Session J12: Multiphase Flows: Computational Methods (8:00am - 8:45am CST)

Conditioned quadrature moment methods for cavitating bubble dispersions

Bubbles dispersed in a fluid flow oscillate in response to pressure changes. These volume changes are sensitive to properties of the surrounding fluid and the bubble contents. The effect of the oscillations on the carrier fluid flow can be significant for even small void fractions. These interactions can be represented in a simulation framework via phase-averaging. However, this model requires statistical moments of the oscillating bubble dispersion. This can be problematic: for polydisperse bubble dispersions, the computational expense of class-based simulations is dominated by these moment computations. We show that quadrature-based moment methods can alleviate this cost and provide a more general description of the evolving bubble quantities. Conditioned hyperbolic QMOM (CHyQMOM) represents and inverts the internal coordinates of the bubble dynamics model. Comparisons to Monte Carlo simulations show that this approach can represent the evolving moment system, despite extrapolation out of the moment space. Fully-coupled simulations of acoustically-excited bubble screens are compared to class-based solutions. This comparison shows that a QMOM approach can reach a broader range of physical conditions, and thus is expected to be useful for matching experimental configurations.   

A new Ghost Fluid Method for eliminating overheating errors in compressible multi-medium flows

We describe a new version of the Ghost Fluid Method (GFM) called the Efficient GFM (EGFM), capable of eliminating overheating errors observed at fluid interfaces in compressible multi-medium flows. Earlier approaches, while mitigating overheating errors to some extent, have not been able to eliminate them completely. The proposed algorithm makes use of a lemma in [1] to apply the exact boundary conditions at the interface. In addition, the isentropic and shock relations proposed in [2] are applied to fix the density values near the interface, and are also extended here to address velocity corrections. The EGFM algorithm is validated using a wide array of 1D single- as well as multi-medium shock tube problems and shock-interface interactions, shock-bubble interaction problems and the Richtmyer-Meshkov instability. When compared with the original GFM approach and its subsequent variants, the EGFM scheme proposed here is robust, and has been demonstrated to result in highly accurate solutions. $^{\mathrm{1}}$L. Xu {\&} T. Liu, J. Comput. Phys.,~\textbf{230}, 4975 (2011). $^{\mathrm{2}}$L. Xu, C. Feng {\&} T. Liu, Commun. Comput. Phys., \textbf{20(3)}, 619 (2016).   

An arbitrary Lagrangian Eulerian approach with exact mass conservation for the numerical simulation of a Taylor bubble problem

An arbitrary Lagrangian Eulerian approach is developed for incompressible multiphase fluid flows with exact mass conservation. The incompressible Navier-Stokes equations are discretized over unstructured moving quadrilateral/hexahedral meshes using the div-stable side-centered finite volume formulation. The pressure field is treated to be discontinuous across the interface. The surface tension term is considered as a force tangent to the interface. Interface kinematic boundary condition in normal direction is applied by satisfying both local and global discrete geometric conservation law [Guventurk and M. Sahin, 2017]. The resulting algebraic equations are solved in a fully coupled manner and preconditioning is performed by using a new matrix factorization proposed by Cetin and Sahin [2019]. The numerical algorithm is validated by simulating a rising Taylor bubble in a tube in two- and three dimensions. The numerical results are consistent with the results of an experimental work of Bugg and Saad [2017].   

An interface-preserving and conservative phase-field method for the surface tension dynamics

We present an interface-preserving and conservative Allen-Cahn phase-field formulation for the modeling of incompressible two-phase flows with surface tension dynamics. To preserve the hyperbolic tangent interface profile, the mobility coefficient is adjusted adaptively as a function of gradients of the velocity and the order parameter in the diffuse interface region in such a way that the free energy minimization properly opposes the convective distortion. In the phase-field formulation, the mass conservation is achieved by enforcing a Lagrange multiplier with both temporal and spatial dependence on the phase-field function. We integrate the interface-preserving and conservative phase-field formulation with the incompressible Navier-Stokes equations and the continuum surface tension force model for the simulation of incompressible two-phase flows. A positivity preserving scheme designed for the boundedness and stability of the solution is employed for the variational discretization using unstructured finite elements. We examine the convergence and accuracy of the Allen-Cahn phase-field solver through a generic one-dimensional bistable diffusion-reaction system in a stretching flow. We quantify and systematically assess the relative interface thickness error and the relative surface tension force error with respect to the convective distortion parameter. Two- and three-dimensional rising bubble cases are further simulated to examine the effectiveness of the proposed model on the volume-preserving mean curvature flow and the interface-preserving capability.   

