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c***********************************************************************
c> \file orrucbfs.f
c> \brief Solves the Orr-Sommerfeld equation for incompressible backward
c> facing step with a moving reference frame
c> \author S. Scott Collis
c***********************************************************************
program orr_uc_bfs
c***********************************************************************
c
c Purpose: This program solves the Orr-Sommerfeld equation using a
c Chebyshev-collocation method.
c
c Version: This version solves the backward facing step problem
c with moving reference frame to find absolute/
c convective instability.
c
c Author: S. Scott Collis
c
c Date: 3-14-92
c
c Revision: 9-18-92
c
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
integer nar, nai, ians, numdat, ilen
real dat(100,6)
real alphar(100), alphai(100), delta, deltauold
real minar, maxar, incar, minai, maxai, incai, dar, dai
real uo(idim),wor(idim),woi(idim),aor(idim),aoi(idim)
complex eigenvalue(100), eigenvector(0:idim), ctemp, A, B
complex alp(3), as, oldeig(3,100), fd
real oldeigr1, oldeigi1
character*20 filename, rootname
logical print, eof
c***********************************************************************
c
c Setup IMSL workspace
c
#ifdef USE_IMSL
REAL RWKSP(100000)
COMMON /WORKSP/ RWKSP
call IWKIN(100000)
#endif
c
c User input
c
eof = .false.
Lmap = 2.0
write (*,10)
10 format (/,/,10x,'Solve Orr-Sommerfeld (Collocation)')
write (*,20)
20 format (/,1x,'Enter the number of modes ==> ',$)
read (*,*) n
write (*,30)
30 format (/,1x,'Enter Reynolds number ==> ',$)
read (*,*) Re
write (*,100)
100 format (/,1x,'Enter root filename ==> ',$)
read (*,'(a)') rootname
ilen = index (rootname,' ')-1
write (*,50)
50 format (/,1x,'Restart (1/0) ==> ',$)
read (*,*) irun
if (irun.eq.0) then
filename = rootname(1:ilen)//'.dat'
open (unit=10,file=filename,form='formatted',status='unknown')
numdat = 1
do while (.not. eof)
read(10,*,end=99,err=900) (dat(numdat,j), j=1,6)
numdat = numdat + 1
end do
99 close (10)
numdat = numdat - 1
do i = numdat-1,1,-1
if ( ( dat(i,6) .gt. 0 ) .and. ( dat(i+1,6) .lt. 0 ) ) i2 = i
end do
i1 = i2 + 1
uc = dat(i1,5)+(dat(i2,5)-dat(i1,5))/
. (dat(i2,6)-dat(i1,6))*(0.0-dat(i1,6))
alphar(1) = dat(i1,1)+(dat(i2,1)-dat(i1,1))/
. (dat(i2,6)-dat(i1,6))*(0.0-dat(i1,6))
alphai(1) = 0.0
istart = 1
else
filename = rootname(1:ilen)//'.abs'
open (unit=10,file=filename,status='old',form='formatted')
EOF = .false.
i = 1
do while (.not. EOF)
read (10,*,END=200,ERR=900) uo(i),wor(i),woi(i),aor(i),aoi(i)
c write (*,*) i,uo(i),wor(i),woi(i),aor(i),aoi(i)
i = i + 1
end do
200 close(10)
i = i - 1
oldeigr1 = wor(i-2)
oldeigi1 = woi(i-2)
oldeigr2 = wor(i-1)
oldeigi2 = woi(i-1)
deltauold = uo(i-1)-uo(i-2)
uc = uo(i)
alphar(1) = aor(i)
alphai(1) = aoi(i)
do i = 1, 3
oldeig(i,1) = CMPLX(oldeigr1,oldeigi1)
oldeig(i,2) = CMPLX(oldeigr2,oldeigi2)
end do
istart = 3
end if
write (*,45)
45 format (/,1x,'Enter delta ==> ',$)
read (*,*) delta
write (*,46)
46 format (/,1x,'Enter deltau ==> ',$)
read (*,*) deltau
c
c Open output file
c
filename = rootname(1:ilen)//'.abs'
open (unit=10,file=filename,status='unknown',form='formatted',
. position='append')
ians = 0
if (ians.eq.1) then
print = .true.
