A parabolic approximate velocity profile was used in Problem 5.11 to model flow in a laminar incompressible boundary layer on a flat plate. For this profile, find the \(x\) component of acceleration, \(a_{x}\), of a fluid particle within the boundary layer. Plot \(a_{x}\) at location \(x=0.8 \mathrm{~m}\), where \(\delta=1.2 \mathrm{~mm}\), for a flow with \(U=6 \mathrm{~m} / \mathrm{s}\). Find the maximum value of \(a_{x}\) at this \(x\) location.

Data From Problem 5.11

Transcribed Image Text:

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 = 8). The equation for the profile is u/U= 2(y/6)-(y/6), where 8 = cx/2 and c is a constant. Show that the simplest expression for the y component of velocity is v = U x 2 2 1 - 3 8 Plot v/U versus y/8 to find the location of the maximum value of the ratio v/U. Evaluate the ratio where 8=5 mm and x = 0.5 m.