Boundary layer thickness, accelerating flow.

[apc3161]

Hello,

You've all probably seen the classic Blasius solution concerning the thickness of a boundary over a flat plate. This problem though assumes that the free stream velocity is constant.

http://www.see.ed.ac.uk/~johnc/teaching/fluidmechanics4/2003-04/fluids9/2-dboundary.html

I am currently faced with a problem where the free stream velocity is increasing as you go down the plate, so I cannot use the Blasius solutions.

I was wondering if anyone out there knows anything about this subject and if they could reference me to some material concerning this topic.

Thanks

 

[minger]

It sounds to me like you have a Falkner-Skan Wedge flow. Basically this is a similarity solution (as the flat plate boundary layer). Similarity is achieved by the variable $$\eta = Cyx^a$$, which is consistent with a power-law freestream velocity distrubtion:
$$U(x) = Kx^m\,\,\,; m=2a+1$$
The exponent m may be termed the power-law parameter. do some blah blah blah, and the common form of the Falkner-Skan equation for similar flows is:
$$f''' + ff'' + \beta(1-f'^2) = 0$$
Where
$$\beta = \frac{2m}{1+m}$$
The boundary conditions are the same for the flat plate:
$$f(0) = f'(0) = 0; f'(\infty) = 1$$
Where the parameter $$\beta$$ is a measure of the pressure gradient, and is positive for positive for a negative or favorable pressure gradient, and negative for an unfavorable pressure gradient; 0 denotes the flat plate.

I won't type the table out, but you should be able to find a table of solutions online somewhere. Basically they are all non-dimensional, so you'll have to find a reference to dimensionalize them to a real-life problem.