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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()
cb = None
Re = 200
nu = 1 / Re
domain_size = (1, 2)
step = 0.05
N = int(domain_size[0] / step)
M = int(domain_size[1] / step)
shape = (N, M)
alpha = 0.8 # Pressure under-relaxation coefficient
t_m = 1
h_c = 0.3
l_c = 0.6
# Staggered vars
u = np.zeros(shape=shape, dtype=float)
u_star = np.zeros(shape=shape, dtype=float)
v = np.zeros(shape=shape, dtype=float)
v_star = np.zeros(shape=shape, dtype=float)
p = np.zeros(shape=shape, dtype=float)
p_star = np.random.rand(shape[0], shape[1])
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
def assert_rule2(value):
assert value > -0.01, f'Coefficient must be positive: {value}' # > 0
# Loop
error = 1
precision = 10 ** -7
t = 0.4
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
# Backwards-facing step boundary conditions (same as sides)
for i in range(int(h_c / step)):
for j in range(int(l_c / step)):
u_star[i][j] = 0
v_star[i][j] = 0
p_star[i][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
assert_rule2(a_E)
assert_rule2(a_W)
assert_rule2(a_N)
assert_rule2(a_S)
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_star[i][j + 1] - p_star[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
assert_rule2(a_E)
assert_rule2(a_W)
assert_rule2(a_N)
assert_rule2(a_S)
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_star[i][j] - p_star[i + 1][j])
# Pressure correction
p_prime = 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
assert_rule2(a_E)
assert_rule2(a_W)
assert_rule2(a_N)
assert_rule2(a_S)
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_prime[i][j] = (a_E * p_prime[i][j+1] + a_W * p_prime[i][j-1] + a_N * p_prime[i-1][j] + a_S * p_prime[i+1][j] + b[i][j]) / a_P
p = p_star + p_prime * alpha
p_star = p
# Pressure boundaries
for i in range(N - 1):
p[i][0] = p[i][1]
p[i][M - 1] = p[i][M - 2]
for j in range(M - 1):
p[0][j] = p[1][j]
p[N - 1][j] = p[N - 2][j]
# Velocity correction
for i in range(1, N - 1):
for j in range(1, M - 1):
u[i][j] = u_star[i][j] + d_e[i][j] * (p_prime[i][j + 1] - p_prime[i][j])
v[i][j] = v_star[i][j] + d_n[i][j] * (p_prime[i][j] - p_prime[i + 1][j])
# Backwards-facing step boundary conditions enforce
for i in range(int(h_c / step)):
for j in range(int(l_c / step)):
u[i][j] = 0
v[i][j] = 0
p[i][j] = 0
# Continuity residual as error measure
error = 0
for i in range(N):
for j in range(M):
error += abs(b[i][j])
# Plotting
print(error)
x, y = np.meshgrid(
np.linspace(0, domain_size[1], shape[1]),
np.linspace(0, domain_size[0], shape[0]),
)
if iteration % 100 == 0:
pcolormesh = axes.pcolormesh(x, y, p, cmap='PuBuGn')
if (cb):
cb.remove()
cb = plt.colorbar(pcolormesh)
factor = np.sqrt(u ** 2 + v ** 2)
u_normalized = u / factor
v_normalized = v / factor
plt.quiver(x, y, u_normalized, v_normalized, scale=45)
plt.pause(0.0001)
plt.show()
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