An approach to address numerical stiffness in the simulation of cavitating flows

A numerical approach based on the preconditioning and the dual-time stepping methodologies is proposed for LES of cavitating flows at low Mach numbers. The methodology is based on a fully compressible homogeneous mixture model as discussed in Gnanaskandan and Mahesh (IJMF. vol. 70, April 2016) which has shown promising results for capturing sheet to cloud cavitation. The goal of the present work is to allow the method to simulate wetted conditions, incipient cavitation, and cavitation inception regimes that typically require low free-stream nuclei, which imply predominantly low Mach number regions in the water. The numerical formulation of the preconditioning equations is discussed in detail. The method is used to simulate (i) cavitation inception over the unsteady flow over a cylinder at Re$=$200, and (ii) the wetted conditions over a marine propeller at Re$=$894000 using LES. Overall, a significant saving in the total run-time as compared to the original solver is obtained, without compromising accuracy.   

Modeling and simulation of N-phase-M-component incompressible flows

A consistent and conservative model is proposed for N-phase-M-component incompressible flows, where N (N≥1) and M (M≥0) are the numbers of phases and components, respectively. Phases are immiscible with each other, while components are dissolvable in given phases. Several consistency conditions are proposed for multiphase and multicomponent flows to avoid unphysical behaviors, e.g., incompatibility between the mass and momentum equations, generating fictitious phases, local void, or overfilling, and leakage of components at phase boundaries. The mass of individual phases, the amount of each component, and the momentum of the fluid mixture are conserved by the proposed model. The model also satisfies the 2nd law of thermodynamics and Galilean invariance. A 2nd-order scheme is developed, which preserves the consistency and conservation properties of the model, and the numerical solution is demonstrated to preserve the Galilean invariance and energy law. Various challenging multiphase and multicomponent flows, including large density and/or viscosity ratios, can be effectively studied by using the proposed model and scheme. The proposed model and scheme are also applicable for multiphase problems where different phases have different miscibility with each other.   

A vortex-particle mesh method for mixed rigid and soft body fluid--structure interaction

We outline a 2D algorithm for solving incompressible flow--structure interaction problems for rigid and soft bodies, within the framework of remeshed-vortex methods. We adopt a two-fluid model to represent the solid and fluid phases on an Eulerian grid, separated by a diffused interface. Rigid solids are treated using the Brinkman penalization approach while a reference map technique is used to account for elastic stresses in hyperelastic solids. We then test our approach against a variety of increasingly challenging benchmarks, and further illustrate the versatility of our solver by studying elastic effects in the context of self-propelled active swimmers, multi-body contact, heat transfer and rectified viscous streaming. Through these illustrations, we showcase the ability of the solver to deal with different constitutive models and boundary conditions, solve multi-physics problems and achieve fast time-to-solution by sidestepping CFL time-step restrictions.   

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

In the present study, level set formulations are used to track solid-fluid interface as well as to track a dynamic grid which captures solid deformation. The deformable solid is governed by a Neo-Hookean 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. This solid force is diffused into the fluid by a Heaviside function. Linear extrapolation has been used to reconstruct the dynamic grid in prevention of the distortion in the grid by artificial advection outside the solid region. Validation simulations have been performed in comparison with literatures. Implementation to thin layer geometries has also been explored by defining a thin region of solid for a few computational cells where solid shear modulus quickly goes from solid to zero. Numerical results are shown to present the robustness of this method.   

A kinetic energy and entropy preserving scheme for compressible two-phase turbulence simulations

Compressible two-phase turbulent flows have wide range of applications such as high-speed atomization, droplet combustion, bubble cavitation, cloud dynamics, Rayleigh-Taylor and Richtmyer-Meshkov instability flows. Accurate numerical modeling of such flows requires a method that is non-dissipative and is stable for high-density ratios and at high Reynolds (\textit{Re}) numbers. For a compressible flow, it is known that discrete conservation of kinetic energy is not a sufficient condition for numerical stability unlike in incompressible flows (Honein {\&} Moin, \textit{JCP}, 2004; Chandrashekar, \textit{CCP}, 2013; Kuya et al., \textit{JCP}, 2018). In this study, we adopt the recently developed diffuse-interface method (Jain, Mani {\&} Moin, \textit{JCP}, 2020) that is non-dissipative. We propose discrete consistency conditions between the numerical fluxes of mass, momentum, kinetic energy, and internal energy, such that an exact conservation of kinetic energy and approximate conservation of entropy of each phase are achieved in the absence of pressure work, viscosity, and thermal diffusion. To this end, we present numerical simulations of compressible two-phase turbulent flows at finite and infinite \textit{Re}, to illustrate the stability and accuracy of the method in isotropic turbulence and Taylor-Green vortex flows.   

