A model wing section is accelerated from rest along a tank of still water and then…
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow
Fluids and Continuum Physics, Ch Vorticity, circulation and vortex motion
Problem
A model wing section is accelerated from rest along a tank of still water and then held at constant speed. The section lies in the plane of the page, the water streams past it from left to right, and circulations are counted positive anticlockwise as seen on the page. Treat the water as incompressible, of uniform density, with gravity the only body force, and treat the motion as inviscid except in the thin sheet of fluid that rolls up at the trailing edge. A student measures the circulation round a small loop taken tightly round the wing and finds Γwing=−14 m2 s−1, that is 14 m2 s−1 clockwise. In the write-up the student draws the vortex left behind at the trailing edge turning clockwise too, and writes: "Both circulations were made by the same motion of the wing, so both must turn the same way; the water behind the wing therefore carries −14 m2 s−1 as well."
((a)) Name the theorem the student's drawing breaks, and state precisely which loop it must be applied to.
((b)) Give the circulation of the starting vortex, with its sign under this convention.
((c)) The student goes on to claim that the circulation round a large loop of fluid particles which surrounded the wing while everything was still at rest is now −28 m2 s−1. What is it really?