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fd_spatial_m.f90
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fd_spatial_m.f90
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!=============================================================================!
subroutine fd_spatial(name, ind)
!
! Driver for spatial parallel stability solver, adiabatic temperature BC
!
! This version uses Carpenter's third order stencil near the wall
!
! This version is for Ted's mixing layer profiles
!
! Revised: 2-9-96
!=============================================================================!
use stuff
use material
use stencils
implicit none
integer i, j, k, ix, ier, ind
integer i0, idof, j0, jdof
character*80 name
real vm(ny,ndof,nx)
real eta(ny), y(ny), deta(ny), d2eta(ny)
real A(ny,ndof,ndof), B(ny,ndof,ndof), C(ny,ndof,ndof)
real D(ny,ndof,ndof), G(ny,ndof,ndof)
real Vxx(ny,ndof,ndof), Vxy(ny,ndof,ndof), Vyy(ny,ndof,ndof)
real Vxz(ny,ndof,ndof), Vyz(ny,ndof,ndof), Vzz(ny,ndof,ndof)
real g1vm(ny,ndof), g2vm(ny,ndof), g3vm(ny,ndof)
real g11vm(ny,ndof), g12vm(ny,ndof), g13vm(ny,ndof)
real g21vm(ny,ndof), g22vm(ny,ndof), g23vm(ny,ndof)
real g31vm(ny,ndof), g32vm(ny,ndof), g33vm(ny,ndof)
real gum(ny,nsd,nsd), grhom(ny,nsd), gtm(ny,nsd), gpm(ny,nsd)
real divum(ny), g1divum(ny), g2divum(ny), g3divum(ny)
real rhom(ny), u1m(ny), u2m(ny), u3m(ny), tm(ny), pm(ny)
real rmu(ny), dmu(ny), d2mu(ny)
real rlm(ny), dlm(ny), d2lm(ny)
real con(ny), dcon(ny), d2con(ny)
real g1mu(ny), g2mu(ny), g3mu(ny)
real g1lm(ny), g2lm(ny), g3lm(ny)
real g1con(ny), g2con(ny), g3con(ny)
real g1dmu(ny), g2dmu(ny), g3dmu(ny)
real g1dlm(ny), g2dlm(ny), g3dlm(ny)
real g1dcon(ny), g2dcon(ny), g3dcon(ny)
real S1jj(ny), S2jj(ny), S3jj(ny), S(ny,nsd,nsd), Lapt(ny)
real fact
complex Dh(ny,ndof,ndof), Ah(ny,ndof,ndof), Bh(ny,ndof,ndof)
complex :: scale
complex A0(2*ndof*ny,2*ndof*ny), B0(2*ndof*ny,2*ndof*ny), &
D0(2*ndof*ny,2*ndof*ny)
complex C0(ndof*ny,ndof*ny), C1(ndof*ny,ndof*ny), C2(ndof*ny,ndof*ny)
complex Cinv0(ndof*ny,ndof*ny)
complex evec(2*ndof*ny,2*ndof*ny), alp(2*ndof*ny), bet(2*ndof*ny)
complex omg(2*ndof*ny), cs(2*ndof*ny)
real temp1(2*ndof*ny), temp2(2*ndof*ny)
integer index(2*ndof*ny)
!.... stuff for LAPACK eigensolver
integer info, lwork
complex, allocatable :: work(:)
real, allocatable :: rwork(:)
integer, allocatable :: ipvt(:)
!.... metrics for the curved wall
real :: h(ny), dhds(ny), dhdr(ny), dhdsr(ny), dhdrr(ny)
!=============================================================================!
if (Re.ge.1.0e98 .or. Re.eq.zero) Navier = .false.
