Model of fourth-order cumulants for description of turbulent transport by large-scale vortex structures

Abstract

Introduction. New directions of turbulent-flow investigations based on constructing closed equations of turbulent transport and their numerical implementation on computers have been intensely developed lately. Tile approach based on the use of a two-parametric model of turbulence gained wide application, as well as second-order closure models, which are effective from the computational viewpoint and yield results whose accuracy is sufficient for many practical applications, ttowever, the use of these models for description of turbulent transport in stratified flows gives a qualitatively incorrect result in some cases (see, for example, [1]). The anisotropic character of the buoyancy effect on the structure of turbulence is manifested in the long-wave range of the spectrum of turbulent oscillations [2]. This spectral range corresponds to large-scale vortex structures (LVS) containing the main portion of turbulence energy. According to experimental and theoretical studies, the following LVS are formed in stratified flows: turbulent spots in the case of stable stratification and coherent structures in the case of unstable stratification, which are mainly responsible for turbulent transport. The effects of intermittency and asymmetry of vertical turbulent transport caused by the influence of LVS make the probability distributions of turbulent fluctuations significantly non-Gaussian. The turbulence structure in these flows is usually described by third-order closure models, where the triple correlations (asymmetry) are calculated from differential transport equations (sec, for example, [3, 4]). A refinement of such models is proposed in the present paper.Applicability of Millionshchikov's Quasinormality Hypothesis. Millionshchikov's quasinormality hypothesis for calculation of the fourth moments of statistical characteristics of turbulent flows .(velocity fluctuations and mixed fourth-order covariations of velocity, temperature, and concentration fluctuations) is often used for closure in constructing semi-empirical models of turbulent transport of the second and third order. According to this hypothesis, fourth-order cumulants can be ignored in comparison with the corresponding correlation functions. As applied to the moments of hydrodynamic fields, this hypothesis means that we can use the equality that expresses the fourth moments through the second moments

The Use of Differential Constraints for Analyzing Turbulence Models

“…r It follows from the formula for C [4] that the contribution of the second term in the algebraic model for the cumulant C is essential. Thus the governing equations are:…”

“…Different closing approximations give different models. This general approach has been employed in [4] to obtain a closed model of turbulent transport that does not imply equality to zero of the fourth-order cumulants. The closure procedure was performed at the level of the fifth moments, i.e.…”

“…As a consequence, in the former case the triple-correlation equations are of the first-order without a dumping mechanism for triple correlations that leads (in some cases) to physically contradictory results [5] (e. g. the appearance of negative portions of the spectrum of the turbulent kinetic energy, [6], [7]). The approach proposed in [3], [4] allows us to overcome this obstacle; the technique also includes a physically reasonable way for constructing approximate algebraic parametrizations of higher moments.…”

Invariant Manifolds in Parametric turbulent Models

“…it is assumed that the fifth-order cumulants are equal to zero. Therefore (see, [1], [24]), the following third-order mathematical model can be used Model 2:…”

“…It is well-known that the contribution of the second term in the algebraic model for the cumulant C is essential [24].…”