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
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
|
import numpy as np
import matplotlib.pyplot as plt
PI = np.pi
np.set_printoptions(precision=2, floatmode="maxprec", suppress=True)
figure, axes = plt.subplots()
Re = 100
nu = 1 / Re
domain_size = (1, 2)
step = 0.05
N = int(domain_size[1] / step)
M = int(domain_size[0] / step)
shape = (N, M)
alpha = 0.8 # Coefficient
t_m = 1
h_c = 0.5
# Staggered vars
u = np.zeros(shape=shape, dtype=float)
u_star = np.zeros(shape=shape, dtype=float)
u_new = np.zeros(shape=shape, dtype=float)
v = np.zeros(shape=shape, dtype=float)
v_star = np.zeros(shape=shape, dtype=float)
v_new = np.zeros(shape=shape, dtype=float)
p = np.zeros(shape=shape, dtype=float)
p_star = np.zeros(shape=shape, dtype=float)
p_new = np.zeros(shape=shape, dtype=float)
d_e = np.zeros(shape=shape, dtype=float)
d_n = np.zeros(shape=shape, dtype=float)
b = np.zeros(shape=shape, dtype=float)
def u_boundary(t, y):
def f(t):
return 1 if t_m < t else 0.5 * (np.sin(0.5 * PI * (2 * t / t_m - 1)) + 1)
return 6 * f(t) * (y - h_c) * (1 - y) / (1 - h_c)**2
# Loop
error = 1
precision = 10 ** -7
t = 0.15
iteration = 0
while error > precision:
iteration += 1
# Inflow boundary condition
for i in range(N):
y = i * step
u_star[i][0] = u_boundary(t, y)
v_star[i][0] = 0
# Sides boundary conditions
for j in range(M):
v_star[0][j] = 0
v_star[N - 1][j] = 0
u_star[0][j] = 0
u_star[N - 1][j] = 0
# x-momentum
for i in range(1, N - 1):
for j in range(1, M - 1):
u_E = 0.5 * (u[i][j] + u[i][j + 1])
u_W = 0.5 * (u[i][j] + u[i][j - 1])
v_N = 0.5 * (v[i - 1][j] + v[i - 1][j + 1])
v_S = 0.5 * (v[i][j] + v[i][j + 1])
a_E = -0.5 * u_E * step + nu
a_W = +0.5 * u_W * step + nu
a_N = -0.5 * v_N * step + nu
a_S = +0.5 * v_S * step + nu
a_e = 0.5 * step * (u_E - u_W + v_N - v_S) + 4 * nu
A_e = -step
d_e[i][j] = A_e / a_e
u_star[i][j] = (a_E * u[i][j + 1] + a_W * u[i][j - 1] + a_N * u[i - 1][j] + a_S * u[i + 1][j]) / a_e
+ d_e[i][j] * (p[i][j + 1] - p[i][j])
# y-momentum
for i in range(1, N - 1):
for j in range(1, M - 1):
u_E = 0.5 * (u[i][j] + u[i + 1][j])
u_W = 0.5 * (u[i][j - 1] + u[i + 1][j - 1])
v_N = 0.5 * (v[i - 1][j] + v[i][j])
v_S = 0.5 * (v[i][j] + v[i + 1][j])
a_E = -0.5 * u_E * step + nu
a_W = +0.5 * u_W * step + nu
a_N = -0.5 * v_N * step + nu
a_S = +0.5 * v_S * step + nu
a_n = 0.5 * step * (u_E - u_W + v_N - v_S) + 4 * nu
A_n = -step
d_n[i][j] = A_n / a_n
v_star[i][j] = (a_E * v[i][j + 1] + a_W * v[i][j - 1] + a_N * v[i - 1][j] + a_S * v[i + 1][j]) / a_n
+ d_n[i][j] * (p[i][j] - p[i + 1][j])
# Pressure correction
p_c = np.zeros(shape=shape, dtype=float)
for i in range(1, N - 1):
for j in range(1, M - 1):
a_E = -d_e[i][j] * step
a_W = -d_e[i][j-1] * step
a_N = -d_n[i-1][j] * step
a_S = -d_n[i][j] * step
a_P = a_E + a_W + a_N + a_S
b[i][j] = step * (-(u_star[i][j] - u_star[i][j-1]) + (v_star[i][j] - v_star[i-1][j]))
p_c[i][j] = (a_E * p_c[i][j+1] + a_W * p_c[i][j-1] + a_N * p_c[i-1][j] + a_S * p_c[i+1][j] + b[i][j]) / a_P
p_new = p + p_c * alpha
# Pressure boundaries
for i in range(N - 1):
p_new[i][0] = p_new[i][1]
p_new[i][M - 1] = p_new[i][M - 2]
for j in range(M - 1):
p_new[0][j] = p_new[1][j]
p_new[N - 1][j] = p_new[N - 2][j]
# Velocity correction
for i in range(1, N - 1):
for j in range(1, M - 1):
u_new[i][j] = u_star[i][j] + alpha * d_e[i][j] * (p_c[i + 1][j] - p_c[i][j])
v_new[i][j] = v_star[i][j] + alpha * d_n[i][j] * (p_c[i][j] - p_c[i + 1][j])
# Continuity residual as error measure
error = 0
for i in range(N):
for j in range(M):
error += abs(b[i][j])
u = u_new
v = v_new
p = p_new
# Plotting
print(u)
print(v)
print(p)
print(error)
x, y = np.meshgrid(
np.linspace(0, domain_size[1], shape[1]),
np.linspace(0, domain_size[0], shape[0]),
)
if iteration % 5 == 0:
axes.pcolormesh(x, y, p, cmap='hot')
plt.quiver(x, y, u, v, scale=1)
plt.pause(0.0001)
plt.show()
|