Quantification of preferential concentration of colliding particles in a homogeneous isotropic turbulent flow
Introduction
Suspensions of dust, impurities, droplets, bubbles, and other finite-size particles advected by turbulent flows are commonly encountered in many natural phenomena and industrial processes. Industrial applications encompass, for instance, optimization of spray combustion in diesel engines (Ning et al., 2009) and in rocket propellers (Gosman and Ioannides, 1983), separation of solid and droplets in cyclones (Sommerfeld and Laín, 2009) or pneumatic transport of coal dust (Laín and Sommerfeld, 2008, Laín and Sommerfeld, 2012, Laín and Sommerfeld, 2013). Examples of natural and environmental processes include spreading of atmospheric aerosols (Ayala et al., 2008a, Ayala et al., 2008b), formation of raindrops (Falkovich et al., 2002), evolution of cumulus clouds (Shaw et al., 1998) or dynamics of sand storms (Westphal et al., 1987).
The behaviour of inertial particles is much more complex than that of fluid particles. This is due to their finite size and to their mass density being different from that of the carrier fluid. As a consequence of their inertia, the dynamics of such particles is dissipative leading to inhomogeneities in their spatial distribution (Bec et al., 2005). This phenomenon, referred usually as preferential concentration, has been observed in experiments (Eaton and Fessler, 1994) and is a consequence of the interaction of particles with the eddies of the carrier flow. As a result, heavy particles tend to accumulate in regions of low vorticity and high strain rate (Wang and Maxey, 1993) leading, under certain conditions, to the formation of clusters, understood here as regions with significantly higher concentration of particles surrounded by areas of low concentration (Squires and Eaton, 1991). Clusters may be formed in both, homogeneous and inhomogeneous turbulent flows. The phenomenon of clustering of heavy particles in inhomogeneous turbulent flows is explained by the effect of turbulent migration (turbophoresis) from regions of high intensity of turbulent velocity fluctuations to regions of low turbulence (Reeks, 1983). Clustering of inertial particles also takes place in homogeneous turbulence, where the gradients of the mean velocity fluctuations of the carrier flow are zero and, consequently, particle transport via turbophoresis does not take place in the conventional sense. However, despite the stochastic character of turbulence, the distribution of heavy inertial particles in turbulent flows is not random, and their interaction with the coherent vortex structures of the turbulent flow may give rise to significant clustering.
An alternative explanation of particle clustering of point-particles in homogeneous and isotropic turbulence was introduced by Chen et al. (2006) and Goto and Vassilicos (2006) in two dimensions and later by Goto and Vassilicos (2008) and Coleman and Vassilicos (2009) in the three-dimensional setting. These authors show visualizations that demonstrate a very strong correlation (in a wide range of scales) between the accumulation of heavy particles and the distribution of local properties of the fluid acceleration field, the zero acceleration points. This effect has been called ``sweep-stick mechanism” and, following Coleman and Vassilicos (2009), it ``explains this coincidence in terms of the sweeping of regions of low acceleration by the local fluid velocity and the fact that particles on zero-acceleration points move together with these points (stick), whereas they move away from nonzero-acceleration points”. Behaviours compatible with this mechanism have been found in experiments as well as in numerical computations of finite size particles (e.g. Monchaux et al., 2010, Uhlmann and Chouippe, 2017). Moreover, Bragg et al. (2015) found that in Navier–Stokes turbulence both, inertial particle positions and zero acceleration points, are clustered in regions of high coarse-grained strain, resulting in a correlation between the distributions of the two sets of points.
Preferential concentration of particles in turbulent flows has important practical consequences because affects the probability to find close particle pairs and thus influence their possibility to collide, or to have biological and chemical interactions. It is therefore very important in a large spectrum of applications to quantify the effects of inertia on these interactions. For instance, the problem of estimating the time scales of rain initiation in warm clouds (Shaw, 2003) can be mentioned. Other examples are the problem of microorganisms predator-prey encounters in turbulent flows (Mann et al., 2002) and the enhancement of chemical reaction rates for active particles suspended in fluid flows (Motter et al., 2003) especially in fast fluidised beds (Carlos Varas et al., 2017, Cahyadi et al., 2017). Moreover, models for planet formation involve aggregation through collisions of dust grains in the circumstellar disc (Safranov, 1969).
