Long time statistics of prolate spheroids dynamics in a turbulent channel flow

Highlights

• Long time simulations of the dispersion of prolate spheroids are performed in a turbulent channel flow.

• The steady-state particle distribution depends on particle aspect ratio and relaxation time.

• The steadiness of the distribution can strongly influence the particle statistics.

• A slight effect of the averaging duration is noticed for some statistics.

Abstract

Long time simulations of the dispersion of prolate spheroids in a turbulent channel are performed to study the effect of the averaging procedure on particle statistics. The turbulent flow at $R{e}_{\tau }=180$ is directly simulated (DNS) and $300000$ ellipsoids, modelled as point-particles are tracked to produce statistics. After a long time, an initially uniform particle distribution reaches a steady state in which almost all particles are segregated in the viscous sublayer. The final state of the distribution depends on the particle shape and relaxation time. Statistics of the particle velocity (translational and angular), orientation and concentration are then computed, using different averaging durations, before and after the particle distribution reaches the equilibrium. The steadiness of the distribution may have a significant effect on the statistics, especially for those strongly tied to the particle wall-normal motion. A non negligible influence of the averaging duration is also noticed for some statistics of high inertia particles.

Introduction

Particle-laden flows are of interest for the comprehension of a variety of applications. These range from the marine dispersion of plastic microparticles (Barboza and Gimenez, 2015), cloud formation (Devenish et al., 2012) as well as industrial applications such as papermaking (Lundell et al., 2011). One of the main challenges in predicting the behavior of such flows is linked to the particle shape, whose complex dynamics results from their coupled translational and rotational motion. Elongated or fiber shaped particles can be modelled by prolate spheroids, and provide a better understanding of the two-phase flow characteristics than a spherical model, as described by Voth and Soldati (2017).

Dynamics of spheroids was theoretically studied by Jeffery (1922) and Happel and Brenner (1965) who provided formulas for the computation of the hydrodynamic actions (force and torque) exerted by the flow, applicable for small particle Reynolds numbers (creeping flow regime). Experimentally, Krushkal and Gallily (1984) showed that brownian fibers orient preferentially if the shear magnitude of the flow is greater than the particle brownian diffusivity. In a duct, Bernstein and Shapiro (1994) found that fibers orient along the mean fluid velocity if the flow is laminar, while no preferential orientation occurs if the flow is turbulent. Analoguous experiments conducted by Newsom and Bruce (1998); Parsheh et al. (2005) showed that the turbulent intensity of the flow has a randomizing effect on the particle orientation. Also, particle rotation rate depends on their orientation relative to the fluid vorticity, and angular velocity statistics are strongly affected by particle length as shown by Parsa et al. (2012); Sabban et al. (2017).

In addition to experimental and theoretical studies, direct numerical simulation (DNS) has become a convenient way to study turbulent flows and was pioneered by the work of Riley and Patterson (1974) on isotropic turbulence. The first DNS of a particle-laden channel flow coupled with the Lagrangian tracking of particles (LPT) was used to investigage the dispersion of small spherical particles by McLaughlin (1989). In this simulation, the fluid velocity and pressure fields are known from the DNS and are used to model the forces necessary to track the motion of particles approximated as material points. The first study of this kind for elongated particles is reported by Zhang et al. (2001) who modelled fibers as prolate ellipsoids and avoided Euler’s angles orientation singularity (the gimbal lock) using unit quaternions to track particle orientation. Numerical studies of Mortensen et al. (2008); Marchioli et al. (2010) compared orientation and velocity statistics of prolate spheroids of different aspect ratios and relaxation times. They concluded that particle preferential orientation was especially strong in the viscous sublayer, where the mean shear is strong compared to the turbulent fluctuations. Also, preferential orientation was decreased with increasing particle inertia. More recently Challabotla et al. (2016a) reported that gravity is responsible for increased fluxes of oblate and prolate particle towards walls in upward flow while the contrary occured in downward flow. Also, Arcen et al. (2017) showed that gravity increases the preferential orientation of inertial particles in the channel core. Recently, Voronoi diagrams were used by Yuan et al. (2018) to study the clustering of prolate spheroids, and the transition between the ordered particle orientation near the wall and the nearly isotropic orientation in the bulk flow were described by Zhao et al. (2019) using Laplace triangle.

A difficulty encountered in the study of particle-laden flows is that particles drift toward the walls at a slow pace. The statistics presented are thus generally computed while the particle spatial distribution is not steady. A direct consequence is that these statistics can depend on the time window chosen to collect data and their uniqueness is not guaranteed. The long-time evolution of the distribution of spherical particles in a turbulent channel was studied by Sardina et al. (2012); Bernardini (2014), but no such study exists for prolate spheroids at the time being. The issue with this void is that the accuracy of the statistics given in the aforementioned studies (e.g. Zhang, Ahmadi, Fan, McLaughlin, 2001, Mortensen, Andersson, Gillissen, Boersma, 2008, Marchioli, Fantoni, Soldati, 2010, Zhao, Marchioli, Andersson, 2014, Arcen, Ouchene, Khalij, Tanière, 2017) cannot be assessed, thus restraining their employability to develop macroscopic model for instance. We choose to address this problem by performing long-time numerical simulations of a spheroid-laden channel flow at $R{e}_{\tau }=180$. In the same manner as Sardina et al. (2012) did for spheres, the temporal evolution of the particle distribution is first investigated to obtain the time a which the steady state is reached. Following this investigation, particle statistics are computed in and out of the steady state, using time intervals of different length to examine the effect of these two conditions.