Session P26: Multiphase Flows: Computational Methods I

Accurate conservative diffuse-interface method for the simulation of two-phase flows

We present a novel phase-field model for the simulation of two-phase flows that is accurate, conservative, bounded, and stable. The proposed model conserves the volume/mass of each of the phases, and results in bounded transport of the volume fraction. We present results from the canonical test cases of a drop advection and a drop in a shear flow, showing the improvement in the accuracy over the commonly used second-order conservative phase-field method. Furthermore, we also derive consistent momentum transport equation for the proposed phase-field model and show that the proposed model when coupled with the consistent momentum transport equation results in discrete conservation of kinetic energy, which is a sufficient condition for the numerical stability of incompressible flows, in the absence of dissipative mechanisms. To illustrate the robustness of the method to simulate high-density ratio turbulent two-phase flows, we present the numerical simulation of an infinite Reynolds number high-density ratio droplet-laden isotropic turbulence.

Authors acknowledge Predictive Science Academic Alliance Program III at Stanford University   

A Dual-Scale Modeling Approach for Turbulent Interfaces with Finite Weber Numbers

Direct Numerical Simulation remains a prohibitively expensive task, especially for cases involving atomization. Instead of DNS, a dual-scale modeling approach (Gorokhovski and Herrmann, 2008) that describe turbulent phase interface dynamics in a Large Eddy Simulation spatial filtering context is proposed. Spatial filtering of the equations of fluid motion introduce several sub-filter terms that require modeling. Instead of developing individual closure models for the interface associated terms, the dual-scale approach uses an exact closure by explicitly filtering a fully resolved realization of the phase interface. This resolved realization is maintained using a Refined Local Surface Grid approach (Herrmann, 2008) employing an unsplit geometric Volume-of Fluid method (Owkes and Desjardins, 2014). Advection of the phase interface on this DNS scale requires a reconstruction of the fully resolved interface velocity. In this work, results from the dual-scale LES model employing sub-filter velocity reconstruction by combined approximate deconvolution and non-linear spectral enrichment (Bassenne et al. 2019) including a sub-grid surface dynamics model (Herrmann 2013) are compared to DNS results for a phase interface in a homogeneous isotropic turbulent flow at several Weber numbers.   

Computing foaming flows: from microfluidic crystals to breaking waves

Foamy flows are critical in numerous  natural and industrial processes and remain notoriously difficult to compute as they involve coupled, multiscale physical processes. The simulation of foamy flows, involving non-coalescing bubbles, presents a number of formidable challenges, in addition to those associated with resolving the bubble interactions with the fluid flow and the solid boundaries.  We present a novel multilayer simulation framework (Multi-VOF)  that advances the state of the art in simulation capabilities of foamy flows. The framework  introduces a novel scheme for the distinct handling of multiple neighboring bubbles and a new regularization method that produces sharp interfaces and removes spurious fragments. Multi-VOF is verified and validated with experimental results and complemented with open source, efficient scalable software. We demonstrate capturing of bubble crystalline structures in realistic microfluidics devices and foamy flows involving tens of thousands of bubbles.   

A Numerical Formulation to Study Interactions Between Fluids and Deformable Solid in Extension to Thin Film Geometries

In the present work, we use a fully Eulerian Level set method to track locations of the solid, solid strain, and calculate solid stress on the Eulerian grid. Stress-strain relationship of the solid is governed by the Neo-Hookean solid model. A unified framework of equation of motion is used to solve for both fluid and solid dynamics. Fluid-Structure Interaction is accounted for by adding a volumetric body force term in the solid region and the solid force is diffused into the fluid by a Heaviside function. Linear extrapolation has been used to reconstruct the solid strain field at the solid-fluid interface in prevention of the discontinuities. We first Validated our implementations by fully resolving the solid and comparing with existing literature results. We have then extended our implementations to thin film geometries by defining a thin region of solid of an order of a few computational cells where solid shear modulus quickly goes from solid value to zero. Numerical results are shown in this presentation.   

