# Boundary layer thickness, accelerating flow.

• apc3161
In summary, the conversation discusses the Falkner-Skan Wedge flow, which is a similarity solution for a variable free stream velocity. The equation for similar flows is given, along with the boundary conditions. The parameter \beta is a measure of the pressure gradient and can be used to find solutions for different pressure gradients. Overall, more information on this topic can be found online along with tables of non-dimensional solutions.

#### 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 [Broken]

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

Last edited by a moderator:
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.