Kinetic and dynamic probability-density-function descriptions of disperse turbulent two-phase flows

Abstract

This article analyzes the status of two classical one-particle probability density function (PDF) descriptions of the dynamics of discrete particles dispersed in turbulent flows. The first PDF formulation considers only the process made up by particle position and velocity Z(p)=(x(p),U(p)) and is represented by its PDF p(t; y(p),V(p)) which is the solution of a kinetic PDF equation obtained through a flux closure based on the Furutsu-Novikov theorem. The second PDF formulation includes fluid variables into the particle state vector, for example, the fluid velocity seen by particles Z(p)=(x(p),U(p),U(s)), and, consequently, handles an extended PDF p(t; y(p),V(p),V(s)) which is the solution of a dynamic PDF equation. For high-Reynolds-number fluid flows, a typical formulation of the latter category relies on a Langevin model for the trajectories of the fluid seen or, conversely, on a Fokker-Planck equation for the extended PDF. In the present work, a new derivation of the kinetic PDF equation is worked out and new physical expressions of the dispersion tensors entering the kinetic PDF equation are obtained by starting from the extended PDF and integrating over the fluid seen. This demonstrates that, under the same assumption of a Gaussian colored noise and irrespective of the specific stochastic model chosen for the fluid seen, the kinetic PDF description is the marginal of a dynamic PDF one. However, a detailed analysis reveals that kinetic PDF models of particle dynamics in turbulent flows described by statistical correlations constitute incomplete stand-alone PDF descriptions and, moreover, that present kinetic-PDF equations are mathematically ill posed. This is shown to be the consequence of the non-Markovian characteristic of the stochastic process retained to describe the system and the use of an external colored noise. Furthermore, developments bring out that well-posed PDF descriptions are essentially due to a proper choice of the variables selected to describe physical systems and guidelines are formulated to emphasize the key role played by the notion of slow and fast variables.

A Lagrangian probability-density-function model for collisional turbulent fluid–particle flows

“…where A p is the acceleration due to fluid-particle interactions, g is the gravity acceleration and C is the collision operator. It is worth underlining that if the fluid field is not known, the problem is not well posed and the kinetic approach, that is only x and V p are considered as variables, is incomplete (Minier & Profeta 2015). The kinetic level of description can be considered valid in a wide range of situations.…”

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

“…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).…”

Is the kinetic equation for turbulent gas-particle flows ill posed?

“…Following the analysis of M&P, we consider the case for dispersion of an instantaneous point source in statistically stationary homogeneous and isotropic turbulence with a zero external force F ext = 0 , in which case g(t) = where R(s) is the autocorrelation 1 3 U s (0) · U s (s) of the flow velocity fluctuations U (s) measured along a particle trajectory. (2),(3)), (4) correspond to equations (65), (66), (67) in [12]. M&P claim that equation (2) is ill-posed in the sense that solutions to this can (will) exhibit unphysical behaviour except in special or, to use their phrase, 'lucky' cases.…”

“…It has been claimed in recent studies of PDF methods [11,12], that the kinetic PDF is the marginal of the GLM PDF. This claim is based on analysis that purports to show that the dispersion tensors appearing in a kinetic PDF equation derived from the GLM PDF equation are 'strictly identical' to the corresponding tensors emerging directly from the kinetic modeling approach.…”

“…As such the associated PDF equation is described by a Fokker-Planck equation. This GLM PDF equation has sometimes been inappropriately referred to as the dynamic PDF equation [12] implying that it is a more general PDF approach from which the kinetic equation can in general be derived. However it is important to appreciate the kinetic equation is not a standard Fokker-Planck equation, since it captures the non-Markovian features of the underlying flow velocities.…”