In General > s.a. chaos; Reynolds Number.
* Idea: An eddy-like state of fluid motion where the inertial-vortex forces of the eddies are larger than any of the other forces that tend to damp the eddies out; Characteristics are the apparently random local eddies and whirlpools, diffusion, and dissipation.
* History: The first serious study began with Reynolds, who proposed that its onset is due to instabilities in the laminar flow, that can be characterized (in a classical fluid) by the Reynolds number; This is now thought to be too simplistic; Reaching the critical value for R is only a sufficient condition, and there are some flows that do not have a critical Reynolds number; Very little is understood from first principles, and it has famously been called the last great unsolved problem of classical physics.
* Goal: There is no consensus even on what finding a solution to the problem means; According to engineers, finding mean velocity profiles, wall stresses, and p gradients (use statistical theory, from O Reynolds on), and the motivation is reducing the energy spent in overcoming the drag caused by turbulence; For physicists, the goal is understanding the non-linear processes and the details of motions at various scales.
* Basic concepts: Randomness, eddy viscosity, cascade, scaling.
* And chaos: The approach to turbulence has been shown to be chaotic for certain systems, following the predictions of Ruelle & Takens (1971), and contrary to the Landau-Hopf theory; Turbulent flow itself is thought to be chaotic, but this cannot be experimentally tested.
> Online resources: see Wikipedia page.

Related Topics > s.a. approaches to quantum gravity; magnetism [megnetohydrodynamics]; Scale Invariance; sound [analog metric viewpoint]; Transport.
* Superfluid turbulence: Shows quantized vortices (& Onsager, Feynman).
* Magnus effect: A turbulence and viscosity effect; For a moving ball, a region of turbulence develops downstream; If the ball spins, the region is asymmetric, more on the side of the trailing edge, and exerts a force on the ball in the same direction as the Bernoulli effect; Applications: Golf balls, it explains why dimples are effective.
* Batchelor's Law: A law that describes the size and distribution of the swirls and eddies that form as fluids mix, forming a complex structure similar to a fractal.
@ Superfluid turbulence: Donnelly SA(88)nov; > s.a. Superfluids.
@ Magnus effect: Nathan AJP(08)feb [and flight of baseball].
@ Applications: Leung & Gibson CJOL(04)ap/03 [in geophysics and astrophysics]; Ghosh et al PRS(05) [enhancing particle coalescence].
@ Quantum fluids: Fisher & Pickett pw(06)apr; Vinen & Donnelly PT(07)apr; Tsubota CP(09) [superfluid helium and Bose-Einstein condensates]; Nemirovskii PRP(13) [rev].
@ Numerical simulations: Smits & Marusic PT(13)sep [wall-bounded]; > s.a. computational physics.
@ Other approaches: Canuto & Dubovikov IJMPA(97); Kozyrev TMP(08) [ultrametric theory]; Jejjala ert al IJMPD(10)-a1005-GRF [string theory].
@ And chaos: Ruelle 95; Li a1306 [Reynolds number and the distinction between turbulence and chaos].
@ In astrophysics and cosmology: Low ap/03-in; Leubner et al AiG(06)ap [plasma fluctuations, non-extensive entropy]; Esquivel & Lazarian ApJ(10)-a0905 [Tsallis statistics approach]; Gaite a1202 [cosmic structure]; > s.a. black-hole phenomenology; interstellar matter; Intergalactic Matter.
@ Other topics: Gurarie ht/95 [and statistical physics, field theory]; Gotoh & Nakano JSP(03) [role of pressure]; Galanti & Tsinober PLA(04) [ergodicity]; Choi et al mp/04 [wave turbulence, rev]; Lück et al PLA(06) [coherence length]; Hof et al PRL(08) [evidence for transient nature of all turbulence]; news cosmos(20)jan [proof of Batchelor's Law]; Migdal a2005 [stationary vorticity distribution]; Monsalve et al PRL(20) [observing weak turbulence].

@ Historical: Reynolds PTRS(1883); Darrigol HSPBS(02) [XIX century]; Eyink & Sreenivasan RMP(06) [Onsager]; Bodenschatz & Eckert in(11)-a1107 [Prandtl].
@ Intros, reviews: Deissler RMP(84); Dwoyer et al ed-85; Frisch & Orszag PT(90)jan; Kadanoff PT(95)sep; L'vov & Procaccia PW(96); Gawedzki cd/96 [intro]; Gibson AMR(96)ap/99 [review]; Nelkin AJP(00)apr [RL]; Bernard cm/00-ln; Tabeling PRP(02) [2D]; Barenghi pw(04)dec; Falkovich & Sreenivasan PT(06)apr [universal properties].
@ Texts: Mathieu & Scott 00; Davidson 04 [r PT(05)oct].
@ General references: Muriel PhyA(09) [proposed definitions]; Benzi & Biferale JSP(09) [and the Parisi-Frisch multifractal conjecture]; Eling et al CP(11)-a1004 [geometrization]; Smart PT(11)jan; Bardos & Titi JoT(13)-a1301-conf [mathematical tools]; Calzetta a2010 [relativistic].
@ Statistical approach: Ruelle JSP(14) [non-equilibrium statistical mechanics]; Leschziner 15 [graduate text]; Iyer et al PRX(19) [velocity circulation as a bifractal].
@ Scaling: Gawedzki ht/97; Falkovich et al RMP(01); Carbone et al RNC(04); Bershadskii JSP(07) [finite-size corrections]; Flandoli et al CMP(08); Renner & Peinke JSP(12) [scaling models]; Berera & Clark PRE-a1909 [and attractor dimension].
@ Other systems: Naulin et al PLA(04) [plasma, statistical]; Zakharov et al PRP(04) [1D waves]; Wyngaard 10 [Earth's atmosphere, r PT(11)jan, PRS(11)]; news Phy(12)jul [droplets throwing out a fine spray as they hit a liquid surface]; Green et al PRX(14) [turbulence and the gravity-fluid correspondence].

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