Knowlpedia

Identity

Vorticity equation for incompressible flow

The curl of the momentum equation, with transport, vortex stretching, diffusion, and curl of forcing.

Core idea

For a smooth three-dimensional solution of the with constant ν\nu, the satisfies

∂tω+(u⋅∇)ω=(ω⋅∇)u+νΔω+∇×f.\partial_t\omega+(u\cdot\nabla)\omega =(\omega\cdot\nabla)u+\nu\Delta\omega+\nabla\times f.

The term (ω⋅∇)u(\omega\cdot\nabla)u is vortex stretching and tilting. For the Euler equations, take ν=0\nu=0.

Derivation

Take curl of the momentum equation. The pressure gradient has zero curl, curl commutes with time derivatives and the Laplacian, and

∇×((u⋅∇)u)=(u⋅∇)ω−(ω⋅∇)u\nabla\times((u\cdot\nabla)u) =(u\cdot\nabla)\omega-(\omega\cdot\nabla)u

when ∇⋅u=0\nabla\cdot u=0. The identity follows by expanding components, or by writing (u⋅∇)u=∇(∣u∣2/2)−u×ω(u\cdot\nabla)u=\nabla(|u|^2/2)-u\times\omega.

Planar flows

For a two-dimensional velocity independent of the third coordinate, the stretching term vanishes. The scalar vorticity then obeys a transport-diffusion equation with the corresponding scalar curl of the force.

References