# User:Oleg Schilling/Proposed/Turbulence: Axisymmetric

**Axisymmetric turbulence** refers to high Reynolds number flows with an axis of symmetry imposed by an external distorsion, i.e. solid body rotation, buoyancy effect, external magnetic field acting over a conducting fluid, axisymmetric mean velocity gradients *etc*.

## Introduction

In the classical phenomenology of fluid turbulence, which is mainly inherited from Kolmogorov (1941) (K41), it is assumed that the flow is statistically homogeneous and isotropic at sufficiently small scale, even if the energy injected at large scale results from an inhomogeneous and anisotropic process, via instabilities and/or effects of boundaries. It is expected that, at intermediate scales, the cascade process leads to forget the conditions of energy injection. This introductory viewpoint immediately calls into plays the following statements or remarks:

- The study of anisotropic, possibly restricted to axisymmetric, turbulence cannot exclude a multi-scale approach.

- In some cases, where the anisotropy is created by a body force (Coriolis, buoyancy, Lorentz) and/or very large scale gradients (velocity, density, pressure), a typical length scale is suggested, as a threshold delineating the range of (smaller) scales where the isotropy is restored. The example of the Oszmidov scale in stably stratified turbulence is the best known.

- Anisotropy is not necessarily maximum at the largest scales: Turbulence subjected to solid body rotation offers a case "without production" (
*i.e.*the Coriolis force produces no energy) in which anisotropy can be dominant in an inertial range of scales.

- The classical K41 spectral scheme, injection/cascade/dissipation, with no link of large-scale injection to small-scale dissipation, except the dissipation rate equal to a constant energy flux, is valid at very high Reynolds number. Is is questioned in real-life turbulence. The spreading of anisotropy in both inertial and dissipative zone remains often an open problem.

It is therefore important to delineate the range of scales in which isotropy can be assumed given various parameters, as the Reynolds number and parameters related to the (large-scale) anisotropizing effect. In real-life turbulence, it can be shown that anisotropy can extend towards the smallest scales. Flows dominated by production and flow dominated by waves are shown to be very different in their anisotropic dynamics.

Another important point is the universality: As soon as Homogeneous isotropic turbulence (HIT) is abandoned, it is expected that specific inhomogeneous anisotropic behaviours might result from specific flow cases for energy (or anisotropy) injection, via specific flow patterns, with their own body forces, large-scale gradients, and boundary conditions. A significant part of our survey will address Homogeneous Anisotropic Turbulence (HAT), in which the most significant effects of specific boundary effects can be ignored, so that at least some classes of flows can be described with a relative universality. These classes are illustrated by rotating flows, stably-stratified flows, uniformly sheared flow, for instance, following the survey in a recent book. MHD flows give other instances of HAT, with possibly coupled fields.

In addition to statistical and phenomenological approach, we will also
consider an important dynamical approach for explaining the emergence
of anisotropy, its spreading at various scale, and its possible asymptotic
sustainance. Pressure fluctuation in Navier-Stokes equations, with and
without coupled fields, is often assumed to be responsible for a
generic "return-to-isotropy" (RTI) effect. This aspect merits to be seriously
analyzed. To check the relevance of the pressure mechanism, compressible
turbulence with its interesting ``pressure-less* limit will be touched upon.*

Finally, a large community of physicists of turbulence is interested in scaling and intermittency, with very marginal interest for anisotropy. Their approach is supported by investigation of two-point statistics for velocity increments, using \(n^{th}-\)order structure functions and "anomalous exponents". We think that the curvature of the line of anomalous exponents is a relevant descriptor of intermittency in HIT (if it exists), but that these exponents reflect both anisotropy and intermittency in an intricate way in non-HIT turbulence. Attention will be restricted to second-order and third-order statistics in this entry, because the impact of anisotropy, and anisotropy versus intermittency, is too controversial for higher moments (only the case of rotating turbulence will be discussed for intermittency and high order structure functions, because it is well documented.) Looking at third-order moments, for instance, in terms of velocity, velocity increments, velocity gradients, it is important to point out some features:

- The \(4/5\) Kolmogorov law can be deeply altered in axisymmetric turbulence. This law, considered as exact (at very high Reynolds number), is much more constrained by the assumption of isotropy than usually admitted.

- Some triple correlations are zero because of isotropy and not because of Gaussianity. This is the case of single-point triple vorticity correlations, single-point triple vorticity correlations. The emergence of the latter in axisymmetric turbulence, for instance induced by the Coriolis force in rotating turbulence, calls into play various anisotropic and non-Gaussian aspects, which must be carefully analyzed and distinguished.

## Essentials for the description. Representation theorems, invariance groups, symmetries, frames

## Typical examples. Rotating, Stably-Stratified and MagnetoHydroDynamics Turbulence, with and without mean shear

## Dynamics and statistics. Physical space description

### Engineering models. Single-point and structure-based modelling

### Two-point approach

#### Kármán-Howarth. Second- and third-order statistics

#### Angular harmonics and SO(3) symmetry group

## Fourier space description. Homogeneous Anisotropic Turbulence. Dynamics and statistics

### Lin-type equations. Second- and third-order statistics

## Studies in progress. Perspectives

## References

- Kolmogorv, A N (1941). The local structure of turbulence in incompressible viscous fluid for very large Reynolds number.
*Dkl. Akad. Nauk. SSSR*31: 538-540.

- Kassinos, Stavros C; Reynolds, William C and Rogers, Mike M (2001). One-point turbulence structure tensors.
*Journal of Fluid Mechanics*428: 213-248.

- Chandrasekhar, S (1961). Hydrodynamics and hydrodynamic stability. Dover, New York.

- Sagaut, P and Cambon, C (2009). Homogeneous turbulence dynamics. Cambridge University Press, Cambridge.