On the importance of the history force in dispersion of particles in von Kármán vortex street

Highlights

• Impact of History force on particles distribution is investigated numerically.

• For all cases presence of the history force suppresses the particle inertia.

• History force increases particle clustering for large particles.

• At moderate Stokes numbers history force decreases particle clustering.

• Deposition rate of particles decreases when the history force is taken into account.

Abstract

Effects of the history (Basset) force on the dispersion of solid particles over a cylinder inside a two-dimensional channel flow associated with von Kármán vortex shedding are investigated. Particles with Stokes numbers (${\mathit{St}}_k$) of 0.1, 0.5, 1.0 and 5.0 and particle-to-fluid density ratios of 1.1, 10 and 1000 are considered. Clustering (i.e. preferential concentration) of particles is studied qualitatively by visualising particle distributions, and quantitatively by analysing the preference of particles in sampling the vorticity field and by using Voronoϊ tessellation analyses. It is found that the effects of history force on the distribution of particles are noticeable only at moderate particle-to-fluid density ratios and Stokes numbers, though not very significant. Nonetheless, our results suggest that for flows with strong vortices the impact of the history force can be significant. It has also been observed for the first time that inclusion of the history force at high Stokes numbers, i.e. ${\mathit{St}}_k=5.0$, can increase particle clustering at both moderate and high particle-to-fluid density ratios. At lower Stokes numbers, however, particle clustering decreases when the history force is taken into account. In general, the history force suppresses the influence of particle inertia in all the investigated cases. Finally, the deposition rate of particles, mainly on the front wall of the cylinder, decreases when the history force is included for all cases except when ${\mathit{St}}_k=0.1$.

Introduction

The transport of particles in fluid flows occurs in many natural and biological processes, and engineering applications, such as atmospheric clouds, volcanic eruptions, pollen dispersion, cloud microphysics, rain formations, pollutant dispersions, human respiratory systems and combustion systems. One can find numerous studies dealing with the subject of the particle-laden flows, such as [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [25], [26], [27], [28], [29], [30], [31], [32]. Notable experimental methods are mainly based on particle tracking velocimetry techniques, in which particles trajectories are calculated from high speed videos recorded by several cameras (usually three or more) [5], [7], [8], [11]. Among notable numerical studies we can refer to those based on Eulerian and Lagrangian methods. In these methods fluid field is solved with Navier-Stokes equations in an Eulerian frame, whereas particles are tracked in a Lagrangian frame by solving the Particle Equation of Motion (PEM) [1], [2], [3], [4], [9]. In cases where the particle size is comparable or larger than the smallest scales in the flow (e.g. Kolmogorov scale in turbulent flows), immersed boundary methods are typically used to fully resolve particle fluid interactions [6].

A general form of PEM has been formulated by Maxey and Riley [15], which determines particle acceleration resulted from various forces based on Newton’s second law. The original Maxey and Riley equation includes the Stokes drag, pressure gradient, added mass, buoyancy and history terms. One of the most difficult terms in the PEM considering computational requirements is the history force (also known as Boussinesq–Basset force or memory effect), which is the viscous force caused by the diffusion of vorticity around particle boundary due to the relative acceleration between the fluid and the particle. Calculation of history force requires performing a time-consuming integral over the whole history of the particle. Another problem with the history force is the presence of a singularity in its formulation. These are the main reasons that the history force has not been considered in many studies.

Although in some cases the impact of the history force on the particle dispersion is not noticeable [3], several numerical, experimental and analytical investigations show that in some other cases history force has non-negligible impact on dispersion and clustering of particles [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. For example, Yannacopoulos et al. [17] reported that history force can influence the magnitude of the particle drift towards either a vortex centre or a separatix. Similar phenomenon has also been reported by Candlier et al. [22], who reported that particles tend to stay longer inside vortices when history force was taken into account. Oliveri et al. [28] observed that the history force can reduce intensity of particle clustering. Furthermore, they have also found that the highest level of clustering occurs for particles with Stokes number close to unity and particles with particle-to-fluid density ratio of 10 are the most affected ones by the history force.

Daitche et al. [29] investigated the advection of particles in a Von Kármán flow with and without considering the history term and reported a weaker tendency for clustering of particles in the presence of the history force. According to Guseva et al. [26], horizontal diffusion of particles is significantly weakened in a two-dimensional convection flow, when the history force is included. Recently, Daitche [30] studied the impact of the history force on preferential concentration of particles in an isotropic turbulence. He found that the influence of history force becomes more significant at low particle-to-fluid density ratios (0.5–10) such that particles behave more similarly to tracers in the presence of the history force.

Above-mentioned studies indicate that the history force can be an important term in the PEM, which has the potential to influence particle clustering, especially at low particle-to-fluid density ratios. However, a quantitative study on the impact of the history force on distribution and clustering of particles in a Von Kármán flow has not yet been carried out. Current study conducts a detailed investigation on the effect of history force on particle clustering qualitatively and quantitatively. Solid particles investigated in this study are much smaller than the smallest scales of the flow and, therefore, are simulated through an Eulerian-Lagrangian approach. A high-order numerical scheme developed by Daitche [25], is implemented to be able to detect the minute differences in the particle dynamics exerted by history force more accurately. A wide range of particle Stokes numbers and particle-to-fluid density ratios are investigated, and their influence on the distribution and advection of particles are investigated thoroughly. Furthermore, the history force role in the deposition of particles on the walls of the channel/cylinder is also investigated.

In the following sections, first we introduce the problem under investigation and methods used. Then results are presented in two sections, i) a short presentation of flow-related features and ii) a detailed presentation of particle-related results including distribution of particles with various Stokes numbers and density ratios, and clustering analyses of particles. Finally a summary of results is presented and implications of our findings in particle dispersion are presented and discussed.