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conteucbl.f
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conteucbl.f
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C> \file conteucbl.f
C> \brief Solve OS saddle point problem shifting reference frame
C> \author S. Scott Collis
c***********************************************************************
program conte_uc_bl
c***********************************************************************
c
c Purpose: Solve the Orr-Sommerfeld equation using 4th order Runge-
c Kutta explicit integration with orthogonalization using
c the method of Conte.
c
c This routine tracks the saddle point in the complex
c wavenumber plane by shifting the reference frame
c velocity, u_c.
c
c This routine works rather well but to get good
c eigenvalues you must really resolve the boundary-layer
c profile well. The OS shooting solution appears to be
c quite sensitive to the profile.
c
c Author: Scott Collis
c
c Date: 7-17-92
c
c Revised: 1-1-2020
c
c***********************************************************************
c Common variables
c***********************************************************************
parameter (imax=20000)
integer nbl
complex alpha, c, rgamma
real U(0:imax), d2U(0:imax), Re, ymin, ymax, h, beta, uc
real Uspl(0:imax), d2Uspl(0:imax), ydat(0:imax), f2p
real Uorg(0:imax)
common /mean/ nbl, u, d2u, ymin, ymax, h, f2p, uc,
& rgamma, beta, uspl, d2uspl, ydat, Uorg
common /eig/ c, alpha, Re
c***********************************************************************
parameter (neq=4, inunit=20)
complex bc1(neq), bc2(neq)
real testalpha
integer north, nstep, num, omega, saddle, readbl
logical eigfun
character*1 input
character*20 infile
character*80 string
c***********************************************************************
complex alphas,resid,eigenvalue(3),alphasold,A,B,alp(3)
real delta,alphar(3),alphai(3),deltauc
c***********************************************************************
c
c Defaults
c
nbl = 1000
nstep = 1000
north = 50
testalpha = 10.0
Re = 580.
beta = 0.0
alphar(1) = 0.179
alphai(1) = 0.0
cr = 0.36413124E+00
ci = 0.79527387E-02
ymin = 0.0
ymax = 17.0
f2p = 0.5
delta = 0.0
deltauc = 0.0
num = 1
Uc = 0.0
omega = 0
saddle = 0
readbl = 0
c
c Input from keyboard
c
write (*,10)
10 format (/,10x,'Solve Orr-Sommerfeld (Shooting)',/)
write (*,15)
15 format(/,1x,'Read from keyboard, file, default (k,f,d) ==> ',$)
read (*,'(a)') input
if (input .eq. 'k' .or. input .eq. 'K') then
write (*,20)
20 format (/,1x,'Enter number of profile points ==> ',$)
read (*,*) nbl
write (*,25)
25 format (/,1x,'Enter number of integration steps ==> ',$)
read (*,*) nstep
write (*,27)
27 format (/,1x,'Enter Test Alpha ==> ',$)
read (*,*) testalpha
write (*,30)
30 format (/,1x,'Enter Reynolds number ==> ',$)
read (*,*) Re
write (*,35)
35 format (/,1x,'Enter beta ==> ',$)
read (*,*) beta
write (*,37)
37 format (/,1x,'Enter guess for BL second deriv. at wall ==> ',$)
read (*,*) f2p
write (*,40)
40 format (/,1x,'Enter alpha_r ==> ',$)
read (*,*) alphar(1)
write (*,45)
45 format (/,1x,'Enter alpha_i ==> ',$)
read (*,*) alphai(1)
write (*,50)
50 format (/,1x,'Enter guess for (c_r, c_i) ==> ',$)
read (*,*) cr, ci
write (*,55)
55 format (/,1x,'Enter Ymin and Ymax ==> ',$)
read (*,*) Ymin, Ymax
write (*,65)
65 format (/,1x,'Enter reference frame velocity u_c ==> ',$)
read (*,*) uc
write (*,70)
70 format (/,1x,'Enter delta ==> ',$)
read (*,*) delta
write (*,71)
71 format (/,1x,'Enter deltauc ==> ',$)
read (*,*) deltauc
else if (input .eq. 'f' .or. input .eq. 'F') then
write (*,60)
