LBM非平衡外推格式计算圆柱边界的代码

以下是使用LBM非平衡外推格式计算圆柱边界的简单示例代码：

import numpy as np
from numba import jit

# 定义常量
nx = 101  # 网格数
ny = 101
Lx = 1.0  # 区域大小
Ly = 1.0
dx = Lx / (nx - 1)  # 网格间距
dy = Ly / (ny - 1)
dt = 1e-3  # 时间步长
T = 5.0  # 总时间
nu = 0.1  # 动力粘度系数
Re = 50.0  # 雷诺数
omega = 1.0 / (3.0 * nu + 0.5)  # 松弛频率
cx = np.array([0, 1, 0, -1, 0, 1, -1, -1, 1])  # 每个格子的速度分量
cy = np.array([0, 0, 1, 0, -1, 1, 1, -1, -1])
w = np.array([4.0 / 9.0, 1.0 / 9.0, 1.0 / 9.0, 1.0 / 9.0, 1.0 / 9.0, 1.0 / 36.0, 1.0 / 36.0, 1.0 / 36.0, 1.0 / 36.0])  # 每个格子的权重

# 初始化数组
f = np.zeros((nx, ny, 9))
rho = np.ones((nx, ny))
ux = np.zeros((nx, ny))
uy = np.zeros((nx, ny))

# 边界条件
def set_boundary(rho, ux, uy):
    # 上下边界
    rho[0, :] = rho[1, :]
    rho[-1, :] = rho[-2, :]
    ux[0, :] = 0.0
    uy[0, :] = 0.0
    ux[-1, :] = 0.0
    uy[-1, :] = 0.0
    
    # 左右边界
    rho[:, 0] = rho[:, 1]
    rho[:, -1] = rho[:, -2]
    ux[:, 0] = 0.0
    uy[:, 0] = 0.0
    ux[:, -1] = 0.0
    uy[:, -1] = 0.0
    
    # 圆柱边界
    for j in range(1, ny-1):
        rho[0, j] = 1.0
        ux[0, j] = 0.0
        uy[0, j] = 0.0
        rho[-1, j] = 1.0
        ux[-1, j] = 0.0
        uy[-1, j] = 0.0
    for i in range(1, nx-1):
        rho[i, 0] = 1.0
        ux[i, 0] = 0.0
        uy[i, 0] = 0.0
        rho[i, -1] = 1.0
        ux[i, -1] = 0.0
        uy[i, -1] = 0.0
        for k in range(9):
            cu = 3.0 * (cx[k] * ux[i, j] + cy[k] * uy[i, j])
            f[i, j, k] = rho[i, j] * w[k] * (1.0 + cu + 0.5 * cu**2 - 1.5 * (ux[i, j]**2 + uy[i, j]**2))

# LBM非平衡外推格式计算
@jit(nopython=True)
def lbm_nbe(f, rho, ux, uy):
    # 预测
    for i in range(1, nx-1):
        for j in range(1, ny-1):
            # 计算宏观量
            rho[i, j] = 0.0
            ux[i, j] = 0.0
            uy[i, j] = 0.0
            for k in range(9):
                rho[i, j] += f[i, j, k]
                ux[i, j] += f[i, j, k] * cx[k]
                uy[i, j] += f[i, j, k] * cy[k]
            ux[i, j] /= rho[i, j]
            uy[i, j] /= rho[i, j]
            
            # BGK更新
            for k in range(9):
                cu = 3.0 * (cx[k] * ux[i, j] + cy[k] * uy[i, j])
                f_star = rho[i, j] * w[k] * (1.0 + cu + 0.5 * cu**2 - 1.5 * (ux[i, j]**2 + uy[i, j]**2))
                f[i, j, k] = omega * f_star + (1.0 - omega) * f[i, j, k]
    
    # 残差修正
    for i in range(1, nx-1):
        for j in range(1, ny-1):
            # 计算宏观量
            rho[i, j] = 0.0
            ux[i, j] = 0.0
            uy[i, j] = 0.0
            for k in range(9):
                rho[i, j] += f[i, j, k]
                ux[i, j] += f[i, j, k] * cx[k]
                uy[i, j] += f[i, j, k] * cy[k]
            ux[i, j] /= rho[i, j]
            uy[i, j] /= rho[i, j]
            
            # 计算残差
            resx = 0.0
            resy = 0.0
            for k in range(9):
                cu = 3.0 * (cx[k] * ux[i, j] + cy[k] * uy[i, j])
                f_star = rho[i, j] * w[k] * (1.0 + cu + 0.5 * cu**2 - 1.5 * (ux[i, j]**2 + uy[i, j]**2))
                resx += cx[k] * (f[i, j, k] - f_star)
                resy += cy[k] * (f[i, j, k] - f_star)
            
            # 更新分布函数
            for k in range(9):
                f[i, j, k] -= (resx * cx[k] + resy * cy[k])
    
    # 边界条件
    set_boundary(rho, ux, uy)
    
    return rho, ux, uy

# 主程序
for t in range(int(T / dt)):
    rho, ux, uy = lbm_nbe(f, rho, ux, uy)

需要注意的是，此代码仅为示例，未对计算结果进行处理和可视化。完整的圆柱边界计算代码需要考虑更多细节和优化。