# Renormalization Group Analysis of Turbulent Hydrodynamics

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Referee this paper: arXiv:1012.0461 by Dirk Barbi, Gernot Munster

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Turbulent hydrodynamics is characterised by universal scaling properties of its structure functions. The basic framework for investigations of these functions has been set by Kolmogorov in 1941. His predictions for the scaling exponents, however, deviate from the numbers found in experiments and numerical simulations. It is a challenge for theoretical physics to derive these deviations on the basis of the Navier-Stokes equations. The renormalisation group is believed to be a very promising tool for the analysis of turbulent systems, but a derivation of the scaling properties of the structure functions has so far not been achieved. In this work, we recall the problems involved, present an approach in the framework of the exact renormalisation group to overcome them, and present first numerical results.

summarized
paper authored Dec 2, 2010
retagged Jul 6, 2014

## 1 Review

+ 3 like - 1 dislike

This is not a review in the narrow sense as I feel not properly entitled to judge the paper, but let me write down what I like about it and why anyway.

Applications of  renormalization group like methods  are not unknown in fluid dynamic turbulence theory. For example Yakhot & Orszag (1986) introduced the small scale removal procedure to investigate statistically homogeneous, stationary  and isotropic Navier-Stokes turbulence, and calculate for example an effective scale dependent viscosity, derive Kolmogorov's constant from first principles, and calculate the slop of the turbulent kinetic energy spectrum among other things, see also Smith & Woodruff (1998) for a review and further applications of this method. Applying this scale removal procedure not only to the Navier-Stokes equation but to the extended system of governing equations that describe for example a weakly stable stratified fluid flows and discerning between horizontal and vertical directions, Sukorianski et al. (2005) derive a coupled system (some kind of "Callan-Symanzik") equations. Their solutions describe the "running" of the turbulent diffusion coefficients for horizontal and vertical momentum and temperature diffusion.

However, these specific scale elimination methods, widespread in the fluid dynamics community are similar in spirit but not exactly equivalent to what theoretical physicists mean by renormalization group or RG flow analysis.  Due to a number of more or less restricting assumptions, their range of possible applications is rather limited compared to the renormalization methods and concepts applied in theoretical physics:

• A specific scale invariant fixed point (Kolmogorov inertial subrange) is explicitly assumed right from the start, due to the assumption of a balance between a large scale stochastic forcing with a power-law spectrum of a power-law spectrum.
• The DIA (direct interaction approximation) is often assumed, which means that
- not all possible interactions are taken into account
- a gap in the spectrum of fluctuations considered is present
- it corresponds to some kind of a "Reynolds decomposition"  which is equivalent to
a first order closure, and at most 2-point functions (second order structure functions)
of the fluctuations can be computed.
• Moving away from the assumed fixed point is not possible, as the scale removal method does not allow for new operators (couplings) to become (ir)relevant in the course of the RG flow.
• No rescaling step is included in the scale elimination transformation, therefore the fixed point investigated is strictly speaking not a "true" scale invariance of the fixed point.
• No vertex (coupling constant) renormalization is applied in the scale removal procedure.

Due to the above limitations, the scale elimination method alows only to study the fixed point (scale invariant turbulent subrange) explicitly assumed. There are other investigations that take anomalous scaling into account to allow for deviations from the Kolmogorov inertial subrange, however going beyond a single fixed point is still not possible.

Conversely, the authors of the present paper have developped theoretical and numerical methods to apply the Exact Renormalization Group (ERG)  to hydrodynamic turbulence, which allow in principle to study the behavior of RG flows with no or multiple fixed points at different scales.

To apply the ERG to turbulent hydrodynamics, the authors had to derive an appropriate action that corresponds to the incompressible Navier Stokes equation, which they subsequently inserted into the RG (Wegner-Houghton) equation and solved it numerically:

• To derive a generating functional from which the action can be read off, the Navier-Stokes equations are cast into the solenoidal form which is U(1) gauge invariant.
• Similar to the case of the small scale elimination procedure, a stochastic forcing is needed to put energy into the system at large scales.
• The generating functional is the integral over all solutions of the Navier-Stokes equation averaged over all realizations of the stochastic forcing.
• The constraint of incompressibility is taken into account by making use of the Faddeev-Popov method
• The functional determinant is rewritten by introducing new Grassman fields which allows one to simplify the numerical calculations.
• Getting rid of the non-local interactions by introducing a new field and its corresponding propagator, the final form of the action corresponding to the incompressible Navier-Stokes equations is obtained.

The numerical methods developed by the authors for analyzing the RG flow are tested by investigating “toy systems”, such as scalar and O(3) symmetric field theories. Different kinds of fixed points can successfully be characterized and located. By making use of the Local Potential Approximation (LPA), some characteristics of Navier-Stokes turbulence are successfully retrieved by analyzing numerically the RG flow:

• The trivial fixed point corresponds to Kolmogorov turbulence.
•  The well known Kolmogorov scaling of lower order correlation functions is successfully retrieved.
• The scaling of higher order $n$-point functions, which can not be obtained by Kolmogorov's theory or Orszag & Yakhot's scale elimination procedure are successfully computed by the RG flow.

Applying the methods presented in this paper to hydrodynamic turbulence, has several advantages compared to the scale removal and related procedures:

• Scale invariance is not explicitly assumed, no fixed point has to be present a priori in the RG flow
• In principle, all interactions are included and can freely evolve in the course of the RG flow, and it can "escape" in principle from (intermediate) fixed points.
• No particular form of the input energy spectrum is assumed.
• The direct interaction approximation (DIA) is not invoked, such that the scaling of higher order $n$-point functions at a specific fixed point can be computed.
• Coupling constant renormalization is taken into account.
• Rescaling  implicitly included -> fixed points are exactly scale invariant.
• RG flows with multiple fixed points can be investigated..

In summary, I think that this paper is very interesting and I have not seen before such an attempt to apply the ERG to hydrodynamic turbulence. This in theoretical physics originating approach seems to be mostly unknown to the fluid dynmics community, even to those people who apply renormalization group like methods to turbulence theory. Due to the advantages listed above, it seems that the methods to analyse the RG flow of hydrodynamic systems developed by the authors of the present paper open a whole new range of possibilities to study open questions in turbulence theory, which cannot be settled by scale removal and other similar in the fluid dynamics community more well known renormalization group like methods.

Update: the paper is also published here in Physics Research International

reviewed Jul 14, 2014 by (6,040 points)
edited Jul 22, 2014 by Dilaton

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