Vorticity is generated at the no-slip surface of a submarine's hull as it cruises…
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
Vorticity is generated at the no-slip surface of a submarine's hull as it cruises through open water, and is assumed to spread outward from the hull afterwards by viscous diffusion alone, into water that is otherwise undisturbed. Water has ν=1.00×10−6m2s−1.
((a)) Starting from the incompressible Navier-Stokes equation for a fluid of uniform density and viscosity, take the curl to derive the viscous vorticity equation Dω/Dt=(ω⋅∇)u+ν∇2ω, explaining why neither the pressure term nor the gravity term survives the operation.
((b)) For a component ω of vorticity with no stretching term present, show that the equation can be written ∂ω/∂t=−∇⋅(ωu−ν∇ω). Using the divergence theorem, show that the total amount of that vorticity component inside any fixed volume can change only through a flux of vorticity across the boundary, carried by the flow or diffused, never by production in the interior.
((c)) Estimate the time for vorticity generated at the hull to diffuse outward, by this mechanism alone, to a radius of 1.5m, and compare it with the time the submarine, cruising at 8ms−1, takes to travel one hull-length of 70m. State what the comparison implies about whether molecular diffusion alone can account for a wake 1.5m wide over that travel distance.