Refinement of a previous hypothesis of the Lyapunov analysis of isotropic turbulence

The purpose of this brief comunication is to improve a hypothesis of the previous work of the author (de Divitiis, Theor Comput Fluid Dyn, doi:10.1007/s00162-010-0211-9) dealing with the finite--scale Lyapunov analysis of isotropic turbulence. There, the analytical expression of the structure function of the longitudinal velocity difference $\Delta u_r$ is derived through a statistical analysis of the Fourier transformed Navier-Stokes equations, and by means of considerations regarding the scales of the velocity fluctuations, which arise from the Kolmogorov theory. Due to these latter considerations, this Lyapunov analysis seems to need some of the results of the Kolmogorov theory. This work proposes a more rigorous demonstration which leads to the same structure function, without using the Kolmogorov scale. This proof assumes that pair and triple longitudinal correlations are sufficient to determine the statistics of $\Delta u_r$, and adopts a reasonable canonical decomposition of the velocity difference in terms of proper stochastic variables which are adequate to describe the mechanism of kinetic energy cascade.


Introduction
In the previous work [2], the author applies the finite-scale Lyapunov theory to analyse the homogeneous isotropic turbulence. In particular, this theory leads to the analytical closure of the von Kármán-Howarth equation, giving the longitudinal triple velocity correlation k in terms of the longitudinal velocity correlation f and ∂f /∂r (see also the Appendix), and shows that the structure function of the longitudinal velocity difference is expressed by where ∆u r = (u(x + r) − u(x)) · r/r, ξ, η and ζ are uncorrelated centered gaussian random variables, with ξ 2 = η 2 = ζ 2 = 1, and ψ is a function of the Taylor scale Reynolds number R λ = uλ T /ν and of the separation distance r ≡ |r|, according to being u = u 2 r the longitudinal velocity standard deviation, λ T is the Taylor microscale, and χ = χ(R λ ) = 1 is a proper function of R λ which provides nonzero skewness of ∆u r [2], [3].
In Ref. [2], the demonstration of Eq. (1) is carried out through statistical elements regarding the Fourier transformed Navier-Stokes equations, whereas the proof of Eq. (2) is based on the fact that, according to the Kolmogorov theory, the ratio (small scale velocity)-(large scale velocity) depends on λ T /ℓ ≈ √ R λ , where ℓ is the Kolmogorov microscale. Therefore, the analysis of Ref. [2] seems to require the adoption of ℓ whose definition is based on another theory.
Here, instead of using the Kolmogorov scale, we obtain Eqs. (1) and (2), starting from the canonical decomposition of the fluid velocity in terms of proper centered random variables ξ k . In order to describe the mechanism of kinetic energy cascade, the variables ξ k are properly chosen in such a way that each of them exhibits a non-symmetric distribution function. Moreover, due to the isotropy, we postulate that the knowledge of f and k represents a sufficient condition to determine the statistics of ∆u r .

Lyapunov analysis of the velocity fluctuations
This section renews the procedure for calculating the velocity fluctuations, which is based on the Lyapunov analysis of the fluid strain [2], and on the momentum Navier-Stokes equations being u ≡ (u 1 , u 2 , u 3 ), T kh and ρ the fluid velocity, stress tensor and density, respectively. In order to obtain the velocity fluctuation, consider now the relative motion between two contiguous particles, expressed by the infinitesimal separation vector dx which obeys to the equation where dx varies according to the velocity gradient which in turn follows the Navier-Stokes equations.
As observed in Ref. [2], dx is much faster than the fluid state variables, and the Lyapunov analysis of Eq. (4) provides the expression of the local deformation in terms of maximal Lyapunov exponent The map χ : x 0 → x, is the function which gives the current position x of a fluid particle located at the referential position x 0 at t = t 0 [8]. Equation (3) can be written in terms of the referential position The adoption of the referential coordinates allows to factorize the velocity fluctuation and to express it in Lyapunov exponential form of the local deformation. As this deformation is assumed to be much more rapid than −∂u k /∂x 0p u h + 1/ρ ∂T kh /∂x 0p , the velocity fluctuation can be obtained integrating Eq. (6) with respect to the time, where −∂u k /∂x 0p u h + 1/ρ ∂T kh /∂x 0p is considered to be constant This assumption is justified by the fact that, according to the classical formulation of motion of continuum media [8], the terms into the circular brackets of Eq. (6) are considered to be smooth functions of t -at least during the period of a fluctuation-whereas the fluid deformation varies very rapidly according to Eqs. (4)-(5).

