Assessment of an energy-based surface tension model for simulation of two-phase flows using second-order phase field methods

Second-order phase field models have emerged as an attractive option for capturing the advection of interfaces in two-phase flows. Prior to these, state-of-the-art models based on the Cahn-Hilliard equation, which is a fourth-order equation, allowed for the derivation of surface tension models through thermodynamic arguments. In contrast, the second-order phase field models do not follow a known energy law, and deriving a surface tension term for these models using thermodynamic arguments is not straightforward. In this work, we justify that the energy-based surface tension model from the Cahn-Hilliard context can be adopted for second-order phase field models as well and assess its performance. We test the surface tension model on three different second-order phase field equations; the conservative diffuse interface model of Chiu and Lin [1], and two models based on the modified Allen-Cahn equation introduced by Sun and Beckermann [2]. Additionally, we draw the connection between the energy-based model with a localized variation of the continuum surface force (CSF) model. Using canonical tests, we illustrate the lower magnitude of spurious currents, better accuracy, and superior convergence properties of the energy-based surface tension model compared to the CSF model, which is a popular choice used in conjunction with second-order phase field methods, and the localized CSF model. Importantly, in terms of computational expense and parallel efficiency, the energy-based model incurs no penalty compared to the CSF models.Comment: 13 pages, 5 figures, Revision submitted to Journal of Computational Physic

Effect of interpolation kernels and grid refinement on two way-coupled point-particle simulations

The predictive capability of two way--coupled point-particle Euler-Lagrange model in accurately capturing particle-flow interactions under grid refinement, wherein the particle size can be comparable to the grid size, is systematically evaluated. Two situations are considered, (i) uniform flow over a stationary particle, and (ii) decaying isotropic turbulence laden with Kolmogorov-scale particles. Particle-fluid interactions are modeled using only the standard drag law, typical of large density-ratio systems. A zonal, advection-diffusion-reaction (Zonal-ADR) model is used to obtain the undisturbed fluid velocity needed in the drag closure. Two main types of interpolation kernels, grid-based and particle size--based, are employed. The effect of interpolation kernels on capturing the particle-fluid interactions, kinetic energy, dissipation rate, and particle acceleration statistics are evaluated in detail. It is shown that the interpolation kernels whose width scales with the particle size perform significantly better under grid refinement than kernels whose width scales with the grid size. Convergence with respect to spatial resolution is obtained with the particle size--based kernels with and without correcting for the self-disturbance effect. While the use of particle size--based interpolation kernels provide spatial convergence and perform better than kernels that scale based on grid size, small differences can still be seen in the converged results with and without correcting for the particle self-disturbance. Such differences indicate the need for self-disturbance correction to obtain the best results, especially when the particles are larger than the grid size.Comment: Submitted to International Journal of Multiphase Flo

Direct numerical simulation of electrokinetic transport phenomena: variational multi-scale stabilization and octree-based mesh refinement

Finite element modeling of charged species transport has enabled the analysis, design, and optimization of a diverse array of electrochemical and electrokinetic devices. These systems are represented by the Poisson-Nernst-Planck (PNP) equations coupled with the Navier-Stokes (NS) equation. Direct numerical simulation (DNS) to accurately capture the spatio-temporal variation of ion concentration and current flux remains challenging due to the (a) small critical dimension of the electric double layer (EDL), (b) stiff coupling, large advective effects, and steep gradients close to boundaries, and (c) complex geometries exhibited by electrochemical devices. In the current study, we address these challenges by presenting a direct numerical simulation framework that incorporates: (a) a variational multiscale (VMS) treatment, (b) a block-iterative strategy in conjunction with semi-implicit (for NS) and implicit (for PNP) time integrators, and (c) octree based adaptive mesh refinement. The VMS formulation provides numerical stabilization critical for capturing the electro-convective instabilities often observed in engineered devices. The block-iterative strategy decouples the difficulty of non-linear coupling between the NS and PNP equations and allows using tailored numerical schemes separately for NS and PNP equations. The carefully designed second-order, hybrid implicit methods circumvent the harsh timestep requirements of explicit time steppers, thus enabling simulations over longer time horizons. Finally, the octree-based meshing allows efficient and targeted spatial resolution of the EDL. These features are incorporated into a massively parallel computational framework, enabling the simulation of realistic engineering electrochemical devices. The numerical framework is illustrated using several challenging canonical examples

