matlab求解平板边界层问题.doc

粘性流体力学程序平板边界层问题求解1.1编程思路平面边界层问题可以归结为在已知边界层条件下解一个高阶微分方程,即解。Matlab提供了解微分方程的方法，运用换元法将高阶微分方程降阶，然后运用“ode45”函数进行求解。函数其难点在于如何将边界条件中运用好，由四阶龙格-库塔方法知其核心是换元试算匹配，故在运用函数时通过二分法实现是可行的。1.2m函数function dy = rigid(x,y)dy = zeros(3,1);dy(1) = y(2);dy(2) = y(3);dy(3) = -0.5*y(1)*y(3);%main程序X, Y = ode45(rigid,0 5,0 0 0);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+) %二分法试算f的初始值以满足f趋向无穷时的边界条件，图像上可以清晰看出f无穷时的结果 X, Y = ode45(rigid,0 5,0 0 1);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+) X, Y = ode45(rigid,0 5,0 0 0.5);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+) X, Y = ode45(rigid,0 5,0 0 0.25);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+) X, Y = ode45(rigid,0 5,0 0 0.375);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+) grid on X, Y = ode45(rigid,0 5,0 0 0.3125);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+) grid on X, Y = ode45(rigid,0 5,0 0 0.34375);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+)grid on X, Y = ode45(rigid,0 5,0 0 0.328125);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+)grid on X, Y = ode45(rigid,0 10,0 0 0.328125);%当f为0.328125时，逼近结果已经很好，在0到5的变化范围内已经非常接近精确解plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+)grid on X, Y = ode45(rigid,0 5,0 0 0.335975);plot(X, Y(:,1),-,X, Y(:,2),*,X, Y(:,3),+)grid on1.3 数据展示选取f=0.335975时的数据展示，X代表的变化，Y的第一二三列代表的值。数据如下表： 0 0 0 0.33600.0001 0.0000 0.0001 0.33600.0003 0.0000 0.0001 0.33600.0004 0.0000 0.0002 0.33600.0006 0.0000 0.0002 0.33600.0013 0.0000 0.0005 0.33600.0021 0.0000 0.0007 0.33600.0028 0.0000 0.0010 0.33600.0036 0.0000 0.0012 0.33600.0073 0.0000 0.0025 0.33600.0111 0.0000 0.0037 0.33600.0148 0.0000 0.0050 0.33600.0185 0.0001 0.0062 0.33600.0372 0.0002 0.0125 0.33600.0559 0.0005 0.0188 0.33600.0746 0.0009 0.0251 0.33600.0933 0.0015 0.0313 0.33600.1868 0.0059 0.0627 0.33590.2802 0.0132 0.0941 0.33580.3737 0.0235 0.1255 0.33550.4671 0.0366 0.1568 0.33500.5921 0.0589 0.1987 0.33400.7171 0.0863 0.2403 0.33250.8421 0.1189 0.2818 0.33040.9671 0.1567 0.3229 0.32761.0921 0.1996 0.3636 0.32401.2171 0.2476 0.4038 0.31951.3421 0.3006 0.4434 0.31411.4671 0.3584 0.4823 0.30771.5921 0.4211 0.5203 0.30031.7171 0.4885 0.5573 0.29191.8421 0.5604 0.5932 0.28251.9671 0.6367 0.6279 0.27212.0921 0.7173 0.6612 0.26082.2171 0.8020 0.6931 0.24872.3421 0.8905 0.7234 0.23592.4671 0.9828 0.7520 0.22252.5921 1.0785 0.7790 0.20862.7171 1.1774 0.8042 0.19442.8421 1.2794 0.8276 0.18012.9671 1.3843 0.8492 0.16573.0921 1.4917 0.8690 0.15143.2171 1.6014 0.8871 0.13753.3421 1.7134 0.9034 0.12403.4671 1.8272 0.9181 0.11103.5921 1.9428 0.9312 0.09873.7171 2.0600 0.9428 0.08713.8421 2.1785 0.9530 0.07633.9671 2.2982 0.9619 0.06634.0921 2.4189 0.9696 0.05724.2171 2.5405 0.9762 0.04904.3421 2.6629 0.9819 0.04164.4671 2