A small vortex-generator fin on an aircraft wing sheds a trailing vortex of…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity

Fluid Dynamics, Ch The Navier-Stokes equations and unsteady exact solutions

Problem

A small vortex-generator fin on an aircraft wing sheds a trailing vortex of circulation Γ\Gamma into the surrounding air (ν=1.51×10−5 m2 s−1\nu=1.51\times10^{-5}\ \mathrm{m^2\,s^{-1}}). Once shed, the vortex is stretched along its own axis by the straining flow of the surrounding wake, modelled as the steady, axisymmetric, incompressible strain ur=−12Sru_r=-\tfrac12 Sr, uz=Szu_z=Sz, with S=25 s−1S=25\ \mathrm{s^{-1}} a constant set by the wake geometry. ((a)) Confirm that ur=−12Sru_r=-\tfrac12Sr, uz=Szu_z=Sz satisfies ∇⋅u=0\nabla\cdot\mathbf u=0 in cylindrical coordinates. ((b)) For a purely axisymmetric vortex ω=ωz(r)z^\boldsymbol\omega=\omega_z(r)\hat{\mathbf z} with no time dependence, add this strain to the viscous vorticity equation, using (ω⋅∇)u=ωz ∂uz/∂z z^(\boldsymbol\omega\cdot\nabla)\mathbf u=\omega_z\,\partial u_z/\partial z\,\hat{\mathbf z}, to obtain the steady balance −12Sr dωzdr=Sωz+ν 1rddr(rdωzdr).-\frac12 Sr\,\frac{\mathrm d\omega_z}{\mathrm dr}=S\omega_z+\nu\,\frac1r\frac{\mathrm d}{\mathrm dr}\left(r\frac{\mathrm d\omega_z}{\mathrm dr}\right). Verify by direct substitution that ωz=Ce−Sr2/4ν\omega_z=Ce^{-Sr^2/4\nu}, with CC a constant, satisfies this balance for any CC. ((c)) The corresponding swirl is the Burgers vortex uθ=(Γ/2πr)[1−e−Sr2/4ν]u_\theta=(\Gamma/2\pi r)\left[1-e^{-Sr^2/4\nu}\right]. From duθ/dr=0\mathrm d u_\theta/\mathrm dr=0, find the radius at which the air moves fastest around this vortex, and explain, contrasting with the unstretched Lamb-Oseen vortex, why this radius stays fixed for as long as the strain SS persists rather than growing with time.

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