kpiyz)*p(kvz)*p(kvz)*g_cov(3)+p(kvz)*p(kvy)*g_cov(1)*g_cov(1)*&
&p(kpixy)*p(kvx)+p(kvz)*p(kvy)*p(kvy)*g_cov(1)*p(kpiyy)*g_cov(2))
pizz=1./(g_cov(3)*(1.-p(kvz)*p(kvz)*g_cov(3)))*(p(kvx)*g_cov(1)*p(kpixy)*p(kvy)*g_cov(2)-p(kpixx)*g_cov(1)-p(kpiyy)*g_cov(2)+&
&p(kvx)*p(kvx)*g_cov(1)*g_cov(1)*p(kpixx)+p(kvy)*g_cov(1)*p(kpiyz)*p(kvz)*g_cov(3)+2.*p(kvx)*g_cov(1)*p(kpixz)*p(kvz)*&
&g_cov(3)+p(kvy)*g_cov(1)*g_cov(1)*p(kpixy)*p(kvx)+p(kvy)*p(kvy)*g_cov(1)*p(kpiyy)*g_cov(2)+p(kvz)*g_cov(3)*p(kpiyz)*&
&p(kvy)*g_cov(2))
end subroutine get_derived_pi_zz
! ***********************************************************************************
subroutine get_sigma(ix,iy,iz,u,sigma, dutdt, dutdx, dutdy, dutdz, duxdx, duydy, duzdz, duzdt, duzdx, duzdy, duxdt,&
&duxdy, duxdz, duydt, duydx, duydz)
implicit none
integer, intent(in) :: ix,iy,iz
real(8), dimension(:) :: sigma
real(8), dimension(:,:,:,:), allocatable :: u
real(8) :: ut, ux, uy, uz
real(8), intent(out) :: dutdt, dutdx, dutdy, dutdz, duxdx, duydy, duzdz, duzdt, duzdx, duzdy, duxdt, duxdy, duxdz, duydt, duydx,&
& duydz
real(8) :: ortho_check, trace_check
!these u are the contravariant primitive v
ux=u(ix,iy,iz,kvx)
uy=u(ix,iy,iz,kvy)
uz=u(ix,iy,iz,kvz)
ut=1./sqrt(1.-g_cov(1)*ux*ux-g_cov(2)*uy*uy-g_cov(3)*uz*uz)
!not these us are the contravariant u
ux=ux*ut
uy=uy*ut
uz=uz*ut
dutdt=deriv(ix,iy,iz,dtt)
dutdx=deriv(ix,iy,iz,dtx)
dutdy=deriv(ix,iy,iz,dty)
dutdz=deriv(ix,iy,iz,dtz)
duxdx=deriv(ix,iy,iz,dxx)
duydy=deriv(ix,iy,iz,dyy)
duzdz=deriv(ix,iy,iz,dzz)
duzdt=deriv(ix,iy,iz,dzt)
duzdx=deriv(ix,iy,iz,dzx)
duzdy=deriv(ix,iy,iz,dzy)
duxdt=deriv(ix,iy,iz,dxt)
duxdy=deriv(ix,iy,iz,dxy)
duxdz=deriv(ix,iy,iz,dxz)
duydt=deriv(ix,iy,iz,dyt)
duydx=deriv(ix,iy,iz,dyx)
duydz=deriv(ix,iy,iz,dyz)
!AAA please, compute only sigmas which are really used (for example, only the spatial part)
if(coordinates .eq. MINKOWSKI) then !Minkoski coordinates
!THIS SIGMAS ARE NOT USED, BUT THEY ARE LEFT HERE FOR A POSSIBLE FUTURE USE
!sigma(stt)=-(2./3.)*dutdt +(2./3.)*dutdt*(ut**2.)+ut*ux*dutdx+ut*uy*dutdy+ut*uz*dutdz+duxdx/3.+duydy/3.+duzdz/3.-(ut**2.)*duxdx&
! &/3.-(ut**2.)*duydy/3.-(ut**2.)*duzdz/3.
!sigma(stx)=-0.5*duxdt+0.5*duxdt*(ut**2.)+ut*ux*duxdx/6.+ut*uy*duxdy/2.+ut*uz*duxdz/2.+ut*ux*dutdt/6.+0.5*dutdx+0.5*dutdx*ux**2.&
! &+0.5*ux*uy*dutdy+0.5*ux*uz*dutdz-ut*ux*duydy/3.-ut*ux*duzdz/3.