Modeling scalar transport in two-phase flows with a diffuse-interface method

Transport of scalar quantities in two-phase flows is an important problem that finds applications in wide range of natural phenomena and industrial processes. In a wide range of applications transport coefficients on the two sides of the interface can be drastically different resulting in an effective confinement of the scalar in one of the phases in the time scales of interest. This results in the formation of sharp gradients in the scalar concentration at the interface, which is a numerically challenging problem to simulate. To overcome this challenge, we developed a novel transport model (Jain {\&} Mani, \textit{CTR Annual Research Briefs}, 2019) for the simulation of scalar quantities in two-phase flows with a conservative diffuse-interface method. The provable strengths of the model are that: (a) it maintains the positivity property of the scalar concentration field despite using the central-difference scheme, (b) the transport of the scalar field is consistent with the transport of the volume fraction field, and therefore, prevents the artificial leakage of the scalar across the interface. We present numerical simulations using the proposed model in a wide range of two-phase flow regimes, spanning laminar to turbulent flows.   

A Deep Neural Network-Based Approach for the Force Predictions of Particulate Multiphase Flows

The hydrodynamics forces within random distributions of fixed monodisperse spherical particles are predicted using deep neural network-based technique. The test data are obtained from particle resolved-direct numerical simulations. The fully resolved simulations are based on direct forcing immersed boundary method. The methodology is examined for wide ranges of Reynolds number and mean volume fraction. A data augmentation strategy is implemented to achieve 3D rotation-invariant trained network and to improve the accuracy of the predictions. To evaluate the performance of the model, direct force predictions are compared with the corresponding fully resolved solutions.   

Large eddy simulation of sprays using the spectral-element method.

A novel approach for modeling sprays from high-speed liquid injection is presented here. The liquid phase is modeled stochastically by Lagrangian parcels, while the gas phase is modeled in an Eulerian reference frame using the spectral-element method. The coupling between gas and liquid phase is described. The gas-phase solution is stabilized using a high-pass filter relaxation term, which is suitable for LES using spectral methods and bypasses the need for Smagorinsky-type sub-grid scale models. Sensitivity of the solution to grid resolution, temporal resolution, and spray parameters is analyzed. Simulation results are compared quantitatively and qualitatively against experimental data from the Engine Combustion Network. Some of the challenges and advantages of the current approach for modeling sprays are discussed. The aim of this work is to enable high-fidelity internal combustion engine simulations.   

Verification of a Dual-Scale Approach to Modeling Sub-Filter Shear-Induced Instabilities

A method to predict sub-filter shear-induced velocities on a liquid-gas material interface for use in a dual-scale LES-DNS method is verified with Direct Numerical Simulations. The dual-scale method maintains both an LES flow solver grid and an additional Refined Local Surface Grid (RLSG) in flow solver cells that contain a material interface. The RLSG is tasked with maintaining a fully resolved realization of the material interface and transporting the interface with sub-grid velocities that model sub-grid physical effects. The sub-grid model reconstructs velocities due the shear-induced instabilities using the Orr-Sommerfeld equations and appropriate boundary and interface conditions. The Orr-Sommerfeld equations are then solved by a Chebyshev collocation method and the results are compared against DNS solutions of the same flow conditions. The sub-grid model is shown to accurately predict the dynamics of the filtered material interface in the presence of unstable sub-filter corrugations within a linear regime of growth, and the model is shown to be well suited in predicting the dynamics of an interface with small random fluctuations.   

Simulating Primary Atomization of a Liquid Jet in a Supersonic Crossflow

Numerical simulations can provide valuable insight to understanding the atomization process of fuel within the combustor of a scramjet engine. A high-fidelity extension to a diffuse-interface approach for simulating compressible multiphase flows (Garrick et al., 2017) was recently developed (Fritz et al., 2019) targeting the accurate simulation of primary atomization. To this end, a novel, unstructured, cell-based adaptive mesh refinement (AMR) framework (Ballesteros, 2019) is used to achieve higher mesh resolution in regions of liquid and shock discontinuities. Furthermore, low dissipation, high spatial accuracy is obtained by using a WENO-Z (Borges et al., 2008) reconstruction, while the numerical smearing of the material interface is controlled using a THINC reconstruction scheme (Shyue and Xiao, 2014). Curvature is computed using a stretched variant of the standard height function method (Cummins et al., 2005) to account for the smearing of the material interface. Simulations of liquid jet in supersonic crossflow are performed to study the jet penetration height, droplet size and velocities in the near field of the injected plume.   

