Partitioning of particle velocities in gas–solid turbulent flows into a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study

Abstract

The velocity distribution of dilute suspensions of heavy particles in gas–solid turbulent flows is investigated. A statistical approach – the mesoscopic Eulerian formalism (MEF) – is developed in which an average conditioned on a realization of the turbulent carrier flow is introduced and enables a decomposition of the instantaneous particle velocity into two contributions. The first is a contribution from an underlying continuous turbulent velocity field shared by all the particles – the mesoscopic Eulerian particle velocity field (MEPVF) – that accounts for all particle–particle and fluid–particle two-point correlations. The second contribution corresponds to a distribution – the quasi-Brownian velocity distribution (QBVD) – that represents a random velocity component satisfying the molecular chaos assumption that is not spatially correlated and identified with each particle of the system. The MEF is used to investigate properties of statistically stationary particle-laden isotropic turbulence. The carrier flow is computed using direct numerical simulation (DNS) or large-eddy simulation (LES) with discrete particle tracking employed for the dispersed phase. Particle material densities are much larger than that of the fluid and the force of the fluid on the particle is assumed to reduce to the drag contribution. Computations are performed in the dilute regime for which the influences of inter-particle collisions and fluid-turbulence modulation are neglected. The simulations show that increases in particle inertia increase the contribution of the quasi-Brownian component to the particle velocity. The particle velocity field is correlated at larger length scales than the fluid, with the integral length scales of the MEPVF also increasing with particle inertia. Consistent with the previous work of Abrahamson (1975), the MEF shows that in the limiting case of large inertia, particle motion becomes stochastically equivalent to a Brownian motion with a random spatial distribution of positions and velocities. For the current system of statistically stationary isotropic turbulence, both the DNS and LES show that the fraction of the kinetic energy residing in the mesoscopic field decreases with particle inertia as the square root of the ratio of the total particulate-phase kinetic energy to that of the fluid.

Droplet growth in warm turbulent clouds

“…This means that the particle velocity contains an element that is non-local and uncorrelated with the local fluid. Based on their studies of particle motion in DNS turbulence, Février et al (2005) and Masi et al (2010) partitioned the particle motion into a spatially random uncorrelated motion (RUM) and a mesoscopic motion derived from a smoothly varying particle velocity field which is responsible for the spatially correlated part of the particle motion. By measuring the spatial velocity correlation between pairs of particles, they were able to calculate the contribution the RUM and the mesoscopic velocity fields make to the particle's turbulent kinetic energy and how this varied with St.…”

Eulerian and Lagrangian Large-Eddy Simulations of an evaporating two-phase flow

“…They are called the mesoscopic liquid properties. The monodisperse equation system used in this study is a simplification of the model by Février et al [5]. There is no account for the random uncorrelated motion [15].…”

“…Therefore, LES seems a natural candidate for the investigation of the complex physical phenomena involved in reacting two-phase flows, for which two numerical strategies can be applied: -In Euler-Lagrange (EL) simulations, the gas is modeled by a classical Eulerian approach whereas particles are tracked in a Lagrangian framework [3,4]. -Euler-Euler (EE) simulations use the Eulerian description for the gaseous and the dispersed phases [5,6]. The Euler-Lagrange approach is commonly used as individual droplet physical mechanisms can be easily implemented: polydispersion of a spray, crossing trajectories, bouncing on walls, group and wake combustion of droplets.…”

On the application of Mesoscopic Eulerian Formalism to modulation of turbulence by solid phase

“…In this regard, Fevrier et al [3] have proposed the Mesoscopic Eulerian Formalism (MEF) to have comprehensive understanding on the distribution of finite inertia particles in a turbulent flow. The particle spatial velocity correlations are assumed to be induced only via the interactions with the fluid.…”