-
Notifications
You must be signed in to change notification settings - Fork 4
/
UniformFlow.py
102 lines (92 loc) · 5.11 KB
/
UniformFlow.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
#####################################################################################
# UniformFlow.py #
# Script for simulation of steady convection-diffusion heat problem on two- #
# dimensional square domain using the Finite Volume Method on a cartesian, #
# structured mesh with square cells. Central difference fluxes applied for #
# the diffusive terms, and either central of upwinded difference fluxes #
# applied for the convective terms. #
# Implementation does not include source terms and is limited to the #
# following sets of convective velocity fields and boundary conditions: #
# (problem 1) uniform flow [u,v] = [1,0], homogeneous Neumann BC. at north #
# and south walls (dTn/dy=dTs/dy=0), homogeneous Dirichlet BC at west wall #
# (Tw=0), inhomogeneous Dirichlet BC at east wall (Te=1). #
# (problem 2) see StagnationPointFlow.py #
# Linear system of equations solved either directly using Matlab's backslash #
# operator, or iteratively using either Jacobi, Gauss-Seidel or SOR stationary #
# iterative methods. #
# #
# Input : #
# n : Number of cells along x,y-axis #
# L : Size of square in x,y-direction #
# Pe : Global Peclet number #
# problem : Problem #: 1 or 2, selects case of convective field and BCs #
# fvscheme : Finite volume scheme for convection-diffusion, #
# either 'cds-cds' or 'uds-cds' #
# #
# Output : #
# T : Temperature at cell nodes, T(1:n,1:n) #
# A : Convection-diffusion system matrix, A(1:n^2,1:n^2) #
# s : Source array with BC contributions, s(1:n,1:n) #
# TT : Temperature field extrapolated to walls, TT(1:n+2,1:n+2) #
# CF,DF : Conv. and diff. fluxes through walls, CF=[CFw,CFe,CFs,CFn] #
# GHC : Global heat conservation, scalar (computed from wall fluxes) #
# Plots of the temperature field and convective velocity field #
#####################################################################################
import FVConvDiff2D
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from scipy.sparse.linalg import spsolve
## Input
n = 100 # number of cells along x,y-axis
L = 1.0 # size of square in x,y-direction
Pe = 10.0 # global Peclet number
problem = 1 # problem to solve
fvscheme = 'uds-cds' # finite volume scheme ('cds-cds' or 'uds-cds')
## Assemble system matrix
A, s = FVConvDiff2D.preprocess(n, L, Pe, problem, fvscheme)
## Do direct solution
T = spsolve(A, s.reshape(n*n, order="F")).reshape(n, n, order='F')
## Extend T-field to domain walls and get GHC-residual
TT, GHC, _, _ = FVConvDiff2D.postprocess(
T, n, L, Pe, problem, fvscheme)
## Plot solution and streamlines of the flow
plt.ion() # turn on interactive mode
f, axarr = plt.subplots(1, 2, sharey=True)
f.suptitle('Convection-diffusion by %s for Pe = %d, \
flux-error = %0.3e' % (fvscheme, Pe, GHC))
# Coordinate arrays
dx = L/n # cell size in x,y-direction
xf = np.arange(0., L+dx, dx) # cell face coordinate vector along x,y-axis
xc = np.arange(dx/2., L, dx) # cell center coordinates along x-axis
xt = np.hstack([0., xc, 1.]) # extended cell center coor. vector, incl. walls
Xc, Yc = np.meshgrid(xc, xc) # cell center coordinate arrays
Xt, Yt = np.meshgrid(xt, xt) # extended cell center coor. arrays, incl. walls
# Generate convective velocity field at cell faces
if problem == 1: # problem 1 - uniform flow
Ut = np.ones((np.size(Xc, 0), np.size(Xc, 1)))
Vt = np.zeros((np.size(Xc, 0), np.size(Xc, 1)))
elif problem == 2: # problem 2 - corner flow
Ut = -Xc.copy()
Vt = Yc.copy()
else:
print('problem not implemented')
axarr[0].streamplot( # only supports an evenly spaced grid
xc, xc, Ut, Vt, density=1, linewidth=2,
color=T, cmap=cm.coolwarm, norm=None, arrowsize=1,
arrowstyle='-|>', minlength=0.3)
axarr[0].set_title('Streamlines')
axarr[0].set_xlabel('x')
axarr[0].set_ylabel('y')
axarr[0].grid(True)
axarr[0].set_xlim(0, 1)
axarr[0].set_ylim(0, 1)
# Temperature field
p = axarr[1].pcolor(Xt, Yt, TT, cmap=cm.coolwarm, vmin=0, vmax=1)
axarr[1].set_title('Temperature')
axarr[1].set_xlabel('x')
axarr[1].set_ylabel('y')
axarr[1].grid(True)
axarr[1].set_xlim(0, 1)
axarr[1].set_ylim(0, 1)
f.colorbar(p, ax=axarr[1])