!.... initialize the gradients
g1vm = zero
g2vm = zero
g3vm = zero
g11vm = zero
g12vm = zero
g13vm = zero
g21vm = zero
g22vm = zero
g23vm = zero
g31vm = zero
g32vm = zero
g33vm = zero
!.... make a mixing layer mesh
call mgengrid( y, eta, deta, d2eta, vm )
!.... read Ted's mean profile
call mgetmean( vm, y, eta)
!.... Initialize the metric terms for curvature
if (curve.eq.1) then
call calch( x, ny, y, h, dhds, dhdr, dhdsr, dhdrr )
else
h = one
dhds = zero
dhdr = zero
dhdsr = zero
dhdrr = zero
end if
!.... loop over the streamwise stations
do ix = 1, nx
!.... initialize
G = zero
A = zero
B = zero
C = zero
D = zero
Vxx = zero
Vxy = zero
Vyy = zero
Vxz = zero
Vyz = zero
Vzz = zero
rhom = vm(:,1,ix)
u1m = vm(:,2,ix)
u2m = zero
u3m = vm(:,4,ix)
tm = vm(:,5,ix)
pm = one / (gamma * Ma**2) * rhom * tm
if (ider) then
!.... Compute first derivatives of mean field
call grad(ndof, nx, ny, nz, vm, g1vm, g2vm, g3vm, ix, one, dy, one)
!.... Compute second derivatives of mean field
call grad(ndof, nx, ny, nz, g1vm, g11vm, g12vm, g13vm, &
ix, one, dy, one)
call grad(ndof, nx, ny, nz, g2vm, g21vm, g22vm, g23vm, &
ix, one, dy, one)
call grad(ndof, nx, ny, nz, g3vm, g31vm, g32vm, g33vm, &
ix, one, dy, one)
!.... transform the gradients to physical space
do k = 1, ndof
g22vm(:,k) = g22vm(:,k) * deta**2 + g2vm(:,k) * d2eta
g2vm(:,k) = g2vm(:,k) * deta
end do
end if
!.... write out the mean field and its gradients
open(10,file='rho.out')
do j = 1, ny
write (10,13) y(j), vm(j,1,ix), g2vm(j,1), g22vm(j,1)
end do
close(10)
open(10,file='u.out')
do j = 1, ny
write (10,13) y(j), vm(j,2,ix), g2vm(j,2), g22vm(j,2)
end do
close(10)
open(10,file='v.out')
do j = 1, ny
write (10,13) y(j), vm(j,3,ix), g2vm(j,3), g22vm(j,3)
end do
close(10)
open(10,file='w.out')
do j = 1, ny
write (10,13) y(j), vm(j,4,ix), g2vm(j,4), g22vm(j,4)
end do
close(10)
open(10,file='t.out')
do j = 1, ny
write (10,13) y(j), vm(j,5,ix), g2vm(j,5), g22vm(j,5)
end do
close(10)
13 format(4(1pe20.13,1x))
!.... initialize gradient of mean velocity
gum(:,1,1) = g1vm(:,2)
gum(:,1,2) = g2vm(:,2)
gum(:,1,3) = g3vm(:,2)
gum(:,2,1) = g1vm(:,3)
gum(:,2,2) = g2vm(:,3)
gum(:,2,3) = g3vm(:,3)
gum(:,3,1) = g1vm(:,4)
gum(:,3,2) = g2vm(:,4)
gum(:,3,3) = g3vm(:,4)
!.... compute the divergence (curve*)
divum = ( gum(:,1,1) + u2m * dhdr ) / h + gum(:,2,2) + gum(:,3,3)
!.... initialize gradient of rho and T in the mean
grhom(:,1) = g1vm(:,1)
grhom(:,2) = g2vm(:,1)
grhom(:,3) = g3vm(:,1)
gtm(:,1) = g1vm(:,5)
gtm(:,2) = g2vm(:,5)
gtm(:,3) = g3vm(:,5)
fact = one / (gamma * Ma**2)
gpm(:,1) = fact * ( grhom(:,1) * tm + rhom * gtm(:,1) )
gpm(:,2) = fact * ( grhom(:,2) * tm + rhom * gtm(:,2) )
gpm(:,3) = fact * ( grhom(:,3) * tm + rhom * gtm(:,3) )
!.... compute the gradient of the divergence of um (curve*)
g1divum = -dhds / h**3 * ( gum(:,1,1) + u2m * dhdr ) + &
one / h**2 * ( g11vm(:,2) + gum(:,2,1) * dhdr + &
u2m * dhdsr ) + one / h * ( g12vm(:,3) + g13vm(:,4) )
g2divum = -dhdr / h**2 * ( gum(:,1,1) + u2m * dhdr ) + &
one / h * ( g12vm(:,2) + gum(:,2,2) * dhdr + &
