A tornado is modelled as a Rankine vortex: inside a core of radius the air turns as 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 tornado is modelled as a Rankine vortex: inside a core of radius a=80 ma = 80\ \mathrm{m} the air turns as a solid body, uθ=Ωru_\theta = \Omega r, and outside the core it moves as a line vortex, uθ=Γ/(2πr)u_\theta = \Gamma/(2\pi r), the two expressions matching at r=ar = a. The measured tangential wind speed at the edge of the core is 40 m s−140\ \mathrm{m\ s^{-1}}. The flow is treated as planar and horizontal. ((a)) Find the circulation Γ\Gamma round any horizontal loop enclosing the core. ((b)) Find the core angular velocity Ω\Omega and the vorticity ωz\omega_z inside the core. ((c)) Find the tangential wind speed at radius r=200 mr = 200\ \mathrm{m}.

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