| Fluid dynamics involves a wide variety of stochastic phenomena, ranging from kinetic theory to turbulence. The concept of closure relations is recurrent in the statistical treatment of such phenomena. The idealized equilibrium condition is associated with the Gaussian probability density function (pdf). On the other hand, the modeling of non-equilibrium phenomena has prompted the need for alternative non-Gaussian pdf representations. Over the years, many pdf representations have been developed for that purpose.; Among them, the maximum entropy method stands out as a specially attractive alternative, for its conceptual foundations. However, the strong nonlinearity of maximum entropy pdfs poses extraordinary mathematical difficulties in solving for analytic relations between moments and pdf parameters, and this is a crucial step in deriving closure relations.; In this work, we develop a systematic study of some well-known pdf representations, to investigate the closure relations associated with them. Then, we shift our focus to maximum entropy pdfs, with the aim of deriving closure relations associated with them. In order to pursue this task, we research the properties of these pdfs. The results of this research enable us to develop effective small disturbance approximations that express pdf parameters in terms of moments. When these results are combined with exact moment equations, which were previously derived by Baganoff, they yield successful approximations to the closure relations aforementioned. |
