Session Q6: Multiphase Flows: Modeling and Theory

Diffuse-Interface Modelling of Interfacial Flows

Multiphase processes are ubiquitous in many engineering applications and their modeling can help us to understand phenomena occurring across different scales, from microfluidic devices to oil recovery. However, this is not without difficulties as for instance the presence of complexities on the solid boundaries of the system, such as topological and/or chemical heterogeneities, and the inherent separation of length scales in the medium, impose a number of challenges. Here, we study several examples of interfacial flows by making use of a diffuse-interface approach. In particular, we use the Cahn-Hilliard equations for the evolution of an order parameter (the phase field) coupled with the Navier-Stokes equation for fluid flow. These equations are then solved using FreeFem++. The examples considered in our work include the effects of the solid geometry and wetting properties on the process of confined phase separation, and the process of fluid flow through a porous medium. For the latter in particular, we also make use of a homogenized theoretical model with the aim to determine in which regimes an effective macroscopic description is able to reproduce the full multiscale system.   

Three-dimensional linear stability analysis of a rising Taylor Bubble

The stability of a rising Taylor bubble in liquids is studied using the finite-element method. For different flow conditions, characterised by aspect ratio, dimensionless E${\ddot{\rm o}}$tv${\ddot{\rm o}}$s and inverse viscosity numbers, we compute the steady state solution of a rising three-dimensional axisymmetric Taylor bubble. The linear stability of the steady state solutions to three-dimensional infinitesimal perturbations is then investigated. The analysis enables us to determine the region in parameter spaces at which a Taylor bubble becomes unstable. In this talk, we present the finite element formulations derived from the full Navier-Stokes equations for the steady-state calculations and the linear stability analysis. The methods used in solving these formulations, the validations carried out, and the results obtained for the different flow conditions studied are discussed.   

Adjoint-based control of a surfactant-free drop suspended in an axisymmetric straining flow

Optimal control theory has been remarkably successful in controlling a wide variety of single-phase flows. However, to date there has been no fundamental study on its application to multiphase problems. In this study, the continuous adjoint method is employed for the first time to control a clean, neutrally buoyant droplet of viscosity $\mu_{I} $suspended in a low Reynolds number axisymmetric straining flow of viscosity $\mu_{E}$. The control variable is the non-dimensional Capillary number $C$. For the forward problem, each C generates a distinct steady-state droplet. This information is utilized by the inverse problem to compute the adjoint velocities and the corresponding control gradient at the drop interface. The resultant gradient is subsequently used to update $C$. The algorithm is repeated until the desired shape is realized. To reduce computational cost, both the forward and adjoint velocities are obtained by solving Fredholm integral equations of the second kind. The theory is tested for three viscosity ratios; $\lambda $ $= \quad \mu_{I}/\mu_{E} \quad =$ 0.1, 1 and 10. For all three cases, the cost functional is successfully minimized. The adjoint gradient is shown to be in good agreement with the corresponding finite-difference approximation of it.   

Macroscopic Modeling of Micro-Inertia Effects in Multiphase Flows

A new approach to introduce micro-inertia into macroscopic models of multiphase flows is developed using the Non-Equilibrium Thermodynamics bracket formalism (Beris and Edwards, Thermodynamics of Flowing Systems, Oxford U. Press, 1994). The approach relies in the use on an internal contravariant conformation tensor variable to describe the multiphase micro-structure. Current applications include those of dilute emulsions, and concentrated colloidal suspensions. In the first case, the resulting constitutive equations are consistent with literature-developed asymptotic theories of the flow and deformation around isolated droplets in the limit of small capillary, Ca, and/or small particle Reynolds, Re, numbers. These asymptotic solutions are also used to uniquely determine all the model parameters. Structural predictions of the ellipsoid droplet morphology obtained with the new model are compared against classic experiments by Torza et al. (J. Colloid Interf. Sci., 38:395, 1972) and by Raja et al. (J. Fluid Mech., 646:255, 2010) showing good agreement. In the second case, the developed model significantly improves the conformation-based model developed by Phan-Thien (J. Rheol., 39:679, 1994) allowing for the prediction of both negative first and second normal stresses.   

