Chapter 5, Problem 24P

A parabolic velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5.11. Derive the stream function for this flow field. Locate streamlines at one-quarter and one-half the total volume flow rate in the boundary layer.

5.11 A useful approximation for the .x component of velocity in an incompressible laminar boundary layer is a parabolic variation from u = 0 at the surface (y = 0) to the freestream velocity, U, at the edge of the boundary layer (y = δ). The equation for the profile is u/U = 2(y/δ) − (y/δ)², where δ = cx¹/² and c is a constant. Show that the simplest expression for the y component of velocity is

$\frac{\upsilon }{U}=\frac{\delta }{x}\left[\frac{1}{2}{\left(\frac{y}{\delta }\right)}^2-\frac{1}{3}{\left(\frac{y}{\delta }\right)}^3\right]$

Plot υ/U versus y/δ to find the location of the maximum value of the ratio υ/U. Evaluate the ratio where δ = 5 mm and x = 0.5 m.