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Oldroyd-B2-reduced.py
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#----------------------------------------------------------------------------#
# Fully Spectral Linear stability analysis Oldroyd B Model
# (Reduced profiles)
# Last modified: Thu 18 Apr 2013 16:58:57 BST
#----------------------------------------------------------------------------#
""" Perform Linear stability analysis to find eigenvalues for the stability
of the streaky flow. Set all components of the profile to zero except first
5 Fourier Modes."""
# MODULES
import sys
import time
from scipy import *
from scipy import linalg
from scipy.sparse import linalg as sparselinalg
import cPickle as pickle
import ConfigParser
#FUNCTIONS
def mk_single_diffy():
"""Makes a matrix to differentiate a single vector of Chebyshev's,
for use in constructing large differentiation matrix for whole system"""
# make matrix:
mat = zeros((M, M), dtype='d')
for m in range(M):
for p in range(m+1, M, 2):
mat[m,p] = 2*p*oneOverC[m]
return mat
def mk_diff_y():
"""Make the matrix to differentiate a velocity vector wrt y."""
D = mk_single_diffy()
MDY = zeros( (vecLen, vecLen) )
for cheb in range(0,vecLen,M):
MDY[cheb:cheb+M, cheb:cheb+M] = D
del cheb
return MDY
def mk_diff_z():
"""Make matrix to do fourier differentiation wrt z."""
MDZ = zeros( (vecLen, vecLen), dtype='complex')
n = -N
for i in range(0, vecLen, M):
MDZ[i:i+M, i:i+M] = eye(M, M, dtype='complex')*n*gamma*1.j
n += 1
del n, i
return MDZ
def cheb_prod_mat(velA):
"""Function to return a matrix for left-multiplying two matrices
of velocities."""
D = zeros((M, M), dtype='complex')
#failcount = 0
for n in range(M):
for m in range(-M+1,M): # Bottom of range is inclusive
itr = abs(n-m)
if (itr < M):
D[n, abs(m)] += 0.5*oneOverC[n]*CFunc[itr]*CFunc[abs(m)]*velA[itr]
del m, n, itr
return D
def prod_mat(velA):
"""Function to return a matrix ready for the left dot product with another
velocity vector"""
MM = zeros((vecLen, vecLen), dtype='complex')
#First make the middle row
midMat = zeros((M, vecLen), dtype='complex')
for n in range(2*N+1): # Fourier Matrix is 2*N+1 cheb matricies
yprodmat = cheb_prod_mat(velA[n*M:(n+1)*M])
endind = 2*N+1-n
midMat[:, (endind-1)*M:endind*M] = yprodmat
del n
#copy matrix into MM, according to the matrix for spectral space
# top part first
for i in range(0, N):
MM[i*M:(i+1)*M, :] = column_stack((midMat[:, (N-i)*M:], zeros((M, (N-i)*M))) )
del i
# middle
MM[N*M:(N+1)*M, :] = midMat
# bottom
for i in range(0, N):
MM[(i+N+1)*M:(i+2+N)*M, :] = column_stack((zeros((M, (i+1)*M)), midMat[:, :(2*N-i)*M] ))
del i
return MM
def mk_bigM():
""" makes the matrix to appear on the left hand side of the generalised
eigenvalue problem"""