Flash nano-precipitation of polymer blends: a role for fluid flow?

Porous structures can be formed by the controlled precipitation of polymer blends; ranging from porous matrices, with applications in membrane filtration, to porous nano-particles, with applications in catalysis, targeted drug delivery and emulsion stabilisation. Under a diffusive exchange of solvent for non-solvent, prevailing conditions favour the decomposition of polymer blends into multiple phases. Interestingly, dynamic structures can be `trapped' via vitrification prior to thermodynamic equilibrium. A promising mechanism for large-scale polymer processing is flash nano-precipitation (FNP). FNP particle formation has recently been modelled using spinodal decomposition theory, however the influence of fluid flow on structure formation is yet to be clarified. In this study, we couple a Navier-Stokes equation to a Cahn-Hilliard model of spinodal decomposition. The framework is implemented using Code BLUE, a massively scalable fluid dynamics solver, and applied to flows within confined impinging jet mixers. The present method is valid for a wide range of mixing timescales spanning FNP and conventional immersion precipitation processes. Results aid in the fabrication of nano-scale polymer particles with tuneable internal porosities.   

Predicting the Agglomeration of Cohesive Particles in a Gas-Solid Flow and its Effect on the Solids Flow

In flows of cohesive particles, agglomerates will readily form and break. These agglomerates are expected to complicate how particles interact with the surrounding fluid in multiphase flows, and consequently how the solids flow. In this work, a dilute flow of particles driven by gas against gravity is studied. A continuum framework, composed of a population balance to predict the formation of agglomerates, and kinetic-theory-based balances, is used to predict the flow of particles. The closures utilized for the birth and death rates due to aggregation and breakage in the population balance take into account how the impact velocity (the granular temperature) affects the outcome of a collision as aggregation, rebound, or breakage. The agglomerate size distribution and solids velocity predicted by the continuum framework are compared to discrete element method (DEM) simulations, as well to experimental results of particles being entrained from the riser of a fluidized bed.   

Hydrodynamic force and torque models for a particle moving near a wall at finite particle Reynolds numbers

We present models for the hydrodynamic force and torque experienced by a spherical particle moving near a solid wall in a viscous fluid at finite particle Reynolds numbers. In order to account for the effects of finite particle Reynolds number, we use four types of simple motions at different particle Reynolds numbers. Using the lattice Boltzmann method, we fully resolve the flow field near the particle and obtain the models for hydrodynamic force and torque as functions of particle Reynolds number and the dimensionless gap between the particle and the wall. The resolution is up to 50 grids per particle diameter. After comparing numerical results of the coefficients with conventional results based on Stokes flow, we propose new models for hydrodynamic force and torque at different particle Reynolds numbers. Furthermore, the models are validated against general motions of a particle and available modeling results from literature. The proposed models could be used as subgrid scale models where the flows between particle and wall can not be fully resolved, or be used in Lagrangian simulations of particle-laden flows when particles are close to a wall instead of the currently used models for an isolated particle. Details please refer to Int. J. Multiphase Flow, 92:1-19 (2017).   

Compressibility Effects on Particle-Fluid Interaction Force for Eulerian-Eulerian Simulations.

Particle-fluid interaction forces are essential in modeling multiphase flows. Several models can be found in the literature based on empirical, numerical, and experimental results from various simplified flow conditions. Some of these models also account for finite Mach number effects. Using these models is relatively straightforward with Eulerian-Lagrangian calculations if the model for the total force on particles is used. In Eulerian-Eulerian simulations, however, there is the pressure gradient terms in the momentum equation for particles. For low Mach number flows, the pressure gradient force is negligible if the particle density is much greater than that of the fluid. For supersonic flows where a standing shock is present, even for a steady and uniform flow, it is unclear whether the significant pressure-gradient force should to be separated out from the particle force model. To answer this conceptual question, we perform single-sphere fully-resolved DNS simulations for a wide range of Mach numbers. We then examine whether the total force obtained from the DNS can be categorized into well-established models, such as the quasi-steady, added-mass, pressure-gradient, and history forces.   