Statistical analysis of velocity difference
As explained in this section, the Lyapunov analysis of the local deformation and some plausible assumptions about the statistics of u lead to determine the structure function of ∆u r and its PDF. The statistical properties of ∆u r are here investigated expressing the fluid velocity through the following canonical decomposition [9] u = kÛ k ξ k , whereÛ k (k = 1, 2, ...) are proper coordinate functions of t and x, and ξ k (k = 1, 2, ...) are certain dimensionless independent stochastic variables which satisfy where ̟ ijk = 1 for i = j = k, else ̟ ijk =0. It is worth to remark that the variables ξ k are adequately chosen in such a way that they can describe properly the mechanism of energy cascade. Specifically, the adoption of ξ k with |p| >>> 1 is justified by the fact that the evolution equation of the velocity correlation (see for instance the von Kármán-Howarth equation, appendix) includes also the third order velocity correlation k(r) which is responsible for the intensive mechanism of energy cascade and (∆u r ) 3 / (∆u r ) 2 3/2 = 0. As the result, it is reasonable that the canonical decomposition (8), . This has very important implications for what concerns the statistics of the fluctiations of ∆u r . In order to analyze this question, consider now the dimensionless velocity fluctuationû. This is obtained in terms of ξ k substituting Eq. (8) into Eq. (7) where r =rλ T and u h =û h u. Therefore ij A hij ξ i ξ j arises from the inertia and pressure terms, whereas 1/R λ k b hk ξ k is due to the fluid viscosity. Now, thanks to the local isotropy, u h is a gaussian stochastic variable [5], [9], accordingly, ξ k satisfy into Eq. (10), the Lindeberg condition, a very general necessary and sufficient condition for satisfying the central limit theorem [5]. This condition does not apply to the velocity difference. In fact, as ∆u is the difference between two correlated gaussian variables, its PDF could be a non gaussian distribution function. To study this, the fluctuation ∆u r is first expressed in terms of ξ k This fluctuation can be reduced to the contributions L, S, G + and G − , appearing into Eq. (11) [6]: in particular, L is the sum of all linear terms due to the fluid viscosity, S ≡ S ij ξ i ξ j is the sum of all bilinear forms arising from the inertia and pressure terms, whereas G + and G − are, respectively, definite positive and negative quadratic forms of centered gaussian variables, which derive from the inertia and pressure terms. The quantity L + S tends to a gaussian random variable being the sum of statistically orthogonal terms [6], [5], while G + and G − are determined by means of the hypotheses of isotropy and of fully developed flow Observe that, due to these hypotheses, G + and G − are uncorrelated, thus η, ζ are two independent centered gaussian variables, with η 2 = ζ 2 =1. Furthermore, as the knowledge of f and k is considered to be a sufficient condition for determining the statistics of ∆u r , ψ 2 and ψ 3 are assumed to be proportional with each other through a constant which depends only on R λ Therefore, the longitudinal velocity difference can be written as where ξ is a centered gaussian random variable with ξ 2 = 1, that thanks to the hypotheses of fully developed flow and of isotropy, is considered to be statistically independent from η and ζ.
Comparing the terms of Eqs. (14) and (11), we obtain that ψ 1 and ψ 2 are related with each other and that their ratio ψ ≡ ψ 1 /ψ 2 depends on R λ and r Now, the divisor at the R. H. S. of Eq. (15) is the sum of the following three terms: Hence, taking into account the properties (9) of ξ k , |B| >>> |A|, |C|, thus ψ tends to a quantity arising only from the terms ξ 3 k which appear in Eq. (15).
This expression corresponds to that obtained in Ref. [2] ψ(r, and the dimensionless longitudinal velocity difference is given by Eq. (1) It is worth to remark that ψ expresses the fluctuations ratio (large scale velocity)-(small scale velocity), that is ψ ≈ u/u s ≈ (u 2 /λ T )(l s /u 2 s ), where l s and u s are, respectively the characteristic small scale and the corresponding velocity. This implies that u/u s ≃ λ T /l s ≈ √ R λ , thus l s identifies the Kolmogorov scale and u s l s /ν ≈ 1 is the corresponding Reynolds number.
We conclude this brief comunication by observing that the mechanism of energy cascade acts on ∆u r whose expression, here calculated with the finite-scale Lyapunov theory and Eq. (9), provides a non-symmetric PDF, where the absolute values of the dimensionless moments |H n (0)|, rise with the Taylor scale Reynolds number for n > 3.

Acknowledgments
This work was partially supported by the Italian Ministry for the Universities and Scientific and Technological Research (MIUR).