else
print = .false.
end if
c
c Set initial cluster to find the saddle point
c
alphar(2) = alphar(1) + delta
alphar(3) = alphar(1) - delta
alphai(2) = alphai(1) - delta
alphai(3) = alphai(1) - delta
call MAKE_DERIVATIVES
C call MAKE_BL_METRICS
call INIT_BFS_PROFILE
as = cmplx(10,10)
kmax = 30
do k = istart, kmax+istart-1
resid = 10.
if (k .le. 2) then
do i = 1, 3
oldeig(i,k) = 999.
end do
else if (k .eq. 3 .and. istart .ne. 1) then
do i = 1, 3
fd = (oldeig(i,k-1)-oldeig(i,k-2))/deltauold
oldeig(i,k) = oldeig(i,k-1) + fd*deltauold
end do
else
do i = 1, 3
fd = (oldeig(i,k-1)-oldeig(i,k-2))/deltau
oldeig(i,k) = oldeig(i,k-1) + fd*deltau
end do
end if
call SHIFT_BL
icount = 0
do while (abs(resid) .gt. 1e-7 )
icount = icount + 1
do i = 1, 3
aso = as
alpha = cmplx (alphar(i), alphai(i))
alp(i) = alpha
call MAKE_MATRIX
ctemp = oldeig(i,k)
call SOLVE(ctemp, eigenvector, print)
eigenvalue(i) = ctemp
oldeig(i,k) = ctemp
write (*,60) real(alpha),aimag(alpha),real(eigenvalue(i)),
. aimag(eigenvalue(i))
60 format (1x,4(e17.10,4x))
end do
A = cmplx(0.,-1.)*(eigenvalue(2)-eigenvalue(1))
B = cmplx(0.,-1.)*(eigenvalue(3)-eigenvalue(1))
as = (A*(alp(3)**2-alp(1)**2)-B*(alp(2)**2-alp(1)**2))/
. (2.*B*(alp(1)-alp(2))-2.*A*(alp(1)-alp(3)))
resid = as-aso
if (k .eq. istart .or. icount .gt. 0) then
write (*,80) real(as),aimag(as), abs(resid)
80 format (1x,'==> ',3(e17.10,4x))
end if
alphar(1) = real(as)
alphai(1) = aimag(as)
alphar(2) = alphar(1) + delta
alphar(3) = alphar(1) - delta
alphai(2) = alphai(1) - delta
alphai(3) = alphai(1) - delta
end do
write (*,90) uc, real(eigenvalue(1)),aimag(eigenvalue(1)),
. real(as),aimag(as)
write (*,*)
write (10,90) uc, real(eigenvalue(1)),aimag(eigenvalue(1)),
. real(as),aimag(as)
90 format (1x,5(e17.10,4x))
uc = uc - deltau
end do
close (10)
goto 910
900 continue
write (*,*) 'Error reading file...'
910 stop
end
C***********************************************************************
subroutine MAKE_DERIVATIVES
C***********************************************************************
C
C Make the required matrices that take derivatives in Chebyshev
C space
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
integer m, p, LDD, i, j, k
real Identity(0:idim,0:idim), D1hat(0:idim,0:idim)
LDD = idim
call CHEBYD (D1hat, LDD, n)
call CHEBYD (D1 , LDD, n)
c
c We enforce noslip indirectly by setting v' = 0 at -1.
c NOTE THAT D1-D4 SHOULD ONLY BE APPLIED TO THE v FIELD ONLY.
c
do i = 0, n
c D1(0,i) = 0.0
D1(N,i) = 0.0
end do
C
C To get higher derivatives just do matrix multiplication
C
do i = 0, n
do j = 0, n
D2(i,j) = 0.0
do k = 0, n
D2(i,j) = D2(i,j) + D1hat(i,k)*D1(k,j)
end do
end do
end do
c
c At the upper surface we allow the flow to slip by setting v'' = 0
c at 1. NOTE THAT D1-D4 SHOULD ONLY BE APPLIED TO THE v FIELD ONLY.
