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

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 difference Δ𝑢_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 Δ𝑢_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.

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 Δ𝑢_𝑟Δ𝑢_𝑟²=𝜉+𝜓𝜒𝜂²−1−𝜁²−11+2𝜓²1+𝜒², (1) where Δ𝑢_𝑟=(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 𝑅_𝜆=𝑢𝜆_𝑇/𝜈 and of the separation distance 𝑟≡|𝐫|, according to 𝜓𝑟,𝑅_𝜆=𝑅_𝜆15√𝜓(𝑟),15 (2) where 𝑢=⟨𝑢²_𝑟⟩ is 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 Δ𝑢_𝑟 [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 𝜆_𝑇/ℓ≈𝑅_𝜆, 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 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 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 Δ𝑢_𝑟.

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 𝜕𝑢_𝑘𝜕𝑡=−𝜕𝑢_𝑘𝜕𝑥_ℎ𝑢_ℎ+1𝜌𝜕𝑇_𝑘ℎ𝜕𝑥_ℎ, (3) where 𝐮≡(𝑢₁,𝑢₂,𝑢₃), 𝑇_𝑘ℎ, 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 vector 𝑑𝐱 which obeys 𝑑𝐱̇=∇𝐮𝑑𝐱, (4) 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𝜕𝐱𝜕𝐱₀≈𝑒^{Λ(𝑡−𝑡0)}. (5)

The map  𝝌: 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] 𝜕𝑢_𝑘=−𝜕𝑡𝜕𝑢_𝑘𝜕𝑥_0𝑝𝑢_ℎ+1𝜌𝜕𝑇_𝑘ℎ𝜕𝑥_0𝑝𝜕𝑥_0𝑝𝜕𝑥_ℎ. (6)

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 −𝜕𝑢_𝑘/𝜕𝑥_0𝑝𝑢_ℎ+1/𝜌𝜕𝑇_𝑘ℎ/𝜕𝑥_0𝑝, the velocity fluctuation can be obtained by integrating (6) with respect to the time, where −𝜕𝑢_𝑘/𝜕𝑥_0𝑝𝑢_ℎ+1/𝜌𝜕𝑇_𝑘ℎ/𝜕𝑥_0𝑝 is considered to be constant 𝑢_𝑘≈1Λ−𝜕𝑢_𝑘𝜕𝑥_0𝑝𝑢_ℎ+1𝜌𝜕𝑇_𝑘ℎ𝜕𝑥_{0𝑝𝑡=𝑡0}. (7)

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 of 𝐮 lead to determination of the structure function of Δ𝑢_𝑟 and its PDF.

The statistical properties of Δ𝑢_𝑟 are here investigated expressing the fluid velocity through the following canonical decomposition [4]: 𝐮=_𝑘𝐔_𝑘𝜉_𝑘, (8) where 𝐔_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 𝜉_𝑘=0,𝜉_𝑖𝜉_𝑗=𝛿_𝑖𝑗,𝜉_𝑖𝜉_𝑗𝜉_𝑘=𝜛_𝑖𝑗𝑘𝑝,𝑝⋙1,𝑝⋙𝜉⁴_𝑘, (9) where 𝜛_𝑖𝑗𝑘 = 1 for 𝑖=𝑗=𝑘, else 𝜛_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, 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 ⟨(Δ𝑢_𝑟)³⟩/⟨(Δ𝑢_𝑟)²⟩^3/2≠ 0. As a result, it is reasonable that the canonical decomposition (8) (Δ𝑢_𝑟=(𝐫/𝑟)⋅∑_𝑘(𝐔_𝑘(𝐱+𝐫)−𝐔_𝑘(𝐱))𝜉_𝑘) includes variables 𝜉_𝑘 with |⟨𝜉³_𝑘⟩|⋙1 [5]. This has very important implications for what concerns the statistics of the fluctiations of Δ𝑢_𝑟. In order to analyze this question, consider now the dimensionless velocity fluctuation 𝐮. This is obtained in terms of 𝜉_𝑘 by substituting (8) into (7) 𝑢_ℎ=_𝑖𝑗𝐴_ℎ𝑖𝑗𝜉_𝑖𝜉_𝑗+1𝑅_𝜆𝑘𝑏_ℎ𝑘𝜉_𝑘, (10) where 𝑟=𝑟𝜆̂_𝑇 and 𝑢_ℎ=𝑢_ℎ𝑢. Therefore ∑_𝑖𝑗𝐴_ℎ𝑖𝑗𝜉_𝑖𝜉_𝑗 arises from the inertia and pressure terms, whereas 1/𝑅_𝜆∑_𝑘𝑏_ℎ𝑘𝜉_𝑘 is due to the fluid viscosity. Now, thanks to the local isotropy, 𝑢_ℎ is a Gaussian stochastic variable [4, 5]; accordingly, 𝜉_𝑘 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 Δ𝑢_𝑟 is first expressed in terms of 𝜉_𝑘Δ𝑢_𝑟(𝐫)=_𝑖𝑗Δ𝐴_𝑟𝑖𝑗(𝐫)𝜉_𝑖𝜉_𝑗+1𝑅_𝜆𝑘Δ𝑏_𝑟𝑘(𝐫)𝜉_𝑘≡𝐿+𝑆+𝐺⁺+𝐺⁻. (11)

