Analyzing the distribution of settling spherical particles without vertical periodicity by means of particle-resolved simulations

Particle-laden flows are a branch of multiphase flows in which the dispersed phase is composed of solid particles and the continuum phase of fluid [1]. The dilute regime is characterized by low particle concentration, but non-negligible particle-particle interactions. The complexity of this problem is further increased if the particles are large enough such that wake effects become important and point-particle models are not accurate anymore. In this scenario one must rely on particle-resolved direct numerical simulations.

One of the main issues when simulating particles settling under gravity is that the vertical velocity of the particles is not known a-priori. Therefore, one must find a way to overcome the fact that particles could, eventually, leave the computational domain. One possible solution is to solve the governing equations on a grid which is moving at some predetermined translational velocity. This is typically used, among others, by the group of Dušek for single particles [2, 3] and can also be implemented for many particles. However, in this work we propose a algorithm that is based on an inertial reference frame, allowing the simulation of many particles in suspension without relying on vertical periodicity.

More specifically, we consider the settling of a set of particles under the action of gravity in a Newtonian fluid with density ρ_f and kinematic viscosity ν. Each particle is a rigid sphere of uniform density ρ_p with diameter D. The algorithm uses two inertial reference frames which we refer as global and observer, respectively. 

As a proof of the success of the algorithm we show results for a case with with  particles in a computational domain of size [64.6 × 64.6 × 192] D³. The particles have the same density ratio (ρ_p/ρ_f = 1.5) and the Galileo number Ga = 178 is such that: i) a single particle follows a steady-oblique path and ii) many particles do form clusters in a triply periodic configuration. In the talk we will show how the initial condition (particles uniformly distributed with a horizontally-averaged solid volume fraction of 〈φ〉_xy = 5 · 10^-3 along a length of approximately 180D) and the final state clearly differ from each other.

1. [1]  L. Brandt and F. Coletti, Annu. Rev. Fluid Mech., 2022.
   
   [2]  M. Uhlmann and J. Dušek, Intl. J. Multiphase Flow,  2014.
   
   [3]  W. Zhou, M. Chrust, and J. Dušek, J. Fluid Mech., 2017.