Session Z36: Particle-Laden Flows: Modeling, Theory, and Experimentation II

Effects of the relative humidity on the particle-laden jet with crossflow

Particle-laden airflows are attracting more attention driven by the on-going environmental issues. Our previous studies have shown that the particle-laden jet under a crossflow, being governed by the organized vortices, demonstrate different particle dispersion pattern according to the velocity ratio (R) between the jet and crossflow. Meanwhile, the relative humidity (RH) of the background airflow changes significantly depending on the time and space; however, the mechanism of the particle dynamics driven by the RH change has not yet been elucidated. In the present study, we experimentally investigate the particle concentration and dispersion by controlling the air RH within the range of 30 – 70 % in the particle-laden jet with crossflow. The R (jet/crossflow) is 1.1 – 2.85. Considering the wettability of particles as a major factor, we used hydrophilic particles of Silicon (contact angle (CA) ~ 33⁰) and hydrophobic particles of PTFE (CA ~ 100⁰). As a result, the concentration of the hydrophilic particles decays more rapidly for lower RH and low R. Conversely, the concentration decay rate of hydrophobic particles is independent of the RH.   

Second-order inertial forces and torques on a spherical particle in a viscous linear flow

Making use of matched asymptotic expansions and of a coordinate system co-moving with the background flow, we compute the force and torque acting on a small rigid sphere translating and rotating unsteadily in a general linear flow field. The loads are determined up to second order with respect to the relevant small parameter, namely the square root of the shear Reynolds number. The outer solution (which at first order is responsible for the Basset-Boussinesq history force at short time and for shear-induced forces such as the Saffman lift force at long time) is expressed via a flow-dependent tensorial kernel. The second-order inner solution brings a number of different contributions to the force and torque. Some are proportional to the relative translational or rotational acceleration of the particle and fluid, while others take the form of quadratic terms combining the relative translational/rotational velocity between the particle and fluid with the rotation/strain rate of the background flow. The complete set of second-order contributions is obtained by adding the outer and inner contributions. Classical effects such as the added-mass force or the spin-induced lift force are recovered, and new effects involving the rotation/strain rate of the background flow are revealed. The resulting force and torque equations provide the first rational extension of the classical Basset-Boussinesq-Oseen equation incorporating all first- and second-order fluid inertia effects resulting from both unsteadiness and velocity gradients of the carrying flow.

 

    

Explaining and modeling the effect of particle distribution on mean forces and torques using hierarchical machine learning

Hydrodynamic force and torque experienced by each individual particle in a distribution, even at fixed mesoscale variables such as Reynolds number and particle volume fraction, is significantly influenced by the deterministic location of nearby particles (neighbors). Particle-resolved (PR) simulations provide the most accurate computational representation of these unique particle forces and torques. However, the associated computational cost typically enables consideration of only O(10⁴) static particle systems. The present work develops robust neural models with this limited available data using a physics-based hierarchical framework and symmetry-preserving networks. Among considered mesoscale conditions the models achieve a maximum accuracy of 85% and 96% in the prediction of neighbor-induced force and torque perturbations respectively. Furthermore, the generalizability of these models is thoroughly investigated using PR data of distinct particle distributions that are not involved in the training process. Upon establishing satisfactory generalizability, these relatively inexpensive models are deployed on several very large-scale, statistically-different particle distributions. A rigorous analysis and outcomes of this investigation will be discussed in explaining the effects of clustering and anisotropy on the observed mean and higher order statistics of force and torque.   

Towards filter-dependent closure models for dilute and moderately dense particle-laden flow

The tendency of inertial particles to cluster in turbulent flows introduces challenges in developing accurate subgrid-scale models. Filtered equations of particle-laden flows depend on accurately modeling the difference between the filtered and `true' field. Efficient computations of certain two point statistics from `true' particle data may be used to develop or validate closure models across different flow regimes. In particular, any introduction of a mesh or grid results in some amount of implicit filtering at the sub-grid scale. It is common to utilize radial distribution functions (RDFs) as a characterization of clustering and it is well understood how to compute RDFs from particle positions using neighbor finding. We propose an efficient method for computing filter-dependent statistics, thus revealing the effects of filtering on two point statistics. In this talk, we will present a novel algorithm for computing two point statistics from particle fields and their filtered equivalents, provide benchmarks for an implementation of this algorithm, and show how this metric differentiates between different flow regimes (from dilute suspensions of particles in homogeneous turbulence to moderately dense cluster-induced turbulence) and varies as a function of filter width.   

The determination of the critical Reynolds number for particle-wall collisions using the lattice-Boltzmann method

Crystallisation, a key unit operation for the separation and purification of pharmaceuticals, is typically performed in mechanically agitated vessels. The particle size distribution (PSD) from these processes must be carefully controlled as crystal size affects both physical properties, e.g. dissolution rate or bioavailability, as well as aspects such as the efficiency of downstream purification. As increased particle collisions lead to decreased particle size, determining the conditions under which particle collisions occur allows for increased control of the PSD.

In this work, a lattice-Boltzmann method solver is used to model particle collisions with a rigid surface in an impinging jet flow. Through the variation of jet velocity, a critical Reynolds number above which particle-wall collisions occur is obtained. Particle shape and orientation are also varied, and their influence on the critical particle Reynolds number is determined. Results are presented for both 2D and 3D geometries, in addition to benchmark test cases that are compared to experimental values.   

