Clustering of aerosol particles in isotropic turbulence

Abstract

It has been recognized that particle inertia throws dense particles out of regions of high vorticity and leads to an accumulation of particles in the straining-flow regions of a turbulent flow field. However, recent direct numerical simulations (DNS) indicate that the tendency to cluster is evident even at particle separations smaller than the size of the smallest eddy. Indeed, the particle radial distribution function (RDF), an important measure of clustering, increases as an inverse power of the interparticle separation for separations much smaller than the Kolmogorov length scale. Motivated by this observation, we have developed an analytical theory to predict the RDF in a turbulent flow for particles with a small, but non-zero Stokes number. Here, the Stokes number ($\hbox{\it St}$) is the ratio of the particle's viscous relaxation time to the Kolmogorov time. The theory approximates the turbulent flow in a reference frame following an aerosol particle as a local linear flow field with a velocity gradient tensor and acceleration that vary stochastically in time. In monodisperse suspensions, the power-law dependence of the pair probability is seen to arise from a balance of an inward drift caused by the particles' inertia that scales linearly with the particle separation distance and a pairwise diffusion owing to the random nature of the flow with a diffusivity that scales quadratically with the particle separation distance. The combined effect leads to a power law behaviour for the RDF with an exponent, $c_1$, that is proportional to $\hbox{\it St}^2$. Predictions of the analytical theory are compared with two types of numerical simulation: (i) particle pairs interacting in a local linear flow whose velocity varies according to a stochastic velocity gradient model; (ii) particles interacting in a flow field obtained from DNS of isotropic turbulence. The agreement with both types of simulation is very good. The theory also predicts the RDF for unlike particle pairs (particle pairs with different Stokes numbers). In this case, a second diffusion process occurs owing to the difference in the response of the pair to local fluid accelerations. The acceleration diffusivity is independent of the pair separation distance; thus, the RDF of particles with even slightly different viscous relaxation times undergoes a transition from the power law behaviour at large separations to a constant value at sufficiently small separations. The radial separation corresponding to the transition between these two behaviours is predicted to be proportional to the difference between the Stokes numbers of the two particles. Once again, the agreement between the theory and simulations is found to be very good. Clustering of particles enhances their rate of coagulation or coalescence. The theory and linear flow simulations are used to obtain predictions for the rate of coagulation of particles in the absence of hydrodynamic and colloidal particle interactions.

Droplet growth in warm turbulent clouds

“…Numerous simulations of particle motion for small St in isotropic turbulence have led to confirmation of this longterm equilibrium form (e.g. Chun et al, 2005;Kerstein and Krueger, 2006). * For details of how this equilibrium is approached and the need for an ever-increasing grid refinement to observe the RDF accurately as r → 0 (since more particles are required to preserve statistical accuracy) the reader is referred to IJzermans et al (2009).…”

“…However, several studies (e.g. Sundaram and Collins, 1997;Wang et al, 1998b;Chun et al, 2005;Ammar and Reeks, 2009) have shown that this is not the case. The persistence or finite lifetime of turbulent structures can make droplet collision rates very much ratelimited by diffusion in the vicinity of the droplets, as is the case of particle transport in a turbulent boundary layer.…”

“…This multi-scale clustering has been accounted for in terms of the sweep-stick mechanism, which explains why droplet clustering mimics the multi-scale clustering of vanishing fluid acceleration points in a turbulent flow (Coleman and Vassilicos, 2009). Analytical expressions for the RDF have been derived by Chun et al (2005) for St 1 and Zaichik and Alipchenkov * Note that in both these simulations the simulated particle velocity field is assumed to be zero. This neglects the component due to random uncorrelated motion (see section 4.2.5), which manifests itself as a mean drift velocity proportional to the gradient of the particle kinetic stresses (referred to as turbophoresis).…”

A method for the computation of turbulent polymeric liquids including hydrodynamic interactions and chain entanglements

“…Notably, even in dilute systems, such turbulence-polymer couplings would most probably be entwined with both chain entanglement and hydrodynamic-interactions effects between chains in locally polymer-dense areas, as is the case, for example, in between turbulent coherent vortices where polymers might be expected to concentrate [14,24]. Similar ideas are also valid for turbulent flows in colloidal dispersions and aerosols [25], that feature particle aggregation/clustering phenomena [26,27] which require the capturing of hydrodynamic interactions between particles at high concentration areas, as is the case, for example, during rain initiation processes [28][29][30].…”

“…Notably, even in dilute systems, such turbulence-polymer couplings would most probably be entwined with both chain entanglement and hydrodynamic-interactions effects between chains in locally polymer-dense areas, as is the case, for example, in between turbulent coherent vortices where polymers might be expected to concentrate [14,24]. Similar ideas are also valid for turbulent flows in colloidal dispersions and aerosols [25], that feature particle aggregation/clustering phenomena [26,27] which require the capturing of hydrodynamic interactions between particles at high concentration areas, as is the case, for example, during rain initiation processes [28][29][30].Therefore, there is a need for mesoscopic, physical formulations/numerical methods that allow the direct computation of turbulent polymeric (and colloidal) fluids. Such methods have to overcome a number of challenges: (a) the forcing of the fluid by the particles is delta-function type (i.e., pointwise), hence standard methods for computational fluid dynamics need an extremely fine grid in order to capture the microscopic flow field in between the particles that corresponds to their hydrodynamic interactions, (b) efforts to average the forced Navier-Stokes equations in order to overcome this problem, lead, due to nonlinearity, to the appearance of subgrid scale stresses that need to be taken into account via some type of modeling, (c) the accompanying numerical method needs to handle the Brownian, i.e., stochastic motion of polymer and colloidal particles; this adds an additional level of complexity to standard computational methods for suspensions [31], (d) in polymeric liquids, the formulation and numerics need to describe the formation and dynamics of entanglements between macromolecular chains, in order for the approach to be applicable to arbitrary polymer volume fractions.…”