Question

A crude approximation for the x-component of velocity in an incompressible laminar boundary layer flow is a linear variation from u = 0 at the surface (y = 0) to the freestream velocity, U, at the boundary layer edge (y = δ). The equation for he profile is u = Uy/δ, where δ = cx1/2 and c is a constant. Show that the simplest expression for the y component of velocity is v = uy/4x. Evaluate the maximum value of the ratio v /U, at a location where x = 0.5 m and δ = 5 mm.

Answer 1

Answer:

v/U=0.79

Explanation:

Given u=$u=\frac{Uy}{\partial }=\frac{Uy}{cx^{1/2}}$

Now for the given flow to be possible it should satisfy continuity equation

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

Applying values in this equation we have

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\\\\\frac{\partial u}{\partial x}=\frac{\partial (\frac{Uy}{cx^{1/2}})}{\partial x}\\\\\frac{\partial u}{\partial x}=\frac{-1}{2}\frac{Uy}{cx^{3/2}}$

Thus we have

$\frac{\partial v}{\partial y}=\frac{1}{2}\frac{Uy}{cx^{3/2}}\\\\\therefore \int \partial v=\int \frac{1}{2}\frac{Uy}{cx^{3/2}}\partial x\\\\v=\frac{1}{4}\frac{Uy^{2}}{cx^{3/2}}\\\\v=\frac{1}{4}\frac{uy}{x}$ Hence proved $\because u=\frac{Uy}{cx^{1/2}}$

For maximum value of v/U put y =$\partial$

$v=\frac{1}{4}\frac{Uy^{2}}{cx^{3/2}}$

$v=\frac{1}{4}\frac{Uy^{2}}{cx^{3/2}}\\\\\frac{v}{U}=\frac{\partial ^{2}}{4cx^{3/2}}$

Thus solving we get using the given values

v/U=0.79