60 format (/,1x,'Enter filename ==> ',$)
read (*,'(a)') infile
open (unit=inunit, file=infile, form='formatted')
string(1:1) = '#'
do while (string(1:1) .eq. '#')
read(inunit,'(a)',err=99) string
end do
call readi (string,nbl)
read(inunit,'(a)') string
call readi (string,nstep)
read(inunit,'(a)') string
call readr (string,testalpha)
read(inunit,'(a)') string
call readr (string,Re)
read(inunit,'(a)') string
call readr (string,beta)
read(inunit,'(a)') string
call readr (string,f2p)
read(inunit,'(a)') string
call readr (string,alphar(1))
read(inunit,'(a)') string
call readr (string,alphai(1))
read(inunit,'(a)') string
call readr (string,cr)
read(inunit,'(a)') string
call readr (string,ci)
read(inunit,'(a)') string
call readr (string,Ymin)
read(inunit,'(a)') string
call readr (string,Ymax)
read(inunit,'(a)') string
call readr (string,Uc)
read(inunit,'(a)') string
call readr (string,delta)
read(inunit,'(a)') string
call readr (string,deltauc)
read(inunit,'(a)') string
call readi (string,num)
read(inunit,'(a)') string
call readi (string,omega)
read(inunit,'(a)') string
call readi (string,saddle)
end if
c
c Check fixed dimensions
c
if (nbl .gt. imax) then
write (*,300)
300 format (/,/,1x,'N > Idim...Stopping',/,/)
goto 210
end if
c
c Set constants
c
alpha = CMPLX (alphar(1), alphai(1))
c = CMPLX (cr,ci)
h = (ymax-ymin)/float(nbl)
if (omega .eq. 1) c = c/alpha
c
c Echo input
c
write (*,100) nbl
100 format (/,1x,'nbl = ',i5)
write (*,102) nstep
102 format (1x,'nstep = ',i5)
write (*,105) testalpha
105 format (1x,'Test Alpha = ',e20.10)
write (*,110) Re
110 format (1x,'Re = ',e20.10)
write (*,115) beta
115 format (1x,'Beta = ',e20.10)
write (*,117) f2p
117 format (1x,'f2p = ',e20.10)
write (*,140) ymin, ymax
140 format (1x,'Ymin = ',e20.10,' Ymax = ',e20.10)
write (*,120) alphar(1), alphai(1)
120 format (1x,'alpha = (',e20.10,', ',e20.10,')')
write (*,130) REAL(c), AIMAG(c)
130 format (1x,'c = (',e20.10,', ',e20.10,')')
write (*,145) Uc
145 format (1x,'Uc = ',e20.10)
write (*,150) delta
150 format (1x,'delta = ',e20.10)
write (*,155) deltauc
155 format (1x,'deltauc = ',e20.10)
write (*,160) num
160 format (1x,'num = ',i5,/)
c
c set the boundary conditions
c
do i = 1, neq
bc1(i) = 0.0
bc2(i) = 0.0
end do
c
c Fix to make my nondimensionalization match Mack's
c
Re = Re*SQRT(2.)
alphar(1) = alphar(1)
alphai(1) = alphai(1)
alphar(2) = alphar(1) + delta
alphar(3) = alphar(1) - delta
alphai(2) = alphai(1) - delta
alphai(3) = alphai(1) - delta
ymin = ymin
ymax = ymax
c
c Loop for different convection speeds
c
if (readbl.eq.1) then
call READ_BL
else
call SOLVE_BL
end if
if (saddle .eq. 1) then
eigfun = .false.
alphas = cmplx(10.,10.)
do k = 1, num
c write (*,*) uc
call SHIFT_BL
resid = 10.
icount = 0
do while ( abs(resid) .gt. 1.E-5 )
icount = icount + 1
do i = 1, 3
alphasold = alphas
alpha = cmplx (alphar(i), alphai(i))*SQRT(2.)
alp(i) = alpha/SQRT(2.)
call CONTE(nstep,testalpha,neq,2,bc1,bc2,ymin,ymax,eigfun)
eigenvalue(i) = c*alp(i)
c write (*,61) real(alp(i)),aimag(alp(i)),real(eigenvalue(i))
c & ,aimag(eigenvalue(i))
61 format (1x,4(ES17.9E3,1x))
end do
A = cmplx(0.,-1.)*(eigenvalue(2)-eigenvalue(1))
B = cmplx(0.,-1.)*(eigenvalue(3)-eigenvalue(1))
alphas = (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 = alphas-alphasold
if (k .eq. 1 .or. icount .gt. 4) then
write (*,80) real(alphas), aimag(alphas), abs(resid)
80 format (1x,'==> ',3(e17.10,4x))
end if
alphar(1) = real(alphas)
alphai(1) = aimag(alphas)
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(alphas),aimag(alphas)
90 format (1x,f6.4,2x,4(e17.10,4x))
uc = uc - deltauc
end do
else
eigfun = .true.
