A dynamic subgrid scale model for Large Eddy Simulations based on the Mori–Zwanzig formalism

Abstract

The development of reduced models for complex multiscale problems remains one of the principal challenges in computational physics. The optimal prediction framework of Chorin et al. [1], which is a reformulation of the Mori-Zwanzig (M-Z) formalism of non-equilibrium statistical mechanics, provides a methodology for the development of mathematically-derived reduced models of dynamical systems. Several promising models have emerged from the optimal prediction community and have found application in molecular dynamics and turbulent flows. In this work, a new M-Z-based closure model that addresses some of the deficiencies of existing methods is developed. The model is constructed by exploiting similarities between two levels of coarse-graining via the Germano identity of fluid mechanics and by assuming that memory effects have a finite temporal support. The appeal of the proposed model, which will be referred to as the 'dynamic-MZ-τ' model, is that it is parameter-free and has a structural form imposed by the mathematics of the coarse-graining process (rather than the phenomenological assumptions made by the modeler, such as in classical subgrid scale models). To promote the applicability of M-Z models in general, two procedures are presented to compute the resulting model form, helping to bypass the tedious error-prone algebra that has proven to be a hindrance to the construction of M-Z-based models for complex dynamical systems. While the new formulation is applicable to the solution of general partial differential equations, demonstrations are presented in the context of Large Eddy Simulation closures for Burgers equation, decaying homogeneous turbulence, and turbulent channel flow. The performance of the model and validity of the underlying assumptions are investigated in detail.

The Adjoint Petrov–Galerkin method for non-linear model reduction

“…To gain a reduction in computational cost, an approximation to the memory must be devised. A variety of such approximations exist, and here we outline the τ-model [52,64]. The τ-model can be interpreted as the result of assuming that the memory is driven to zero in finite time and approximating the integral with a quadrature rule.…”

“…Here, τ ∈ R is a stabilization parameter that is sometimes referred to as the "memory length." It is typically static and user-defined, though methods of dynamically calculating it have been developed in [52]. The a priori selection of τ and sensitivity of the model output to this selection are discussed later in this manuscript.…”

“…Most notably, Stinis and co-workers [44,45,46,47,48,49] have developed several models for approximating the memory, including finite memory and renormalized models, and examined their application to the semi-discrete systems emerging from Fourier-Galerkin and Polynomial Chaos Expansions of Burgers' equation and the Euler equations. Application of MZ-based techniques to the classic POD-ROM approach has not been undertaken.This manuscript leverages work that the authors have performed on the use of the MZ formalism to develop closure models of partial differential equations [50,51,52,53,54]. In addition to focusing on the development and analysis of MZ models, the authors have examined the formulation of the MZ formalism within the context of the VMS method [54].…”

“…This manuscript leverages work that the authors have performed on the use of the MZ formalism to develop closure models of partial differential equations [50,51,52,53,54]. In addition to focusing on the development and analysis of MZ models, the authors have examined the formulation of the MZ formalism within the context of the VMS method [54].…”

Generalized Langevin Equations for Systems with Local Interactions

“…The first effective method developed within this class is the continued fraction expansion of Mori [43], which can be conveniently formulated in terms of recurrence relations [35,20]. Other methods based on first-principles include perturbation methods [63,60], mode coupling techniques, [51,22], optimal prediction methods [11,14,53,8], and various series expansion [55,47,46,67]. First-principle calculation methods can effectively capture non-Markovian memory effects, e.g., in coarse-grained particle simulations [66,24].…”