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Wednesday, November 13, 2013

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

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


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: \[\delta_{90} = \eta_{90} \delta = \eta_{90} \sqrt{\frac{\mu x}{U_{\infty}}} \]


2 comments:

  1. السلام عليكم اخي ، اخي والله محتاج مساعدة عندي بحث عن معادلة ، والاستاذ المشرف علي استاذ صارم والله اخي ، عندي اخر سنة في الجامعة التقنية وهذا البحث هو اللي حينقذني من الطرد ، محتاج مساعدتك الله يجازيك بالخير ، والله نفسيتي تعبانة المعادلة طلعتلي روحي

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    1. أرسل لي عنوان بريدك الالكتروني وان شاء أقدر اساعد

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