According to the review article of Monchaux et al. (2012), there is a tremendous need to identify, characterize and quantify preferential concentration and clustering of inertial particles. This task is usually addressed by the comparison of the actual particle number concentration field to that of uniformly distributed particles which is given by a Poisson distribution. Qualitative analysis of clustering is often given through visualizations, a technique widely used in numerical papers. The segregation parameter (Soldati and Marchioli, 2009) or clustering index (Monchaux et al., 2012) Σp compares the standard deviation for the actual particle distribution and that of a Poisson distribution; positive values of Σp indicate segregation of particles. Box counting, also called pdf particle concentration method (Cihonski et al., 2013), compare the actual number distribution of particles in the domain with that of the same number of particles randomly distributed (i.e., the Poisson distribution); then, the distance between both distributions is calculated by defining some appropriate norm to obtain a single scalar characterizing how far the particle distribution is from a uniform one. Another possibility is to determine a typical length scale for the structure of particle clusters; in this respect the correlation dimension D2 is frequently used: it is defined by the exponent of the power-law behaviour at small scales of the probability P2(r) to find two-particles at a distance less or equal to r; when particles cluster the value of D2 is lower than the dimension of the space (Bec et al., 2005). All these indicators try to describe the clustering of particles using just one value.
One of the widely used indicators for particle clustering in the literature is the radial distribution function, RDF, also called pair correlation function (Monchaux et al., 2012). It measures the number of particles that are at given distances (r < d ≤ r + dr, where dr is the bin width) away from a base particle and normalizes that number by the number of particles that would be at that distance if all particles were dispersed uniformly about the domain. The RDF itself is the average of this measure over all particles. In this measure, values marginally greater than 1 indicate the existence of a cluster at that length scale, whereas values less than 1 indicate a lack of particles at that length scale. Typically, in presence of particle clusters, the RDF grows as r tends to zero being such growth more pronounced as clustering increases. This measure has the advantage of providing scale by scale quantification of clustering (Wang et al., 2000; Février et al., 2005; De Jong et al., 2010).
Voronoï diagrams have been proposed by Monchaux et al. (2010) to identify and characterize particle clusters. Mathematically, a Voronoï diagram is a way of dividing the space in a number of non-overlapping cells associated to each particle. The Voronoï cell of a particle is defined as the volume consisting in all points closer to that particle than to any other, being inversely proportional to the local particle concentration field. The standard deviation of the actual pdf of the Voronoï volumes is compared with that of a Poisson distribution, allowing a link with the clustering level: if the standard deviation of the Voronoï volumes is exceeding that of the uniform distribution there is particle clustering. Also, Voronoï diagrams can be used efficiently to identify clusters from data, at least in two dimensions (Monchaux et al., 2012).
Another approach used to characterize the topological and morphological characteristics of particle clusters is the Minkowski functionals (Calzavarini et al., 2008), a tool of integral geometry, previously used in cosmology to study the distribution of galaxies in the Universe (Kerscher, 2000). Also, they have been used to characterize colloidal particle networks (Hütter, 2003), analysis of microstructure of materials (Reuteler, 2012), digital picture analysis (Serra, 1982) or even to distinguish mammary gland tissue with cancer from healthy one (Mattfeldt et al., 2007). The Minkowski functionals span the linear space of isotropic measures on compact convex sets in d-dimensional Euclidean space. This theorem was first proven by Hadwiger (1957). In the three-dimensional space the morphological quantities of interest are the volume, surface, mean curvature and Euler characteristic of the body composed by the disjoint union of balls of radius r centred at each particle, the so-called parallel body. As very different sets of particles can have the same correlation dimension or clustering index, the progression of the Minkowski functionals with r allows getting detailed insight into the topological and morphological structure of particle distribution in the underlying turbulent flow field. Particularly, in the context of homogeneous and isotropic turbulence laden with point particles, Calzavarini et al. (2008) demonstrated that tracers, heavy particles and bubbles cluster in a different way: tracers (fluid particles) keep uniformly distributed, particles tend to cluster in 2D-like structures whereas bubbles accumulate in 1D-like filaments. A comprehensive introduction to Minkowski functionals and their applications can be found in Mecke (2000).