alpha = cmplx (alphar(1), alphai(1))*SQRT(2.)
call CONTE(nstep,testalpha,neq,2,bc1,bc2,ymin,ymax,eigfun)
write (*,61) real(alpha),aimag(alpha),real(c),
& aimag(c)
end if
c
c Read error
c
goto 210
99 write (*,200)
200 format(/,/1x,'Error in input file...',/,/)
210 continue
stop
end
C***********************************************************************
subroutine CONTE(nstep, testalpha, n, r, yo, yf, to, tf, eigfun)
C***********************************************************************
C
C First order linear boundary value problem solver using Conte's
C method. Fourth order Runge-Kutta is used for time advancement
C
c***********************************************************************
complex c, alpha
real Re
common /eig/ c, alpha, Re
C***********************************************************************
integer i, m, q, r, s, mi, mj, qend, IPVT(n-r), icount, north
real t, tq(0:nstep), to, tf, h
complex yo(n), yf(n), B(n-r,0:nstep), err, cold, ctemp
complex U(n,n-r,0:nstep), P(n-r,n-r,0:nstep), z(n,n-r)
complex y(n,0:nstep), v(n,0:nstep), w(n-r), omega(n,0:nstep)
complex eta(n),A(n,n-r),x(n),FAC(n-r,n-r), det1, det
complex olderr, INPROD, ut(n,n-r), fd, max, rgamma
complex cm1, cm2, errm1, errm2, qt, At, Bt, Ct, Utemp(n)
real aa, bb, cc
real test, testalpha, pi, det2, AI(n,n-r)
real fdxr, fdxi, fdyr, fdyi
logical norm, eigfun
external INPROD, FHOMO, FPART, RKQC
real errtol, maxcount, eigtol
c
c initialize variables
c
errtol = 1.0e-14
maxcount = 20
eigtol = 1.0e-14
eigfun = .true.
c
c set the normalization constraint
c
pi = ACOS(-1.0)
c
c compute the step size
c
h = (tf-to)/nstep
c
c Begin the eigenvalue iteration loop
c
icount = 1
err = 1.0
cm1 = c + 1.0
do while ( ((abs(err) .ge. errtol) .or. (abs(c-cm1) .ge. eigtol))
& .and. (icount .le. maxcount) )
q = 0
tq(0) = tf
c
c Set the initial conditions
c
k = 0
U(1,1,0) = CEXP(-alpha*tf)
U(2,1,0) = (-alpha)*CEXP(-alpha*tf)
U(3,1,0) = (alpha**2)*CEXP(-alpha*tf)
U(4,1,0) = (-alpha**3)*CEXP(-alpha*tf)
rgamma = SQRT(alpha**2+(0.,1.)*alpha*Re*(1.0-c))
U(1,2,0) = CEXP(-rgamma*tf)
U(2,2,0) = (-rgamma)*CEXP(-rgamma*tf)
U(3,2,0) = (rgamma**2)*CEXP(-rgamma*tf)
U(4,2,0) = (-rgamma**3)*CEXP(-rgamma*tf)
c
c Gram-Schmidt
c
w(1) = SQRT(inprod(n, U(1,1,k), U(1,1,k)))
do i = 1, n
z(i,1) = U(i,1,k)/w(1)
end do
do mi = 2, (n-r)
do i = 1, n
eta(i) = U(i,mi,k)
end do
do mj = mi-1, 1, -1
do i = 1, n
eta(i) = eta(i) - inprod(n, U(1,mi,k), z(1,mj))*z(i,mj)
end do
end do
w(mi) = SQRT(inprod(n, eta, eta))
do i = 1, n
z(i,mi) = eta(i)/w(mi)
end do
end do
c
c Now update the U matrix with the orthonormal values
c
do i = 1, n
do m = 1, n-r
U(i,m,k) = z(i,m)
end do
end do
#ifdef CHECK_ORTHOGONALITY
aa = ABS(inprod(n, U(1,1,k), U(1,1,k)))
bb = ABS(inprod(n, U(1,2,k), U(1,2,k)))
cc = ABS(inprod(n, U(1,1,k), U(1,2,k)))
test = ACOS(cc/SQRT(aa*bb))*180./pi
write (*,*) k, mi, mj, test
#endif
c
c Integrate the homo. and particular equations
c
do k = 1, nstep
t = tf - h*k
c
c Loop thru all homogeneous solutions
c
do m = 1, n-r
#ifdef USE_NR_ODEINT
do i = 1, n
Utemp(i) = U(i,m,k-1)
end do
call ODEINTR(Utemp,n,t+h,t,1.E-6,-h,1.e-20,nok,nbad,
& FHOMO,RKQC)
do i = 1, n
U(i,m,k) = Utemp(i)
end do
#else
call CRK4(n, U(1,m,k-1), U(1,m,k), t+h, -h, FHOMO)
#endif
end do
c
c Test to see if normalization is required
c
norm = .false.
