Session P29: Multiphase Flows: Modeling and Theory

Large Eddy Simulation (LES) for Multiphase Flows based on Interface Retaining Coarsening

Large Eddy Simulation (LES), where the unsteady motion of the large scales is simulated and models are used to describe the average motion of the small scales, is a promising way for predicting the dynamics of single phase flow. It is likely that a similar strategy is useful for multiphase, but not in a conventional way. For multiphase flows where sharp moving phase boundaries separate different fluids or phases, the dynamics of the interface often determines the behavior of the flow. In a coarse, or reduced order model, it may therefore be important to retain a sharp interface for the resolved scales, in a similar way that modeling of disperse flows often retain bubbles or drops as point particles. We describe a systematic process to coarsen fully resolved numerical solutions for multiphase flows while retaining a sharp interface. The different phases are identified by an index function that takes different values in the different phases and is coarsened by solving a constant coefficient diffusion equation, while tracking the interface contour. Small flow scales of one phase, left behind when the interface is moved, are embedded in the other phase by solving another diffusion equation with a modified diffusion coefficient that is zero at the interface location to prevent diffusion across the interface, along with a pseudo pressure equation to preserves the incompressibility of the coarsened volumetric velocity field. Several examples of different levels of coarsening are shown. The dynamics of the small scales in the mixed regions can be modeled in many different ways, including using homogeneous mixture, drift flux, and two fluid Euler-Euler models, as well as Euler-Lagrange models. We are currently applying simple homogeneous mixture model and the evolution equation for the coarsened flow field is derived. Those subgrid terms are determined based on machine learning and preliminary results of predictions for the closure terms on a 2D jet are shown.    

Thermodynamically Consistent Phase-Field Cahn-Hilliard Navier-Stokes Models for Aqueous Phase Separating Multiphase Flow Systems

Generation of multicomponent microdroplets is an important step for such applications as development of polymeric nanoparticles for drug delivery and emulsions for cosmetics, where strategies based on phase separation of flowing mixtures are posited to control the droplet creation process. From a theoretical standpoint, such phase separating flows are mediated by the coupled Cahn-Hilliard Navier-Stokes (CHNS) system, for which a variety of models exist that describe the coupling terms between the two equations, depending on the system. In this work, we describe the application of a class of these models which can be formally derived ensuring thermodynamic consistency. After that, we also explore how these thermodynamically consistent models can be applied to understand systems that consist of flows of aqueous solutions in microchannels to understand how diffusive mechanisms associated with phase separation interact with viscous and convective processes associated to hydrodynamic flow. Such an understanding holds the potential to predict phenomena that could inform experimental realizations of phase separating flow systems.   

A multiphase theory for transient osmotic swelling of chemically responsive hydrogels

The osmotic pressure due to a concentration difference of identical solute molecules across a semi-permeable membrane can be determined by van't Hoff's formula for chemical potential equilibration when the number of solute molecules remain constant on both sides of the membrane. This condition can be inherently relaxed at the interface between an aqueous supernatant domain and a chemically responsive hydrogel, when a chemical stimulus freed from inside the gel slowly diffuses into the supernatant while creating a dynamic osmotic pressure balanced by the poroelastic diffusion of water into the gel. We introduce a continuum poroelastic theory for the dynamic build-up and relaxation of osmotic pressure due to an interplay between copper cations as osmosis-driving agents and acid in a polyacrylic acid hydrogel thin film. Our theory rigorously relates the non-equilibrium osmotic pressure to the vertical gradients of the solute concentration across the interface, in contrast with the concentration dependence in van't Hoff's formula. The theory quantitatively captures the osmosis induced swelling and contraction of the gel film and the threshold concentrations for their emergence, in agreement with experiments.   

Meshfree dispersed multiphase flow prediction using equivariant neural network

Deterministic influence of neighboring particles on a particle of interest is not accounted for in a typical Euler-Lagrange simulation.Recent efforts have shown that Deep Learning methods hold promise in capturing these influences.The considered problem of flow around a particle in a unique neighbor configuration involves rotational symmetry about the mean-flow direction.However, conventional neural networks do not implicitly satisfy symmetries.Thus, this work makes use of equivariant neural networks that inherently preserve rotational symmetry.We believe that accurate flow prediction is a crucial intermediate step to eventually obtain robust particle forces. Achieving flow predictions using a Convolutional Neural Network at the resolution of Particle-Resolved Direct Numerical Simulation is computationally intensive.Hence, a meshfree flow method based on point-cloud equivariant neural network to obtain flow field predictions is proposed.This computationally efficient methodology produces results that are 75-90% accurate for particle volume fraction and Reynolds number in the range of [0.11,0.45] & [2.69,172.96] respectively.   

