Fluid Mechanics seventh edition by Frank M. White · Problem P3.25
Integral Relations for a Control Volume — Frank M. White
Physics — Fluid Mechanics · continuity-equation-control-volume
Problem
As will be discussed in Chaps. 7 and 8, the flow of a stream past a blunt flat plate creates a broad low-velocity wake behind the plate. A simple model is given in Fig. P3.25, with only half of the flow shown due to symmetry. The velocity profile behind the plate is idealized as ``dead air'' (near-zero velocity) behind the plate, plus a higher velocity, decaying vertically above the wake according to the variation , where is the plate height and is the top of the wake. Find as a function of stream speed .Intuitive explanation
This problem tests your understanding of momentum conservation in a control volume with a non-uniform velocity profile. The key insight is that the net horizontal momentum flux entering the control volume must equal the net momentum flux leaving it—and the difference tells you how much momentum the plate has "stolen" from the flow. By carefully accounting for the momentum at the inlet (uniform ) versus the outlet (the decaying wake profile), you can solve for the velocity deficit that characterizes how much slower the wake is compared to the free stream.
Intuitive explanation
This problem tests your understanding of momentum conservation in a control volume with a non-uniform velocity profile. The key insight is that the net horizontal momentum flux entering the control volume must equal the net momentum flux leaving it—and the difference tells you how much momentum the plate has "stolen" from the flow. By carefully accounting for the momentum at the inlet (uniform ) versus the outlet (the decaying wake profile), you can solve for the velocity deficit that characterizes how much slower the wake is compared to the free stream.