ta = pi*testalpha/180.0
do mi = 1, n-r
do mj = 1, n-r
if (mi .ne. mj) then
aa = ABS(inprod(n, U(1,mi,k), U(1,mi,k)))
bb = ABS(inprod(n, U(1,mj,k), U(1,mj,k)))
cc = ABS(inprod(n, U(1,mi,k), U(1,mj,k)))
test = ACOS(cc/SQRT(aa*bb))
if (test .lt. ta) norm = .true.
c write (*,*) k, mi, mj, test, testalpha, norm
end if
end do
end do
c
c Perform normalization
c
if ( norm .or. (k .eq. nstep) ) then
q = q + 1
tq(q) = t
if (k .eq. nstep) then
qend = q
c write (*,*) 'qend = ',qend
end if
c
c Gram-Schmidt
c
w(1) = SQRT(inprod(n, U(1,1,k), U(1,1,k)))
do i = 1, n
z(i,1) = U(i,1,k)/w(1)
end do
do mi = 2, (n-r)
do i = 1, n
eta(i) = U(i,mi,k)
end do
do mj = mi-1, 1, -1
do i = 1, n
eta(i) = eta(i) - inprod(n, U(1,mi,k), z(1,mj))*z(i,mj)
end do
end do
w(mi) = SQRT(inprod(n, eta, eta))
do i = 1, n
z(i,mi) = eta(i)/w(mi)
end do
end do
c
c Now I have the orthonormal basis in z and
c the norms in w so I can compute the P orthonormalization
c matrix
c
do j = 1, n-r
do i = 1, j
if (j .eq. i) then
P(j,i,q) = 1.0/w(j)
else
P(j,i,q) = 0.0
do s = i, j-1
P(j,i,q) = P(j,i,q)-inprod(n,U(1,j,k),z(1,s))/w(j)*
& P(s,i,q)
end do
end if
end do
end do
c
c Check the P matrix
c
if (.false.) then
do i = 1, n
do m = 1, n-r
ut(i,m) = 0.0
do j = 1, n-r
ut(i,m) = ut(i,m) + U(i,j,k)*P(m,j,q)
end do
end do
end do
do i = 1,n
write (*,*) i,(ut(i,m), m = 1, n-r)
end do
write (*,*)
do i = 1,n
write (*,*) i,( z(i,m), m = 1, n-r)
end do
write (*,*)
write (*,*)
end if
c
c Now update the U matrix with the orthonormal values
c
do i = 1, n
do m = 1, n-r
U(i,m,k) = z(i,m)
end do
end do
end if
end do
c
c check boundary conditions
c
if (.true.) then
c
c strictly enforce the zero BC
c
B(1,qend) = -U(1,2,nstep)/U(1,1,nstep)
B(2,qend) = 1.0
else
c
c strictly enforce the zero slope BC
c
B(1,qend) = 1.0
B(2,qend) = -U(2,1,nstep)/U(2,2,nstep)
olderr = err
err = U(1,1,nstep)*B(1,qend) + U(1,2,nstep)*B(2,qend)
end if
ctemp = c
if (icount .eq. 1) then
cm2 = c
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
errm2 = err
c = CMPLX( REAL(c)*.9999, AIMAG(c) )
else if (icount .eq. 2) then
cm1 = c
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
errm1 = err
c c = CMPLX( REAL(c), AIMAG(c)*.9999 )
fdxr = REAL(err-errm2) / REAL(c-cm2)
fdxi = AIMAG(err-errm2) / REAL(c-cm2)
fd = CMPLX( fdxr, fdxi )
c = c - err/fd
else
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
qt = (c-cm1)/(cm1-cm2)
At = qt*err-qt*(1.+qt)*errm1+qt**2*errm2
Bt = (2.*qt+1.)*err-(1.+qt)**2*errm1+qt**2*errm2
Ct = (1.+qt)*err
if ( ABS(Bt+SQRT(Bt**2-4.*At*Ct)) .gt.
