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

This paper is about the kinetic equation for gas-particle flows, in particular its well-posedness and realizability and its relationship to the generalized Langevin model (GLM) probability density function (PDF) equation. Previous analyses, e.g. [J.-P. Minier and C. Profeta, Phys. Rev. E 92, 053020 (2015)], have concluded that this kinetic equation is ill posed, that in particular it has the properties of a backward heat equation, and as a consequence, its solution will in the course of time exhibit finite-time singularities. We show that this conclusion is fundamentally flawed because it ignores the coupling between the phase space variables in the kinetic equation and the time and particle inertia dependence of the phase space diffusion tensor. This contributes an extra $\mathit{positive}$ diffusion that always outweighs the $\mathit{negative}$ diffusion associated with the dispersion along one of the principal axes of the phase space diffusion tensor. This is confirmed by a numerical evaluation of analytic solutions of these $\mathit{positive}$ and $\mathit{negative}$ contributions to the particle diffusion coefficient along this principal axis. We also examine other erroneous claims and assumptions made in previous studies that demonstrate the apparent superiority of the GLM PDF approach over the kinetic approach. In so doing, we have drawn attention to the limitations of the GLM approach, which these studies have ignored or not properly considered, to give a more balanced appraisal of the benefits of both PDF approaches.

Figure 1

Dispersion of an instantaneous point source of particles in a simple turbulent shear flow. Comparison of the analytic solution of the kinetic equation for the particle spatial concentration and a random walk simulation based on Stokes drag with a Gaussian process for the aerodynamic driving force. Panels (a) and (b) for the concentration contours are taken from Ref. [11], where the analytic solution is also given. Panel (c) is taken from Ref. [10], where analytic solutions are also given. See also Refs. [7, 12].

Figure 2

Plots of $\frac{1}{2}\left|{\omega }_1\right|/{(\tilde{\mathbf{\Gamma }}·\tilde{\mathrm{\Theta }})}_{11}$ Eq. (19) for the ratio of $\mathit{negative}$/$\mathit{positive}$ contributions to the particle diffusion coefficient ${\tilde{D}}_1$ of the transformed variable ${\tilde{Z}}_{p1}$(with a $\mathit{negative}$ eigenvalue) in the moving frame of reference, as a function of time $t$ for a range of values of the particle response time ${\tau }_p$. Both $t$ and ${\tau }_p$ are scaled on $T_L$, the Lagrangian integral timescale of the carrier flow measured along a particle trajectory.

Figure 3

Evolution of the particle diffusion coefficient ${\tilde{D}}_1\left(t\right)$ evaluated using Eq. (19) in the moving frame of reference for a range of values of ${\tau }_p$ (the particle response time normalized by the Lagrangian integral time scale, $T_L$). Time is real time $t$ normalized on $T_L.$

Figure 4

Moments $〈{\tilde{Z}}_{pi}{\tilde{Z}}_{pj}〉$ in the moving frame of reference based on the moments $〈{\mathit{Z}}_p{\mathit{Z}}_p〉$ for ${\tau }_p/T_L$ in the fixed frame of reference as solutions of the fixed frame kinetic Eq. (3) or equivalently by evaluating $\langle {\mathit{Z}}_p{\mathit{Z}}_p\rangle$ from solutions of the particle equation of motion Eq. (1).