The present work studies the isotropic and homogeneous turbulence for incompressible fluids through a specific Lyapunov analysis. The analysis consists in the calculation of the velocity fluctuation through the Lyapunov theory applied to the local deformation using the Navier-Stokes equations, and in the study of the mechanism of energy cascade through the finite scale Lyapunov analysis of the relative motion between two particles. The analysis provides an explanation for the mechanism of energy cascade, leads to the closure of the von Kármán-Howarth equation, and describes the statistics of the velocity difference. Several tests and numerical results are presented.

A new approach to obtain the closure formulas for the von Kármán-Howarth and Corrsin equations is presented, which is based on the Lagrangian representation of the fluid motion, and on the Liouville theorem associated to the kinematics of a pair of fluid particles. This kinematics is characterized by the finite-scale separation vector which is assumed to be statistically independent from the velocity field. Such assumption is justified by the hypothesis of fully developed turbulence and by the property that this vector varies much more rapidly than the velocity field. This formulation leads to the closure formulas of von Kármán-Howarth and Corrsin equations in terms of longitudinal velocity and temperature correlations following a demonstration completely different with respect to the previous works. Some of the properties and the limitations of the closed equations are discussed. In particular, we show that the times of evolution of the developed kinetic energy and temperature spectra are finite quantities which depend on the initial conditions.

This study analyzes the temperature fluctuations in incompressible homogeneous isotropic turbulence through the finite scale Lyapunov analysis of the relative motion between two fluid particles. The analysis provides an explanation of the mechanism of the thermal energy cascade, leads to the closure of the Corrsin equation, and describes the statistics of the longitudinal temperature derivative through the Lyapunov theory of the local deformation and the thermal energy equation. The results here obtained show that, in the case of self-similarity, the temperature spectrum exhibits the scaling laws κ n , with n ≈ −5/3, −1 and −17/3 ÷ −11/3 depending upon the flow regime. These results are in agreement with the theoretical arguments of Obukhov-Corrsin and Batchelor and with the numerical simulations and experiments known from the literature. The PDF of the longitudinal temperature derivative is found to be a non-gaussian distribution function with null skewness, whose intermittency rises with the Taylor scale Péclet number. This study applies also to any passive scalar which exhibits diffusivity.

This note studies the mechanism of turbulent energy cascade through an opportune bifurcations analysis of the Navier–Stokes equations, and furnishes explanations on the more significant characteristics of the turbulence. A statistical bifurcations property of the Navier–Stokes equations in fully developed turbulence is proposed, and a spatial representation of the bifurcations is presented, which is based on a proper definition of the fixed points of the velocity field. The analysis first shows that the local deformation can be much more rapid than the fluid state variables, then explains the mechanism of energy cascade through the aforementioned property of the bifurcations, and gives reasonable argumentation of the fact that the bifurcations cascade can be expressed in terms of length scales. Furthermore, the study analyzes the characteristic length scales at the transition through global properties of the bifurcations, and estimates the order of magnitude of the critical Taylor-scale Reynolds number and the number of bifurcations at the onset of turbulence

In this work, we calculate the self-similar longitudinal velocity correlation function and the statistics of the velocity difference, using the results of the Lyapunov analysis of the fully developed isotropic homogeneous turbulence just presented by the author in a previous work (de Divitiis, Theor Comput Fluid Dyn, doi:10.1007/s00162-010-0211-9). There, a closure of the von Kármán-Howarth equation is proposed and the statistics of velocity difference is determined through a specific statistical analysis of the Fourier-transformed Navier-Stokes equations. The longitudinal correlation functions correspond to steady-state solutions of the von Kármán-Howarth equation under the self-similarity hypothesis introduced by von Kármán. These solutions and the corresponding statistics of the velocity difference are numerically determined for different Taylorscale Reynolds numbers. The obtained results adequately describe the several properties of the fully developed isotropic turbulence.