Industrial scale large eddy simulations (LES) with adaptive octree meshes using immersogeometric analysis

We present a variant of the immersed boundary method integrated with octree meshes for highly efficient and accurate Large-Eddy Simulations (LES) of flows around complex geometries. We demonstrate the scalability of the proposed method up to $\mathcal{O}(32K)$ processors. This is achieved by (a) rapid in-out tests; (b) adaptive quadrature for an accurate evaluation of forces; (c) tensorized evaluation during matrix assembly. We showcase this method on two non-trivial applications: accurately computing the drag coefficient of a sphere across Reynolds numbers $1-10^6$ encompassing the drag crisis regime; simulating flow features across a semi-truck for investigating the effect of platooning on efficiency.Comment: Accepted for publication at Computer and Mathematics with Application

Energy stable and conservative numerical schemes for simulating two-phase flows using Cahn-Hilliard Navier Stokes equations

Developing accurate, stable and thermodynamically consistent numerical methods to simulate two-phase flows is critical for many applications. We develop numerical methods to simulate two-phase flows with deforming interfaces at various density contrasts by solving thermodynamically consistent Cahn-Hilliard Navier-Stokes equations. We develop three essentially unconditionally energy-stable Crank-Nicolson-type time integration schemes. The first two time integration schemes are fully implicit based on pressure-stabilization techniques. The third approach utilizes projection method to decouple the pressure to improve the efficiency of the fully implicit method. We rigorously prove energy stability of the semi-discrete schemes for the approaches with pressure-stabilization approaches. We also prove the existence of solutions of the advective-diffusive Cahn-Hilliard operator. In the first approach we use a pressure coupled approach with both Navier-Stokes and Cahn-Hilliard equations are solved using fully implicit non-linear schemes in a block iterative manner. In the second approach we extend the block iterative method in the first approach, to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The fully coupled approach method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The first two methods are based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure in the first two approaches. In the third approach we present a projection based framework extending the fully implicit method in the second approach, to a block iterative, hybrid semi-implicit-fully-implicit in time method. We use a semi-implicit time discretization for Navier-Stokes and a fully-implicit time discretization for Cahn-Hilliard equations. This third approach method requires Newton iteration only for Cahn-Hilliard resulting faster solution time at relatively larger time steps. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure. However, pressure is decoupled using a projection step resulting in two linear positive semi-definite systems for velocity and pressure instead of the saddle point system of a pressure-stabilized method. We deploy all three approaches on a massively parallel numerical implementation using fast octree-based adaptive meshes. All the linear systems are solved using efficient and scalable algebraic multigrid (AMG) method. A detailed scaling analysis of all three solvers is presented. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems for a large range of density ratios

Energy stable and conservative numerical schemes for simulating two-phase flows using Cahn-Hilliard Navier Stokes equations

Developing accurate, stable and thermodynamically consistent numerical methods to simulate two-phase flows is critical for many applications. We develop numerical methods to simulate two-phase flows with deforming interfaces at various density contrasts by solving thermodynamically consistent Cahn-Hilliard Navier-Stokes equations. We develop three essentially unconditionally energy-stable Crank-Nicolson-type time integration schemes. The first two time integration schemes are fully implicit based on pressure-stabilization techniques. The third approach utilizes projection method to decouple the pressure to improve the efficiency of the fully implicit method. We rigorously prove energy stability of the semi-discrete schemes for the approaches with pressure-stabilization approaches. We also prove the existence of solutions of the advective-diffusive Cahn-Hilliard operator. In the first approach we use a pressure coupled approach with both Navier-Stokes and Cahn-Hilliard equations are solved using fully implicit non-linear schemes in a block iterative manner. In the second approach we extend the block iterative method in the first approach, to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The fully coupled approach method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The first two methods are based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure in the first two approaches. In the third approach we present a projection based framework extending the fully implicit method in the second approach, to a block iterative, hybrid semi-implicit-fully-implicit in time method. We use a semi-implicit time discretization for Navier-Stokes and a fully-implicit time discretization for Cahn-Hilliard equations. This third approach method requires Newton iteration only for Cahn-Hilliard resulting faster solution time at relatively larger time steps. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure. However, pressure is decoupled using a projection step resulting in two linear positive semi-definite systems for velocity and pressure instead of the saddle point system of a pressure-stabilized method. We deploy all three approaches on a massively parallel numerical implementation using fast octree-based adaptive meshes. All the linear systems are solved using efficient and scalable algebraic multigrid (AMG) method. A detailed scaling analysis of all three solvers is presented. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems for a large range of density ratios

