I am unable to converge the blasius equation using shooting technique

clear all

clc

eta = 10;

dt = 0.01;

N = eta/dt + 1;

t = linspace(0,eta, N);

K = 1001;

tol = 1e-5;

fid1 = fopen('blasius_eta_data.txt', 'wt');

fid2 = fopen('blasius_f_data.txt', 'wt');

fid3 = fopen('blasius_g_data.txt', 'wt');

fid4 = fopen('blasius_h_data.txt', 'wt');

fid5 = fopen('guessdata.txt', 'at');

f = zeros(N,K);

g = zeros(N,K);

h = zeros(N,K);

f(1,:) = 0;

g(1,:) = 0;

h(1,1) = 5;

h(1,2) = 7;

for k = 1:K-1

for i = 2:N

f(i,k) = f(i-1, k) + g(i-1, k)*dt;

g(i,k) = g(i-1, k) + h(i-1, k)*dt;

h(i,k) = h(i-1, k) - (f(i-1, k)*h(i-1, k)*dt)/2;

end

% if tolerance is met, iter = k and break out of the loop

if abs(g(N,k) - 1) < tol

iter = k

break

end

% want to skip this case

if k == 1

continue;

else

% applying shooting method to determine the next guess

h(1, k+1) = h(1,k) + (1 - g(N,k))*(h(1,k) - h(1, k-1))/(g(N, k) - g(N, k-1));

% condition on h(1,:) - cannot be a negative value

if sign(h(1, (k+1))) == 1

h(1, k+1) = -h(1, k+1);

end

end

end

%transporting for data collection purposes

t_col = transpose(t);

fprintf(fid1, '%.5f\n', t_col);

fprintf(fid1, '%.5f\n', f(:, iter));

fprintf(fid1, '%.5f\n', g(:, iter));

fprintf(fid1, '%.5f\n', h(:, iter));

fprintf(fid1, '%.5f %.5f\n', [dt; h(1, iter)]);

%

% plotting

plot(f(:, iter), t);

hold on

plot(g(:, iter), t);

plot(h(:, iter), t);

Answer 1

[Divyam]

Hi @Mouli Bhaskar Duddupudi,

To improve the chances of convergence, you must first improve the initial guesses & based on your data. You can also further decrease the step size and the tolerance of the function.

Upon examining the code for the shooting star method, the implementation for the sign flip seems to be incorrect as the fluid acceleration at the boundary layer should be positive due to the nature of flow. Here is the correct implementation for the sign flip condition,

if h(1, k+1) < 0
    h(1, k+1) = abs(h(1, k+1)); % Make it positive if it's negative
end

You can also utilize this file exchange code for solving the Blasius equation boundary layer flow over a flat plate: https://www.mathworks.com/matlabcentral/fileexchange/69310-solving-blasius-equation-with-the-shooting-method