Large eddy simulation of flow in porous media: Analysis of the commutation error of the double-averaged equations

Abstract

The continuum approach employing porous media models is a robust and efficient solution method in the area of the simulation of fixed-bed reactors. This paper applies the double-averaging methodology to refine the continuum approach, opening a way to alleviate its main limitations: space-invariant averaging volume and inaccurate treatment of the porous/fluid interface. The averaging operator is recast as a general space–time filter allowing for the analysis of commutation errors in a classic large eddy simulation (LES) formalism. An explicit filtering framework has been implemented to carry out an a posteriori evaluation of the unclosed terms appearing in the double-averaged Navier–Stokes (DANS) equations, also considering a space-varying filter width. Two resolved simulations have been performed. First, the flow around a single, stationary particle has been used to validate derived equations and the filtering procedure. Second, an LES of the turbulent flow in a channel partly occupied with a porous medium has been realized and filtered. The commutation error at the porous–fluid interface has been evaluated and compared to the prediction of two models. The significance of the commutation error terms is also discussed and assessed. Finally, the solver for DANS equations has been developed and used to simulate both of the studied geometries. The magnitude of the error associated with neglecting the commutation errors has been investigated, and an LES simulation combined with a porous drag model was performed. Very encouraging results have been obtained indicating that the inaccuracy of the drag closure overshadows the error related to the commutation of operators.

Large‐eddy simulation of a channel flow over an irregular porous matrix

“…Additionally, up-scaled fluid properties like the averaged velocity or pressure are governed by a set of averaged Navie-Stokes equations that include the influence of the solid phase, for example, by introducing the drag force induced by the solid phase on the fluid. Further, when an unsteady flow is considered, or when the solid phase changes topology over time, the volume average can be combined with a time average, resulting in the double-averaged Navier-Stokes (DANS) equations [3,4].…”

“…The process of volume averaging is de facto equivalent to a typical filtering procedure encountered in LES turbulence modelling [4]. Moreover, the drag force term in the DANS equations is given by a non-local integral based on the microscopic (sub-filter) flow properties, which, from the practical, simulation-based point of view, creates a closure problem similar to the one present in the theory of LES.…”

“…The closure models for drag and mechanical dispersion induced by the presence of the solid phase have been historically developed based on a priori analytical considerations yielding drag laws fitted to experimental data and semi-empirical correlations [5][6][7]. The increased availability of computational power allowed for studies of multi-phase flows with greater detail, employing particle-resolved simulations (PRS) combined with accurate turbulence modelling [4,8,9]. In these studies, regular and periodically repeating geometries allow for accessible, efficient and accurate simulations.…”

“…Another example of this approach is in Breugem and Boersma [9], where the authors used a regular three-dimensional array of cubes to study the flow over a permeable and porous wall. In Sadowski et al [4], the same configuration was used to investigate the effects of commutation errors arising from combining LES with the DANS system, and validate the LES approach for such a configuration.…”

“…For that, we simulate a turbulent flow in a channel half-filled with a porous medium, represented by an array of cubes, where an irregularity is introduced by shifting the position of each cube in a random direction. The quality and resolution provided by LES are assessed for each simulation, and the results are compared to the regular array results from Sadowski et al [4]. Lastly, we test the influence of the geometry on macroscopic turbulence and provide recommendations for improving relevant closure models.…”