& ABS(Bt-SQRT(Bt**2-4.*At*Ct)) ) then
c = ctemp-(ctemp-cm1)*2.*Ct/(Bt+SQRT(Bt**2-4.*At*Ct))
else
c = ctemp-(ctemp-cm1)*2.*Ct/(Bt-SQRT(Bt**2-4.*At*Ct))
end if
cm2 = cm1
cm1 = ctemp
errm2 = errm1
errm1 = err
end if
write (*,30) icount,real(ctemp),aimag(ctemp),real(err),
& aimag(err)
30 format (1x,i4,2(e17.8,e17.8,3x))
icount = icount + 1
end do
if (eigfun) then
c
c Second Pass
c
icount = icount - 1
write (*,40) real(ctemp), aimag(ctemp), qend
40 format (/,'Eigenvalue = ',e17.8,1x,e17.8,2x,i5/)
write (*,45) icount, abs(err), abs(c-cm1)
45 format (' Eigenvalue iterations = ', i5/,' |error| = ',
& es17.8/,' |c-cm1| = ', es17.8/)
q = qend
max = 0.0
k = nstep
do i = 1, n
y(i,k) = v(i,k)
do m = 1, n-r
y(i,k) = y(i,k) + U(i,m,k)*B(m,q)
end do
end do
do m = 1, n-r
B(m,q-1) = 0.0
do j = 1, n-r
B(m,q-1) = B(m,q-1) + P(j,m,q)*B(j,q)
end do
end do
do i = 1, n
y(i,nstep) = 0.0
do m = 1, n-r
y(i,nstep) = y(i,nstep) + U(i,m,nstep)*B(m,q)
end do
end do
do k = nstep-1, 0, -1
t = tf - h*k
if ( t .gt. tq(q-1) ) then
q = q - 1
do m = 1, n-r
B(m,q-1) = 0.0
do j = 1, n-r
B(m,q-1) = B(m,q-1) + P(j,m,q)*B(j,q)
end do
end do
end if
do i = 1, n
y(i,k) = 0.0
do m = 1, n-r
y(i,k) = y(i,k) + U(i,m,k)*B(m,q-1)
end do
end do
end do
arg = ATAN2( AIMAG(y(1,0)), REAL(y(1,0)) )
do k = 0, nstep
do i = 1, n
y(i,k) = y(i,k)*CEXP( (0.,-1.)*arg )
end do
if ( ABS( REAL( y(1,k) ) ) .gt. ABS(REAL(max)) ) then
max = y(1,k)
end if
end do
do k = 0, nstep
t = tf - h*k
write (11,20) t, REAL(y(1,k)/max),AIMAG(y(1,k)/max)
write (12,20) t, REAL(y(2,k)/max),AIMAG(y(2,k)/max)
write (13,20) t, REAL(y(3,k)/max),AIMAG(y(3,k)/max)
write (14,20) t, REAL(y(4,k)/max),AIMAG(y(4,k)/max)
20 format ( 1x, 3(ES16.8E3,1x) )
end do
end if
return
end
C***********************************************************************
function INPROD(n, v1, v2)
C***********************************************************************
C
C Perform and inner product on two comples vectors, v1 and v2
C
c***********************************************************************
complex v1(n), v2(n)
complex INPROD
integer n, i
#if USE_ANALYTIC_INPROD
c
c Lets define a different complex inner product so that the
c iteration is analytic
c
INPROD = 0.0
do i = 1, n
INPROD = INPROD + v1(i)*v2(i)
end do
#else
INPROD = 0.0
do i = 1, n
INPROD = INPROD + v1(i)*conjg(v2(i))
end do
#endif
return
end
C***********************************************************************
subroutine FHOMO(neq, yo,t,yf)
C***********************************************************************
C
C Function evaluation for the Orr-Sommerfeld equation
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (imax=20000)