bigM = zeros((10*vLen, 10*vLen), dtype=complex)
################Navier Stokes x direction:###############
#*u
bigM[0:vLen, 0:vLen] = - Re*GRAD + beta*LAPLACIAN
#*v
bigM[0:vLen, vLen:2*vLen] = - Re*MMDYU
#*w
bigM[0:vLen, 2*vLen:3*vLen] = - Re*MMDZU
#*p
bigM[0:vLen, 3*vLen:4*vLen] = - 1.j*k*eye(vLen,vLen)
#cxx
bigM[0:vLen, 4*vLen:5*vLen] = (1-beta)*oneOverWeiss*1.j*k*eye(vLen,vLen)
#cyy
#czz
#cxy
bigM[0:vLen, 7*vLen:8*vLen] = (1-beta)*oneOverWeiss*MDY
#cxz
bigM[0:vLen, 8*vLen:9*vLen] = (1-beta)*oneOverWeiss*MDZ
#cyz
################Navier Stokes y direction:###############
#*u
#*v
bigM[vLen:2*vLen, vLen:2*vLen] = - Re*GRAD - Re*MMDYV \
+ beta*LAPLACIAN
#*w
bigM[vLen:2*vLen, 2*vLen:3*vLen] = - Re*MMDZV
#*p
bigM[vLen:2*vLen, 3*vLen:4*vLen] = - MDY
#cxx
#cyy
bigM[vLen:2*vLen, 5*vLen:6*vLen] = (1-beta)*oneOverWeiss*MDY
#czz
#cxy
bigM[vLen:2*vLen, 7*vLen:8*vLen] = (1-beta)*oneOverWeiss*1.j*k*eye(vLen,vLen)
#cxz
#cyz
bigM[vLen:2*vLen, 9*vLen:10*vLen] = (1-beta)*oneOverWeiss*MDZ
################Navier Stokes z direction:###############
#*u
#*v
bigM[2*vLen:3*vLen, vLen:2*vLen] = -Re*MMDYW
#*w
bigM[2*vLen:3*vLen, 2*vLen:3*vLen] = - Re*GRAD - Re*MMDZW \
+ beta*LAPLACIAN
#*p
bigM[2*vLen:3*vLen, 3*vLen:4*vLen] = - MDZ
#cxx
#cyy
#czz
bigM[2*vLen:3*vLen, 6*vLen:7*vLen] = (1-beta)*oneOverWeiss*MDZ
#cxy
#cxz
bigM[2*vLen:3*vLen, 8*vLen:9*vLen] = (1-beta)*oneOverWeiss*1.j*k*eye(vLen,vLen)
#cyz
bigM[2*vLen:3*vLen, 9*vLen:10*vLen] = (1-beta)*oneOverWeiss*MDY
################Incompressability equation:###############
#*u
bigM[3*vLen:4*vLen, 0:vLen] = 1.j*k*eye(vLen,vLen)
#*v
bigM[3*vLen:4*vLen, vLen:2*vLen] = MDY
#*w
bigM[3*vLen:4*vLen, 2*vLen:3*vLen] = MDZ
#*p
#cxx
#cyy
#czz
#cxy
#cxz
#cyz
################cxx equation:####################
#*u
bigM[4*vLen:5*vLen, 0:vLen] = 2.j*k*MMCXX + 2*dot(MMCXY,MDY) \
+ 2*dot(MMCXZ,MDZ)
#*v
bigM[4*vLen:5*vLen, vLen:2*vLen] = -prod_mat(dot(MDY,conxx))
#*w
bigM[4*vLen:5*vLen, 2*vLen:3*vLen] = -prod_mat(dot(MDZ,conxx))
#*p
#cxx
bigM[4*vLen:5*vLen, 4*vLen:5*vLen] = -oneOverWeiss*eye(vLen,vLen) - GRAD
#cyy
#czz
#cxy
bigM[4*vLen:5*vLen, 7*vLen:8*vLen] = 2*MMDYU
#cxz
bigM[4*vLen:5*vLen, 8*vLen:9*vLen] = 2*MMDZU
#cyz
################cyy equation:####################
#*u
#*v
bigM[5*vLen:6*vLen, vLen:2*vLen] = +2j*k*MMCXY +2*dot(MMCYY,MDY) \
+ 2*dot(MMCYZ,MDZ) \
- prod_mat(dot(MDY,conyy))
#*w
bigM[5*vLen:6*vLen, 2*vLen:3*vLen] = -prod_mat(dot(MDZ,conyy))
#*p
#cxx
#cyy
bigM[5*vLen:6*vLen, 5*vLen:6*vLen] = -oneOverWeiss*eye(vLen,vLen) - GRAD \
+ 2*MMDYV
#czz
#cxy
#cxz
#cyz
bigM[5*vLen:6*vLen, 9*vLen:10*vLen] = 2*MMDZV
################czz equation:####################
#*u
#*v
bigM[6*vLen:7*vLen, vLen:2*vLen] = -prod_mat(dot(MDY,conzz))
#*w
bigM[6*vLen:7*vLen, 2*vLen:3*vLen] = -prod_mat(dot(MDZ,conzz)) + 2.j*k*MMCXZ \