u2m * dhdrr ) + ( g22vm(:,3) + g23vm(:,4) )
g3divum = one / h * ( g13vm(:,2) + gum(:,2,3) * dhdr ) + &
g23vm(:,3) + g33vm(:,4)
!.... compute some stuff that is useful for the viscous terms (curve*)
S(:,1,1) = ( gum(:,1,1) + u2m * dhdr ) / h
S(:,1,2) = pt5 * ( ( gum(:,2,1) - u1m * dhdr ) / h + gum(:,1,2) )
S(:,1,3) = pt5 * ( gum(:,3,1) / h + gum(:,1,3) )
S(:,2,1) = S(:,1,2)
S(:,2,2) = gum(:,2,2)
S(:,2,3) = pt5 * ( gum(:,3,2) + gum(:,2,3) )
S(:,3,1) = S(:,1,3)
S(:,3,2) = S(:,2,3)
S(:,3,3) = gum(:,3,3)
S1jj = -pt5 * ( dhdr**2 + dhdrr * h ) / h**2 * u1m + &
pt5 * dhdr * gum(:,1,2) / h + pt5 * g22vm(:,2) - &
dhds * gum(:,1,1) / h**3 + g11vm(:,2) / h**2 + &
pt5 * g33vm(:,2) + ( h * dhdsr - dhdr * dhds ) / h**3 * u2m + &
3.0 * dhdr * gum(:,2,1) / (two * h**2) + &
pt5 * g12vm(:,3) / h + pt5 * g13vm(:,4) / h
S2jj = pt5 * ( dhdr * dhds - h * dhdsr ) / h**3 * u1m - &
3.0 * dhdr * gum(:,1,1) / (two * h**2) + &
pt5 * g12vm(:,2) / h - dhdr**2 * u2m / h**2 + &
dhdr * gum(:,2,2) / h + g22vm(:,3) - &
pt5 * dhds * gum(:,2,1) / h**3 + pt5 * g11vm(:,3) / h**2 + &
pt5 * g33vm(:,3) + pt5 * g23vm(:,4)
S3jj = pt5 * g13vm(:,2) / h + pt5 * g23vm(:,3) + &
pt5 * dhdr * gum(:,2,3) / h + pt5 * dhdr * gum(:,3,2) / h + &
pt5 * g22vm(:,4) + pt5 * g11vm(:,4) / h**2 + g33vm(:,4) - &
pt5 * dhds * gum(:,3,1) / h**3
!.... Laplacian of Tm (curve*)
LapT = one/h * ( -dhds/h**2 * gtm(:,1) + one/h * g11vm(:,5) + &
h * g22vm(:,5) + gtm(:,2) * dhdr + h * g33vm(:,5) )
!.... compute mean material properties
call getmat(tm*te, rmu, rlm, con, dmu, d2mu, &
dlm, d2lm, dcon, d2con)
!.... nondimensionalize
rmu = rmu / rmue
dmu = dmu * Te / rmue
d2mu = d2mu * Te**2 / rmue
con = con / cone
dcon = dcon * Te / cone
d2con = d2con * Te**2 / cone
rlm = rlm / rlme
dlm = dlm * Te / rlme
d2lm = d2lm * Te**2 / rlme
!.... compute gradients of viscosity using chain-rule
g1mu = dmu * gtm(:,1)
g2mu = dmu * gtm(:,2)
g3mu = dmu * gtm(:,3)
g1dmu = d2mu * gtm(:,1)
g2dmu = d2mu * gtm(:,2)
g3dmu = d2mu * gtm(:,3)
!.... compute gradients of conductivity using chain-rule
g1con = dcon * gtm(:,1)
g2con = dcon * gtm(:,2)
g3con = dcon * gtm(:,3)
g1dcon = d2con * gtm(:,1)
g2dcon = d2con * gtm(:,2)
g3dcon = d2con * gtm(:,3)
!.... compute gradients of bulk viscosity using chain-rule
g1lm = dlm * gtm(:,1)
g2lm = dlm * gtm(:,2)
g3lm = dlm * gtm(:,3)
g1dlm = d2lm * gtm(:,1)
g2dlm = d2lm * gtm(:,2)
g3dlm = d2lm * gtm(:,3)
!==============================================================================
!.... Continuity equation (curve*)
G(:,1,1) = one
A(:,1,1) = u1m / h
A(:,1,2) = rhom / h
B(:,1,1) = u2m
B(:,1,3) = rhom
C(:,1,1) = u3m
C(:,1,4) = rhom
D(:,1,1) = divum
D(:,1,2) = grhom(:,1) / h
D(:,1,3) = grhom(:,2) + rhom * dhdr / h
D(:,1,4) = grhom(:,3)
!.... Momentum equation -- x_1 (convective + pressure) (curve*)
G(:,2,2) = rhom
A(:,2,1) = tm/(h * gamma * Ma**2)
A(:,2,2) = rhom * u1m / h
A(:,2,5) = rhom/(h * gamma * Ma**2)
B(:,2,2) = rhom * u2m
C(:,2,2) = rhom * u3m
D(:,2,1) = u1m / h * ( gum(:,1,1) + u2m * dhdr ) + &
u2m * gum(:,1,2) + u3m * gum(:,1,3) &
