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rayleigh_benard.py
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rayleigh_benard.py
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from matplotlib import pyplot as plt
import numpy as np
from scipy.linalg import eig
from chebdiff import chebdiff
class EigValueProblem():
def __init__(self, N, L, bc1, bc2, Ra, Pr, k):
self.N = N
self.L = L
self.bc1 = bc1
self.bc2 = bc2
self.Ra = Ra
self.Pr = Pr
self.k = k
self.A = None
self.B = None
self.C = None
self.rr = None
self.kk = None
self.G = None
self.A_k = None
self.B_k = None
self.sW = None
self.sT = None
self.W = None
self.T = None
self.x, self.DM = chebdiff(N, 4)
self.x = self.x / L
for i in range(len(self.DM)):
self.DM[i] = self.DM[i]/L**(i+1)
def eval_AB(self):
Ra = self.Ra
Pr = self.Pr
k = self.k
N = self.N
D2 = self.DM[1]
D4 = self.DM[3]
K2 = np.eye(N) * k**2
K4 = np.eye(N) * k**4
LHS00 = D4 + K4 - 2 * np.dot(D2, K2)
LHS01 = -Ra * K2
LHS10 = np.eye(N)
LHS11 = D2 - K2
self.A = np.vstack((np.hstack((LHS00, LHS01)),
np.hstack((LHS10, LHS11))))
RHS00 = (D2 - K2)/Pr
RHS01 = np.zeros_like(RHS00)
RHS10 = np.zeros_like(RHS00)
RHS11 = np.eye(N)
self.B = np.vstack((np.hstack((RHS00, RHS01)),
np.hstack((RHS10, RHS11))))
def setBC(self):
N = self.N
bc1 = self.bc1
bc2 = self.bc2
D1 = self.DM[0]
D2 = self.DM[1]
I = np.eye(N)
C1 = np.zeros((3, 2*N))
if (bc1.lower() == 'free'):
C1[0, 0:N] = I[0, :] # w = 0 at z = 0
C1[1, 0:N] = D2[0, :] # D2w = 0 at z = 0
C1[2, N:2*N] = I[0, :] # T = 0 at z = 0
elif (bc1.lower() == 'rigid'):
C1[0, 0:N] = I[0, :] # w = 0 at z = 0
C1[1, 0:N] = D1[0, :] # D2w = 0 at z = 0
C1[2, N:2*N] = I[0, :] # T = 0 at z = 0
else:
raise Exception('bc1 = %s not coded' % bc1)
rr1 = np.array([0, 1, N])
C2 = np.zeros((3, 2*N))
if (bc2.lower() == 'free'):
C2[0, 0:N] = I[N-1, :] # w = 0 at z = L
C2[1, 0:N] = D2[N-1, :] # D2w = 0 at z = L
C2[2, N:2*N] = I[N-1, :] # T = 0 at z = L
elif (bc2.lower() == 'rigid'):
C2[0, 0:N] = I[N-1, :] # w = 0 at z = L
C2[1, 0:N] = D1[N-1, :] # D2w = 0 at z = L
C2[2, N:2*N] = I[N-1, :] # T = 0 at z = L
else:
raise Exception('bc1 = %s not coded' % bc1)
rr2 = np.array([N-1, N-2, 2*N-1])
self.C = np.vstack((C1, C2))
self.rr = np.concatenate([rr1, rr2])
self.kk = np.setdiff1d(np.arange(2*N), self.rr)
def eval_constrained_mat(self):
rr = self.rr
kk = self.kk
C = self.C
self.G = G = - np.linalg.solve(C[:, rr], C[:, kk])
self.A_k = self.A[np.ix_(kk, kk)] + np.dot(self.A[np.ix_(kk, rr)], G)
self.B_k = self.B[np.ix_(kk, kk)] + np.dot(self.B[np.ix_(kk, rr)], G)
def solve(self):
self.eval_AB()
self.setBC()
self.eval_constrained_mat()
N = self.N
A_k = self.A_k
B_k = self.B_k
s, WT_k = eig(A_k, B_k)
n_eigvecs = np.shape(WT_k)[1]
WT = np.zeros((2*N, n_eigvecs), dtype=np.complex)
WT[self.kk, :] = WT_k
WT[self.rr, :] = np.dot(self.G, WT_k)
n = (int)(n_eigvecs/2)
self.W = WT[0:N, :n]
self.T = WT[N:, n:]
self.sW = s[0:n]
self.sT = s[n:]
self.plot_mins_mode('W')
self.plot_mins_mode('T')
def plot_mins_mode(self, var):
index = 0
plt.figure()
if var == 'W':
index = np.argmin(np.abs(self.sW))
plt.plot(self.x, np.real(self.W[:,index]))
plt.plot(self.x, np.imag(self.W[:,index]))
plt.title('%f+i%f mode plot of %s for %s - %s'
%(np.real(self.sW[index]),np.imag(self.sW[index]),
var, self.bc1, self.bc2))
elif var == 'T':
index = np.argmin(np.abs(self.sT))
plt.plot(self.x, np.real(self.T[:,index]))
plt.plot(self.x, np.imag(self.T[:,index]))
plt.title('%f+i%f mode plot of %s for %s-%s'
%(np.real(self.sT[index]),np.imag(self.sT[index]),
var, self.bc1, self.bc2))
plt.xlabel('x')
plt.ylabel(var)
plt.savefig('%s-%s_%s_plot.pdf'%(self.bc1, self.bc2, var))
def plotStreamLine():
k = 2.221
x = np.linspace(0, 2 * 2*np.pi/k, 100)
z = np.linspace(0, 1, 100)
X , Z = np.meshgrid(x, z)
f = np.sin(np.pi*0.5)* np.sin(k*0.5) - np.sin(np.pi*Z)* np.sin(k*X)
plt.figure()
plt.contour(X, Z, f)
plt.grid()
plt.xlabel('x')
plt.ylabel('z')
plt.title('Streamlines for Rayleigh Benard free-free problem')
plt.savefig("streamlines.pdf")
pass
if __name__ == '__main__':
rb1 = EigValueProblem(200, 0.5, 'free', 'free', 657.5, 7.56, 2.221)
rb1.solve()
rb1 = EigValueProblem(200, 0.5, 'rigid', 'rigid', 1708, 7.56, 3.117)
rb1.solve()
rb1 = EigValueProblem(200, 0.5, 'free', 'rigid', 1101, 7.56, 2.682)
rb1.solve()
rb1 = EigValueProblem(200, 0.5, 'rigid', 'free', 1101, 7.56, 2.682)
rb1.solve()
plotStreamLine()