A wide open tank of water carries a vertical vortex which is well described by a…
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 wide open tank of water carries a vertical vortex which is well described by a Rankine vortex: inside a core of radius a the water turns as a solid body with angular velocity Ω, and outside the core the swirl is uθ=Γ/(2πr). Initially the core radius is a1=8.0 cm and the tangential speed at the edge of the core is 0.12 m s−1. The swirling column is then stretched vertically to 3.0 times its original length by a slow gentle axial flow (suction above, inflow below). Treat the water as incompressible and the motion as inviscid, so that the circulation round every ring of fluid particles is unchanged, and assume the core remains a uniform cylinder of uniform vorticity. Take g=9.8 m s−2.
((a)) Find the circulation Γ of the vortex.
((b)) Predict the core radius after the stretching.
((c)) Predict the core vorticity ωz and the tangential speed at the edge of the core after the stretching.
((d)) Predict the depth of the free-surface dip on the axis (below the level of the distant surface) after the stretching.