## shooting method solving compressible boundary layer equations

### zhou liu (view profile)

on 13 Oct 2019
Latest activity Commented on by darova

on 16 Oct 2019

### darova (view profile)

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.

darova

### darova (view profile)

on 13 Oct 2019
Equations for image and code don't look similar
First equation can be simplified to second order?
zhou liu

### zhou liu (view profile)

on 15 Oct 2019
the equations deduction is come from https://flylib.com/books/en/4.201.1.212/1/
The author also offered a JAVA code using his own ODE solver.
package TechJava.MathLib;
import TechJava.Gas.*;
public class CompressODE extends ODE
{
double he, ue, Te, mach, Pr, C;
double ratio1, ratio2;
// The CompressODE constructor calls the ODE
// constructor passing it some compressible
// boundary layer specific values. There are five
// first order ODEs and two free variables.
public CompressODE(double Te, double mach) {
super(5,2);
this.Te = Te;
this.mach = mach;
he = 3.5*AbstractGas.R*Te/0.02885;
ue = mach*Math.sqrt(1.4*AbstractGas.R*Te/0.02885);
ratio1 = ue*ue/he;
ratio2 = 110.4/Te;
Pr = 0.75;
}
// The getFunction() method returns the right-hand
// sides of the five first-order compressible
// boundary layer ODEs
// y = delta(C*f'') = delta(n)*(-f*f'')
// y = delta(f') = delta(n)*(f'')
// y = delta(f) = delta(n)*(f')
// y = delta(Cg') = delta(n)*(-Pr*f*g'
// Pr*C*(ue*ue/he)*f''*f'')
// y = delta(g) = delta(n)*(g')
public void getFunction(double x, double dy[],
double ytmp[]) {
C = Math.sqrt(ytmp)*(1.0+ratio2)/
(ytmp+ratio2);
dy = -ytmp*ytmp/C;
dy = ytmp/C;
dy = ytmp;
dy = -Pr*(ytmp*ytmp/C +
ratio1*ytmp*ytmp/C);
dy = ytmp/C;
}
// The getE() method returns the error, E[], in
// the free variables at the end of the range
// that was integrated.
public void getError(double E[], double endY[]) {
E = endY - 1.0;
E = endY - 1.0;
}
// This method initializes the dependent variables
// at the start of the integration range. The V[]
// contains the current guess of the free variable
// values
public void setInitialConditions(double V[]) {
setOneY(0, 0, V);
setOneY(0, 1, 0.0);
setOneY(0, 2, 0.0);
setOneY(0, 3, 0.0);
setOneY(0, 4, V) ;
setOneX(0, 0.0);
}
}
import TechJava.MathLib.*;
public class ShootingCompress
{
public static void main(String args[]) {
// Create a CompressODE object
CompressODE ode = new CompressODE(218.6, 8.0);
// Solve the ODE over the desired range with the
// specified tolerance and initial step size.
// The V[] array holds the initial conditions of
// the free variables.
double dx = 0.1;
double range = 5.0;
double tolerance = 1.0e-6;
double V[] = {0.0826, 25.8};
// Solve the ODE over the desired range
int numSteps =
ODESolver.ODEshooter(ode, V, range, dx, tolerance);
// Print out the results
System.out.println("i eta Cf'' f' f");
for(int i=0; i<numSteps; ++i) {
System.out.println(
""+i+" "+ode.getOneX(i)+" "+ode.getOneY(i,0)+
" "+ode.getOneY(i,1)+" "+ode.getOneY(i,2));
}
System.out.println();
System.out.println("i eta Cg' g");
for(int i=0; i<numSteps; ++i) {
System.out.println(
""+i+" "+ode.getOneX(i)+
" "+ode.getOneY(i,3)+" "+ode.getOneY(i,4));
}
}
}
The equations and BS settings are same as that in his code, but I cann't get the correct answer using MATLAB.

### darova (view profile)

on 15 Oct 2019

bvp4c should work
see attached script

darova

### darova (view profile)

on 15 Oct 2019
But can i have one more try?
I differentiated C function and it looks like below According to this origin equations can be re-written as: Unable to solve the collocation equations -- a singular Jacobian encountered.
I just changed initial conditions: all ones. See attached script. This time it should be cool!
zhou liu

### zhou liu (view profile)

on 16 Oct 2019
Thanks again!
You have different function structure from mine. I define Cf'' as a function of eta, but you define f'' as a function. So both of us are correct in math, right?
In addition, I tried your code, and indeed the five parameters(f, f', f'', g, g' ) are consistent with the boundary values. But it seems not consistent with the correct answer as follows: PS：
I I differentiated C function and found: which is different from you :(
However, both of our equations can not get the correct answer, and I found that the "nspan" is quite sensitive. When I increase the span number from [0 1] to [0 25], it will get "a singular Jacobian encountered". Sad...
darova

### darova (view profile)

on 16 Oct 2019
Indeed, you differentiated C function in a correct way
The shape of results looks similar  