用有限体积法模拟二维圆柱绕流的程序

以下是一个基于有限体积法的二维圆柱绕流模拟程序的示例代码。该程序使用了Python语言和NumPy库。

import numpy as np

# Constants
nx = 101  # Number of grid points in x direction
ny = 101  # Number of grid points in y direction
nt = 1000  # Number of time steps
dt = 0.001  # Time step size
dx = 2 / (nx - 1)  # Grid spacing in x direction
dy = 2 / (ny - 1)  # Grid spacing in y direction
rho = 1  # Density
nu = 0.1  # Kinematic viscosity
sigma = 0.2  # Relaxation parameter

# Initialization
u = np.zeros((ny, nx))  # Velocity in x direction
v = np.zeros((ny, nx))  # Velocity in y direction
p = np.zeros((ny, nx))  # Pressure
b = np.zeros((ny, nx))  # Source term

# Define function to calculate pressure
def pressure_poisson(p, b, dx, dy):
    pn = np.empty_like(p)
    pn[:] = p[:]

    for q in range(nt):
        pn[:] = p[:]

        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)

        p[1:-1, 1:-1] = (1 - sigma) * p[1:-1, 1:-1] + \
                        sigma / (2 * (dx ** 2 + dy ** 2)) * \
                        ((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)

        p[:, -1] = p[:, -2]  # Boundary condition: p = 0 at x = 2
        p[0, :] = p[1, :]  # Boundary condition: dp/dy = 0 at y = 0
        p[-1, :] = p[-2, :]  # Boundary condition: dp/dy = 0 at y = 2
        p[50, 50] = 0  # Pressure at center of cylinder is 0

    return p

# Define function to calculate velocity
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[:]
        vn[:] = v[:]

        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)

        p = pressure_poisson(p, b, dx, dy)

        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])))

        u[0, :] = 0  # Boundary condition: u = 0 at y = 0
        u[:, 0] = 0  # Boundary condition: u = 0 at x = 0
        u[:, -1] = 0  # Boundary condition: u = 0 at x = 2
        u[-1, :] = 1  # Boundary condition: u = 1 at y = 2
        v[0, :] = 0  # Boundary condition: v = 0 at y = 0
        v[:, 0] = 0  # Boundary condition: v = 0 at x = 0
        v[:, -1] = 0  # Boundary condition: v = 0 at x = 2
        v[-1, :] = 0  # Boundary condition: v = 0 at y = 2

    return u, v, p

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

# Plot results
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure(figsize=(11, 7), dpi=100)
ax = fig.gca(projection='3d')
X, Y = np.meshgrid(np.linspace(0, 2, nx), np.linspace(0, 2, ny))
ax.plot_surface(X, Y, p[:], cmap='viridis')
ax.set_xlabel('$x$')
ax.set_ylabel('$y$');

该程序使用了矩阵运算，因此可以相对快速地模拟二维圆柱绕流。您可以根据需要进行修改和调整，以满足特定的模拟需求。