Level Set Modeling of Supersonic Liquid Breakup Using The Method of Characteristics

Numerous studies have been carried out that model the breakup of liquid jets by subsonic air flow. However, its supersonic equivalent has not received enough attention due to computational complexities involved. In this study, a Method of Characteristics (MOC) based algorithm is proposed to simulate the breakup of liquid jets under supersonic conditions. A special version of the Level Set method is required to track the gas-liquid interface in the system. In the MOC-based approach, first-order hyperbolic Euler equations in the gaseous phase are reduced to the characteristic form and then compatibility relations, which do not contain derivatives in the direction normal to the characteristic surfaces, are obtained. Since the MOC reduces dimensionality of the problem by one, faster, yet accurate, solutions can be achieved in comparison to finite difference or finite volume methods. Preliminary results from our ongoing investigations will be presented.   

High fidelity single framework simulations of acoustic wave--bubble cloud--elastic solid interactions

High-fidelity simulations of interacting acoustic waves, bubble clouds and elastic or viscoelastic materials are challenging. Indeed, these simulations typically require two separate numerical frameworks for compressible fluids and solids. The addition of cavitating bubble clouds further complicates the problem by introducing a wide range of spatio-temporal scales. However, such simulations are of particular interest in the development of ultrasound therapies (e.g. lithotripsy and histotripsy) and in understanding traumatic brain injury prediction. We introduce a single framework method to simulate these physics. It is based on a sub-grid model for the cavitating bubble clouds and a hypoelastic solid material valid for small strains. These were implemented in our open-source Multi-component Flow Code (MFC) (Bryngelson et al., Comp. Phys. Comm., 2020). This hypoelastic solid model enables the introduction of elastic and viscoelastic solids into these simulations, as long as we remain in the linear (small strain) regime. We demonstrate the capabilities of the solver for these problems, with example 3D simulations of burst-wave lithotripsy including cavitating bubble clouds. Furthermore, we present the computational advantages of the most recent GPU-enabled version of the solver.   

A robust approach for numerical modeling of non-isothermal multiphase flows

Several manufacturing processes such as casting, welding, and additive manufacturing are inherently multiphysics problems with a wide range of thermo-physical properties. These processes involve fluid flow, heat transfer, solidification/melting, and fluid-structure interaction. Numerical simulation of these multiphysics problems is quite challenging due to several length and time scales. In this presentation, we present a robust approach for the numerical modeling of non-isothermal multiphase flows with phase change. To ensure the numerical stability of the scheme in presence of high-contrasting thermo-physical properties, we consistently treat the mass, momentum, and energy transport in the conservative form of discrete equations. We combine the phase-field model with a level-set advection equation to track the solid, liquid, and gas phases in the computational domain. The proposed model discretely satisfies mass, momentum, and energy conservation, which is tested and presented in our problems. We also present results for several benchmarking cases, including the Stefan problem, melting of ice in water, and thermocapillary droplet migration in liquid metal.   

A segregated algorithm for the solution of the Cahn-Hilliard equation in multiphase flows

A segregated algorithm for the solution of the Cahn-Hilliard equation in multiphase flowsPhase field methods are gaining momentum in science and engineering to model multicomponent and multiphase systems thanks to their thermodynamically consistent formulation and the general smoothness of the fields. In fact, they provide a framework to include complex physical processes (such as phase-change) and result in less spurious oscillations when dealing with surface tension, compared to other methods like the volume of fluid. The Cahn-Hilliard equation is the principal governing equation in the phase field method as it results from a minimization of the free energy functional and thus includes all the relevant physical phenomena such as phase-change and surface tension forces. However, its solution is not straightforward as it is a fourth-order non linear partial differential equation. A number of explicit methods have been proposed in literature together with an implicit mixed formulation. Segregated implicit algorithms are seldom used due to stability issues.In this work, we present a novel segregated algorithm for the solution of the Cahn-Hilliard equation based on the incomplete block-Schur preconditioning technique. Performance and accuracy of the algorithm are compared against a block-coupled mixed formulation for a number of cases. We also illustrate several applications of the method to multiphase flows with phase change.   

