A smooth cylinder of radius m is mounted across a wind tunnel and spun about its own…

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 Potential flow, complex potentials and lift

Problem

A smooth cylinder of radius a=0.080a = 0.080 m is mounted across a wind tunnel and spun about its own axis at Ω=90 rad s−1\Omega = 90\ \mathrm{rad\,s^{-1}}, anticlockwise as seen with zz towards the viewer. The tunnel wind is U=12 m s−1U = 12\ \mathrm{m\,s^{-1}} along +x+x, the air has density ρ=1.2 kg m−3\rho = 1.2\ \mathrm{kg\,m^{-3}}, and the flow outside a thin layer on the surface is treated as steady, incompressible, inviscid and irrotational. For this idealised comparison, take the circulation of the outer flow, round any circuit enclosing the cylinder, to be that of the surface motion itself, the value it would have if the surface layer passed the surface's own motion on in full; a real surface layer passes on less than this. On the surface the tangential velocity is uθ=−2Usin⁡θ+Γ/(2πa)u_\theta = -2U\sin\theta + \Gamma/(2\pi a), with θ\theta measured anticlockwise from the downstream direction +x+x, and the Kutta-Joukowski theorem gives L′=−ρUΓL' = -\rho U\Gamma. ((a)) Find the circulation Γ\Gamma, with its sign. ((b)) Find the largest surface speed, say where on the cylinder it occurs, and find the pressure there minus the free-stream static pressure at the same height. ((c)) Find the force per unit span L′L' and state its direction. ((d)) Chapter 7 wrote the sideways force on a spinning body with a measured lift coefficient. Using CL=∣L′∣/(12ρU2(2a))C_L = |L'|/(\tfrac12\rho U^{2}(2a)), find the coefficient this ideal-flow calculation predicts, and compare it with a measured value of 1.61.6 at this spin rate.

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