Turbulence |

**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.

@ __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].

**References**

@ __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].

@ __Statistical approach__: Ruelle JSP(14) [non-equilibrium statistical mechanics];
Leschziner 15 [graduate text].

@ __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].

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 20 jan 2019