孤立翼型面元法 MATLAB 程序：示例与代码解析

以下是一个用 MATLAB 编写的简单孤立翼型面元法程序的示例：

% 孤立翼型面元法 MATLAB 程序示例

% 翼型几何形状定义
% 以NACA 0012翼型为例
x = [0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 1];
yc = [0, 0.0015, 0.006, 0.0135, 0.024, 0.0375, 0.054, 0.0735, 0.096, 0.1486, 0.2264, 0.3157, 0.4145, 0.519, 0.5588, 0.5652];
yt = [0, 0.0011, 0.0046, 0.0103, 0.0184, 0.0286, 0.0411, 0.0561, 0.0735, 0.1486, 0.2264, 0.3157, 0.4145, 0.519, 0.5588, 0.5652];

% 创建面元
numPanels = length(x) - 1;
panels = zeros(numPanels, 4); % 每个面元的坐标 [x1, y1, x2, y2]
for i = 1:numPanels
    panels(i, :) = [x(i), yc(i), x(i+1), yc(i+1)];
end

% 边界条件定义
V_inf = 1; % 无穷远处的远场速度
alpha = 5; % 迎角
alpha_rad = deg2rad(alpha);
U_inf = V_inf * cos(alpha_rad);
W_inf = V_inf * sin(alpha_rad);

% 影响系数矩阵构建
A = zeros(numPanels);
for i = 1:numPanels
    for j = 1:numPanels
        A(i,j) = calcInfluenceCoeff(panels(i,:), panels(j,:));
    end
end

% 右侧项构建
b = zeros(numPanels, 1);
for i = 1:numPanels
    b(i) = -U_inf * cos(alpha_rad) - W_inf * sin(alpha_rad);
end

% 解线性方程组并得到面元强度
gamma = A\b;

% 计算速度分布
x_plot = linspace(0, 1, 100); % 用于绘制速度分布的点
y_plot = zeros(size(x_plot));
for i = 1:length(x_plot)
    y_plot(i) = calcVelocity(x_plot(i), alpha_rad, gamma, panels);
end

% 绘制速度分布曲线
figure
plot(x_plot, y_plot)
xlabel('x')
ylabel('Velocity')
title('Velocity Distribution around an Airfoil')

% 计算压力分布
cp = 1 - (y_plot ./ V_inf).^2;

% 绘制压力分布曲线
figure
plot(x_plot, cp)
xlabel('x')
ylabel('Cp')
title('Pressure Coefficient Distribution around an Airfoil')


% 计算面元对点的影响系数函数
function A_ij = calcInfluenceCoeff(Panel_i, Panel_j)
    x1_i = Panel_i(1);
    y1_i = Panel_i(2);
    x2_i = Panel_i(3);
    y2_i = Panel_i(4);

    x1_j = Panel_j(1);
    y1_j = Panel_j(2);
    x2_j = Panel_j(3);
    y2_j = Panel_j(4);

    A_ij = -0.5 * (log(dist(x1_i, y1_i, x2_i, y2_i) / dist(x1_j, y1_j, x2_j, y2_j)) / (2 * pi));
end

% 计算点的速度函数
function V = calcVelocity(x, alpha, gamma, panels)
    V = 0;
    for i = 1:length(panels)
        V = V + (gamma(i) / (2 * pi)) * calcLift(x, panels(i,:)) * cos(alpha);
    end
end

% 计算面元的升力函数
function L = calcLift(x, panel)
    x1 = panel(1);
    y1 = panel(2);
    x2 = panel(3);
    y2 = panel(4);

    dx = x2 - x1;
    dy = y2 - y1;

    L = dx * (x - x1) + dy * (y - y1);
end

% 计算两点之间的距离函数
function d = dist(x1, y1, x2, y2)
    d = sqrt((x2 - x1)^2 + (y2 - y1)^2);
end

请注意，这只是一个简化的示例，可能无法适应所有情况，并且还有一些细节需要根据您的具体需求进行调整。此外，还可以根据需要进行扩展和改进，以适应更复杂的问题。

希望这个示例能够帮助您开始编写孤立翼型面元法的 MATLAB 程序！如有任何进一步的问题，请随时提问。