+ gtm(:,1) / (h * gamma * Ma**2)
D(:,2,2) = rhom * ( gum(:,1,1) + u2m * dhdr ) / h
D(:,2,3) = rhom * ( gum(:,1,2) + u1m * dhdr / h )
D(:,2,4) = rhom * gum(:,1,3)
D(:,2,5) = grhom(:,1) / (h * gamma * Ma**2)
!.... (viscous lambda) (curve*)
if (Navier) then
fact = rlme / (rmue * Re)
A(:,2,2) = A(:,2,2) - fact * ( g1lm / h**2 - rlm / h**3 * dhds )
A(:,2,3) = A(:,2,3) - fact * rlm / h**2 * dhdr
A(:,2,5) = A(:,2,5) - fact * dlm * divum / h
B(:,2,3) = B(:,2,3) - fact * ( g1lm / h )
C(:,2,4) = C(:,2,4) - fact * ( g1lm / h )
D(:,2,3) = D(:,2,3) - fact * ( g1lm * dhdr / h**2 - &
rlm / h**3 * dhds * dhdr + &
rlm / h**2 * dhdsr )
D(:,2,5) = D(:,2,5) - fact * ( g1dlm * divum / h + dlm * g1divum )
Vxx(:,2,2) = fact * rlm / h**2
Vxy(:,2,3) = fact * rlm / h
Vxz(:,2,4) = fact * rlm / h
!.... (viscous mu) (curve*)
fact = one / Re
A(:,2,2) = A(:,2,2) - fact * ( two * g1mu / h**2 - &
two * rmu * dhds / h**3 )
A(:,2,3) = A(:,2,3) - fact * ( g2mu / h + &
rmu * 3.0 * dhdr / h**2 )
A(:,2,4) = A(:,2,4) - fact * g3mu / h
A(:,2,5) = A(:,2,5) - fact * dmu * two * S(:,1,1) / h
B(:,2,2) = B(:,2,2) - fact * ( g2mu + rmu * dhdr / h )
B(:,2,5) = B(:,2,5) - fact * dmu * two * S(:,1,2)
C(:,2,2) = C(:,2,2) - fact * g3mu
C(:,2,5) = C(:,2,5) - fact * dmu * two * S(:,1,3)
D(:,2,2) = D(:,2,2) - fact * ( g2mu / h * (-dhdr) - &
rmu * ( dhdr**2 + dhdrr * h ) / h**2 )
D(:,2,3) = D(:,2,3) - fact * ( two * g1mu / h**2 * dhdr + &
two * rmu * ( dhdsr * h - dhds * dhdr ) / h**3 )
D(:,2,5) = D(:,2,5) - fact * two * ( g1dmu / h * S(:,1,1) + &
g2dmu * S(:,1,2) + g3dmu * S(:,1,3) + &
dmu * S1jj )
Vxx(:,2,2) = Vxx(:,2,2) + fact * two * rmu / h**2
Vxy(:,2,3) = Vxy(:,2,3) + fact * rmu / h
Vyy(:,2,2) = Vyy(:,2,2) + fact * rmu
Vxz(:,2,4) = Vxz(:,2,4) + fact * rmu / h
Vzz(:,2,2) = Vzz(:,2,2) + fact * rmu
end if
!.... Momentum equation -- x_2 (convective + pressure) (curve*)
G(:,3,3) = rhom
A(:,3,3) = rhom * u1m / h
B(:,3,1) = tm/(gamma * Ma**2)
B(:,3,3) = rhom * u2m
B(:,3,5) = rhom/(gamma * Ma**2)
C(:,3,3) = rhom * u3m
D(:,3,1) = u1m / h * ( gum(:,2,1) - u1m * dhdr ) + &
u2m * gum(:,2,2) + u3m * gum(:,2,3) + &
gtm(:,2) / (gamma * Ma**2)
D(:,3,2) = rhom * ( gum(:,2,1) - two * u1m * dhdr ) / h
D(:,3,3) = rhom * gum(:,2,2)
D(:,3,4) = rhom * gum(:,2,3)
D(:,3,5) = grhom(:,2) / (gamma * Ma**2)
!.... (viscous lambda) (curve*)
if (Navier) then
fact = rlme / (rmue * Re)
A(:,3,2) = A(:,3,2) - fact * ( g2lm / h - rlm * dhdr / h**2 )
B(:,3,3) = B(:,3,3) - fact * ( g2lm + rlm * dhdr / h )
B(:,3,5) = B(:,3,5) - fact * dlm * divum
C(:,3,4) = C(:,3,4) - fact * ( g2lm )
D(:,3,3) = D(:,3,3) - fact * ( g2lm / h * dhdr - &
rlm * dhdr / h**2 * dhdr + &
rlm / h * dhdrr )
D(:,3,5) = D(:,3,5) - fact * ( g2dlm * divum + dlm * g2divum )
Vxy(:,3,2) = fact * rlm / h
Vyy(:,3,3) = fact * rlm
Vyz(:,3,4) = fact * rlm
!.... (viscous mu) (curve*)
fact = one / Re
A(:,3,2) = A(:,3,2) + fact * rmu * 3.0 * dhdr / h**2
A(:,3,3) = A(:,3,3) - fact * ( g1mu / h**2 - &
rmu * dhds / h**3 )
A(:,3,5) = A(:,3,5) - fact * dmu * two * S(:,2,1) / h