Multicomponent Near-critical Flow Simulations: Reducing Spurious Pressure Oscillations

Spurious pressure oscillations are the most common numerical instabilities observed in multiphase and multicomponent flow problems near the critical point. A diffuse-interface model is developed to simulate transcritical mixing in multispecies and multiphase systems where spurious pressure oscillations are problematic. To reduce the spurious pressure oscillations, three methods have been proposed and implemented: (1) artificially thickening the interface between different species, (2) reconstruction of the primitive variables in the characteristics space, and (3) developing a hybrid method that switches between quasi-conservative double-flux and the classical fully-conservative numerical procedures based on the changes in the effective specific heat ratio and the effective reference internal energy. Characteristic-wise reconstruction tends to reduce the spurious pressure oscillations compared to primitive-wise reconstruction. The hybrid model is found to effectively reduce the magnitude of the spurious pressure oscillations, the loss of energy conservation, and provides more accurate results for all tested cases.   

Eulerian Label Advection method for bubble tracking in two-fluid Navier-Stokes simulations

The dynamics of bubbles near the air-sea interface have an important impact on many free-surface entrainment driven problems including air-sea gas exchange, waves in the surf-zone, and wake around a vessel. The dynamics are driven by breakup, coalescence, degassing, entrainment, and dissolution, and understanding these processes individually in numerical simulations requires a reliable method to track individual bubbles through time. Tracking methods that solve an inverse problem based Lagrangian properties can be problematic in the presence of multiple processes, which is inherent near the air-sea interface. We present a forward method that uses an Eulerian advection step built upon the conservative Volume of Fluid (cVOF) method to identify the sources of fluid in a grid cell based on labels previously assigned using informed connected-component labeling (ICL). Eulerian Label Advection (ELA) allows us to identify a bubble's history from its air sources, making Lagrangian tracking achievable using a forward Eulerian method. We demonstrate using 3D numerical simulations of two-fluid turbulent flow that ELA is able to track bubbles in the presence of multiple processes, ultimately enabling accurate and independent quantification of the processes involved bubble dynamics.   

Methodology for DNS Data-driven Machine Learning Bubble Drag Model and Its Integration to OpenFOAM

This work aims to develop a two-phase DNS data-driven bubble drag model and to implement it into a multiphase CFD simulation. To achieve the goal, a Tensorflow (TF)-OpenFOAM(OF) integration interface has been established. The interface can call and make machine learning model predict quantities of interest on the fly. A benchmark case for the bubble drag coefficient is proposed to validate the interface. A feedforward neural network was utilized to approximate the drag correlation (Tomiyama et al., 1998) using artificially generated data. Results of the integration show good consistency in radial void fraction and velocity profiles. Next, actual DNS bubble tracking datasets are used as a data source (Fang et al., 2017, Cambareri et al., 2019). The data segments where bubbles have quasi-stable main-stream velocity were filtered out for drag coefficient calculation. The DNS-informed model predicts drag coefficient by taking bubble Reynolds and Eötvös number as input. The model is applied in an Euler-Euler two-phase flow simulation of a bubbly pipe flow in OF. The closure terms, except the drag model, utilize the baseline model of Liao et al. (2020). The results of radial void fraction and velocity profiles are compared to reference by baseline.   

A multiscale numerical model for the multiphase flow in porous media

Accurate prediction of multiphase flow dynamics in porous media is difficult due to the challenges associated with capturing the interfacial deformations, resolving the subgrid thin films, and accounting for the complex solid geometries. Here, we propose an accurate and efficient multiscale numerical framework for modeling the pore scale multiphase flow in the presence of complex solid geometries and thin films. Within this framework, we solve the Navier-Stokes equations using a projection method with approximate factorization and fractional time stepping on a staggered Cartesian grid, and employ a topologically preserved level set method to capture the evolution of immiscible fluid-fluid interfaces. In order to account for the effect of complex solid boundaries and capture the evolution of thin films, we couple an immersed boundary method based on a direct forcing approach with a sub-grid thin-film model. We evaluate the accuracy of our multiscale framework by studying the immiscible oil-water displacement in a converging-diverging capillary tube for both drainage and imbibition scenarios, and compare the results with a fully resolved solution on a much finer grid resolution.   