integer nbl
complex alpha, c, rgamma
real U(0:imax), d2U(0:imax), Re, ymin, ymax, h, beta, uc
real Uspl(0:imax), d2Uspl(0:imax), ydat(0:imax), f2p
real Uorg(0:imax)
common /mean/ nbl, u, d2u, ymin, ymax, h, f2p, uc,
& rgamma, beta, uspl, d2uspl, ydat, Uorg
common /eig/ c, alpha, Re
c***********************************************************************
integer neq
complex yo(neq), yf(neq)
real t
real UU, d2UU
real Pi
pi = acos(-1.0)
c
c Get the velocity field
c
#ifdef USE_SPINE_DERIVATIVE
call SPEVAL(nbl+1,ydat,U,Uspl,t,UU)
call SPEVAL(nbl+1,ydat,d2U,d2Uspl,t,d2UU)
#else
call SPDER(nbl+1,ydat,U,Uspl,t,UU,dUU,d2UU)
#endif
do j = 1 , neq-1
yf(j) = yo(j+1)
end do
yf(neq) = 2.*alpha**2*yo(3)-alpha**4*yo(1) +
& (0.,1.)*alpha*Re*((UU-c)*(yo(3)-alpha**2*yo(1))-
& d2UU*yo(1))
return
end
C***********************************************************************
subroutine FPART(neq, yo,t,yf)
C***********************************************************************
C
C Function evaluation for the Orr-Sommerfeld equation
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (imax=20000)
integer nbl
complex alpha, c, rgamma
real U(0:imax), d2U(0:imax), Re, ymin, ymax, h, beta, uc
real Uspl(0:imax), d2Uspl(0:imax), ydat(0:imax), f2p
real Uorg(0:imax)
common /mean/ nbl, u, d2u, ymin, ymax, h, f2p, uc,
& rgamma, beta, uspl, d2uspl, ydat, Uorg
common /eig/ c, alpha, Re
c***********************************************************************
integer neq
complex yo(neq), yf(neq)
real t
real UU, d2UU
real Pi
pi = acos(-1.0)
c
c Get the velocity field
c
#ifdef USE_SPINE_DERIVATIVE
call SPEVAL(nbl+1,ydat,U,Uspl,t,UU)
call SPEVAL(nbl+1,ydat,d2U,d2Uspl,t,d2UU)
#else
call SPDER(nbl+1,ydat,U,Uspl,t,UU,dUU,d2UU)
#endif
do j = 1 , neq-1
yf(j) = yo(j+1)
end do
yf(neq) = 2.*alpha**2*yo(3)-alpha**4*yo(1) +
& (0.,1.)*alpha*Re*((UU-c)*(yo(3)-alpha**2*yo(1))-
& d2UU*yo(1))
return
end
C***********************************************************************
subroutine READ_BL
C***********************************************************************
C
C Read an arbitrary profile from .
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (imax=20000)
integer nbl
complex alpha, c, rgamma
real U(0:imax), d2U(0:imax), Re, ymin, ymax, h, beta, uc
real Uspl(0:imax), d2Uspl(0:imax), ydat(0:imax), f2p
real Uorg(0:imax)
common /mean/ nbl, u, d2u, ymin, ymax, h, f2p, uc,
& rgamma, beta, uspl, d2uspl, ydat, Uorg
common /eig/ c, alpha, Re
c***********************************************************************
integer i
character*20 filename
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 = -1
do while (.true.)