This fluctuation can be reduced to the contributions 𝐿, 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≡𝑆_𝑖𝑗𝜉_𝑖𝜉_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 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 𝐺⁻=−𝜁²−1𝜓₂(𝑟),𝐺⁺=𝜂²−1𝜓₃(𝑟). (12)

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 Δ𝑢_𝑟, 𝜓₂ and 𝜓₃ are assumed to be proportional with each other through a constant which depends only on 𝑅_𝜆𝜒𝑅_𝜆=𝜓₃(𝑟)𝜓₂(𝑟). (13)

Therefore, the longitudinal velocity difference can be written as Δ𝑢_𝑟=𝜓₁𝜉+𝜓₂𝜒𝜂²−1−𝜁²−1, (14) 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 𝑅_𝜆 and 𝑟21+𝜒²𝜓²₂𝜓²₁=𝐺⁺+𝐺⁻²𝑆_𝑖𝑗𝜉_𝑖𝜉_𝑗+1/𝑅_𝜆Δ𝑏_𝑟𝑘𝜉_𝑘². (15)

Now, the divisor at the R. H. S. of (15) is the sum of the following three terms: 𝐴=𝑆_𝑖𝑗𝑆_𝑝𝑞⟨𝜉_𝑖𝜉_𝑗𝜉_𝑝𝜉_𝑞2⟩,𝐵=𝑅_𝜆𝑆_𝑖𝑗⟨𝜉_𝑖𝜉_𝑗𝜉_𝑘⟩Δ𝑏_𝑟𝑘,1𝐶=𝑅²_𝜆Δ𝑏²_𝑟𝑘𝜉²_𝑘. (16)

Hence, taking into account the properties (9) of 𝜉_𝑘, |𝐵|⋙|𝐴|, |𝐶|; thus 𝜓 tends to a quantity arising only from the terms ⟨𝜉³_𝑘⟩ which appear in (15): 𝜓𝜓≡²𝜓₁=𝜑(𝑟)𝑅_𝜆. (17)

This expression corresponds to that obtained in [1] 𝜓𝑟,𝑅_𝜆=𝑅_𝜆15√𝜓(𝑟),𝜓(0)(1),15=𝑂 (18) and the dimensionless longitudinal velocity difference is given by (1) Δ𝑢_𝑟Δ𝑢_𝑟²=𝜉+𝜓𝜒𝜂²−1−𝜁²−11+2𝜓²1+𝜒². (19)

It is worth to remark that 𝜓 expresses the fluctuations ratio (large scale velocity)–(small scale velocity); that is, 𝜓≈𝑢/𝑢_𝑠≈(𝑢²/𝜆_𝑇)(𝑙_𝑠/𝑢²_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_𝑠𝑙_𝑠/𝜈≈1 is the corresponding Reynolds number.

