Vortex stretching

Studies of vortices in turbulent fluid motion byLeonardo da Vinci.

Influid dynamics,vortex stretchingis the lengthening ofvorticesin three-dimensional fluid flow, associated with a corresponding increase of the component ofvorticityin the stretching direction—due to theconservation of angular momentum.

Vortex stretching is associated with a particular term in thevorticity equation.For example, vorticity transport in an incompressible inviscid flow is governed by

${D{\vec {\ Omega }} \over Dt}=\left({\vec {\ Omega }}\cdot {\vec {\nabla }}\right){\vec {v}},$

whereD/Dtis thematerial derivative.The source term on the right hand side is the vortex stretching term. It amplifies the vorticity${\vec {\ Omega }}$when the velocity is diverging in the direction parallel to${\vec {\ Omega }}$.

A simple example of vortex stretching in a viscous flow is provided by theBurgers vortex.

Vortex stretching is at the core of the description of the turbulenceenergy cascadefrom the large scales to the small scales inturbulence.In general, in turbulencefluid elementsare more lengthened than squeezed, on average. In the end, this results in more vortex stretching thanvortex squeezing.Forincompressible flow—due to volume conservation of fluid elements—the lengthening implies thinning of the fluid elements in the directions perpendicular to the stretching direction. This reduces the radial length scale of the associated vorticity. Finally, at the small scales of the order of theKolmogorov microscales,the turbulencekinetic energyis dissipated into heat through the action ofmolecularviscosity.

References

• Chorin, A.J.(1994),Vorticity and turbulence(2nd ed.), Springer,ISBN0-387-94197-5
• Tennekes, H.;Lumley, J.L.(1972),A First Course in Turbulence,Cambridge, MA: MIT Press,ISBN0-262-20019-8