read (11,*,end=100) ydat(i),uorg(i),junk
i = i + 1
end do
close (11)
100 nbl = i
call SHIFT_BL
write (*,30)
30 format (1x,'Velocity Profile completed...',/)
return
end
C***********************************************************************
subroutine SOLVE_BL
C***********************************************************************
C
C Integrate the boundary layer similarity equation to get velocity
C profile.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (imax=20000)
integer nbl
complex alpha, c, rgamma
real U(0:imax), d2U(0:imax), Re, ymin, ymax, h, beta, uc
real Uspl(0:imax), d2Uspl(0:imax), ydat(0:imax), f2p
real Uorg(0:imax)
common /mean/ nbl, u, d2u, ymin, ymax, h, f2p, uc,
& rgamma, beta, uspl, d2uspl, ydat, Uorg
common /eig/ c, alpha, Re
c***********************************************************************
integer i, j, k, p
real xi(3,0:imax), f(3), eta(3), y
real k1(3), k2(3), k3(3), k4(3), err, x2old, f1old
external BLASIUS, RKQCR
do j = 1,3
do i = 0,nbl
xi(j,i) = 0.0
end do
f(j) = 0.0
eta(j) = 0.0
end do
c
c Set the boundary conditions including guess for f"(0)
c
xi(1,0) = 0
xi(2,0) = 0
xi(3,0) = f2p
err = 1.0
p = 1
do while ( abs(err) .gt. 1.e-10)
do i = 1, nbl
y = ymin + float(i)*h
#ifdef USE_NR_ODEINT
do j = 1, 3
eta(j) = xi(j,i-1)
end do
call ODEINTR(eta,3,y-h,y,1.E-7,h/2.,1.e-20,nok,nbad,
& BLASIUS,RKQCR)
do j = 1, 3
xi(j,i) = eta(j)
end do
#else
call SRK4(3, xi(1,i-1), xi(1,i), y-h, h, BLASIUS)
#endif
end do
c
c Check f'(ymax)
c
if (p .eq. 1) then
xi3old = xi(3,0)
xi(3,0) = xi(3,0)*.99
else
xi3temp = xi(3,0)
xi(3,0) = xi(3,0)+((xi3old-xi(3,0))/(xi2old-xi(2,nbl)))*
& (1.0 - xi(2,nbl))
xi3old = xi3temp
end if
p = p + 1
xi2old = xi(2,nbl)
err = 1.0 - xi2old
c write (*,*) p, err
end do
c
c Now assemble the velocity field
c
do i = 0, nbl
Uorg(i) = xi(2,i)
ydat(i) = ymin + i*h
end do
c
c Compute 2nd order finite difference approximation to
c second derivative
c
c do i = 1, nbl-1
c d2u(i) = (u(i+1)-2*u(i)+u(i-1))/(h)**2
c end do
c d2u(0) = (-u(4)+4.*u(3)-5.*u(2)+2.*u(1))/(h)**2
c d2u(nbl) = (2.*u(nbl)-5.*u(nbl-1)+4.*u(nbl-2)-u(nbl-3))/(h)**2
call SHIFT_BL
write (*,20)
20 format (1x,'Velocity Profile completed...',/)
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 (imax=20000)
integer nbl
complex alpha, c, rgamma
real U(0:imax), d2U(0:imax), Re, ymin, ymax, h, beta, uc
real Uspl(0:imax), d2Uspl(0:imax), ydat(0:imax), f2p
real Uorg(0:imax)
common /mean/ nbl, u, d2u, ymin, ymax, h, f2p, uc,
& rgamma, beta, uspl, d2uspl, ydat, Uorg
common /eig/ c, alpha, Re
c***********************************************************************
c
c Need to interpolate the velocity profile to evaluate it at
c arbitrary y
c
do i = 0, nbl
U(i) = Uorg(i) - Uc
end do
call SPLINE(nbl+1,ydat,u,uspl)
c
c Use the spline result for the second derivative
c
do i = 0, nbl
d2u(i) = uspl(i)
end do
call SPLINE(nbl+1,ydat,d2u,d2uspl)
do i = 0, nbl
y = ymin + i*h
call SPEVAL(nbl+1,ydat,u,uspl,y,us)
call SPEVAL(nbl+1,ydat,d2u,d2uspl,y,d2us)
write (10,10) y, us, d2us, u(i), d2u(i)
10 format (1x,5(ES16.8E3,1x))
end do
return
end
C***********************************************************************
subroutine BLASIUS(neq,xi,y,f)
C***********************************************************************
C
C Function evaluation for the boundary-layer similarity equation
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (imax=20000)
integer nbl
complex alpha, c, rgamma
real U(0:imax), d2U(0:imax), Re, ymin, ymax, h, beta, uc
real Uspl(0:imax), d2Uspl(0:imax), ydat(0:imax), f2p
real Uorg(0:imax)
common /mean/ nbl, u, d2u, ymin, ymax, h, f2p, uc,
& rgamma, beta, uspl, d2uspl, ydat, Uorg
common /eig/ c, alpha, Re
c***********************************************************************
integer neq
real xi(neq), f(neq), y
do j = 1 , 2
f(j) = xi(j+1)
end do
f(3) = -xi(1)*xi(3)-beta*(1.-xi(2)**2)
return
end