Assessment of pseudo-potential lattice Boltzmann method for multiphase flows with real fluid properties

The multirange pseudo-potential lattice Boltzmann model (PP-LBM) for multiphase flows was implemented along with various strategies, including the multi-relaxation time collision operator and the so-called ??-scheme, to diminish the parasitic currents and enhance numerical stability. The PP-LBM predictions show an excellent agreement with the theoretical coexistence curve, given by Maxwell’s equal-area rule, for both Carnahan-Starling (CS) and Peng-Robinson (PR) equations of state (EOS). Increasing the isotropic order of the interaction force has a stronger impact on reducing the magnitude of the spurious velocities, as compared to the implementation of more sophisticated collision operators. Due to a satisfactory representation of the vapor-liquid coexistence curve and acentric factor dependence, the PR EOS is chosen to determine real fluid properties such as equilibrium densities and surface tension for a number of compounds, including n-alkanes (carbon number up to 10), ammonia, and hydrogen. For such paraffinic hydrocarbons, the average surface tension error lies in a range of 2.3-6.2%. Although PP-LBM suffers to accurately predict experimental equilibrium density values, the error is inherent to the PR EOS, as PP-LBM predicts well those given by Maxwell’s equal-area rule.   

A robust low-Mach solver for phase-changing flows

We extended a previously developed VoF-based numerical method for phase-changing flows (Scapin N. et al., J. Comput. Physics, 2020) to solve the governing equations in the limit of zero Mach number. Compared to the fully compressible formulation, the proposed methodology has three distinct advantages. First, it is able to relax the assumption of constant thermophysical properties while effectively filtering the acoustic effects and the associated stringent numerical time-step restrictions. Next, the hydrodynamic pressure is still governed by an elliptic equation for which FFT-based solvers can be employed. Finally, the divergence constrain of the velocity field can be imposed up to machine precision provided that the thermodynamic pressure and the thermophysical properties are updated consistently. The algorithm is implemented on top of a second order accurate two-fluid Navier-Stokes solver coupled with an algebraic volume of fluid method (MTHINC), and extended with the corresponding transport equations for the vaporized liquid mass and thermal energy. Finally, the robustness of the method for more demanding simulations is assessed in two configurations: evaporating droplets in homogeneous turbulent flows and evaporating two-layer Rayleigh-Bernard turbulence.   

Reducing volume and shape errors in front tracking

Volume conservation and shape preservation are two well-known issues related to the advection and remeshing in front tracking. To address these issues, we propose a divergence-preserving velocity interpolation method and a parabolic fit vertex positioning method for remeshing operations for three dimensional front tracking. Errors in preserving the divergence of the velocity field when interpolating the velocity from the fluid mesh to the vertices of the triangles of the front are a primary reason for volume conservation errors when advecting the front. The proposed interpolation method preserves the discrete divergence of the fluid velocity by construction and is compared with other common interpolation methods. Additionally, the parabolic fit vertex positioning method for remeshing operations locally approximates the front with a smooth polynomial surface, improving volume conservation and shape preservation. Results of representative test-cases are presented, which demonstrate that the proposed methods can conserve the volume and shape of the front up to an order of magnitude better than conventionally used methods.   

Discrete Exterior Calculus Discretization of Two-phase Incompressible Navier-Stokes Equations with a Conservative Phase Field Method

We extend the discrete exterior calculus (DEC)  method  for simulating single phase flow (Jagad et al. Phys. Fluid 2021) to two-phase flows of immiscible fluids with surface tension and discontinuous jumps in density and viscosity. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The differential forms, exterior derivatives and Hodge star operators are then discretized using DEC.  With appropriate choice of phase-field parameters, boundedness of the phase field is proved for the DEC method for both a first order forward Euler and a second order time integration scheme without employing a mass redistribution technique. Several standard advection tests on planar domain and a curved surface are conducted for investigating the applicability of interface capturing in a surface flow. Our results show excellent boundedness, mass conservation and convergence properties. Various physical simulations of incompressible two-phase flows (e.g. falling drop, rising bubble, drop impact on a liquid pool) under large density and viscosity jumps as well as surface tension demonstrate the positive features and advantages of the proposed DEC scheme.