Session T35: Multiphase Flows: Modeling and Theory II

Volume Fraction Effects on the Added Mass Force

The added mass or inviscid unsteady force is a well-known phenomenon in multiphase flow arising from acceleration effects. Modeling of the added mass force is of interest in a multitude of industrial, engineering, and environmental applications. The present work seeks an improved understanding of the added mass force in the incompressible regime by accounting for volume fraction effects. We explore the effects of volume fraction on the added mass force in different triply periodic configurations consisting of volume fractions ranging from 0 to 0.4 using computational tools. This is achieved by impulsively applying a spatially uniform body force to the fluid while holding the particle in place, thereby accelerating the flow in a spatially homogeneous manner. We extract the added mass coefficient by separating the pressure gradient and added mass forces. In the dilute limit, we recover the analytical value of the added mass coefficient of 0.5 and observe substantial deviation from this in the finite volume fraction regime.   

A novel mass-conserving contact line boundary condition for second-order phase-field models

The phase-field method is a widely adopted technique for simulating multiphase flows. One popular class consists of models based on second-order phase-field equations, which offer advantages over higher-order models in certain aspects, including better bound preservation and milder timestep restrictions. However, an ongoing challenge with such models is the treatment of contact lines. Because a second-order phase-field equation admits only one constraint on the phase-field variable at each boundary, it is unclear how to simultaneously conserve mass and prescribe a contact angle model. In this presentation, we introduce a novel solution to this problem: a local mass conservation (no-flux) boundary condition on the phase-field equation in conjunction with the generalized Navier boundary condition on the momentum equation. The result is a second-order phase-field model that conserves mass while also accurately modeling contact lines in systems with arbitrary (non-90 degree) equilibrium contact angles. We describe the formulation of the model as applied to a conservative second-order phase-field model and present numerical results of canonical contact line test cases.   

Surface-tension driven flows: self-similar DNS

We revisit the scale-invariant recoil of an inviscid liquid wedge driven by surface tension, which was first theoretically studied and numerically solved with boundary integral methods by Keller and Miksis (KM) [SIAM J. Appl. Math, vol. 43, n°2, 1983, pp. 268-277].

We reformulate the problem in self-similar coordinates X = (ρ/σt²)^1/3 x  and Y = (ρ/σt²)^1/3 y, where σ is surface tension, ρ mass density, x and y the physical space variables, t time and using τ = ln t as time variable. This transformation simply adds two new terms to the usual Navier-Stokes equations: a second advection term, and a source term.

The direct numerical simulation of this reformulated problem actually converges towards KM solution, which is stationary in (X, Y) coordinates, and demonstrates the stability of the scale-invariant solution. The numerical advantage of the use of (X, Y, τ) variables over (x, y, t) variables is that the characteristic length and time scales displayed by the solution are O(1) over the whole computation time span. This allows us to study the perturbation of the route towards self-similar behavior, caused by viscosity.   

The nucleation process: Cavitation, Boiling, and coupling with the macroscale

Nucleation of a second phase in a mother fluid is a ubiquitous phenomenon whose prediction proved a formidable task, particularly in the case of water. Here a self-contained model is discussed which is shown able to accurately reproduce data for bulk water over the most extended range of temperatures for which accurate experiments are available [1]. The computations are based on a Ginzburg-Landau model which, as only inputs, requires a reliable equation of state for the bulk free energy and the interfacial tension of the water-vapor system. Rare event techniques borrowed from statistical mechanics allow the determination of the free-energy barrier and the nucleation rate. By consistently including thermal fluctuations [2] in the spirit of Fluctuating Hydrodynamics, the approach is extended to dynamic conditions in the presence of solid walls of different wettability [4] to allow coupling with fluid motion [4]. The talk will discuss a wide range of possible applications, such as homogeneous and heterogeneous cavitation, coupling with fluid flow, and possibly condensation.   