Lagrangian stochastic modelling in Large-Eddy Simulation of turbulent particle-laden flows

Large-Eddy Simulation (LES) in Eulerian-Lagrangian studies of particle-laden flows is one of the most promising and viable approaches when Direct Numerical Simulation (DNS) is not affordable. However applicability of LES to particle-laden flows is limited by the modeling of the Sub-Grid Scale (SGS) turbulence effects on particle dynamics. These effects may be taken into account through a stochastic SGS model for the Equations of Particle Motion (EPM) that extends the Velocity Filtered Density Function method originally developed for reactive flows, to two-phase flows. The underlying filtered density function is simulated through a Lagrangian Monte Carlo procedure, where a set of Stochastic Differential Equations (SDE) is solved along the trajectory of a particle. The resulting Lagrangian stochastic model has been tested for the reference case of turbulent channel flow. Tests with inertial particles have been performed focusing on particle preferential concentration and segregation in the near-wall region: upon comparison with DNS-based statistics, our results show improved accuracy with respect to LES with no SGS model in the EPM for different Stokes numbers. Furthermore, statistics of the particle velocity recover well DNS levels.   

Prediction of a Densely Loaded Particle-Laden Jet using a Euler-Lagrange Dense Spray Model

Modeling of a dense spray regime using an Euler-Lagrange discrete-element approach is challenging because of local high volume loading. A subgrid cluster of droplets can lead to locally high void fractions for the disperse phase. Under these conditions, spatio-temporal changes in the carrier phase volume fractions, which are commonly neglected in spray simulations in an Euler-Lagrange two-way coupling model, could become important. Accounting for the carrier phase volume fraction variations, leads to zero-Mach number, variable density governing equations. Using pressure-based solvers, this gives rise to a source term in the pressure Poisson equation and a non-divergence free velocity field. To test the validity and predictive capability of such an approach, a round jet laden with solid particles is investigated using Direct Numerical Simulation and compared with available experimental data for different loadings. Various volume fractions spanning from dilute to dense regimes are investigated with and without taking into account the volume displacement effects. The predictions of the two approaches are compared and analyzed to investigate the effectiveness of the dense spray model.   

The One-Dimensional Turbulence (ODT) Model Applied to Spray Atomization

Liquid jet breakup is an important fundamental multiphase flow, often found in many industrial engineering applications. The breakup process is very complex, involving jets, ligaments, and small droplets, featuring tremendous complexity in interfacial topology and a large range of spatial scales. One computational strategy for capturing micro-scale processes not affordably resolved in multi-dimensional turbulence simulations is to represent these processes by a lower-dimensional formulation. An approach formulated in one spatial dimension, denoted One-Dimensional Turbulence (ODT), is outlined. The one-dimensional turbulence (ODT) model has been proposed recently as a spray primary breakup model. This stochastic modeling approach provides high lateral resolution by affordably resolving all\textunderscore relevant scales in that direction. ~In this paper, we present our latest ODT results for spray simulations including comparisons with~DNS data and ongoing activities to couple ODT with a LES simulation.   

Observable regularization of Navier-Stokes equations for incompressible two-phase flows

A common phenomenon exists in turbulence, shock, and two-phase flow problems in which the tail of the energy spectrum goes to infinity. This results in an anomaly that we term $k_\infty$ irregularity. Many different methods are developed over the past few decades to solve each of such problems as separate issues. Recently, we developed the concept of observability and used it to derive the observable two-phase Euler equations, which demonstrated good results and considerable computational saving compared to available computational methods for Euler equations. Here, we first present the usage of observability concept for solving an incompressible two-phase flow with no surface tension, i.e. the Rayleigh-Taylor instability. This problem shows that the method is capable of regularizing the equations with no viscous term; to avoid any numerical dissipation we use a pseudo-spectral method for computing spatial derivatives. The effect of observability on the interface instability and its rate of growth are studied. Then the surface tension term is derived using observable divergence theorem. First, the parasitic currents are studied using a quiescent bubble in zero gravity. Then, a rising bubble is studied and the effect of observability limit is demonstrated.