General formalism for a reduced description and modelling of momentum and energy transfer in turbulence

Abstract

Based on hierarchies of filter lengths, the large eddy decomposition and the related subgrid stresses are recognized to represent generalized central moments for the study and modelling of the different modes composing turbulence. In particular, the subgrid stresses and the subgrid dissipation are shown to be alternative observables for quantitatively assessing the scale-dependent properties of momentum flux (subgrid stresses) and the energy exchange between the large and small scales (subgrid dissipation). In this work we present a theoretical framework for the study of the subgrid stress and dissipation. Starting from an alternative decomposition of the turbulent stresses, a new formalism for their approximation and understanding is proposed which is based on a tensorial turbulent viscosity. The derived formalism highlights that every decomposition of the turbulent stresses is naturally approximated by a general form of turbulent viscosity tensor based on velocity increments which is then recognized to be a peculiar property of small-scale stresses in turbulence. The analysis in a turbulent channel shows the rich physics of the small-scale stresses which is unveiled by the tensorial formalism and usually missed in scalar approaches. To further exploit the formalism, we also show how it can be used to derive new modelling approaches. The proposed models are based on the second- and third-order inertial properties of the grid element. The basic idea is that the structure of the integration volume for filtering (either implicit or explicit) impacts the anisotropy and inhomogeneity of the filtered-out motions and, hence, this information could be leveraged to improve the prediction of the main unknown features of small-scale turbulence. The formalism provides also a rigorous definition of characteristic lengths for the turbulent stresses, which can be computed in every type of computational elements, thus overcoming the rather elusive definition of filter length commonly employed in more classical models. A preliminary analysis in a turbulent channel shows reasonable results. In order to solve numerical stability issues, a tensorial dynamic procedure for the evolution of the model constants is also developed. The generality of the procedure is such that it can be employed also in more conventional closures.

Structure function tensor equations in inhomogeneous turbulence

“…The AGKE are a tool with several potential applications. Thanks to the relationship between δu i δu j and the unresolved stresses (Cimarelli, Abbà & Germano 2019), the AGKE can be useful to develop large-eddy turbulence models. Indeed, Cimarelli & De Angelis (2014) already used the GKE a posteriori to improve modelling, and the AGKE could further this approach, by fully accounting for anisotropy, an essential property of wall-bounded turbulent flows.…”

Spatially evolving cascades in temporal planar jets

“…This reduced theory has been used in Cimarelli & De Angelis (2012 to develop new physically based modelling approaches for large-eddy simulation. Indeed, as shown in Germano (2007a,b), the second-order moment of the two-point velocity increment is strictly related to turbulent stresses at different scales and can be directly used to derive anisotropic and nonlinear turbulence closures as highlighted in Cimarelli, Abbà & Germano (2019).…”

Dynamical p-adaptivity for LES of compressible flows in a high order DG framework

“…where • represents the expected value operator. The structure function has been widely used to study turbulence statistics and it is known to be directly related to subgrid stresses [6,7,14]. Larger values of the structure function calculated inside the element denote a poorly correlated velocity field, and the need of a higher resolution, while lower values denote a highly correlated velocity field, which is an indication of a very well resolved turbulent region or laminar conditions, hence the possibility of employing a lower resolution.…”