Blasius Equation: Some Explorations Part 4

Boundary Layer Thickness

There is no sharp outer edge of the boundary layer. Rather it merges smoothly and gradually with the outer free stream. However, there are several rational ways of defining a boundary layer thickness, and we consider one of them here. The idea is determining where the horizontal velocity is nearly equal to the free stream velocity or a constant ratio of it. The thickness , is the y-value at which the u velocity is 0.90 of the free stream velocity. In order to find that value we have need to wrap the previous pieces of code into a single function which is listed below

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function [eta, fun] = BLThickness(x, lim)
    function B= BlasVal(y)
        options = odeset('RelTol', 1e-10, 'AbsTol', 1e-12);
        sol=ode45(@BlasiusFunc, [0,15], [0.0 0.0 1.0], options);
        alpha = sqrt(sol.y(2,end));
        f = (deval(sol, (y./alpha)));
        B = (f(2)./alpha^2)-lim;
       
        function df = BlasiusFunc(~,f)
            df = zeros(3,1);
            df(1) = f(2);
            df(2) = f(3);
            df(3) = -f(1)*f(3)/2;
        end
    end

    [eta, fun]= fzero(@BlasVal, x);
end

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So, in order to compute the value of the \(\eta_{90}\)

BLThickness(3, 0.9)

This gives the value of that corresponds to \(\frac{u}{U_{\infty}}\)=0.90. So the thickness of the boundary layer is given by: \[\delta_{90} = \eta_{90} \delta = \eta_{90} \sqrt{\frac{\mu x}{U_{\infty}}} \]