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spatial.f90
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spatial.f90
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!==============================================================================
subroutine spatial(name, ind)
!==============================================================================
!
! Driver for spatial parallel stability solver with curvature
! This is the Chebyshev Collocation version.
!
! I have both IMSL and LAPACK routines. Currently using LAPACK
!
! Revised: 10-3-96
!==============================================================================
use stuff
use material
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 D1(ny,ny), D2(ny,ny), Dt1(ny,ny), Dt2(ny,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 g22vm(ny,ndof), g23vm(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)
complex C0(ndof*ny,ndof*ny), C1(ndof*ny,ndof*ny), C2(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(:)
!.... stuff for IMSL LU factorization
integer, allocatable :: ipvt(:)
real :: rcond
!.... some parameters
real, parameter :: big = 1.0e98
!.... metrics for the curved wall
real :: h(ny), dhds(ny), dhdr(ny), dhdsr(ny), dhdrr(ny)
!==============================================================================
if (verbose) write(*,*) "Starting Spatial eigenproblem"
!.... determine if inviscid
if (Re.ge.big .or. Re.eq.zero) then
write(*,*) 'A T T E N T I O N: Inviscid flow'
Navier = .false.
end if
!.... make the grid
call sgengrid(y, eta, deta, d2eta)
!.... read the mean field
if (ider) then
call getmean(vm, y, eta, ny, ind)
else
call getmean2(vm, y, eta, g2vm, g22vm, ny, ind)
end if
if (verbose) write(*,*) "Finished mean flow"
!.... Initialize the metric terms for curvature
if (curve.eq.1) then
call calch( x, ny, y, h, dhds, dhdr, dhdsr, dhdrr )
else if (curve.eq.2) then
! h = x + y ! note that x is the radius here
! dhds = zero
! dhdr = one
! dhdsr = zero
! dhdrr = zero
call circh( 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 (obsolete)
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
!.... setup parallel flow
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
!.... Compute the Chebyshev derivative matrices
call chebyd(D1, ny-1) ! use ny-1 since this routine uses 0:ny
D2 = matmul(D1, D1) ! compute the second derivative
if (wallt.eq.2) then
Dt1 = D1 ! Derivative operator for temperature
Dt1(ny,:) = zero ! adiabatic wall
Dt2 = matmul(D1, Dt1) ! Second Derivative operator for temperature
else
Dt1 = D1
Dt2 = D2
end if
!.... Compute derivatives of mean field using the appropriate difference scheme
g1vm = zero
if (ider) then
do idof = 1, ndof
g2vm(:,idof) = matmul(D1,vm(:,idof,ix))
end do
end if
g3vm = zero
g11vm = zero
g12vm = zero
g13vm = zero
if (ider) then
do idof = 1, ndof
g22vm(:,idof) = matmul(D2,vm(:,idof,ix))
end do
end if
g23vm = zero
g33vm = zero
!.... transform the gradients to physical space
if (ider) then
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
if (.true.) then
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 = ny, 1, -1
write (10,13) y(j), vm(j,2,ix), g2vm(j,2), g22vm(j,2)
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))
end if
!.... 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)
!.... It looks like I'm using the NONLINEAR D(5,5) ??
!.... It's okay, since d\bar\rho/dt = 0
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
!==============================================================================
if(verbose) write(*,*) "Finished building matrices"
!.... initialize
C0 = zero
C1 = zero
C2 = zero
!.... 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
!.... C0
if (top.eq.1) then
i = 1
i0 = (i-1)*ndof
idof = 1 ! continuity at infinity
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
end if
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j) - &
Vyy(i,idof,jdof) * D2(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
end do
end do
i = ny
i0 = (i-1)*ndof
idof = 1 ! continuity equation at the wall
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
if (wallt.eq.2) then
idof = ndof ! energy equation at the wall
do jdof = 1, ndof-1
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j) - &
Vyy(i,idof,jdof) * D2(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
jdof = ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * Dt1(i,j) - &
Vyy(i,idof,jdof) * Dt2(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end if
!.... Add the time term into C0 accounting for BC's
i = 1
if (top.eq.1) then
C0(1,1) = C0(1,1) - im * omega ! continuity equation
else
C0(1,1) = -one ! zero density at infinity
end if
C0(2,2) = -one
C0(3,3) = -one
C0(4,4) = -one
C0(5,5) = -one
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
idof = 1
C0(i0+idof,i0+idof) = C0(i0+idof,i0+idof) - im * omega
idof = 2
C0(i0+idof,i0+idof) = C0(i0+idof,i0+idof) - one
idof = 3
C0(i0+idof,i0+idof) = C0(i0+idof,i0+idof) - one
idof = 4
C0(i0+idof,i0+idof) = C0(i0+idof,i0+idof) - one
idof = ndof
if (wallt.eq.0) then
C0(i0+idof,i0+idof) = C0(i0+idof,i0+idof) - one
else if (wallt.eq.2) then
do jdof = 1, ndof
C0(i0+idof,i0+jdof) = C0(i0+idof,i0+jdof) - im * omega * &
G(i,idof,jdof)
end do
else
write(*,"('Illegal value of wallt: ',i4)") wallt
call exit(1)
end if
!.... 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
if (top.eq.1) then
i = 1
i0 = (i-1)*ndof
idof = 1 ! continuity equation at infinity
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
end if
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
end do
end do
i = ny
i0 = (i-1)*ndof
idof = 1 ! density equation at the wall
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
if (wallt.eq.2) then
i = ny
i0 = (i-1)*ndof
idof = ndof ! energy equation at the wall
do jdof = 1, ndof-1
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
jdof = ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * Dt1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end if
!.... C2 (only viscous contribution)
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
C2(i0+idof,i0+jdof) = Vxx(i,idof,jdof)
end do
end do
end do
if (wallt.eq.2) then
i = ny
i0 = (i-1)*ndof
idof = ndof ! energy equation at the wall
do jdof = 1, ndof
C2(i0+idof,i0+jdof) = Vxx(i,idof,jdof)
end do
end if
!.... form the extended system
if (verbose) write(*,*) "Forming the extended system"
A0 = zero
B0 = zero
evec = zero
alp = zero
bet = zero
omg = zero
allocate (ipvt(ndof*ny))
#ifdef CRAY
call CGETRF(ndof*ny, ndof*ny, C0, ndof*ny, ipvt, info)
if (info.ne.zero) write(*,*) 'CGETRF: ',info
#else
call ZGETRF(ndof*ny, ndof*ny, C0, ndof*ny, ipvt, info)
if (info.ne.zero) write(*,*) 'ZGETRF: ',info
#endif
C1 = -C1
C2 = -C2
#ifdef CRAY
call CGETRS('N', ndof*ny, ndof*ny, C0, ndof*ny, &
ipvt, C1, ndof*ny, info)
if (info.ne.0) then
write(*,*) 'CGETRS: ',info
call exit(1)
end if
call CGETRS('N', ndof*ny, ndof*ny, C0, ndof*ny, &
ipvt, C2, ndof*ny, info)
if (info.ne.0) write(*,*) 'CGETRS: ',info
#else
call ZGETRS('N', ndof*ny, ndof*ny, C0, ndof*ny, &
ipvt, C1, ndof*ny, info)
if (info.ne.0) then
write(*,*) 'ZGETRS: ',info
call exit(1)