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.

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 Taylor-scale Reynolds numbers. The obtained results adequately describe the several properties of the fully developed isotropic turbulence.

The present work deals wit h the dynamics of vehicles that intentionally operate in the ground proximity. The dynamics in ground effect is influenced by the vehicle orientation with respect to the ground because the aerodynamic force and moment coefficients, which in turn depend on height and angle of attack, also vary with the Euler angles. This feature, usually neglected in the applications, can be responsible for sizable variations of the aircraft performance and stability. A further effect, caused by the sink rate, determines the unsteadiness that modifies the aerodynamic coefficients. In this work, an analytical formulation is proposed for the force and moment calculation in the presence of the ground and taking the aircraft attitude and sink rate into account. The aerodynamic coefficients are first calculated for a representative vehicle, and its characteristics in ground effect are investigated. Performance and stability characteristics are then discussed with reference to significant equilibrium conditions, while the nonlinear dynamics is studied through the numerical integration of the equations of motion