Inertial particle focusing in curved ducts: Bifurcations and dynamics

Particles suspended in fluid flow through a curved duct can focus to stable equilibrium positions in the duct cross-section due to the balance of two dominant forces - inertial lift force and secondary drag force. Such particle focusing is exploited in various medical and industrial technologies aimed at separating particles by size. In this talk, I will present results of our numerical investigation of the dynamics of neutrally buoyant particles in fluid flow through curved ducts with rectangular cross-sections. I will show that rich bifurcations take place in the particle equilibria as a function of the duct bend radius. I will also offer insights how these bifurcations in combination with particle dynamics can be exploited to separate particles of different sizes.

    

Explicit Runge-Kutta generalized Exponential Time-Differencing method to solve full Maxey-Riley equation with the history force

Fractional differential equations (FDE) are numerically expensive to solve because fractional derivatives are nonlocal quantites and the associated linear integrator (the function that solves the linear part of the FDE exactly) can lack semigroup property leading to increasing memory storage cost. The Maxey-Riley (MR) equation that models the advection of a particle in viscous flow contains the Basset-Boussinesq history force which is a half-derivative, and thus inherits all the numerical challenges of solving an FDE. We have developed an explicit time-integrator, inspired by generalizing exponential time differencing method, to solve the MR equation (and a more general subclass of FDEs) that incurs a fixed-in-time computational and memory cost with tunable accuracy. We show that the lack of semigroup property of the linear integrator can be dealt with by suitably embedding the MR equation into a larger Markovian system. The resulting extended system can subsequently be evolved locally by suitably adapting the standard ideas from explicit time-integrator methods. We demonstrate the computational performance and accuracy of our method for the exemplar case of a particle advected by the MR equation in Couette flow.   

Influence of particle forces and sharpness factor on erosion characteristics for dilute gas-solid flow system at moderate to high Reynolds number

Accurate estimation of particles impact velocities, impact angles and distribution in a given fluid flow are fundamental in characterising solid particle erosion. The hydrodynamic of forces acting on the particles are a key factor that can influence erosion magnitude, pattern, and the location of maximum material loss. Most numerical simulation studies for dilute gas-solid flows have focused on low to moderate particles Reynolds number (Re) and assumed drag and gravity forces as the most important forces. Here, we use the open-source libraries OpenFOAM to solve the continuity and momentum equations for the fluid phase and the particles whose motion is driven by Newton's second law, are modelled in a Lagrangian way with a Discrete Particle Model. The effects of drag, gravity, pressure gradient, virtual mass, and lift forces at high Re on erosion are studied with two test cases: the direct impingement (DI) of solids on a plate and flows through an elbow with Re > 1000. The results suggest that the particle dynamics strongly affect the angle of impact, velocity of impact, and distribution of particles. The erosion predictions appear to be particularly sensitive to drag and lift. For the DI, lift has shown a significant effect on erosion pattern and magnitude by influencing the particle distribution while the effects of lift appear negligible in the case of elbow erosion. The reason for the negligible effect for elbows may be attributed to higher degree of turbulence and mixing as the particles approach the bend. Our study of three particles sizes, 53µm, 300µm and 710µm, also show a negligible effect of lift on the 53µm for both DI and elbow geometries. The results suggest that the assumptions around individual geometries need to be carefully considered in optimising computational time and to enhance the accuracy of erosion prediction.   

Numerical investigation of finite-time blowup related to particle collision

This study inspired by Falkovich et al. (2002) aims at numerical investigation of finite-time blowup by collision using a new approach. we have adopted Eulerian framework under the assumption that particle velocity is a smooth function in space and is uniquely determined by the particle position until collision between particles occur. In particular, the particle velocity gradient can easily go to blow up within a finite time due to the discontinuity when the first collision happens. Using this approach, the singularities in particle velocity gradient, particle number density, and particle vorticity for numerous Stokes numbers and gravity factors are investigated allowing a rigorous mathematical description of particle collision. Three background flows such as a simple and intuitive two-dimensional Taylor-Green vortex flow, two-dimensional decaying turbulence, and three-dimensional isotropic turbulence are used for direct Eulerian simulation. In addition, we have revealed a flow condition leading to collision of particles wherein very thin shear layer constructed by two parallel same-signed vortical structures. Detailed results will be presented in the meeting.

    

Caustic formation in turbulent aerosols at weak particle inertia

Caustic singularities of the spatial distribution of particles in turbulent aerosols increase collision rates and accelerate coagulation. The dominant mechanisms of caustic formation depend sensitively on the dynamical time scales of the particle motion and of the turbulent flow. For strongly inertial particles in rapidly fluctuating flows, caustic formation is described by a Kramers escape of the particle-velocity gradients. Weakly inertial particles in persistent flows have recently been studied in two spatial dimensions, where caustics are induced by rare excursions of the fluid strain, while vorticity remains small. Here we show that in three dimensions, caustics form by a related, yet different mechanism. As in two dimensions, caustics are induced by a large, strain-dominated excursion of the fluid-velocity gradients. In three dimensions, however, the excursion occurs in the $Q$-$R$-plane, where it must exceed a characteristic threshold line. The most likely way to reach this threshold is by an ``optimal fluctuation'' that propagates along the Vieillefosse line and is unique up to similarity transformations. We explicitly determine the probability and shape of the optimal fluctuation, and find that it is dominant in numerical simulations even for moderate particle inertia.