Notes on INS (22): Vortex dynamics in turbulence (review by Pullin & Saffman, 1998)

Time for some more Navier-Stokes

Main reference for this part:

Pullin & Saffman, Vortex dynamics and turbulence, Annu. Rev. Fluid Mech. 1998, 30: 31–51.

– Role played by vortex dynamics in the understanding of high Reynolds numbers turbulence.

– Batchelor (Book: Theory of homogeneous turbulence, 1953) already remarked that two math contribution would greatly enhance our ability to analyze the decay of turbulence: 1) closed-form solutions of NSE; 2) study of vortex dynamics and vortex interactions

Burgers equation {\partial u \over \partial t} + u {\partial u \over \partial x} = \nu {\partial^2 u \over \partial x^2} has closed-form solution found by Cole-Hopf, but such solutions don’t seem to be particularly helpful (?!). So vortex dynamics would be the (only) way to go ?

– Taylor–Green (1937): growth of vorticity by stretching of vortex lines (which is the basic mechanicsm of energy dissipation)

– Confusion over the definition of vortexes. Here: a vortex is a compact region of vorticity surrounded by irrotational fluid.

– Classical examples: vortex filaments (confined inside a tube), vortex blobs (such as Hill’s spherical vortex).

Vortex ring (a.k.a. toroidal vortex): a region of rotating fluid moving through the same or different fluid where the flow pattern takes on a toroidal (doughnut) shape. The movement of the fluid is about the poloidal or circular axis of the doughnut, in a twisting vortex motion. Examples of this phenomenon are a smoke ring or a microburst.

– Robinson & Saffman (1984): nonaxisymmetric Burgers’ vortices.

– Laminar vortex structures ?

– Intermittency of turbulence in high Reynolds number flow (Landau …)

– Kuo & Corrsin (1972): fine-scale structure more likely to consist of tubes than other structures — confirmed by numerical simulations.

– Brown & Roshko (1974): also large-scale tubes.

– Tennekes (1968) modeled turbulence as vortex tubes of diameter \eta (the Kolmogorov length) strechted by eddies of scale \lambda (the Taylor microscale).

Some pictures of eddies (from www.fas.org):

(a rotor is a vertical eddy, and it can turn a helicopter upside down!)

– Lundgren model (?)

– Attempts to construct singularities using vortices

(to be added)

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