310. Turbulence. Statistical approach 1

Posted by Dmitry

March 17, 2009

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Let me get back again to one of my most favourite topics in physics, that is, to developed turbulence. Last time (oh my, mid February) I have tried to explain what I consider the most important (and probably hard-to-solve) open problems in physics of turbulence. Now let me list quickly several (not too promising ) approaches to those problems we were able to develop during last hundred years or so.

It would be most natural to start by discussing statistical approach to developed turbulence.

2D turbulence highlighted by laser

On the Fig: 2D turbulence highlighted by laser beam. Image by I. Afanassiev (U. of Newfoundland)

1. Reynolds averaging. Correlation functions of Eulerian velocities

The first person who started thinking about turbulence as stochastic phenomenon was probably Reynolds (back in 1880s he studied transition from laminar, regular, motion of water in a pipe to turbulent motion). Since the turbulent flow is essentially stochastic, he said, one needs to deal with averaged quantities.

What do we mean by averaging? It actually depends on a particular problem we are dealing with or a question we want to answer. For example, one can average over fast pulsations of the flow or over small linear scales.

Anyway, Reynolds has managed to write down a set of equations

$\partial_tV_i+\partial_jV_iV_j-\partial_j(T_{ij}+\tau_{ij}-P\delta_{ij})$ ,

where V_i is averaged (in the sense above) velocity of the flow, $T_{ij}=\nu (\partial_iV_j+\partial_jV_i)$ ( $\nu$ is viscosity) and $\tau_{ij}=-\langle{}v_iv_j\rangle$ is so called Reynolds tensor (here v_i is velocity of the flow before averaging). The latter satisfies the equations

$\partial_tv_i+\partial_j(V_iv_j+V_jv_i)+\partial_ip=\partial_j\left(\nu\partial_jv_i-v_iv_j-\tau_{ij}\right)$ .

Reynolds averaging is not self-consistent, since the number of variables is larger than the number of equations. For example, if you want to find the equation for the Reynolds tensor $\tau_{ij}$ , you may try to multiply eq. for v_i by v_k and average subsequently, but this will just introduce higher order tensor $\tau_{ijk}$ , so you’ll need to find a equation for that one etc. etc. In overall, the procedure reminds the one we perform deriving BBGKY hierarchy of equations in kinetics, with only difference that it is absolutely unclear where to truncate hierarchy in the case of developed turbulence. Still, Reynolds equations turn out to be useful when we want to study, say, averaged characteristics of the flow in pipes.

2. Transport properties of turbulent flow. Correlation functions of Lagrangian velocities

It seems that one of the most important properties of turbulence is the ability of turbulent flow to transfer matter density, energy and momentum fast compared to a laminar flow.

Initially, to describe transport properties of the turbulent flow, people used simplified models based on analogies with slow laminar flows. For example, Boussinesq has related Reynolds tensor $\tau_{ij}$ with averaged velocity of the flow V_i in the case when fluctuations of velocity are present only along a direction perpendicular to the vector V_i itself. Roughly, in this case

$\tau\sim\nu_{\rm turb}\frac{dV}{dy}$ .

The quantity $\nu_{\rm turb}\sim\langle{}v^2\rangle^{1/2}L$ is called turbulent viscosity (here is characteristic linear scale of the flow). Usually, $\nu_{\rm turb}\gg{}\nu$ by orders of magnitude in turbulent flows.

Talking about transport properties of turbulent flows, I cannot help mentioning work by Taylor, who has studied transport of Lagrangian markers in turbulent flow and found that

$\langle{}x^2\rangle\sim2Dt$ ,

where x=x(a,t) is a Lagrangian marker (this fancy term just means that you pick a liquid particle, which is located at the position at t=0 and follow its motion with time) and the diffusion coefficient is given by

$D=\frac{1}{3}\int{}ds\langle{}v(a,t)v(a,t+s)\rangle$ ,

i.e., by the correlation function of Lagrange velocity. This expression basically shows how, similar to description in terms of correlation functions of Eulerian velocities introduced by Reynolds, one can describe physics of turbulence in terms of correlation functions of Lagrangian velocities. Of course, this description is also incomplete and we again get a hierarchy of equations for correlation functions, which is unclear how to close in the case of developed turbulence.

That is it for now, folks… Next time I am going to return to the discussion of Kolmogorov spectra (probably, the most beautiful result found by applying statistical approach) and cover briefly the issue of intermittency.

Further reading

1. L. Landau, E. Lifshitz, Fluid mechanics
2. Monin, Yaglom, Statistical fluid mechanics (both Vol. 1 and 2)
3. Frost (Ed.). Handbook of turbulence. Fundamentals and applications

Master Mason, Turbulence

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