Refinement of a Previous Hypothesis of the Lyapunov Analysis of Isotropic Turbulence
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 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 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.
In the previous work , 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 in terms of the longitudinal velocity correlation and (see also the appendix), and shows that the structure function of the longitudinal velocity difference is expressed by where , , , and are uncorrelated centered Gaussian random variables, with = 1, and is a function of the Taylor scale Reynolds number and of the separation distance , according to where is the longitudinal velocity standard deviation, is the Taylor microscale, and is a proper function of which provides nonzero skewness of [1, 2].
In , 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 , where is the Kolmogorov microscale. Therefore, the analysis of  seems to require the adoption of 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 . In order to describe the mechanism of kinetic energy cascade, the variables 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 and represents a sufficient condition to determine the statistics of .
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  and on the momentum Navier-Stokes equations where , , and 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 which obeys where varies according to the velocity gradient which in turn follows the Navier-Stokes equations. As observed in , 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
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 , the velocity fluctuation can be obtained by integrating (6) with respect to the time, where is considered to be constant
This assumption is justified by the fact that, according to the classical formulation of motion of continuum media , the terms into the circular brackets of (6) are considered to be smooth functions of —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 lead to determination of the structure function of and its PDF.
The statistical properties of are here investigated expressing the fluid velocity through the following canonical decomposition : where () are proper coordinate functions of and , and () are certain dimensionless independent stochastic variables which satisfy where = 1 for , else . It is worth to remark that the variables are adequately chosen in such a way that they can describe properly the mechanism of energy cascade. Specifically, the adoption of with 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 which is responsible for the intensive mechanism of energy cascade, and 0. As a result, it is reasonable that the canonical decomposition (8) () includes variables with . This has very important implications for what concerns the statistics of the fluctiations of . In order to analyze this question, consider now the dimensionless velocity fluctuation . This is obtained in terms of by substituting (8) into (7) where and . Therefore arises from the inertia and pressure terms, whereas is due to the fluid viscosity. Now, thanks to the local isotropy, is a Gaussian stochastic variable [4, 5]; accordingly, satisfy into (10), the Lindeberg condition, a very general, necessary, and sufficient condition for satisfying the central limit theorem . This condition does not apply to the velocity difference. In fact, as is the difference between two correlated Gaussian variables, its PDF could be a nonGaussian distribution function. To study this, the fluctuation is first expressed in terms of
This fluctuation can be reduced to the contributions , , , and , appearing into (11) : in particular, is the sum of all linear terms due to the fluid viscosity, and is the sum of all bilinear forms arising from the inertia and pressure terms, whereas and are, respectively, definite positive and negative quadratic forms of centered Gaussian variables, which are derived from the inertia and pressure terms. The quantity tends to a Gaussian random variable being the sum of statistically orthogonal terms [5, 6], while and are determined by means of the hypotheses of isotropy and of fully developed flow
Observe that, due to these hypotheses, and are uncorrelated; thus , are two independent centered Gaussian variables, with . Furthermore, as the knowledge of and is considered to be a sufficient condition for determining the statistics of , and are assumed to be proportional with each other through a constant which depends only on
Therefore, the longitudinal velocity difference can be written as where is a centered Gaussian random variable with = 1, that, thanks to the hypotheses of fully developed flow and of isotropy, is considered to be statistically independent from and .
Now, the divisor at the R. H. S. of (15) is the sum of the following three terms:
It is worth to remark that expresses the fluctuations ratio (large scale velocity)–(small scale velocity); that is, , where and are, respectively the characteristic small scale and the corresponding velocity. This implies that ; thus identifies the Kolmogorov scale, and is the corresponding Reynolds number.
The distribution function of is then expressed through the Frobenius-Perron equation , taking into account that , , and are independent stochastic variables where is determined by (19), is the Dirac delta, and is a centered Gaussian PDF with standard deviation equal to the unity. The dimensionless statistical moments of are easily calculated
In particular, , related to the mechanism of energy cascade, is
In conclusion, is implicitly calculated, in function of , taking into account that this Lyapunov theory gives (see appendix)  where is estimated as in , and and are calculated in function of and .
We conclude this paper by observing that the mechanism of energy cascade acts on whose expression, here calculated with the finite-scale Lyapunov theory and (9), provides a nonsymmetric PDF, where the absolute values of the dimensionless moments rise with the Taylor scale Reynolds number for .
For fully developed isotropic homogeneous turbulence, the pair correlation function satisfies the von Kármán-Howarth equation  the boundary conditions of which are where satisfies the equation of the turbulent kinetic energy  and is the Taylor scale. The function , giving the mechanism of energy cascade, is related to the longitudinal triple velocity correlation function
Thus, the von Kármán-Howarth equation provides the relationship between the statistical moments and in function of .
The Lyapunov theory proposed in  gives the closure of the von Kármán-Howarth equation and expresses in terms of and
Accordingly, the skewness of is 
Therefore, the skewness of is
This work was partially supported by the Italian Ministry for the Universities and Scientific and Technological Research (MIUR).
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