c
do i = 0, n
D2(0,i) = 0.0
c D2(N,i) = 0.0
end do
do i = 0, n
do j = 0, n
D3(i,j) = 0.0
do k = 0, n
D3(i,j) = D3(i,j) + D1hat(i,k)*D2(k,j)
end do
end do
end do
do i = 0, n
do j = 0, n
D4(i,j) = 0.0
do k = 0, n
D4(i,j) = D4(i,j) + D1hat(i,k)*D3(k,j)
end do
end do
end do
c CALL WRRRN ('D1', N+1, N+1, D1, LDd+1, 0)
c CALL WRRRN ('D2', N+1, N+1, D2, LDd+1, 0)
c CALL WRRRN ('D3', N+1, N+1, D3, LDd+1, 0)
c CALL WRRRN ('D4', N+1, N+1, D4, LDd+1, 0)
return
end
C***********************************************************************
subroutine MAKE_BL_METRICS
C***********************************************************************
C
C Setup the collocation points in the mapped coordinate, eta and in
C chebyshev space, th. Also compute the transformation metrics.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
real pi, dth
integer i
pi = ACOS(-1.0)
dth = pi/FLOAT(n)
c
c Make mesh in transformed, eta, and Chebyshev space, th
c
do i = 0, n
th(i) = FLOAT(i)*dth
eta(i) = COS(th(i))
end do
c
c Make transformation metrics
c
do i = 0, n
m1(i) = 2.0/Lmap
m2(i) = 0.0
m3(i) = 0.0
m4(i) = 0.0
if (i.eq.0) then
c(i) = 2.
else
c(i) = 1.
end if
end do
return
end
C***********************************************************************
SUBROUTINE FUNCD(X,F,DF)
C***********************************************************************
REAL X, ETA, GAMMA, ETAOUT
COMMON /map/ ETA
GAMMA = 1.2
ETAOUT = 15.0
F = ETAOUT*(1.-TANH(GAMMA))/2.*(X+1)/
. (1.-TANH(GAMMA/2.*(X+1)))-ETA
A = ETAOUT/2.*(1-TANH(GAMMA))
B = GAMMA/2.
DF = A/2.*(1.+EXP(2.*B*(1.+X)) + 2.*B*EXP(2.*B*(1.+X)) +
. 2.*B*EXP(2.*B*(1.+X))*X)
RETURN
END
C***********************************************************************
subroutine INIT_BL_PROFILE
C***********************************************************************
C
C Setup the initial BL profile
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
parameter (nu = 64)
real utemp(0:nu),utemp1(0:nu),utemp2(0:nu),junk, pi
real gamma, yout, y, x, xi, IU, IV, IW
integer nmode, i, merge
character*15 filename
integer LDD
real D1hat(0:idim,0:idim), D2hat(0:idim,0:idim)
common /map/ x
pi = ACOS(-1.0)
LDD = idim
gamma = 1.2
xout = 15.0
write (*,5)
5 format (/,1x,'Read Mean Profile',/)
write (*,9)
9 format (1x,'Enter filename ==> ',$)
read (*,'(a)') filename
open (unit=11,file=filename,status='unknown')
read (11,*) nmode
do i = 0, nmode
read (11,*) y,utemp(i),junk
end do
close (11)
c
c I need to spectrally interpolate this profile onto the new grid.
c
call CHEBYSHEV (utemp,nmode,1)
c
c y is nondimensionalized by dr = sqrt(2*x*nu/u_inf)
c
i = n
y = Lmap*(1.+eta(i))/(1.-eta(i))
do while (y/sqrt(2.) .le. xout)
u(i) = 0.0
X = Y/SQRT(2.)
xi = RTNEWT(-2.,2.2,1e-12,FUNCD)
do m = 0, nmode
u(i) = u(i)+utemp(m)*COS(float(m)*ACOS(xi))
end do
i = i - 1
y = Lmap*(1.+eta(i))/(1.-eta(i))
end do
merge = i
c write (*,11) merge
11 format(/,1x,'Merging at i = ',i4)
do j = merge, 0, -1
u(j) = 1.0
end do
c
c Compute the collocation derivatives
c
call CHEBYD (D1hat, LDD, n)
do i = 0, n
do j = 0, n
D2hat(i,j) = 0.0
do k = 0, n
D2hat(i,j) = D2hat(i,j) + D1hat(i,k)*D1hat(k,j)
end do
end do
end do
do i = 0, n
d1u(i) = 0.0
d2u(i) = 0.0
do k = 0, n
d1u(i) = d1u(i) + D1hat(i,k)*u(k)
d2u(i) = d2u(i) + D2hat(i,k)*u(k)
end do
Uorg(i) = u(i)
end do
c
c Ensure that the derivatives are zero when they are supposed to be.