On the contact line dynamics of three-phase fluids

We investigate a model of multiphase polymer solutions undergoing fluid flow, phase separation, and buoyancy forcing. The thermodynamically consistent model couples three-phase Cahn-Hilliard dynamics with the Navier-Stokes equations. While the two-phase case is well-understood, the possibility of three-phase contact lines introduces several complications. We build on recent work to derive a reduced model of interface and junction dynamics valid in the physically relevant limit of vanishing interfacial thickness. Multiple numerical benchmarks verify our reduced model. We emphasise the interesting new behaviour possible in three dimensions, and touch on possible applications to microfluidic manufacture of complex microparticles.   

Consistent modeling of scalar transport coupled with two-phase flows using phase-field models

Phase field models have emerged as a competitive alternative to sharp-interface models for simulating two-phase flows due to their simplicity in implementation, cost-efficiency, scalability, and regularity. Phase field methods like the Cahn-Hilliard equation and second-order conservative phase field models are particularly attractive, as they enforce mass conservation on the continuous level. However, any equation coupled to the phase field must consistently account for the conservative right-hand side terms in the phase field equations. For example, a consistency modification of momentum equations achieves robust simulations of high Re numbers and high-density ratios for phase-field methods. Additionally, it also ensures thermodynamic consistency in the case of Cahn-Hilliard-based phase-field models. In this study, we present consistency modifications to scalar transport PDEs when coupled with phase-field models for two-phase flows. In addition, an efficient numerical strategy for solving coupled scalar and two-phase flow phase field equations is presented. We use the numerical framework to demonstrate the effect of the consistency terms on accuracy and robustness.   

A Filtered-Interface Method for Simulations of Transcritical Multiphase Flows

In stark contrast to pure fluids, immiscible multi-species mixtures can retain sharp liquid-vapor interface with surface tension at pressures significantly higher than the critical points of the constituent species. Examples of such a system are commonly found in modern automotive and propulsion engines: the injection of low-temperature liquid fuel into a high-pressure, high-temperature gaseous environment. In this context, the thermodynamic state of the mixture can traverse both sub- and supercritical regimes temporally and spatially. In the past, modeling of the temporal transition requires an ad-hoc switching between a fitted-grid front-tracking two-fluid method for subcritical conditions and a single-fluid method for supercritical conditions. The complexity of this method limits its application to 1D domains. Meanwhile, modeling of the spatial transition from sub- to supercritical conditions is so far not possible. To address this, we introduce a newly developed Filtered-Interface Method (FIM). This single-fluid formulation can naturally describe both sub- and supercritical processes as well as the transition between them. After validation of the model against existing experiments and VOF simulations of subcritical interfacial behavior, we demonstrates the ability of FIM in resolving transcritical phenomena via a suite of transcritical configurations: droplet evaporation, droplet injection, and jet injection.   

A Study into the Pinch-Off Dynamics of Low Viscosity Droplet Drips in a Coaxial Flow

A coaxial droplet consists of two immiscible fluids encapsulated in one another after the point of detachment from a coaxial flow. Scientists are consistently working on newer methods to optimise the existing coaxial flow devices to create smaller, more easily controlled droplets to allow for more precise use in fields like encapsulation for drug delivery and cosmetics. Nevertheless, most droplet pinch-off studies focused on pendant droplet pinching in a still surrounding medium. With the advent of greater computational power, recent novel discoveries showed that for single droplets in a still medium, all three, inertial, viscous, and inertial-viscous, regimes were present within the dynamics of pinch-off, but they varied depending on the initial fluid properties. Unlike single pendant droplets, an extensive study into filament breakup of coaxial filaments has yet to be conducted, despite the research conducted into optimising coaxial flows and, subsequently, coaxial droplets for industrial purposes such as pharmaceutical tablet coatings. Our research effectively studies the pinch-off mechanics of low-viscosity Newtonian coaxial droplets using both experimental and the computational volume of fluid (VOF) method under varying parameters to understand the physics behind these changes further. Our collected data shows a monotonic dependence of the inertial regime scaling perfector on parameters such as the external liquid flow and the nozzle size. These new results open the door for better control over the encapsulation process.