!sigma(sty)=-0.5*duydt+0.5*duydt*(ut**2.)+ut*ux*duydx/2.+ut*uy*duydy/6.+ut*uz*duydz/2.+ut*uy*dutdt/6.+0.5*ux*uy*dutdx+0.5*&
! &dutdy+0.5*dutdy*(uy**2.)+0.5*uy*uz*dutdz-ut*uy*duxdx/3.-ut*uy*duzdz/3.
!sigma(stz)=-0.5*duzdt+0.5*duzdt*(ut**2.)+ut*ux*duzdx/2.+ut*uy*duzdy/2.+ut*uz*duzdz/6.+ut*uz*dutdt/6.+0.5*ux*uz*dutdx+0.5*&
! &dutdz+0.5*dutdz*(uz**2.)+0.5*uy*uz*dutdy-ut*uz*duxdx/3.-ut*uz*duydy/3.
sigma(szz)=ut*uz*duzdt+uz*ux*duzdx+uz*uy*duzdy+(2./3.)*duzdz+(2./3.)*duzdz*(uz**2.)-dutdt/3.-duxdx/3.-duydy/3.-(uz**2.)*dutdt/3.&
&-(uz**2.)*duxdx/3.-(uz**2.)*duydy/3.
sigma(sxz)=0.5*ut*ux*duzdt+0.5*duzdx+0.5*(ux**2.)*duzdx+0.5*ux*uy*duzdy+ux*uz*duzdz/6.+0.5*ut*uz*duxdt+ux*uz*duxdx/6.+0.5*uy*uz*&
&duxdy+0.5*duxdz+0.5*duxdz*uz**2.-ux*uz*dutdt/3.-ux*uz*duydy/3.
sigma(sxy)=0.5*ut*ux*duydt+0.5*duydx+0.5*(ux**2.)*duydx+ux*uy*duydy/6.+ux*uz*duydz/2.+0.5*ut*uy*duxdt+ux*uy*duxdx/6.+0.5*duxdy+&
&0.5*duxdy*(uy**2.)+uy*uz*duxdz/2.-ux*uy*dutdt/3.-ux*uy*duzdz/3.
sigma(syz)=0.5*ut*uy*duzdt+ux*uy*duzdx/2.+duzdy/2.+duzdy*(uy**2.)/2.+uy*uz*duzdz/6.+ut*uz*duydt/2.+ux*uz*duydx/2.+uy*uz*duydy/6.&
&+duydz/2.+duydz/2.*(uz**2.)-uy*uz*dutdt/3.-uy*uz*duxdx/3.
sigma(sxx)=ut*ux*duxdt+(2./3.)*duxdx+(2./3.)*duxdx*(ux**2.)+ux*uy*duxdy+ux*uz*duxdz-dutdt/3.-duydy/3.-duzdz/3.-(ux**2.)*&
&dutdt/3.-(ux**2.)*duydy/3.-(ux**2.)*duzdz/3.
sigma(syy)=ut*uy*duydt+ux*uy*duydx+(2./3.)*duydy+(2./3.)*duydy*(uy**2.)+uy*uz*duydz-dutdt/3.-duxdx/3.-duzdz/3.-(uy**2.)*&
&dutdt/3.-(uy**2.)*duxdx/3.-(uy**2.)*duzdz/3.
else !Bjorken coordinates
!THIS SIGMAS ARE NOT USED, BUT THEY ARE LEFT HERE FOR A POSSIBLE FUTURE USE
! sigma(stt)=-(2./3.)*dutdt +(2./3.)*dutdt*(ut**2.)+ut*ux*dutdx+ut*uy*dutdy+ut*uz*dutdz+duxdx/3.+duydy/3.+duzdz/3.+ut/(3.*t)-&
! &(ut**2.)*duxdx/3.-(ut**2.)*duydy/3.-(ut**2.)*duzdz/3.-ut**3./(3.*t)+t*ut*(uz)**2.