c
do j = merge-2, 0, -1
d1u(j) = 0.0
d2u(j) = 0.0
end do
c
c call CHEBYSHEV (U,n,1)
c call CHEBYSHEV (d1u,n,1)
c call CHEBYSHEV (d2u,n,1)
c
c do i = 0, 256
c X = float(I)*PI/256
c IU = 0.0
c IV = 0.0
c IW = 0.0
c DO M = 0, N
c IU = IU + U(M)*COS(FLOAT(M)*X)
c IV = IV + d1u(M)*COS(FLOAT(M)*X)
c IW = IW + d2u(M)*COS(FLOAT(M)*X)
c END DO
c write (*,10) cos(x),Iu,Iv,Iw
c end do
c
c call CHEBYSHEV (U,n,-1)
c call CHEBYSHEV (d1u,n,-1)
c call CHEBYSHEV (d2u,n,-1)
return
end
C***********************************************************************
subroutine INIT_BFS_PROFILE
C***********************************************************************
C
C Setup the initial BL profile
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
parameter (idim2=1000)
integer i, nbl, LDD
character*20 filename
real junk, us(0:idim2), d2us(0:idim2), pi
real xi(0:idim2), d1us(0:idim2), d1max, d2max
real ydat(0:idim2), ymin, ymax, h, ut(0:idim2)
real uspl(0:idim2), d2Uspl(0:idim2), metric
real D1hat(0:idim,0:idim), D2hat(0:idim,0:idim)
character*1 temp
pi = ACOS(-1.0)
LDD = idim
if (.true.) then
write (*,10)
10 format (/,1x,'Read Mean Profile',/)
write (*,20)
20 format (1x,'Enter filename ==> ',$)
read (*,'(a)') filename
open (unit=11,file=filename,status='unknown')
i = 0
read (11,35) temp, nbl
35 format (a1,i5)
nbl = nbl - 1
do i = 0, nbl
read (11,*,end=100) ydat(i),Ut(i),junk
end do
close (11)
100 continue
else
nbl = 100
do i = 0, nbl
ydat(i) = -1.0+i*1.0/nbl
Ut(i) = (1.0-ydat(i)**2)
end do
end if
c
c Make a mirror image
c
c do i = nbl-1, 0, -1
c ydat(2*nbl-i) = 2.*ydat(nbl)-ydat(i)
c Ut(2*nbl-i) = Ut(i)
c end do
c nbl = 2*nbl
c
c Interpolate data onto a Chebyshev Grid
c
ymin = ydat(0)
ymax = ydat(nbl)
h = (ymax-ymin)/float(nbl)
Lmap = ymax-ymin
call MAKE_BL_METRICS
write (*,11) nbl
11 format (/,1x,'Nbl = ',i5)
call SPLINE(nbl+1,ydat,Ut,uspl)
dth = pi/(nbl)
do i = 0, nbl
thtemp = i*dth
xi(i) = COS(thtemp)*Lmap/2. + Lmap/2. + ymin
call SPEVAL(nbl+1,ydat,Ut,uspl,xi(i),us(i))
c write (9,40) xi(i), us(i), thtemp
end do
write (*,66) nbl
66 format (/,1x,'nbl = ',i5,' Enter ncut ==> ',$)
read (*,*) ifilter
write (*,77) ifilter
77 format (/,1x,'Low pass filter at n = ',i5)
call COSFT(us,nbl,1)
do i = ifilter, nbl
us(i) = 0.0
end do
c call DCHEBYSHEVF(us,d1us,nbl)
c call DCHEBYSHEVF(d1us,d2us,nbl)
c call COSFT(us,nbl,-1)
c call COSFT(d1us,nbl,-1)
c call COSFT(d2us,nbl,-1)
c do i = 0, nbl
c write (10,50) xi(i),us(i),d1us(i)*metric,d2us(i)*metric**2
c 50 format (1x,4(e12.4,1x))
c end do
c
c Convert to theta grid
c
do i = 0, n
call CHEBYINTF(nbl,u(i),th(i),us)
end do
c
c Compute the collocation derivatives
c
call CHEBYD (D1hat, LDD, n)
do i = 0, n
do j = 0, n
D2hat(i,j) = 0.0
do k = 0, n
D2hat(i,j) = D2hat(i,j) + D1hat(i,k)*D1hat(k,j)
end do
end do
end do
do i = 0, n
d1u(i) = 0.0
d2u(i) = 0.0
do k = 0, n
d1u(i) = d1u(i) + D1hat(i,k)*u(k)
d2u(i) = d2u(i) + D2hat(i,k)*u(k)
end do
end do
do i = 0, n
Uorg(i) = u(i)
