# Refinement of a Previous Hypothesis of the Lyapunov Analysis of Isotropic Turbulence

Authors: Nicola de Divitiis
Journal: Journal of Engineering
 (2013)

---

## Abstract

The purpose of this paper is to improve a hypothesis of the previous work of N. de Divitiis (2011) dealing with the finite-scale Lyapunov analysis of isotropic turbulence. There, the analytical expression of the structure function of the longitudinal velocity differenceΔur 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 Δur 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.


---

## Body

## 1. Introduction

In the previous work [1], 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
(1)Δur〈(Δur)2〉=ξ+ψ(χ(η2-1)-(ζ2-1))1+2ψ2(1+χ2),
where Δur=(u(x+r)-u(x))·r/r, ξ, η, and ζ are uncorrelated centered Gaussian random variables, with 〈ξ2〉=〈η2〉=〈ζ2〉 = 1, and ψ is a function of the Taylor scale Reynolds number Rλ=uλT/ν and of the separation distance r≡|r|, according to
(2)ψ(r,Rλ)=Rλ1515ψ^(r),
where u=〈ur2〉 is the longitudinal velocity standard deviation, λT is the Taylor microscale, and χ=χ(Rλ)≠1 is a proper function of Rλ which provides nonzero skewness of Δur [1, 2].In [1], the demonstration of (1) is carried out through statistical elements regarding the Fourier transformed Navier-Stokes equations, whereas the proof of (2) is based on the fact that, according to the Kolmogorov theory, the ratio (small scale velocity)–(large scale velocity) depends on λT/ℓ≈Rλ, where ℓ is the Kolmogorov microscale. Therefore, the analysis of [1] seems to require the adoption of ℓ whose definition is based on another theory.Here, instead of using the Kolmogorov scale, we obtain (1) and (2), starting from the canonical decomposition of the fluid velocity in terms of proper centered random variables ξk. In order to describe the mechanism of kinetic energy cascade, the variables ξk are properly chosen in such a way that each of them exhibits a nonsymmetric 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 Δur.


## 2. The 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 [1] and on the momentum Navier-Stokes equations
(3)∂uk∂t=-∂uk∂xhuh+1ρ∂Tkh∂xh,
where u≡(u1,u2,u3), Tkh, and ρ are 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 vectordx which obeys
(4)dẋ=∇udx,
where dx varies according to the velocity gradient which in turn follows the Navier-Stokes equations. As observed in [1], dx is much faster than the fluid state variables, and the Lyapunov analysis of (4) provides the expression of the local deformation in terms of the maximal Lyapunov exponent Λ>0(5)∂x∂x0≈eΛ(t-t0).The mapχ: x0→x is the function which gives the current position x of a fluid particle located at the referential position x0 at t=t0 [3]. Equation (3) can be written in terms of the referential position x0 [3]
(6)∂uk∂t=(-∂uk∂x0puh+1ρ∂Tkh∂x0p)∂x0p∂xh.The adoption of the referential coordinates allows to factorize the velocity fluctuation and to express it in the Lyapunov exponential form of the local deformation. As this deformation is assumed to be much more rapid than-∂uk/∂x0puh+1/ρ∂Tkh/∂x0p, the velocity fluctuation can be obtained by integrating (6) with respect to the time, where -∂uk/∂x0puh+1/ρ∂Tkh/∂x0p is considered to be constant
(7)uk≈1Λ(-∂uk∂x0puh+1ρ∂Tkh∂x0p)t=t0.This assumption is justified by the fact that, according to the classical formulation of motion of continuum media [3], the terms into the circular brackets of (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 (4)-(5).