As a comment, the configuration of homogeneous isotropic turbulence in a periodic box has been considered recently to study the clustering and preferential concentration of finite size particles (Jin et al., 2013, Uhlmann and Doychev, 2014, Uhlmann and Chouippe, 2017). In those contributions, however, only one clustering measure was employed, the Radial Distribution Function in the case of Jin et al. (2013) and the Voronoï diagrams in the other cases. This is a common approach in the studies that address the subject of particle clustering, where only one or two clustering measures are employed.
The purpose of the present paper is to investigate the effect of particle inertia and inter-particle collisions at increasing particle void fraction on particle preferential concentration. Different measures are applied to quantify particle clustering in both situations, colliding and non-colliding particles. To achieve this goal, the configuration of homogeneous isotropic turbulence in a periodic box has been chosen, where the fluid flow is described by means of direct numerical simulations in the frame of one-way coupling, i.e., particles do not affect the fluid dynamics. In that sense, the box turbulence is thought as a tool where particles evolve and eventually approach together building up clusters whereby inter-particle collisions physically occur. Depending on the conditions, such collisions will modify the structure of particle clusters that will be reflected in a change of the clustering measures regarding the case when inter-particle collisions are disregarded.
This paper is organised as follows: Section 2 presents a brief summary of the performed direct numerical simulations based on the lattice Boltzmann method together with the point-particle Lagrangian tracking. An overview of the particle dynamics immersed in a homogenous and isotropic turbulent flow is provided in Section 3. Section 4 presents the analysis of preferential concentration of particles in the investigated cases. Discrete elements clustering is studied employing scalar measures, such as the particle segregation parameter and correlation dimension, the radial distribution function, which is one-dimensional measure, Voronoï diagrams and morphological measures as the Minkowski functionals. Finally, Section 5 presents the conclusions and perspectives.
Section snippets
Summary of numerical approach
The main features of the employed simulation methodology have been described previously by Ernst and Sommerfeld (2012). Therefore, only a brief description of the main characteristics is given hereafter.
Overview of particle dynamics in hit
The motion of particles immersed in a turbulent flow is controlled by the local velocity fluctuations and by the ordered motion of large-scale turbulent structures. As commented in the introduction, dynamics of inertial particles is dissipative which leads to an inhomogeneous particle distribution along the flow field. Interactions of particles with the turbulent features of the flow is usually described by the Stokes numberwhich characterizes particle inertia and is equal to the ratio
Quantification and morphology of particle clusters
As is has been already stated, in homogeneous isotropic turbulence particles do not distribute homogeneously. As a result, the clustering of inertial particles mainly depends on the particle Stokes number and also on the volume fraction. Moreover, the absence and presence of particle-particle collisions has an effect on the particle motion and, consequently, on their preferential concentration.
The motion of particles with small Stokes numbers is completely correlated with the fluid motion and
Conclusions
In this paper, the effect of particle volume fraction, and therefore inter-particle collisions, on particle clustering in homogeneous isotropic turbulence has been addressed by using direct numerical simulations based on lattice Boltzmann method. To that end, different clustering measures have been computed for increasing particle void fractions. The employed tools have been visualization, the segregation parameter, correlation dimension, radial distribution function, Voronoï diagrams and
Acknowledgement
The financial support by the German Research Council (DFG) through Grants SO 204-33/1-3 (Priority program 1273 “Colloidal Process Engineering”) is gratefully acknowledged. Santiago Lain acknowledges Universidad Autónoma de Occidente for the support and funding through several research projects.
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