shooting method solving compressible boundary layer equations
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zhou liu
am 13 Okt. 2019
Bearbeitet: Mouli Bhaskar Duddupudi
am 28 Feb. 2022
Hi, i hope someone can help me. I want to find the solution to the compressible boundary layer equations, this problem is part of my thesis project, but I'm running into some problems.
here is the ODEs
here is the boundary conditions
here is the matlab code
function [x,y] = shooting
% Use fsolve to ensure the boundary function is zero. The result is the
% unknown initial condition.
opt = optimset('Display','off','TolFun',1E-20);
F = fsolve(@(F) eval_boundary(F),[0.0826,0,0,0,25.8],opt);
% Solve the ODE-IVP with the converged initial condition
[x,y] = solve_ode(F);
end
function [x,y] = solve_ode(F)
%basic value of outer flow
Te = 218.6; mach = 8.0; Pr = 0.75;
he = 1004.68 * Te; ue = mach * sqrt(1.4 * 8.29586 * Te / 0.02885);
ratio1 = ue * ue / he; cm = 110.4 / Te; ck = 110.4 / Te;
% C = sqrt(y(5))*(1.0+ratio2)/(y(5)+ratio2);
% Solve the ODE-IVP with initial condition F on [0 100] (arbitrary upper bound)
[x,y] = ode45(@(x,y) [ -y(3) * y(1) / ( sqrt(y(5)) * (1.0 + cm) / ( y(5) + cm ) );
y(1) / ( sqrt(y(5)) * (1.0 + cm) / ( y(5) + cm ) );
y(2);
-Pr * ( y(3) * y(4) / ( sqrt(y(5)) * (1.0 + ck) / ( y(5) + ck ) ) + ratio1 * y(1) * y(1) / ( sqrt(y(5)) * (1.0 + cm) / ( y(5) + cm ) ) ) ;
y(4) / ( sqrt(y(5)) * (1.0 + ck) / ( y(5) + ck ) ) ] , [0 100] , F); %solve BVP
end
function [g] = eval_boundary(F)
% Get the solution to the ODE with inital condition F
[x,y] = solve_ode(F);
% Get the function values (for BCs) at the starting/end points
y2_start = y(1,2); %f(0) = 0
y3_start = y(1,3); %f'(0) = 0
y4_start = y(1,4); %g'(0) = 0
y1_end = y(end,1); %f''(end) = 0
y2_end = y(end,2); %f'(end) = 1
y4_end = y(end,4); %g'(end) = 0
y5_end = y(end,5); %g(0) = 1
% Evaluate the boundary function
g = [
y2_start
y3_start
y4_start
% y1_end
y2_end-1
% y4_end
y5_end-1
];
end
this code is come from Mohammad A Alkhadra (https://ww2.mathworks.cn/matlabcentral/fileexchange/69310-solving-blasius-equation-with-the-shooting-method) for solving blasius boundary layer equations. I changed the equations and BCs but it can not get correct answers (the code return complex number which must be wrong).
I have checked many times for the code and equations deduction, but did not find the wrong place.
I don't know if it is because these ODEs are differential equation with variable coefficients ? this is one different point from the original code.
2 Kommentare
darova
am 13 Okt. 2019
Equations for image and code don't look similar
First equation can be simplified to second order?
Akzeptierte Antwort
darova
am 15 Okt. 2019
bvp4c should work
see attached script
7 Kommentare
darova
am 16 Okt. 2019
Indeed, you differentiated C function in a correct way
The shape of results looks similar
What about Y axis? Maybe it's about scaling?
Mouli Bhaskar Duddupudi
am 28 Feb. 2022
Bearbeitet: Mouli Bhaskar Duddupudi
am 28 Feb. 2022
I have been looking into this problem. What could be reason for Y-axis? Is it because of scaling?
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