Lamb vector

In fluid dynamics, Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb. The Lamb vector is defined as
where formula_2 is the velocity field and formula_3 is the vorticity field of the flow. It appears in the Navier–Stokes equations through the material derivative term, specifically via convective acceleration term,
In irrotational flows, the Lamb vector is zero, so does in Beltrami flows. The concept of Lamb vector is widely used in turbulent flows. The Lamb vector is analogous to electric field, when the Navier–Stokes equation is compared with Maxwell's equations.
Properties of Lamb vector.
The divergence of the lamb vector can be derived from vector identities,
At the same time, the divergence can also be obtained from Navier–Stokes equation by taking its divergence. In particular, for incompressible flow, where formula_6, with body forces given by formula_7, the Lamb vector divergence reduces to
where
In regions where formula_10, there is tendency for formula_11 to accumulate there and vice versa.