B(:,3,2) = B(:,3,2) - fact * g1mu / h
B(:,3,3) = B(:,3,3) - fact * ( two * g2mu + two * rmu * dhdr / h )
B(:,3,4) = B(:,3,4) - fact * g3mu
B(:,3,5) = B(:,3,5) - fact * dmu * two * S(:,2,2)
C(:,3,3) = C(:,3,3) - fact * g3mu
C(:,3,5) = C(:,3,5) - fact * dmu * two * S(:,2,3)
D(:,3,2) = D(:,3,2) - fact * ( g1mu / h**2 * (-dhdr) + &
rmu * (dhds * dhdr - h * dhdsr) / h**3 )
D(:,3,3) = D(:,3,3) + fact * two * rmu * dhdr**2 / h**2
D(:,3,5) = D(:,3,5) - fact * two * ( g1dmu / h * S(:,2,1) + &
g2dmu * S(:,2,2) + g3dmu * S(:,2,3) + &
dmu * S2jj )
Vxx(:,3,3) = Vxx(:,3,3) + fact * rmu / h**2
Vxy(:,3,2) = Vxy(:,3,2) + fact * rmu / h
Vyy(:,3,3) = Vyy(:,3,3) + fact * two * rmu
Vyz(:,3,4) = Vyz(:,3,4) + fact * rmu
Vzz(:,3,3) = Vzz(:,3,3) + fact * rmu
end if
!.... Momentum equation -- x_3 (convective + pressure) (curve*)
G(:,4,4) = rhom
A(:,4,4) = rhom * u1m / h
B(:,4,4) = rhom * u2m
C(:,4,1) = tm/(gamma * Ma**2)
C(:,4,4) = rhom * u3m
C(:,4,5) = rhom/(gamma * Ma**2)
D(:,4,1) = u1m * gum(:,3,1) / h + u2m * gum(:,3,2) + &
u3m * gum(:,3,3) + gtm(:,3) / (gamma * Ma**2)
D(:,4,2) = rhom * gum(:,3,1) / h
D(:,4,3) = rhom * gum(:,3,2)
D(:,4,4) = rhom * gum(:,3,3)
D(:,4,5) = grhom(:,3) / (gamma * Ma**2)
!.... (viscous lambda) (curve*)
if (Navier) then
fact = rlme / (rmue * Re)
A(:,4,2) = A(:,4,2) - fact * g3lm / h
B(:,4,3) = B(:,4,3) - fact * g3lm
C(:,4,4) = C(:,4,4) - fact * g3lm
C(:,4,3) = C(:,4,3) - fact * rlm / h * dhdr
C(:,4,5) = C(:,4,5) - fact * dlm * divum
D(:,4,3) = D(:,4,3) - fact * ( g2lm / h * dhdr )
D(:,4,5) = D(:,4,5) - fact * ( g3dlm * divum + dlm * g3divum )
Vxz(:,4,2) = fact * rlm / h
Vyz(:,4,3) = fact * rlm
Vzz(:,4,4) = fact * rlm
!.... (viscous mu) (curve*)
fact = one / Re
A(:,4,4) = A(:,4,4) - fact * ( g1mu / h**2 - rmu * dhds / h**3 )
A(:,4,5) = A(:,4,5) - fact * dmu * two * S(:,3,1) / h
B(:,4,4) = B(:,4,4) - fact * ( g2mu + rmu * dhdr / h )
B(:,4,5) = B(:,4,5) - fact * dmu * two * S(:,3,2)
C(:,4,2) = C(:,4,2) - fact * g1mu / h
C(:,4,3) = C(:,4,3) - fact * ( g2mu + rmu * dhdr / h )
C(:,4,4) = C(:,4,4) - fact * two * g3mu
C(:,4,5) = C(:,4,5) - fact * dmu * two * S(:,3,3)
D(:,4,5) = D(:,4,5) - fact * two * ( g1dmu / h * S(:,3,1) + &
g2dmu * S(:,3,2) + g3dmu * S(:,3,3) + &
dmu * S3jj )
Vxx(:,4,4) = Vxx(:,4,4) + fact * rmu / h**2
Vyy(:,4,4) = Vyy(:,4,4) + fact * rmu
Vxz(:,4,2) = Vxz(:,4,2) + fact * rmu / h
Vyz(:,4,3) = Vyz(:,4,3) + fact * rmu
Vzz(:,4,4) = Vzz(:,4,4) + fact * two * rmu
end if
!.... Energy equation (Advection + pressure) (curve*)
G(:,5,5) = rhom
A(:,5,2) = rhom * gamma1 * tm / h
A(:,5,5) = rhom * u1m / h
B(:,5,3) = rhom * gamma1 * tm
B(:,5,5) = rhom * u2m
C(:,5,4) = rhom * gamma1 * tm
C(:,5,5) = rhom * u3m
D(:,5,1) = u1m / h * gtm(:,1) + u2m * gtm(:,2) + u3m * gtm(:,3) + &
gamma1 * tm * divum
D(:,5,2) = rhom * gtm(:,1) / h
D(:,5,3) = rhom * gtm(:,2) + rhom * gamma1 * tm * dhdr / h
D(:,5,4) = rhom * gtm(:,3)
D(:,5,5) = rhom * gamma1 * divum
if (Navier) then
!.... diffusion (curve*)
fact = gamma / (Pr * Re)
A(:,5,5) = A(:,5,5) - fact * (g1con / h**2 + dcon * gtm(:,1) / h**2 - &
