Turbulence  

In General > s.a. chaos.
* 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.
* 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.

Reynolds Number
* Idea: A dimensionless parameter that measures non-linearity in the Navier-Stokes equation and characterizes the conditions for laminar flow or turbulence in terms of speed of flow, density, viscosity and diameter of the pipe; The number of active degrees of freedom in turbulent flow is about R 9/4 per L3.
$ Def: The number R := UL/mol = (typical flow velocity) (typical length scale) / (kinematic molecular viscosity).
* Values: For small R, the flow is laminar; As R increases, instabilities set in; For large R, there is full turbulence (in real life, mol is usually of the order of 10–2–10–1 cm2/sec).
@ References: Reynolds PTRS(1895), reprint PRS(95).

Related Topics > s.a. magnetism [megnetohydrodynamic turbulence]; 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.
@ Superfluid turbulence: Donnelly SA(88)nov; > s.a. Superfluids.
@ Magnus effect: Nathan AJP(08)feb [and flight of baseball].
@ Applications: Leung & Gibson CJOL-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].
@ Approaches: Canuto & Dubovikov IJMPA(97); Kozyrev TMP(08) [ultrametric theory].
@ In astrophysics and cosmology: Low ap/03-in; Leubner et al AiG-ap/06 [plasma fluctuations, non-extensive entropy]; Dutta et al a0905 [interstellar medium].
@ Other topics: Ruelle 95 [and chaos]; 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].

References
@ Historical: Reynolds PTRS(1883); Darrigol HSPBS(02) [XIX century]; Eyink & Sreenivasan RMP(06) [Onsager].
@ 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].
@ 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).
@ Other systems: Naulin et al PLA(04) [plasma, statistical]; Zakharov et al PRP(04) [1D waves].

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