Single-point closures for turbulent multiphase flow that incorporate two-point statistics through filtered fields

Turbulent multiphase flows exhibit rich patterns of particle clustering and preferential concentration that are characterized by two-point spatial (and temporal) correlations. However, tractable models of spatially inhomogeneous flows that are useful for large-scale simulation necessarily rely on single-point theories based on the average measure of fluid (or particle) volume in each geometric volume. Unfortunately, signatures of clustering that appear in the two-point measure of particle volume (or pair correlation) disappear in the single-point limit. Here a new result is presented which shows how filtering the instantaneous volume fraction field permits the signatures of clustering to be retained in the single-point limit, thereby opening the door to modeling the aforementioned phenomena through single-point closures.   

Gas-kinetic Modeling of Fluid-induced Forces in a Monodisperse Array of Spherical Particles

Multiphase flows of dense particulate phases are ubiquitous in nature and industries. For the accurate prediction of dense gas-solid systems, precise force models covering various flow regimes and particle concentrations are essential. The fluid-induced forces experienced by particles in a dense distribution are very different from that of an isolated particle due to the flow modification in the vicinity of clustered particles. Although several drag modifications for dense particulate distributions are available in the literature, the effect of gas rarefaction and compressibility is not widely understood. In this talk, we investigate the flow phenomenology in a dense monodisperse particle bed at various flow Reynolds numbers and Knudsen numbers using our in-house gas-kinetic Direct Simulation Monte Carlo (DSMC) solver CHAOS. The semi-empirical particle force model developed from the study will be applicable in modeling and understanding high-speed gas-solid flows.  Particular emphasis is to incorporate a Knudsen number correction for dense particulate flows in dilute gas regimes.   

Super-resolution analysis: reducing computational cost of high fidelity simulation of flashing flows.

High fidelity numerical simulation incur increased computational cost when directly solving partial differential equations (PDE) using traditional techniques like finite volume method (FVM). With the recent advancement in physics based machine learning, the available flow data  in the coarse grids could help in reconstructing solutions of higher fidelity. In this study, we use the super-resolution analysis technique to obtain higher resolution results of a flash boiling flow from an initial coarse grid simulation. A standard convergent-divergent nozzle as a domain  in  this supervised neural network study. We employ down-sampled skip-connection multi-scale convolutional neural networks to reconstruct solutions on a fine grid. Stochastic gradient descent technique was implemented to update the weights by calculating the loss function.  To maintain the size of the output after  the convolution,  reflective padding was used. The effect of different kernel sizes,  learning rates and hidden layers were also investigated in this study. The results are obtained with reasonable accuracy and at a fraction of computational cost as compared with a conventional numerical  method to solve PDE.    

Dynamic wetting failure in curtain coating by the Volume-of-Fluid method and Navier slip model.

Dynamic wetting failure in a curtain coating setup is studied by solving the 2D two-phase Navier-Stokes equation subjected to a Navier-Slip boundary condition and a constant 90-degree contact angle, using the Volume-of-Fluid method with the surface tension force computed using the Continuous Surface Force method with Height Functions for the curvature (http://basilisk.fr). Sufficient resolution of the slip length is achieved using the quadtree adaptive mesh refinement technique, allowing us to predict the critical parameters of wetting failure. The phenomenon of hydrodynamic assist is seen, wherein, the increasing velocity of the impinging liquid and hence the increasing inertia causes the increase in the value of the critical speed of the solid substrate and hence delaying the wetting failure. The results are validated against previous computations of Liu et al. [C.Y. Liu, E. Vandre, M. Carvalho, S. Kumar, J. Fluid Mech. 808, 290 (2016)]. The logarithmic curvature singularity predicted by theory (and hence a pressure singularity) is discovered at the three-phase contact line, making the slip model weakly singular.   