## 3. Statistical Analysis of Velocity Difference

As explained in this section, the Lyapunov analysis of the local deformation and some plausible assumptions about the statistics ofu lead to determination of the structure function of Δur and its PDF.The statistical properties ofΔur are here investigated expressing the fluid velocity through the following canonical decomposition [4]:
(8)u=∑k‍U^kξk,
where U^k (k=1,2,…) are proper coordinate functions of t and x, and ξk (k=1,2,…) are certain dimensionless independent stochastic variables which satisfy
(9)〈ξk〉=0,〈ξiξj〉=δij,〈ξiξjξk〉=ϖijkp,|p|⋙1,|p|⋙〈ξk4〉,
where ϖijk = 1 for i=j=k, else ϖijk=0. It is worth to remark that the variables ξk are adequately chosen in such a way that they can describe properly the mechanism of energy cascade. Specifically, the adoption of ξk with |p|⋙1 is justified by the fact that the evolution equation of the velocity correlation (see, e.g. the von Kármán-Howarth equation in appendix) includes also the third-order velocity correlation k(r) which is responsible for the intensive mechanism of energy cascade, and 〈(Δur)3〉/〈(Δur)2〉3/2≠ 0. As a result, it is reasonable that the canonical decomposition (8) (Δur=(r/r)·∑k‍(U^k(x+r)-U^k(x))ξk) includes variables ξk with |〈ξk3〉|⋙1 [5]. This has very important implications for what concerns the statistics of the fluctiations of Δur. In order to analyze this question, consider now the dimensionless velocity fluctuation u^. This is obtained in terms of ξk by substituting (8) into (7)
(10)u^h=∑ij‍Ahijξiξj+1Rλ∑k‍bhkξk,
where r=r^λT and uh=u^hu. Therefore ∑ij‍Ahijξiξj arises from the inertia and pressure terms, whereas 1/Rλ∑k‍bhkξk is due to the fluid viscosity. Now, thanks to the local isotropy, uh is a Gaussian stochastic variable [4, 5]; accordingly, ξk satisfy into (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 Δu is the difference between two correlated Gaussian variables, its PDF could be a nonGaussian distribution function. To study this, the fluctuation Δur is first expressed in terms of ξk(11)Δu^r(r)=∑ij‍ΔArij(r)ξiξj+1Rλ∑k‍Δbrk(r)ξk≡L+S+G++G-.This fluctuation can be reduced to the contributionsL, S, G+, and G-, appearing into (11) [6]: in particular, L is the sum of all linear terms due to the fluid viscosity, and S≡Sijξiξj 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 are derived from the inertia and pressure terms. The quantity L+S tends to a Gaussian random variable being the sum of statistically orthogonal terms [5, 6], while G+ and G- are determined by means of the hypotheses of isotropy and of fully developed flow
(12)G-=-(ζ2-1)ψ2(r),G+=(η2-1)ψ3(r).Observe that, due to these hypotheses,G+ and G- are uncorrelated; thus η, ζ are two independent centered Gaussian variables, with 〈η2〉=〈ζ2〉=1.  Furthermore, as the knowledge of f and k is considered to be a sufficient condition for determining the statistics of Δur, ψ2 and ψ3 are assumed to be proportional with each other through a constant which depends only on Rλ(13)χ(Rλ)=ψ3(r)ψ2(r).Therefore, the longitudinal velocity difference can be written as(14)Δur=ψ1ξ+ψ2(χ(η2-1)-(ζ2-1)),
where ξ is a centered Gaussian random variable with 〈ξ2〉 = 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 (14) and (11), we obtain that ψ1 and ψ2 are related with each other and that their ratio ψ≡ψ1/ψ2 depends on Rλ and r(15)2(1+χ2)ψ22ψ12=〈(G++G-)2〉〈(Sijξiξj+(1/RλΔbrkξk))2〉.Now, the divisor at the R. H. S. of (15) is the sum of the following three terms:
(16)A=SijSpq〈ξiξjξpξq〉,B=2RλSij〈ξiξjξk〉Δbrk,C=1Rλ2Δbrk2〈ξk2〉.Hence, taking into account the properties (9) of ξk, |B|⋙|A|, |C|; thus ψ tends to a quantity arising only from the terms 〈ξk3〉 which appear in (15):
(17)ψ≡ψ2ψ1=φ(r)Rλ.This expression corresponds to that obtained in [1]
(18)ψ(r,Rλ)=Rλ1515ψ^(r),ψ^(0)=O(1),
and the dimensionless longitudinal velocity difference is given by (1)
(19)Δur〈(Δur)2〉=ξ+ψ(χ(η2-1)-(ζ2-1))1+2ψ2(1+χ2).It is worth to remark thatψ expresses the fluctuations ratio (large scale velocity)–(small scale velocity); that is, ψ≈u/us≈(u2/λT)(ls/us2), where ls and us are, respectively the characteristic small scale and the corresponding velocity. This implies that u/us≃λT/ls≈Rλ; thus ls identifies the Kolmogorov scale, and usls/ν≈1 is the corresponding Reynolds number.The distribution function ofΔur is then expressed through the Frobenius-Perron equation [7], taking into account that ξ, η, and ζ are independent stochastic variables
(20)F(Δur′)=∫ξ∫η∫ζp(ξ)p(η)p(ζ)δ(Δur′-Δur(ξ,η,ζ))dξdηdζ,
where Δur(ξ,η,ζ) is determined by (19), δ is the Dirac delta, and p is a centered Gaussian PDF with standard deviation equal to the unity. The dimensionless statistical moments of Δur are easily calculated
(21)Hn≡〈(Δur)n〉〈(Δur)2〉n/2=1(1+2(1+χ2)ψ2)n/2×∑k=0n‍(nk)ψk〈ξn-k〉〈(χ(η2-1)-(ζ2-1))k〉.In particular,H3, related to the mechanism of energy cascade, is
(22)H3(r)=8ψ3(χ3-1)(1+2ψ2(1+χ2))3/2.In conclusion,χ=χ(Rλ) is implicitly calculated, in function of ψ(0), taking into account that this Lyapunov theory gives H3(0)=-3/7 (see appendix) [1]
(23)H3(0)=8ψ3(0)(χ3-1)(1+2ψ2(0)(1+χ2))3/2=-37,
where ψ^(0)≃1.075 is estimated as in [1], and 〈(Δur)2〉 and ψ^(r) are calculated in function of f(r) and k(r).We conclude this paper by observing that the mechanism of energy cascade acts onΔur whose expression, here calculated with the finite-scale Lyapunov theory and (9), provides a nonsymmetric PDF, where the absolute values of the dimensionless moments |Hn(0)| rise with the Taylor scale Reynolds number for n>3.



---

*Source: 653027-2013-03-31.xml*