con * dhds / h**3 )
B(:,5,5) = B(:,5,5) - fact * (g2con + dcon * gtm(:,2) + &
con * dhdr / h )
C(:,5,5) = C(:,5,5) - fact * (g3con + dcon * gtm(:,3))
D(:,5,5) = D(:,5,5) - fact * (g1dcon * gtm(:,1) / h**2 + &
g2dcon * gtm(:,2) + g3dcon * gtm(:,3) + &
dcon * Lapt )
Vxx(:,5,5) = fact * con / h**2
Vyy(:,5,5) = fact * con
Vzz(:,5,5) = fact * con
!.... dissipation (lambda) (curve*)
fact = gamma * gamma1 * Ma**2 * rlme / (Re * rmue)
A(:,5,2) = A(:,5,2) - fact * two * rlm * divum / h
B(:,5,3) = B(:,5,3) - fact * two * rlm * divum
C(:,5,4) = C(:,5,4) - fact * two * rlm * divum
D(:,5,3) = D(:,5,3) - fact * two * rlm * divum * dhdr / h
D(:,5,5) = D(:,5,5) - fact * dlm * divum * divum
!.... dissipation (mu) (curve*)
fact = gamma * gamma1 * Ma**2 / Re
A(:,5,2) = A(:,5,2) - fact * four * rmu * S(:,1,1) / h
A(:,5,3) = A(:,5,3) - fact * four * rmu * S(:,2,1) / h
A(:,5,4) = A(:,5,4) - fact * four * rmu * S(:,3,1) / h
B(:,5,2) = B(:,5,2) - fact * four * rmu * S(:,1,2)
B(:,5,3) = B(:,5,3) - fact * four * rmu * S(:,2,2)
B(:,5,4) = B(:,5,4) - fact * four * rmu * S(:,3,2)
C(:,5,2) = C(:,5,2) - fact * four * rmu * S(:,1,3)
C(:,5,3) = C(:,5,3) - fact * four * rmu * S(:,2,3)
C(:,5,4) = C(:,5,4) - fact * four * rmu * S(:,3,3)
D(:,5,2) = D(:,5,2) + fact * four * rmu * S(:,2,1) * dhdr / h
D(:,5,3) = D(:,5,3) - fact * four * rmu * S(:,1,1) * dhdr / h
D(:,5,5) = D(:,5,5) - fact * two * dmu * ( S(:,1,1)**2 + &
S(:,1,2)**2 + S(:,1,3)**2 + S(:,2,1)**2 + &
S(:,2,2)**2 + S(:,2,3)**2 + S(:,3,1)**2 + &
S(:,3,2)**2 + S(:,3,3)**2)
end if
!==============================================================================
!.... form the equations
Dh = D + im * beta * C + beta**2 * Vzz
Bh = B - im * beta * Vyz
!.... account for the mapping
do idof = 1, ndof
do jdof = 1, ndof
Bh(:,idof,jdof) = Bh(:,idof,jdof) * deta - Vyy(:,idof,jdof) * d2eta
Vyy(:,idof,jdof) = Vyy(:,idof,jdof) * deta**2
end do
end do
!.... initialize
C0 = zero
C1 = zero
C2 = zero
!.... C0
i = 1
i0 = (i-1)*ndof
idof = 1 ! continuity at the bottom boundary
do jdof = 1, ndof
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = Dh(i,idof,jdof) + &
(gg1/dy) * Bh(i,idof,jdof)
j0 = (i)*ndof
C0(i0+idof,j0+jdof) = (gg2/dy) * Bh(i,idof,jdof)
j0 = (i+1)*ndof
C0(i0+idof,j0+jdof) = (gg3/dy) * Bh(i,idof,jdof)
j0 = (i+2)*ndof
C0(i0+idof,j0+jdof) = (gg4/dy) * Bh(i,idof,jdof)
j0 = (i+3)*ndof
C0(i0+idof,j0+jdof) = (gg5/dy) * Bh(i,idof,jdof)
j0 = (i+4)*ndof
C0(i0+idof,j0+jdof) = (gg6/dy) * Bh(i,idof,jdof)
end do
do idof = 1, ndof
do jdof = 1, ndof
i = 2
i0 = (i-1)*ndof
j0 = (i-2)*ndof
C0(i0+idof,j0+jdof) = (gh1/dy) * Bh(i,idof,jdof) - &
db1/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = (gh2/dy) * Bh(i,idof,jdof) + &
Dh(i,idof,jdof) - &
db2/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i)*ndof
C0(i0+idof,j0+jdof) = (gh3/dy) * Bh(i,idof,jdof) - &
db3/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i+1)*ndof
C0(i0+idof,j0+jdof) = (gh4/dy) * Bh(i,idof,jdof) - &
db4/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i+2)*ndof
