Class: Fluid Mechanics
I understand all of the math here, what I don't understand is literally the beginning of the solution where they somehow define u/U as y/delta. I genuinely am lost right here and I don't understand how the context of the problem tells us this relationship. Any of the other math I can do, but how are they coming up with this relationship? I need to be able to do other problems like this, so this is really messing me up right now.
603 CHAPTER 1
One dimension of a rectangular flat plate is twice the other. Air at uniform speed flows parallel to the plate, and a laminar boundary layer forms on both sides of the plate. Which orientation - long dimension parallel to the wind (Fig, P10102) or short dimension parallel to the wind (Fig. P10102b) - has the higher drag? Explain.
HW-4
HW#4 Solutions
10-99 Solution We are to generate expressions for * and δ, and compare to Blasius.
Analysis
First, we set U(x) = V = constant for a flat plate. We integrate using the definition of δ,
∫0^δ (1 - u/U) dy = 1
We integrate only to y = δ, since beyond that, the integrand is identically zero. After substituting the limits of integration, we obtain δ as a function of x.
The streamwise velocity component of a steady incompressible, laminar, flat plate boundary layer of boundary layer thickness δ is approximated by the simple linear expression, u = Uy/δ for y ≤ δ, and u = U for y > δ (Fig. P1099). Generate expressions for displacement thickness and momentum thickness as functions of δ, based on this linear approximation. Compare the approximate values of δ*/δ and δ/δ to the values of δ5*/δ and δ6/δ obtained from the Blasius solution. Answers: 0.500, 0.167
FIGURE P10102
Similarly,
U(x) = m
After substituting the limits of integration, we obtain δ as a function of x,
δ = (5/2)√(x/300) = 6, δ = 9√(x/300)
The ratios are δ*/δ = 1/2 = 0.500, and δ/δ = 1/6 = 0.167, to three significant digits. We compare these approximate results to those obtained from the Blasius solution, i.e., δ*/δ = 1.72/4.91 = 0.350, and δ6/δ = 0.664/4.91 = 0.135. Thus, our approximate velocity profile yields δ to about 43% error, and δ to about 23% error.
FIGURE P1099
