Fluid Dynamics | Part IB, 2003

Starting from the Euler equations for incompressible, inviscid flow

ρDuDt=p,u=0\rho \frac{D \mathbf{u}}{D t}=-\nabla p, \quad \nabla \cdot \mathbf{u}=0

derive the vorticity equation governing the evolution of the vorticity ω=×u\boldsymbol{\omega}=\nabla \times \mathbf{u}.

Consider the flow

u=β(x,y,2z)+Ω(t)(y,x,0)\mathbf{u}=\beta(-x,-y, 2 z)+\Omega(t)(-y, x, 0)

in Cartesian coordinates (x,y,z)(x, y, z), where tt is time and β\beta is a constant. Compute the vorticity and show that it evolves in time according to

ω=ω0e2βtk\boldsymbol{\omega}=\omega_{0} \mathrm{e}^{2 \beta t} \mathbf{k}

where ω0\omega_{0} is the initial magnitude of the vorticity and k\mathbf{k} is a unit vector in the zz-direction.

Show that the material curve C(t)C(t) that takes the form

x2+y2=1 and z=1x^{2}+y^{2}=1 \quad \text { and } \quad z=1

at t=0t=0 is given later by

x2+y2=a2(t) and z=1a2(t),x^{2}+y^{2}=a^{2}(t) \quad \text { and } \quad z=\frac{1}{a^{2}(t)},

where the function a(t)a(t) is to be determined.

Calculate the circulation of u\mathbf{u} around CC and state how this illustrates Kelvin's circulation theorem.

Typos? Please submit corrections to this page on GitHub.