The distribution function of Δ𝑢_𝑟 is then expressed through the Frobenius-Perron equation [7], taking into account that 𝜉, 𝜂, and 𝜁 are independent stochastic variables 𝐹Δ𝑢^′_𝑟=_𝜉𝜂𝜁𝑝(𝜉)𝑝𝜂𝑝(𝜁)𝛿Δ𝑢^′_𝑟−Δ𝑢_𝑟𝜉,𝜂,𝜁𝑑𝜉𝑑𝜂𝑑𝜁, (20) where Δ𝑢_𝑟(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 Δ𝑢_𝑟 are easily calculated 𝐻_𝑛≡Δ𝑢_𝑟^𝑛Δ𝑢_𝑟^2𝑛/2=11+21+𝜒²𝜓^2𝑛/2×^𝑛_𝑘=0𝑛𝑘𝜓^𝑘𝜉^𝑛−𝑘𝜒𝜂²−1−𝜁²−1^𝑘. (21)

In particular, 𝐻₃, related to the mechanism of energy cascade, is 𝐻₃(𝑟)=8𝜓³𝜒³−11+2𝜓²1+𝜒^23/2. (22)

In conclusion, 𝜒=𝜒(𝑅_𝜆) is implicitly calculated, in function of 𝜓(0), taking into account that this Lyapunov theory gives 𝐻₃(0)=−3/7 (see appendix) [1] 𝐻₃(0)=8𝜓³(0)𝜒³−11+2𝜓²(0)1+𝜒^23/23=−7, (23) where 𝜓(0)≃1.075 is estimated as in [1], and ⟨(Δ𝑢_𝑟)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 on Δ𝑢_𝑟 whose expression, here calculated with the finite-scale Lyapunov theory and (9), provides a nonsymmetric PDF, where the absolute values of the dimensionless moments |𝐻_𝑛(0)| rise with the Taylor scale Reynolds number for 𝑛>3.

Appendix

For the sake of convenience, this section reports some of the results dealing with the closure of the von Kármán-Howarth equation, obtained in [1, 2].

For fully developed isotropic homogeneous turbulence, the pair correlation function 𝑓(𝑟)=⟨𝑢_𝑟(𝐱)𝑢_𝑟(𝐱+𝐫)⟩𝑢² (A.1) satisfies the von Kármán-Howarth equation [8] 𝜕𝑓=𝐾(𝑟)𝜕𝑡𝑢²𝜕+2𝜈²𝑓𝜕𝑟²+4𝑟𝜕𝑓+𝜕𝑟10𝜈𝜆²_𝑇𝑓 (A.2) the boundary conditions of which are 𝑓(0)=1,lim_𝑟→∞𝑓(𝑟)=0, (A.3) where 𝑢≡⟨𝑢²_𝑟(𝐱)⟩ satisfies the equation of the turbulent kinetic energy [8] 𝑑𝑢²𝑑𝑡=−10𝜈𝜆²_𝑇𝑢², (A.4) and 𝜆_𝑇≡−1/𝑓^′′(0) is the Taylor scale. The function 𝐾(𝑟), giving the mechanism of energy cascade, is related to the longitudinal triple velocity correlation function 𝑘𝐾(𝑟)=𝑢³𝜕+4𝜕𝑟𝑟𝑘(𝑟),(𝑟)=𝑢where𝑘^2𝑟(𝐱)𝑢_𝑟(𝐱+𝐫)𝑢³. (A.5)

Thus, the von Kármán-Howarth equation provides the relationship between the statistical moments ⟨(Δ𝑢_𝑟)²⟩ and ⟨(Δ𝑢_𝑟)³⟩ in function of 𝑟.

The Lyapunov theory proposed in [1] gives the closure of the von Kármán-Howarth equation and expresses 𝐾(𝑟) in terms of 𝑓 and 𝜕𝑓/𝜕𝑟𝐾(𝑟)=𝑢³1−𝑓2𝜕𝑓.𝜕𝑟 (A.6)

Accordingly, the skewness of Δ𝑢_𝑟 is [9] 𝐻₃(𝑟)≡Δ𝑢_𝑟³Δ𝑢_𝑟^23/2=6𝑘(𝑟)21−𝑓(𝑟)^3/2. (A.7)

Therefore, the skewness of 𝜕𝑢_𝑟/𝜕𝑟 is 𝐻₃(0)3=−7. (A.8)

Acknowledgments

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