C0(i0+idof,j0+jdof) = (gh5/dy) * Bh(i,idof,jdof) - &
db5/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i+3)*ndof
C0(i0+idof,j0+jdof) = (gh6/dy) * Bh(i,idof,jdof)
i = 3
i0 = (i-1)*ndof
j0 = (i-3)*ndof
C0(i0+idof,j0+jdof) = (gi1/dy) * Bh(i,idof,jdof) - &
da1/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-2)*ndof
C0(i0+idof,j0+jdof) = (gi2/dy) * Bh(i,idof,jdof) - &
da2/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = (gi3/dy) * Bh(i,idof,jdof) + &
Dh(i,idof,jdof) - &
da3/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i)*ndof
C0(i0+idof,j0+jdof) = (gi4/dy) * Bh(i,idof,jdof) - &
da4/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i+1)*ndof
C0(i0+idof,j0+jdof) = (gi5/dy) * Bh(i,idof,jdof) - &
da5/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i+2)*ndof
C0(i0+idof,j0+jdof) = (gi6/dy) * Bh(i,idof,jdof)
i = 4
i0 = (i-1)*ndof
j0 = (i-4)*ndof
C0(i0+idof,j0+jdof) = (gj1/dy) * Bh(i,idof,jdof)
j0 = (i-3)*ndof
C0(i0+idof,j0+jdof) = (gj2/dy) * Bh(i,idof,jdof) - &
da1/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-2)*ndof
C0(i0+idof,j0+jdof) = (gj3/dy) * Bh(i,idof,jdof) - &
da2/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = (gj4/dy) * Bh(i,idof,jdof) + &
Dh(i,idof,jdof) - &
da3/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i)*ndof
C0(i0+idof,j0+jdof) = (gj5/dy) * Bh(i,idof,jdof) - &
da4/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i+1)*ndof
C0(i0+idof,j0+jdof) = (gj6/dy) * Bh(i,idof,jdof) - &
da5/(dy*dy) * Vyy(i,idof,jdof)
do i = 5, ny-2
i0 = (i-1)*ndof
j0 = (i-3)*ndof
C0(i0+idof,j0+jdof) = (ga1/dy) * Bh(i,idof,jdof) - &
(da1)/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-2)*ndof
C0(i0+idof,j0+jdof) = (ga2/dy) * Bh(i,idof,jdof) - &
(da2)/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = Dh(i,idof,jdof) - &
(da3)/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i)*ndof
C0(i0+idof,j0+jdof) = (ga3/dy) * Bh(i,idof,jdof) - &
(da4)/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i+1)*ndof
C0(i0+idof,j0+jdof) = (ga4/dy) * Bh(i,idof,jdof) - &
(da5)/(dy*dy) * Vyy(i,idof,jdof)
end do
i = ny-1
i0 = (i-1)*ndof
j0 = (i-4)*ndof
C0(i0+idof,j0+jdof) = (-1.0)/(12.0*dy) * Bh(i,idof,jdof) - &
db5/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-3)*ndof
C0(i0+idof,j0+jdof) = (6.0)/(12.0*dy) * Bh(i,idof,jdof) - &
db4/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-2)*ndof
C0(i0+idof,j0+jdof) = (-18.0)/(12.0*dy) * Bh(i,idof,jdof) - &
db3/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = (10.0)/(12.0*dy) * Bh(i,idof,jdof) + &
Dh(i,idof,jdof) - &
db2/(dy*dy) * Vyy(i,idof,jdof)
j0 = (i)*ndof
C0(i0+idof,j0+jdof) = (3.0)/(12.0*dy) * Bh(i,idof,jdof) - &
db1/(dy*dy) * Vyy(i,idof,jdof)
end do
end do
i = ny
i0 = (i-1)*ndof
idof = 1 ! continuity at the top boundary
do jdof = 1, ndof
j0 = (i-5)*ndof
C0(i0+idof,j0+jdof) = (3.0)/(12.0*dy) * Bh(i,idof,jdof)
j0 = (i-4)*ndof
C0(i0+idof,j0+jdof) = (-16.0)/(12.0*dy) * Bh(i,idof,jdof)
j0 = (i-3)*ndof
C0(i0+idof,j0+jdof) = (36.0)/(12.0*dy) * Bh(i,idof,jdof)
j0 = (i-2)*ndof