write (20,12) eta(i), u(i), d1u(i), d2u(i)
end do
12 format (1x,4(e16.8,4x))
write (*,30)
30 format (/,1x,'Velocity Profile completed...',/)
40 format (1x,5(e12.5,2x))
90 format (1x,i5,1x,e12.5)
return
end
C***********************************************************************
subroutine SHIFT_BL
C***********************************************************************
C
C Shift the boundary layer profile to a new frame of reference.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
c
c Need to interpolate the velocity profile to evaluate it at
c arbitrary y
c
do i = 0, n
U(i) = Uorg(i) - Uc
end do
return
end
C***********************************************************************
subroutine MAKE_MATRIX
C***********************************************************************
C
C This routine generates the matrices which are combined to make
C the generalized eigenvalue problem.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
real identity(0:idim,0:idim)
complex work, ai
do i = 0, n
do j = 0, n
A4(i,j) = 0.0
A3(i,j) = 0.0
A2(i,j) = 0.0
A1(i,j) = 0.0
B2(i,j) = 0.0
B1(i,j) = 0.0
B0(i,j) = 0.0
identity(i,j) = 0.0
end do
identity(i,i) = 1.0
end do
C
C Include the independent variable transformation metrics
C
do i = 0, n
work = m1(i)**4
do j = 0, n
A4(i,j) = work*D4(i,j)
end do
end do
do i = 0, n
work = 6.*m1(i)**2*m2(i)
do j = 0, n
A3(i,j) = work*D3(i,j)
end do
end do
do i = 0, n
work = 3.*m2(i)**2+4.*m1(i)*m3(i)-2.*alpha**2*m1(i)**2-
. cmplx(0.,1.)*alpha*Re*U(i)*m1(i)**2
do j = 0, n
A2(i,j) = work*D2(i,j)
end do
end do
do i = 0, n
work = m4(i)-2.*alpha**2*m2(i)-cmplx(0.,1.)*alpha*Re*U(i)*m2(i)
do j = 0, n
A1(i,j) = work*D1(i,j)
end do
end do
do i = 0, n
work = alpha**4+cmplx(0.,1.)*alpha**3*Re*U(i)+
. cmplx(0.,1.)*alpha*Re*(d2u(i)*m1(i)**2+d1u(i)*m2(i))
do j = 0, n
A0(i,j) = work*identity(i,j)
end do
end do
do i = 0, n
work = -1.0*cmplx(0.,1.)*Re*m1(i)**2
do j = 0, n
B2(i,j) = work*D2(i,j)
end do
end do
do i = 0, n
work = -1.0*cmplx(0.,1.)*Re*m2(i)
do j = 0, n
B1(i,j) = work*D1(i,j)
end do
end do
do i = 0, n
work = cmplx(0.,1.)*alpha**2*Re
do j = 0, n
B0(i,j) = work*identity(i,j)
end do
end do
return
end
C***********************************************************************
function CHECKEIG(N,A,LDA,EVAL,EVEC)
C***********************************************************************
C
C Check an eigenvalue and eigenvector
C
C***********************************************************************
integer N
complex A(LDA,N), EVAL, EVEC(N)
complex X(N), Y(N)
real CHECKEIG
#ifdef USE_IMSL
CALL MUCRV (N, N, A, LDA, N, EVEC, 1, N, X)
#else
CALL ZGEMV ('N', N, N, 1.0, A, LDA, EVEC, 1, 0.0, X, 1)
#endif
CHECKEIG = 0.0
DO I = 1, N
CHECKEIG = CHECKEIG + ABS(X(I)-EVAL*EVEC(I))
END DO
CHECKEIG = CHECKEIG/FLOAT(N)
RETURN
END
C***********************************************************************
subroutine HTRAN(N,A,B,LD)
C***********************************************************************
C
