## Tuesday, January 10, 2012

### MrsDrPoe: Momentum Integral Example

As promised last week, we'll look at an example problem using the momentum integral boundary layer equation.

Example: If the boundary layer velocity profile is approximated at u = U*y/delta in the boundary layer and u = U in the free stream, and the displacement thickness is measured at 6.7 cm, determine the boundary layer and momentum thicknesses.

Given: u(y) = U*y/delta, u(y) = U if y > delta, delta* = 6.7 cm

Find: boundary layer thickness, momentum thickness
Solution:

Choose some Y in the free stream:

delta* = int(1 - (u(y)/Uinfinity))dy|0,Y
= int(1 - (u(y)/Uinfinity))dy|0,delta + int(1 - (u(y)/Uinfinity))dy|delta,Y
= int(1 - (U*y/delta)/Uinfinity)dy|0,delta + int(0)dy|delta,Y
= int(1 - (y/delta))dy|0,delta
= (delta - (delta*delta)/(2*delta))
= delta - delta/2 = delta/2

delta = 2*delta* = 13.4 cm
Theta = int((u(y)/Uinfinity)*(1 - u(y)/Uinfinity))dy|0,Y
= int((u(y)/Uinfinity)*(1 - u(y)/Uinfinity))dy|0,delta + int((u(y)/Uinfinity)*(1 - u(y)/Uinfinity))dy|delta,Y
= int((u(y)/Uinfinity)*(1 - u(y)/Uinfinity))dy|0,delta + int(0)dy|delta,Y
=int((y/delta)*(1 - y/delta))dy|0,delta
= (delta*delta)/(2*delta) - (delta*delta*delta)/(3*delta*delta)
= delta/6

Theta = delta/6 = 2.233 cm

Not too bad, huh?  Until next week, happy studying!