C0(i0+idof,j0+jdof) = (-48.0)/(12.0*dy) * Bh(i,idof,jdof)
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = Dh(i,idof,jdof) + &
(25.0)/(12.0*dy) * Bh(i,idof,jdof)
end do
!.... Add the time term into C0 accounting for BC's
do idof = 1, ndof
C0(idof,idof) = C0(idof,idof) - im * omega
end do
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
C0(i0+idof,i0+jdof) = C0(i0+idof,i0+jdof) - &
im * omega * G(i,idof,jdof)
end do
end do
end do
i = ny
i0 = (i-1)*ndof
do idof = 1, ndof
C0(i0+idof,i0+idof) = C0(i0+idof,i0+idof) - im * omega
end do
!.... form the equations
Dh = im * A + beta * Vxz
Bh = -im * Vxy
!.... account for the mapping
do idof = 1, ndof
do jdof = 1, ndof
Bh(:,idof,jdof) = Bh(:,idof,jdof) * deta
end do
end do
!.... C1
i = 1
i0 = (i-1)*ndof
idof = 1 ! continuity at the bottom boundary
do jdof = 1, ndof
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = Dh(i,idof,jdof) + &
(gg1/dy) * Bh(i,idof,jdof)
j0 = (i)*ndof
C1(i0+idof,j0+jdof) = (gg2/dy) * Bh(i,idof,jdof)
j0 = (i+1)*ndof
C1(i0+idof,j0+jdof) = (gg3/dy) * Bh(i,idof,jdof)
j0 = (i+2)*ndof
C1(i0+idof,j0+jdof) = (gg4/dy) * Bh(i,idof,jdof)
j0 = (i+3)*ndof
C1(i0+idof,j0+jdof) = (gg5/dy) * Bh(i,idof,jdof)
j0 = (i+4)*ndof
C1(i0+idof,j0+jdof) = (gg6/dy) * Bh(i,idof,jdof)
end do
do idof = 1, ndof
do jdof = 1, ndof
i = 2
i0 = (i-1)*ndof
j0 = (i-2)*ndof
C1(i0+idof,j0+jdof) = (gh1/dy) * Bh(i,idof,jdof)
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = (gh2/dy) * Bh(i,idof,jdof) + &
Dh(i,idof,jdof)
j0 = (i)*ndof
C1(i0+idof,j0+jdof) = (gh3/dy) * Bh(i,idof,jdof)
j0 = (i+1)*ndof
C1(i0+idof,j0+jdof) = (gh4/dy) * Bh(i,idof,jdof)
j0 = (i+2)*ndof
C1(i0+idof,j0+jdof) = (gh5/dy) * Bh(i,idof,jdof)
j0 = (i+3)*ndof
C1(i0+idof,j0+jdof) = (gh6/dy) * Bh(i,idof,jdof)
i = 3
i0 = (i-1)*ndof
j0 = (i-3)*ndof
C1(i0+idof,j0+jdof) = (gi1/dy) * Bh(i,idof,jdof)
j0 = (i-2)*ndof
C1(i0+idof,j0+jdof) = (gi2/dy) * Bh(i,idof,jdof)
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = (gi3/dy) * Bh(i,idof,jdof) + &
Dh(i,idof,jdof)
j0 = (i)*ndof
C1(i0+idof,j0+jdof) = (gi4/dy) * Bh(i,idof,jdof)
j0 = (i+1)*ndof
C1(i0+idof,j0+jdof) = (gi5/dy) * Bh(i,idof,jdof)
j0 = (i+2)*ndof
C1(i0+idof,j0+jdof) = (gi6/dy) * Bh(i,idof,jdof)
i = 4
i0 = (i-1)*ndof
j0 = (i-4)*ndof
C1(i0+idof,j0+jdof) = (gj1/dy) * Bh(i,idof,jdof)
j0 = (i-3)*ndof
C1(i0+idof,j0+jdof) = (gj2/dy) * Bh(i,idof,jdof)
j0 = (i-2)*ndof
C1(i0+idof,j0+jdof) = (gj3/dy) * Bh(i,idof,jdof)
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = (gj4/dy) * Bh(i,idof,jdof) + &
Dh(i,idof,jdof)
j0 = (i)*ndof
C1(i0+idof,j0+jdof) = (gj5/dy) * Bh(i,idof,jdof)
j0 = (i+1)*ndof
C1(i0+idof,j0+jdof) = (gj6/dy) * Bh(i,idof,jdof)
do i = 5, ny-2
i0 = (i-1)*ndof
j0 = (i-3)*ndof
C1(i0+idof,j0+jdof) = (1.0)/(12.0*dy) * Bh(i,idof,jdof)
j0 = (i-2)*ndof
C1(i0+idof,j0+jdof) = (-8.0)/(12.0*dy) * Bh(i,idof,jdof)
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = Dh(i,idof,jdof)
j0 = (i)*ndof
C1(i0+idof,j0+jdof) = (8.0)/(12.0*dy) * Bh(i,idof,jdof)
j0 = (i+1)*ndof
C1(i0+idof,j0+jdof) = (-1.0)/(12.0*dy) * Bh(i,idof,jdof)
end do
i = ny-1