A Rankine vortex has a core of radius in solid-body rotation at angular velocity for

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Instability, transition and turbulence

Fluid Dynamics, Ch Hydrodynamic instability: the normal-mode method

Problem

A Rankine vortex has a core of radius

a = 2.0\,\mathrm{cm}

in solid-body rotation at angular velocity

\Omega_0 = 50\,\mathrm{rad\,s^{-1}}

for

r < a

, matched to an irrotational flow

V = \Gamma/(2\pi r)

for

r > a

. The flow is taken to be inviscid and axisymmetric. ((a)) Find the circulation

\Gamma

and verify that

V

is continuous at

r=a

. ((b)) Evaluate Rayleigh's discriminant

\Phi(r) = r^{-3}\,\mathrm{d}(rV)^2/\mathrm{d}r

inside the core (at

r=1.0\,\mathrm{cm}

) and outside it (at

r=4.0\,\mathrm{cm}

). ((c)) State whether each region is stable, neutral or unstable to axisymmetric disturbances.

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