C Take the complex conjugate transpose of A and put it in B
C
C***********************************************************************
integer N, LD
complex A(LD,N), B(LD,N)
do i = 1, N
do j = 1, N
B(j,i) = CONJG(A(i,j))
end do
end do
RETURN
END
C***********************************************************************
subroutine CXDOTY (N,X,Y,C)
C***********************************************************************
C
C Take the complex conjugate dot product of vector x and y
C
C***********************************************************************
integer N
complex X(N), Y(N), C
C = CMPLX (0.0,0.0)
do I = 1, N
C = C + CONJG(X(I))*Y(I)
end do
RETURN
END
C***********************************************************************
C S O L V E O R R S O M M E R F E L D
C***********************************************************************
subroutine SOLVE(eigenvalue,eigenvector,print)
C***********************************************************************
C
C This routine generates the discrete eigenvalue problem in
C Chebyshev space for the Orr-Sommerfeld equation.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap, uorg(0:idim)
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim), uc
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,uc,uorg,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
complex A(0:idim,0:idim), B(0:idim,0:idim)
complex dA(0:idim,0:idim), dB(0:idim,0:idim)
complex T1(0:idim,0:idim), T2(0:idim,0:idim)
complex T3(0:idim,0:idim), T4(0:idim,0:idim)
complex eval(0:idim), evec(0:idim,0:idim), tvec(0:idim)
complex aeval(0:idim), aevec(0:idim,0:idim), tavec(0:idim)
complex eigenvalue, eigenvector(0:idim)
complex dwda, dw, dalp, y(0:idim), prod
real temp1(0:idim), temp2(0:idim), residual, CHECKEIG, min
real temp3(0:idim), temp4(0:idim), dalpha, diff(0:idim)
integer lda, ldb, ldevec, p, i, j, which, k, l, m
integer index(0:idim), index2(0:idim)
logical first, print
external CHECKEIG
C***********************************************************************
lda = idim+1
ldb = idim+1
ldevec = idim+1
do i = 1, n-1
do j = 1, n-1
A(i-1,j-1) = A4(i,j)+A3(i,j)+A2(i,j)+A1(i,j)+A0(i,j)
B(i-1,j-1) = B2(i,j)+B1(i,j)+B0(i,j)
end do
end do
c
c Now enforce the v = 0 @ +_ 1 boundary condition instead of the
c Orr-Sommerfeld equation at n = 0 and n = N. The Neumann
c boundary condition is implicitly enforced.
c
c do j = 0, n
c B(0,j) = CMPLX(0.0, 0.0)
c B(n,j) = CMPLX(0.0, 0.0)
c A(0,j) = CMPLX(0.0, 0.0)
c A(n,j) = CMPLX(0.0, 0.0)
c end do
c A(0,0) = CMPLX(1.0, 0.0)
c A(n,n) = CMPLX(1.0, 0.0)
c
c But the top and bottom rows are trivial so that they can be
c removed
c
#ifdef USE_IMSL
write(*,*) "Solving Eigenvalue Problem with IMSL interface"
CALL LINCG (N-1, B, LDA, T1, LDA)
CALL MCRCR (N-1, N-1, T1, LDA, N-1, N-1, A, LDA,
. N-1, N-1, T4, LDA)
CALL EVCCG (N-1, T4, LDA, eval, evec, ldevec)
#else
write(*,*) "Solving Eigenvalue Problem with new LAPACK interface"
LWORK = 2*(N-1)
INFO = 0