+ 2*dot(MMCYZ,MDY) + 2*dot(MMCZZ,MDZ)
#*p
#cxx
#cyy
#czz
bigM[6*vLen:7*vLen, 6*vLen:7*vLen] = - oneOverWeiss*eye(vLen,vLen) - GRAD \
+ 2*MMDZW
#cxy
#cxz
#cyz
bigM[6*vLen:7*vLen, 9*vLen:10*vLen] = 2*MMDYW
################cxy equation:####################
#*u
bigM[7*vLen:8*vLen, 0:vLen] = + dot(MMCYY,MDY) + dot(MMCYZ,MDZ)
#*v
bigM[7*vLen:8*vLen, vLen:2*vLen] = -prod_mat(dot(MDY,conxy)) + 1.j*k*MMCXX \
+ dot(MMCXZ,MDZ)
#*w
bigM[7*vLen:8*vLen, 2*vLen:3*vLen] = - prod_mat(dot(MDZ,conxy)) \
- dot(MMCXY,MDZ)
#*p
#cxx
#cyy
bigM[7*vLen:8*vLen, 5*vLen:6*vLen] = MMDYU
#czz
#cxy
bigM[7*vLen:8*vLen, 7*vLen:8*vLen] = -oneOverWeiss*eye(vLen,vLen) - GRAD\
+ MMDYV
#cxz
bigM[7*vLen:8*vLen, 8*vLen:9*vLen] = MMDZV
#cyz
bigM[7*vLen:8*vLen, 9*vLen:10*vLen] = MMDZU
################cxz equation:####################
#*u
bigM[8*vLen:9*vLen, 0:vLen] = + dot(MMCYZ,MDY) + dot(MMCZZ,MDZ)
#*v
bigM[8*vLen:9*vLen, vLen:2*vLen] = - prod_mat(dot(MDY,conxz)) \
- dot(MMCXZ,MDY)
#*w
bigM[8*vLen:9*vLen, 2*vLen:3*vLen] = - prod_mat(dot(MDZ,conxz)) + 1.j*k*MMCXX\
+ dot(MMCXY,MDY)
#*p
#cxx
#cyy
#czz
bigM[8*vLen:9*vLen, 6*vLen:7*vLen] = MMDZU
#cxy
bigM[8*vLen:9*vLen, 7*vLen:8*vLen] = MMDYW
#cxz
bigM[8*vLen:9*vLen, 8*vLen:9*vLen] = -oneOverWeiss*eye(vLen,vLen) - GRAD\
+ MMDZW
#cyz
bigM[8*vLen:9*vLen, 9*vLen:10*vLen] = MMDYU
###############cyz equation:####################
#*u
bigM[9*vLen:10*vLen, 0:vLen] = -1.j*k*MMCYZ
#*v
bigM[9*vLen:10*vLen, vLen:2*vLen] = - prod_mat(dot(MDY,conyz)) + 1.j*k*MMCXZ \
+ dot(MMCZZ,MDZ)
#*w
bigM[9*vLen:10*vLen, 2*vLen:3*vLen] = - prod_mat(dot(MDZ,conyz)) + 1.j*k*MMCXY \
+ dot(MMCYY,MDY)
#*p
#cxx
#cyy
bigM[9*vLen:10*vLen, 5*vLen:6*vLen] = MMDYW
#czz
bigM[9*vLen:10*vLen, 6*vLen:7*vLen] = MMDZV
#cxy
#cxz
#cyz
bigM[9*vLen:10*vLen, 9*vLen:10*vLen] = - oneOverWeiss*eye(vLen,vLen) - GRAD
#Apply Boundary Conditions for u, v, w:
for i in range(3*(2*N+1)):
bigM[M*(i+1)-2,:] = hstack((zeros(M*i), BTOP, zeros(10*vLen-M*(i+1))))
bigM[M*(i+1)-1,:] = hstack((zeros(M*i), BBOT, zeros(10*vLen-M*(i+1))))
del i
return bigM
#
# MAIN
#
#Start the clock:
startTime = time.time()
config = ConfigParser.RawConfigParser()
fp = open('OB-settings.cfg')
config.readfp(fp)
N = config.getint('settings', 'N')
M = config.getint('settings', 'M')
Re = config.getfloat('settings', 'Re')
beta = config.getfloat('settings','beta')
Weiss = config.getfloat('settings','Weiss')
Amp = config.getfloat('settings', 'Amp')
k = config.getfloat('settings', 'k')
fp.close()
Nselect = 0
print 'k is {k}'.format(k=k)
filename = '-N{N}-M{M}-Re{Re}-b{beta}-Wi{Weiss}-amp{Amp}.pickle'.format(\
N=N,M=M,Re=Re,beta=beta,Weiss=Weiss,Amp=Amp)
# Unpickle the answer from part1 and the V and W vectors
fpickle = open('pf'+filename, 'r')
(U,V,W,conxx,conyy,conzz,conxy,conxz,conyz) = pickle.load(fpickle)
fpickle.close()
