Noncommutative Metafluid Dynamics

Pramana, 2005

Les Houches - Ecole d’Ete de Physique Theorique, 2001

To describe transport of scalar and vector fields by a random flow one needs to apply the methods of statistical mechanics to the motion of fluid particles, i.e. to the Lagrangian dynamics. We first present the propagators describing evolving probability distributions of different configurations of fluid particles. We then use those propagators to describe growth, decay and steady states of different scalar and vector quantities transported by random flows. We discuss both practical questions like mixing and segregation and fundamental problems like symmetry breaking in turbulence. Contents I. Introduction A. Propagators B. Kraichnan model C. Large Deviation Approach II. Particles in fluid turbulence A. Single-particle diffusion B. Two-particle dispersion in a spatially smooth velocity C. Two-particle dispersion in a non-smooth incompressible flow D. Two-particle dispersion in a compressible flow E. Multi-particle configurations and zero modes III. Unforced evolution of passive fields A. Decay of tracer fluctuations B. Growth of density fluctuations in compressible flow C. Vector fields in a smooth velocity IV. Cascades of a passive tracer A. Direct cascade B. Inverse cascade in a compressible flow V. Active tracers A. Activity changing cascade direction B. Two-dimensional incompressible turbulence VI. Conclusion References

2012

Structures and statistics of fluid turbulence / Structures et statistiques de la turbulence des fluides The Lundgren-Monin-Novikov hierarchy: Kinetic equations for turbulence

Nature, 2003

Reviews of Modern Physics, 2001

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scaleinvariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.

Physica Scripta, 2013

This paper presents a parameter-free theory of shear-generated turbulence at asymptotically high Reynolds numbers in incompressible fluids. It is based on a two-fluids concept. Both components are materially identical and inviscid. The first component is an ensemble of quasi-rigid dipole-vortex tubes (vortex filaments, excitations) as quasiparticles in chaotic motion. The second is a superfluid performing evasive motions between the tubes. The local dipole motions follow Helmholtz' law. The vortex radii scale with the energy-containing length scale. Collisions between quasiparticles lead either to annihilation (likewise rotation, turbulent dissipation) or to scattering (counterrotation, turbulent diffusion). There are analogies with birth and death processes of population dynamics and their master equations and with Landau's two-flluid theory of liequid Helium. For free homogeneous decay the theory predicts the TKE to follow t −1 . With an adiabatic wall condition it predicts the logarithmic law with von Kármán's constant as 1/ √ 2 π = 0.399. Likewise rotating couples form dissipative patches almost at rest (→ intermittency) wherein under local quasi-steady conditions the spectrum evolves into an "Apollonian gear" as discussed first by . Dissipation happens exclusively at scale zero and at finite scales this system is frictionless and reminds of Prigogine's (1947) law of minimum (here: zero) entropy production. The theory predicts further the prefactor of the 3D-wavenumber spectrum (a Kolmogorov constant) as 1 3 (4 π) 2/3 = 1.802, well within the scatter range of observational, experimental and DNS results.

Low Temperature Physics, 2019

The hydrodynamic problems of superfluid fluid containing chaotic tangles of quantized vortex filaments are discussed. The construction of such hydrodynamics crucially depends on the statistics of vortex tangles, and two important cases are presented. The first corresponds to a tangle that consists of entirely chaotic vortex filaments. This case is implemented in counterflowing helium, and is referred to as Vinen turbulence. In the construction of macroscopic dynamics, the system of equations is closed by the Vinen equation for the density of vortex filaments. The second, referred to as the Hall-Vinen-Bekarevich-Khalatnikov case, corresponds to a situation where the system contains bundles of polarized vortex filaments. In this instance, the system is closed by the Feynman equation that relates density of vortex filaments with the vorticity of superfluid velocity. Problems related to the application of both approaches are discussed.

Science China Physics, Mechanics & Astronomy, 2015

A turbulent flow is maintained by an external supply of kinetic energy, which is eventually dissipated into heat at steep velocity gradients. The scale at which energy is supplied greatly differs from the scale at which energy is dissipated, the more so as the turbulent intensity (the Reynolds number) is larger. The resulting energy flux over the range of scales, intermediate between energy injection and dissipation, acts as a source of time irreversibility. As it is now possible to follow accurately fluid particles in a turbulent flow field, both from laboratory experiments and from numerical simulations, a natural question arises: how do we detect time irreversibility from these Lagrangian data? Here we discuss recent results concerning this problem. For Lagrangian statistics involving more than one fluid particle, the distance between fluid particles introduces an intrinsic length scale into the problem. The evolution of quantities dependent on the relative motion between these fluid particles, including the kinetic energy in the relative motion, or the configuration of an initially isotropic structure can be related to the equal-time correlation functions of the velocity field, and is therefore sensitive to the energy flux through scales, hence to the irreversibility of the flow. In contrast, for single-particle Lagrangian statistics, the most often studied velocity structure functions cannot distinguish the "arrow of time". Recent observations from experimental and numerical simulation data, however, show that the change of kinetic energy following the particle motion, is sensitive to time-reversal. We end the survey with a brief discussion of the implication of this line of work.

Physica A: Statistical Mechanics and its Applications, 1981

Entropy, 2017

In the last few decades a series of experiments have revealed that turbulence is a cooperative and critical phenomenon showing a continuous phase change with the critical Reynolds number at its onset. However, the applications of phase transition models, such as the Mean Field Theory (MFT), the Heisenberg model, the XY model, etc. to turbulence, have not been realized so far. Now, in this article, a successful analogy to magnetism is reported, and it is shown that a Mean Field Theory of Turbulence (MFTT) can be built that reveals new results. In analogy to compressibility in fluids and susceptibility in magnetic materials, the vorticibility (the authors of this article propose this new name in analogy to response functions, derived and given names in other fields) of a turbulent flowing fluid is revealed, which is identical to the relative turbulence intensity. By analogy to magnetism, in a natural manner, the Curie Law of Turbulence was discovered. It is clear that the MFTT is a theory describing equilibrium flow systems, whereas for a long time it is known that turbulence is a highly non-equilibrium phenomenon. Nonetheless, as a starting point for the development of thermodynamic models of turbulence, the presented MFTT is very useful to gain physical insight, just as Kraichnan's turbulent energy spectra of 2-D and 3-D turbulence are, which were developed with equilibrium Boltzmann-Gibbs thermodynamics and only recently have been generalized and adapted to non-equilibrium and intermittent turbulent flow fields.