Guidelines for the formulation of Lagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent flows

Abstract

In this paper, we establish a set of criteria which are applied to discuss various formulations under which Lagrangian stochastic models can be found. These models are used for the simulation of fluid particles in single-phase turbulence as well as for the fluid seen by discrete particles in dispersed turbulent two-phase flows. The purpose of the present work is to provide guidelines, useful for experts and non-experts alike, which are shown to be helpful to clarify issues related to the form of Lagrangian stochastic models. A central issue is to put forward reliable requirements which must be met by Lagrangian stochastic models and a new element brought by the present analysis is to address the single-and two-phase flow situations from a unified point of view. For that purpose, we consider first the single-phase flow case and check whether models are fully consistent with the structure of the Reynolds-stress models. In the two-phase flow situation, coming up with clear-cut criteria is more difficult and the present choice is to require that the single-phase situation be well-retrieved in the fluidlimit case, elementary predictive abilities be respected and that some simple statistical features of homogeneous fluid turbulence be correctly reproduced. This analysis does not address the question of the relative predictive capacities of different models but concentrates on their formulation since advantages and disadvantages of different formulations are not always clear. Indeed, hidden in the changes from one structure to another are some possible pitfalls which can lead to flaws in the construction of practical models and to physically unsound numerical calculations. A first interest of the present approach is illustrated by considering some models proposed in the literature and by showing that these criteria help to assess whether these Lagrangian stochastic models can be regarded as acceptable descriptions. A second interest is to indicate how future developments can be safely built, which is also relevant for stochastic subgrid models for particle-laden flows in the context of Large Eddy Simulations. C 2014 AIP Publishing LLC. [http://dx.

Lagrangian filtered density function for LES-based stochastic modelling of turbulent particle-laden flows

“…These particles behave like fluid tracers and are characterized by τ p → 0: The particle model must be consistent with a correct model in this limit [51]. When τ p → 0, our model reduces to:…”

Large-eddy simulation of turbulent dispersed flows: a review of modelling approaches

“…From an historical perspective, stochastic models were initially developed for free-shear flows in the context of environmental fluid mechanics [98], and closures were typically formulated for the fluid velocity seen by the particles along their trajectory (referred to as velocity seen hereinafter). There is a wide literature on the subject for RANS, and we refer to [24,85,86,91] for a comprehensive review and critical analysis. In the following, we focus on SGS models specifically developed for LES.…”

“…Reprinted from [89,90] with permission from the authors 4.2 Spherical particles in wall-bounded turbulence Applications of particle SGS models to wall-bounded turbulence are more recent and focus mainly on channel flow configurations. Breuer and Happe [17] tested the influence of the Langevin SGS model of Pozorski and Apte [100], extended for an arbitrary direction of particle motion [85], to bubble-laden and particle-laden turbulent channel flow. In the bubble-laden case, a well-resolved LES was performed, and therefore, the SGS model was found to yield subgrid velocities of small magnitude: Only marginal changes in both the velocity statistics and the volume fraction could be observed.…”

“…These models have been widely employed to predict particle deposition and resuspension, especially in the context of Reynoldsaveraged formulations for industrial applications [83,85,102]. Over the last decade, however, growing efforts have been devoted to the extension of stochastic closures to LES with subgrid modelling [60,61,[82][83][84][85]. From an historical perspective, stochastic models were initially developed for free-shear flows in the context of environmental fluid mechanics [98], and closures were typically formulated for the fluid velocity seen by the particles along their trajectory (referred to as velocity seen hereinafter).…”

“…The formulation of such an equation is made in analogy with single-phase turbulence closures that model the impact of subgrid scales on the resolved ones by means of an additional viscosity [98]. These models have been widely employed to predict particle deposition and resuspension, especially in the context of Reynoldsaveraged formulations for industrial applications [83,85,102]. Over the last decade, however, growing efforts have been devoted to the extension of stochastic closures to LES with subgrid modelling [60,61,[82][83][84][85].…”