# Set all components to zero apart from the selected Fourier mode.
if Nselect == 0:
U[:N*M] = 0
U[(N+1)*M:] = 0
V[:N*M] = 0
V[(N+1)*M:] = 0
W[:N*M] = 0
W[(N+1)*M:] = 0
conxx[:N*M] = 0
conxx[(N+1)*M:] = 0
conyy[:N*M] = 0
conyy[(N+1)*M:] = 0
conzz[:N*M] = 0
conzz[(N+1)*M:] = 0
conxy[:N*M] = 0
conxy[(N+1)*M:] = 0
conxz[:N*M] = 0
conxz[(N+1)*M:] = 0
conyz[:N*M] = 0
conyz[(N+1)*M:] = 0
else:
U[:(N-Nselect)*M] = 0
U[(N-Nselect+1)*M:(N+Nselect)*M] = 0
U[(N+Nselect+1)*M:] = 0
V[:(N-Nselect)*M] = 0
V[(N-Nselect+1)*M:(N+Nselect)*M] = 0
V[(N+Nselect+1)*M:] = 0
W[:(N-Nselect)*M] = 0
W[(N-Nselect+1)*M:(N+Nselect)*M] = 0
W[(N+Nselect+1)*M:] = 0
conxx[:(N-Nselect)*M] = 0
conxx[(N-Nselect+1)*M:(N+Nselect)*M] = 0
conxx[(N+Nselect+1)*M:] = 0
conyy[:(N-Nselect)*M] = 0
conyy[(N-Nselect+1)*M:(N+Nselect)*M] = 0
conyy[(N+Nselect+1)*M:] = 0
conzz[:(N-Nselect)*M] = 0
conzz[(N-Nselect+1)*M:(N+Nselect)*M] = 0
conzz[(N+Nselect+1)*M:] = 0
conxy[:(N-Nselect)*M] = 0
conxy[(N-Nselect+1)*M:(N+Nselect)*M] = 0
conxy[(N+Nselect+1)*M:] = 0
conxz[:(N-Nselect)*M] = 0
conxz[(N-Nselect+1)*M:(N+Nselect)*M] = 0
conxz[(N+Nselect+1)*M:] = 0
conyz[:(N-Nselect)*M] = 0
conyz[(N-Nselect+1)*M:(N+Nselect)*M] = 0
conyz[(N+Nselect+1)*M:] = 0
# Setup variables:
gamma = pi / 2.
zLength = 2.*pi/gamma
vecLen = M*(2*N+1)
vLen = vecLen
print vLen, type(vLen)
oneOverWeiss = 1./Weiss
# Set the oneOverC function: 1/2 for m=0, 1 elsewhere:
oneOverC = ones(M)
oneOverC[0] = 1. / 2.
#set up the CFunc function: 2 for m=0, 1 elsewhere:
CFunc = ones(M)
CFunc[0] = 2.
#Boundary arrays:
BTOP = ones(M)
BBOT = ones(M)
BBOT[1:M:2] = -1
# make some useful matrices
MDY = mk_diff_y()
MDZ = mk_diff_z()
MMU = prod_mat(U)
MMV = prod_mat(V)
MMW = prod_mat(W)
MMCXX = prod_mat(conxx)
MMCYY = prod_mat(conyy)
MMCZZ = prod_mat(conzz)
MMCXY = prod_mat(conxy)
MMCXZ = prod_mat(conxz)
MMCYZ = prod_mat(conyz)
MMDYU = prod_mat(dot(MDY, U))
MMDZU = prod_mat(dot(MDZ, U))
MMDYV = prod_mat(dot(MDY, V))
MMDZV = prod_mat(dot(MDZ, V))
MMDYW = prod_mat(dot(MDY, W))
MMDZW = prod_mat(dot(MDZ, W))
LAPLACIAN = -(k**2)*eye(vLen,vLen) + dot(MDY,MDY) + dot(MDZ,MDZ)
GRAD = 1.j*k*MMU + dot(MMV,MDY) + dot(MMW,MDZ)
# Make the matrix for the generalised eigenvalue problem
equations_matrix= mk_bigM()
# Make the scaling matrix for RHS of equation
RHS = eye(10*vLen,10*vLen)
RHS[:3*vLen, :3*vLen] = Re*eye(3*vLen,3*vLen)
# Zero all elements corresponding to p equation
RHS[3*vLen:4*vLen, :] = zeros((vLen,10*vLen))
# Apply boundary conditions to RHS
for i in range(3*(2*N+1)):
RHS[M*(i+1)-1, M*(i+1)-1] = 0
RHS[M*(i+1)-2, M*(i+1)-2] = 0
del i
# Use library function to solve for eigenvalues/vectors
print 'in linalg.eig time=', (time.time() - startTime)
eigenvals = linalg.eigvals(equations_matrix, RHS, overwrite_a=True)
# Save output
eigarray = vstack((real(eigenvals), imag(eigenvals))).T
#remove nans and infs from eigenvalues
eigarray = eigarray[~isnan(eigarray).any(1), :]
eigarray = eigarray[~isinf(eigarray).any(1), :]
savetxt('ev-k{k}{file}-redn{Nselect}.dat'.format(k=k, file=filename[:-7],
Nselect=Nselect), eigarray)
#stop the clock
print 'done in', (time.time()-startTime)
######################TESTS####################################################
###############################################################################