Phase Separation of a Binary Mixture with an External Force Field

The objective of this work is to simulate the phase separation of a regular binary mixture in the presence of an external force field. In general, when a partially miscible binary mixture is brought from the stable, single-phase region of its phase diagram to the unstable region, it separates into two coexisting phases, corresponding to a minimum of the free energy of the mixture. As the mass of each chemical species is conserved, phase transition consists of a reordering process, called spinodal decomposition. 

Now, assume that an external force field is applied to the mixture and that its effects on the species composing the mixture are different from each other. For example, the force field can be gravity or an electric field, applied to a mixture of fluids having different properties. In these cases, the external force will enhance demixing: for example, gravity will tend to increase the concentration of the heavier species in the bottom of the container and the electric field will tend to separate the two species along the direction of the force field.

Here we show that the non-homogeneity of the composition field induced by the external field may cause, locally, a phase transition even when, globally, the system is in a metastable, and even a stable, thermodynamic condition.   

Three-dimensional hydrodynamic and thermal modeling of a flat grooved heatpipe using three different formulation

Mathematical modeling of the grooved heat pipes is a challenging task since various phys- ical phenomenon such as phase change, free-surface flow and heat transfer are involved. Moreover, the fact that the shape of the liquid-vapor interface is unknown a priori, the shape of the interface needs to be determines as a part of the solution procedure, a ca- pability currently not addressed in commercially available engineering CFD software. In this study, a multi-dimensional and multi-scale computational model is presented to gain comprehensive understanding of underlying psychics of the grooved heat pipes. The com- putational model is based on an iterative scheme for the solution with 3-D heat transfer and liquid flow, interface phase change heat transfer (evaporation and condensation) and the shape of the interface. The model is implemented using three different methodologies: (i) a finite difference based formulation of the heat transfer with a unidirectional fluid flow coupled to the phase-change models; (ii) a finite element based formulation with multi- dimensional heat transfer and fluid flow through COMSOL Multiphysic and coupled to phase-change models models through COMSO -MATLA interface; and (iii) a finite volume based formulation with multi-dimensional heat transfer and fluid flow through ANSYS FLUEN with a PYTHONTM based main driver for the coupling of the phase changer models and the generation of the computational domain through a CAD software. The workflows of the three methodologies are demonstrated and the underlying reasons of the differences between the solution methodologies, advantages and disadvantages of proposed methodologies are discussed.   

Adjoint-based control of multi-droplet systems in Stokes flow

In this work, we present an approach to control multiple droplets by applying Lagrange

multiplier theory to two-phase Stokes flows with sharp interfaces. The focus is on the

deformation of clean, neutrally buoyant droplets with constant surface tension, where the

interface deforms with the local fluid velocity. We make use of shape optimization theory

to derive the linearized system and the corresponding optimality conditions necessary

for gradient descent methods. The forward and adjoint systems are represented through

boundary integral equations and discretized using the Nyström method and Quadrature

by Expansion. We couple the boundary integral method with a spectral representation

of the geometry (based on spherical harmonics) to obtain a highly accurate and stable

solver. We show that optimality conditions are correctly derived by comparison to black-

box finite difference approximations and minimization of select benchmark problems.

Finally, we apply the adjoint-based control to non-trivial multi-droplet systems and

evaluate the performance and robustness of the proposed methodology.   

Not Participating

Linear stability analysis is performed for a radially expanding liquid sheet in the presence of a gas medium. Liquid sheet can break up because of aerodynamic effect as well as its thinning. However, study of the aforementioned effects is usually done separately as the formulation becomes complicated. Present work combines both, aerodynamic effect and thinning effect, ignoring the non-linearity in the system. This is done by taking into account the formation of gas boundary layer whilst neglecting viscosity in the liquid phase. Base state analysis results in a Blasius-type system. To study the stability of the liquid sheet, gas-liquid interface is subjected to small perturbations. The linear model derived here can be applied to investigate the instability for sinuous as well as varicose modes, where the former represents displacement in centerline of the sheet and the latter represents modulation in sheet thickness. Temporal instability analysis is performed for sinuous modes, which are significantly more unstable than varicose modes, for a fixed radial distance implying local stability analysis. The growth rates, measured for fixed wavenumbers, predicated by the present model are compared with those obtained by the inviscid Kelvin-Helmholtz instability.