A cubic approximate velocity profile was used in Problem 5.12 to model flow in a laminar incompressible boundary layer on a flat plate. For this profile, obtain an expression for the \(x\) and \(y\) components of acceleration of a fluid particle in the boundary layer. Plot \(a_{x}\) and \(a_{y}\) at location \(x=3 \mathrm{ft}\), where \(\delta=0.04 \mathrm{in}\)., for a flow with \(U=20 \mathrm{ft} / \mathrm{s}\). Find the maxima of \(a_{x}\) at this \(x\) location.

Data From Problem 5.12

Transcribed Image Text:

5.12 A useful approximation for the x component of velocity in an incompressible laminar boundary layer is a cubic variation from u = 0 at the surface (y = 0) to the freestream velocity, U, at the edge of the boundary layer (y=6). The equation for the profile is u/U= (y/6)-(y/6), where 8 = cx/2 and c is a constant. Derive the simplest expression for v/U, the y component of velocity ratio. Plot u/U and v/U versus y/8, and find the location of the maximum value of the ratio v/U. Evaluate the ratio where 8=5mm and x = 0.5 m.