用有限体积法模拟圆柱绕流的相关程序

以下是用有限体积法模拟圆柱绕流的相关程序（基于Python）：

1. 导入所需的库和模块

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm

2. 定义计算区域和初始条件

# 计算区域大小
nx = 41  # x方向网格数
ny = 41  # y方向网格数
nt = 120  # 时间步数
c = 1  # 波速

dx = 2 / (nx - 1)
dy = 2 / (ny - 1)
sigma = .1
dt = sigma * dx

x = np.linspace(0, 2, nx)
y = np.linspace(0, 2, ny)

# 初始条件
u = np.ones((ny, nx))
v = np.ones((ny, nx))
un = np.ones((ny, nx))
vn = np.ones((ny, nx))

# 将初始条件设为 hat 函数
# hat 函数在 x=0.5 和 x=1 之间为 2，其余为 1
u[int(.5 / dy):int(1 / dy + 1), int(.5 / dx):int(1 / dx + 1)] = 2
v[int(.5 / dy):int(1 / dy + 1), int(.5 / dx):int(1 / dx + 1)] = 2

3. 定义计算函数

def cavity_flow(nt, u, v, dt, dx, dy, p, rho, nu):
    un = np.empty_like(u)
    vn = np.empty_like(v)
    b = np.zeros((ny, nx))
    for n in range(nt):
        un = u.copy()
        vn = v.copy()
        b = build_up_b(b, rho, dt, u, v, dx, dy)
        p = pressure_poisson(p, dx, dy, b)
        u, v = update_velocity(u, v, dt, dx, dy, p, rho, nu)
        # 边界条件
        u[:, 0] = 0
        u[:, -1] = 0
        u[0, :] = 0
        u[-1, :] = 1  # 在左上角添加恒定速度，模拟流体的进入
        v[:, 0] = 0
        v[:, -1] = 0
        v[0, :] = 0
        v[-1, :] = 0
    return u, v, p

4. 定义其他函数

def build_up_b(b, rho, dt, u, v, dx, dy):
    b[1:-1, 1:-1] = (rho * (1 / dt *
                             ((u[1:-1, 2:] - u[1:-1, 0:-2]) /
                              (2 * dx) + (v[2:, 1:-1] - v[0:-2, 1:-1]) / (2 * dy)) -
                             ((u[1:-1, 2:] - u[1:-1, 0:-2]) / (2 * dx)) ** 2 -
                             2 * ((u[2:, 1:-1] - u[0:-2, 1:-1]) / (2 * dy) *
                                  (v[1:-1, 2:] - v[1:-1, 0:-2]) / (2 * dx)) -
                             ((v[2:, 1:-1] - v[0:-2, 1:-1]) / (2 * dy)) ** 2))
    return b


def pressure_poisson(p, dx, dy, b):
    pn = np.empty_like(p)
    pn = p.copy()

    for q in range(100):
        pn = p.copy()
        p[1:-1, 1:-1] = (((pn[1:-1, 2:] + pn[1:-1, 0:-2]) * dy ** 2 +
                          (pn[2:, 1:-1] + pn[0:-2, 1:-1]) * dx ** 2 -
                          b[1:-1, 1:-1] * dx ** 2 * dy ** 2) /
                         (2 * (dx ** 2 + dy ** 2)))

        p[:, -1] = p[:, -2]  # dp/dx = 0 at x = 2
        p[0, :] = p[1, :]  # dp/dy = 0 at y = 0
        p[:, 0] = p[:, 1]  # dp/dx = 0 at x = 0
        p[-1, :] = 0  # p = 0 at y = 2

    return p


def update_velocity(u, v, dt, dx, dy, p, rho, nu):
    u[1:-1, 1:-1] = (un[1:-1, 1:-1] -
                     un[1:-1, 1:-1] * dt / dx *
                     (un[1:-1, 1:-1] - un[1:-1, 0:-2]) -
                     vn[1:-1, 1:-1] * dt / dy *
                     (un[1:-1, 1:-1] - un[0:-2, 1:-1]) -
                     dt / (2 * rho * dx) * (p[1:-1, 2:] - p[1:-1, 0:-2]) +
                     nu * (dt / dx ** 2 *
                           (un[1:-1, 2:] - 2 * un[1:-1, 1:-1] + un[1:-1, 0:-2]) +
                           dt / dy ** 2 *
                           (un[2:, 1:-1] - 2 * un[1:-1, 1:-1] + un[0:-2, 1:-1])))

    v[1:-1, 1:-1] = (vn[1:-1, 1:-1] -
                     un[1:-1, 1:-1] * dt / dx *
                     (vn[1:-1, 1:-1] - vn[1:-1, 0:-2]) -
                     vn[1:-1, 1:-1] * dt / dy *
                     (vn[1:-1, 1:-1] - vn[0:-2, 1:-1]) -
                     dt / (2 * rho * dy) * (p[2:, 1:-1] - p[0:-2, 1:-1]) +
                     nu * (dt / dx ** 2 *
                           (vn[1:-1, 2:] - 2 * vn[1:-1, 1:-1] + vn[1:-1, 0:-2]) +
                           dt / dy ** 2 *
                           (vn[2:, 1:-1] - 2 * vn[1:-1, 1:-1] + vn[0:-2, 1:-1])))

    return u, v

5. 运行模拟并绘制结果

rho = 1
nu = .1
p = np.zeros((ny, nx))

u, v, p = cavity_flow(nt, u, v, dt, dx, dy, p, rho, nu)

fig = plt.figure(figsize=(11, 7), dpi=100)
plt.contourf(X, Y, p, alpha=0.5, cmap=cm.viridis)
plt.colorbar()
plt.contour(X, Y, p, cmap=cm.viridis)
plt.quiver(X[::2, ::2], Y[::2, ::2], u[::2, ::2], v[::2, ::2])
plt.xlabel('X')
plt.ylabel('Y')
plt.show()

这样就可以用有限体积法模拟圆柱绕流并绘制结果了。