A Phase-Field Method for Computing Elasto-Capillary Flows of Liquid Crystals

We propose a phase-field model to compute elasto-capillary flows of nematic liquid crystals. The formulation provides a consistent description of nematic microstructure, in particular topological defects, and recovers macroscopic surface tension and liquid crystal anchoring stress. This is made possible by incorporating the Beris-Edwards theory for nematic hydrodynamics based on a tensor order parameter in a phase-field formalism approximating the sharp-interface limit. We apply the method to the problem of drop retraction in the presence of a nematic-isotropic interface. We characterize a variety of different cases and examine their dynamics. In this regime, our calculation uncovers a proportional relationship between steady-state drop deformation and the elasto-capillary length, signifying competition between bulk distortional elasticity and surface tension, mediated by the anchoring energy. The new computational framework opens doors to a large class of fundamental problems concerning colloidal interaction in coupled elasto-capillary fields.   

Solving compressible, multiphase flows with pressure projection and relaxation

Simulating compressible, multiphase flows presents unique challenges in terms of stability, with a wide range of scales and discontinuities from shocks and material interfaces. A well-rounded solver must address how to represent interfaces, transport discontinuous variables, and limit numerical dissipation. Also, since the sound speed and Mach number can vary significantly, pressure projection schemes are useful in providing stability and computational efficiency by removing the acoustic limit on the time step. However, the application of projection methods to these flows is limited in the literature, leaving open questions as to how the pressure in each phase should contribute to the pressure projection equation. Pressure relaxation frameworks described by Saurel et al. (JCP 2009) and Pelanti and Shyue (JCP 2014) provide a straightforward approach to reconciling the pressures in each phase, but these have been exclusively applied in Riemann-based solvers. In this work, we integrate pressure projection and pressure relaxation in the same framework, enabling improvements in stability via consistency. We demonstrate the capabilities of our scheme by simulating several test cases, including underwater bubble-wave interactions and a liquid jet in supersonic crossflow.   

Simulation and Modeling of Droplet Collisions using a Film-Capturing VOF Method

Due to the presence of the liquid-gas interface and corresponding discontinuities in physical properties, simulating multiphase flows can be difficult. Furthermore, interface instabilities and their interaction with the surrounding flow field can occur over a range of temporal and spatial scales spanning multiple orders of magnitude. Even with improvements in computing power, it is not possible to directly resolve all relevant scales in the flow and sub-grid scale models are needed to enable the simulation of realistic multiphase systems. In this talk, we discuss the modeling of droplet coalescence and rebound during droplet collisions. These models are developed in the context of a novel interface reconstruction algorithm for geometric volume of fluid methods, R2P, which is capable of capturing sub-grid scale fluid films. First, we implement existing droplet collision models for use with R2P and compare their results to experiments. We then go on to discuss opportunities for physics-based modeling informed by resolved flow-features and film-metrics extracted from the R2P interface reconstruction.   

Surface Tension Advancements to a Dual-Scale Approach for Turbulent Phase Interfaces

Direct Numerical Simulation remains an expensive task for atomization simulations. To decrease the burden of DNS, a dual-scale modeling approach (Gorokhovski and Herrmann, 2008) that describes 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 turbulent eddy reconstruction by combined approximate deconvolution and non-linear spectral enrichment (Bassenne et al. 2019) and sub-filter surface tension model (Herrmann 2013) are compared to DNS results for a phase interface in a homogeneous isotropic turbulent flow at two different Weber numbers.   

Paraboloid-based models of liquid-gas interfaces from volume fraction data

In this talk, we investigate the use of semi-analytical volume integration for extracting a paraboloid model of a captured interface in the context of volume of fluid (VOF). In contrast with the height-function (HF) method, which also represents the interface as a paraboloid, we allow for the paraboloid to be arbitrarily rotated, making possible the application of the volume integration to unstructured meshes. We evaluate the performance of the paraboloid model on a range of canonical problems and discuss the calculation of error metrics such as interface normal, curvature, and neighbor-cell volume fraction. Finally, we investigate the robustness of the model to volume fraction error and the reliability of piecewise-linear interface reconstruction (PLIC) information used in the formation of the paraboloid model.