! sigma(stx)=-0.5*duxdt+0.5*duxdt*(ut**2.)+ut*ux*duxdx/6.+ut*uy*duxdy/2.+ut*uz*duxdz/2.+ut*ux*dutdt/6.+0.5*dutdx+0.5*dutdx*&
! &(ux**2.)+0.5*ux*uy*dutdy+0.5*ux*uz*dutdz-ut*ux*duydy/3.-ut*ux*duzdz/3.-ux*(ut**2.)/(3.*t)+t/2.*(uz**2.)*ux
! sigma(sty)=-0.5*duydt+0.5*duydt*(ut**2.)+ut*ux*duydx/2.+ut*uy*duydy/6.+ut*uz*duydz/2.+ut*uy*dutdt/6.+0.5*ux*uy*dutdx+0.5*&
! &dutdy+0.5*dutdy*(uy**2.)+0.5*uy*uz*dutdz-ut*uy*duxdx/3.-ut*uy*duzdz/3.-uy*(ut**2.)/(3.*t)+t/2.*(uz**2.)*uy
! sigma(stz)=-0.5*duzdt+0.5*duzdt*(ut**2.)+ut*ux*duzdx/2.+ut*uy*duzdy/2.+ut*uz*duzdz/6.+ut*uz*dutdt/6.+0.5*ux*uz*dutdx+0.5*&
! &dutdz/(t**2.)+0.5*dutdz*(uz**2.)+0.5*uy*uz*dutdy-ut*uz*duxdx/3.-ut*uz*duydy/3.-(ut**2.)*uz/(3.*t)+0.5*(uz**3.)*t&
! &+(ut**2.)/t*uz
sigma(szz)=ut*uz*duzdt+uz*ux*duzdx+uz*uy*duzdy+(2.*duzdz)/(3.*t**2.)+(2./3.)*duzdz*(uz**2.)-dutdt/(3.*(t**2.))-duxdx/(3.*&
&(t**2.))-duydy/(3.*(t**2.))-(uz**2.)*dutdt/3.-(uz**2.)*duxdx/3.-(uz**2.)*duydy/3.&
&+2./3.*ut/(t**3.)+5./3.*(ut/t)*(uz**2.)
sigma(sxz)=0.5*ut*ux*duzdt+0.5*duzdx+0.5*(ux**2.)*duzdx+0.5*ux*uy*duzdy+ux*uz*duzdz/6.+0.5*ut*uz*duxdt+ux*uz*duxdx/6.+0.5*uy*uz*&
&duxdy+duxdz/(2.*t**2.)+0.5*duxdz*uz**2.-ux*uz*dutdt/3.-ux*uz*duydy/3.+(2./3.)*ut*uz*ux/t
sigma(sxy)=0.5*ut*ux*duydt+0.5*duydx+0.5*(ux**2.)*duydx+ux*uy*duydy/6.+ux*uz*duydz/2.+0.5*ut*uy*duxdt+ux*uy*duxdx/6.+0.5*duxdy+&
&0.5*duxdy*(uy**2.)+uy*uz*duxdz/2.-ux*uy*dutdt/3.-ux*uy*duzdz/3.-ut*ux*uy/(3.*t)
sigma(syz)=0.5*ut*uy*duzdt+ux*uy*duzdx/2.+duzdy/2.+duzdy*(uy**2.)/2.+uy*uz*duzdz/6.+ut*uz*duydt/2.+ux*uz*duydx/2.+uy*uz*duydy/6.&
&+duydz/(2.*(t**2.))+duydz/2.*(uz**2.)-uy*uz*dutdt/3.-uy*uz*duxdx/3.+(2./3.)*ut*uz*uy/t
sigma(sxx)=ut*ux*duxdt+(2./3.)*duxdx+(2./3.)*duxdx*(ux**2.)+ux*uy*duxdy+ux*uz*duxdz-dutdt/3.-duydy/3.-duzdz/3.-ut/(3.*t)-&
&(ux**2.)*dutdt/3.-(ux**2.)*duydy/3.-(ux**2.)*duzdz/3.-(ux**2.)*ut/(3.*t)
sigma(syy)=ut*uy*duydt+ux*uy*duydx+(2./3.)*duydy+(2./3.)*duydy*(uy**2.)+uy*uz*duydz-dutdt/3.-duxdx/3.-duzdz/3.-ut/(3.*t)-&
&(uy**2.)*dutdt/3.-(uy**2.)*duxdx/3.-(uy**2.)*duzdz/3.-(uy**2.)*ut/(3.*t)
end if
end subroutine get_sigma
! ***********************************************************************************
subroutine viscous_renormalize(comp)
implicit none
real(8), dimension(:) :: comp
real(8) :: normalization_term, pitt, pitx, pity, pitz
return
if(comp(kpr) .ne. 0) then
call get_derived_pi(comp,pitt,pitx,pity,pitz)
normalization_term=(1.+4*(pitt**2.+comp(kpixx)**2.+comp(kpiyy)**2.+comp(kpizz)**2.+2.*pitx**2.+2.*pity**2.+2.*&
&pitz**2.+2.*comp(kpixy)**2.+2.*comp(kpixz)**2.+2.*comp(kpiyz)**2.)**2./(9.*comp(kpr)**4.))**0.25
else
normalization_term=1.
end if
comp(kpixy:kpizz)=comp(kpixy:kpizz)/normalization_term
end subroutine viscous_renormalize
! ***********************************************************************************
end module viscous