Breakup dynamics in primary jet atomization using mesh- and interface- refined Cahn-Hilliard Navier-Stokes

We present a technique to perform interface-resolved simulations of complex breakup dynamics in two-phase flows using the Cahn-Hilliard Navier-Stokes equations. The method dynamically decreases the interface thickness parameter in relevant regions and simultaneously increases local mesh resolution, preventing numerical artifacts. We perform a detailed numerical simulation of pulsed jet atomization that shows a complex cascade of break-up mechanisms involving sheet rupture and filament formation. To understand the effect of refinement on the breakup, we analyze the droplet size distribution. The proposed approach opens up resolved simulations for various multiphase flow phenomena.This is a pre-print of the article Khanwale, Makrand A., Kumar Saurabh, Masado Ishii, Hari Sundar, and Baskar Ganapathysubramanian. "Breakup dynamics in primary jet atomization using mesh-and interface-refined Cahn-Hilliard Navier-Stokes." arXiv preprint arXiv:2209.13142 (2022). DOI: 10.48550/arXiv.2209.13142. Attribution 4.0 International (CC BY 4.0). Copyright 2022 The Authors. Posted with permission

Scalable adaptive algorithms for next-generation multiphase simulations

The accuracy of multiphysics simulations is strongly contingent up on the finest resolution of mesh used to resolve the interface. However, the increased resolution comes at a cost of inverting a larger matrix size. In this work, we propose algorithmic advances that aims to reduce the computational cost without compromising on the physics by selectively detecting the key regions of interest (droplets/filaments) that requires significantly higher resolution. The overall framework uses an adaptive octree-based mesh generator, which is integrated with PETSc's linear algebra solver. We demonstrate the scaling of the framework up to 114,688 processes on TACC Frontera. Finally we deploy the framework to simulate primary jet atomization on an \textit{equivalent} 35 trillion grid points - 64× greater than the state-of-the-art simulations.This is a pre-print of the article Saurabh, Kumar, Masado Ishii, Makrand A. Khanwale, Hari Sundar, and Baskar Ganapathysubramanian. "Scalable adaptive algorithms for next-generation multiphase simulations." arXiv preprint arXiv:2209.12130 (2022). DOI: 10.48550/arXiv.2209.12130. Attribution 4.0 International (CC BY 4.0). Copyright 2022 The Authors. Posted with permission

Direct numerical simulation of electrokinetic transport phenomena in fluids: variational multi-scale stabilization and octree-based mesh refinement

Computational modeling of charged species transport has enabled the analysis, design, and optimization of a diverse array of electrochemical and electrokinetic devices. These systems are represented by the Poisson-Nernst-Planck (PNP) equations coupled with the Navier-Stokes (NS) equation. Direct numerical simulation (DNS) to accurately capture the spatio-temporal variation of ion concentration and current flux remains challenging due to the (a) small critical dimension of the diffuse charge layer (DCL), (b) stiff coupling due to fast charge relaxation times, large advective effects, and steep gradients close to boundaries, and (c) complex geometries exhibited by electrochemical devices. In the current study, we address these challenges by presenting a direct numerical simulation framework that incorporates (a) a variational multiscale (VMS) treatment, (b) a block-iterative strategy in conjunction with semi-implicit (for NS) and implicit (for PNP) time integrators, and (c) octree based adaptive mesh refinement. The VMS formulation provides numerical stabilization critical for capturing the electro-convective flows often observed in engineered devices. The block-iterative strategy decouples the difficulty of non-linear coupling between the NS and PNP equations and allows the use of tailored numerical schemes separately for NS and PNP equations. The carefully designed second-order, hybrid implicit methods circumvent the harsh timestep requirements of explicit time steppers, thus enabling simulations over longer time horizons. Finally, the octree-based meshing allows efficient and targeted spatial resolution of the DCL. These features are incorporated into a massively parallel computational framework, enabling the simulation of realistic engineering electrochemical devices. The numerical framework is illustrated using several challenging canonical examples.This is a preprint from Kim, Sungu, Kumar Saurabh, Makrand A. Khanwale, Ali Mani, Robbyn K. Anand, and Baskar Ganapathysubramanian. "Direct numerical simulation of electrokinetic transport phenomena: variational multi-scale stabilization and octree-based mesh refinement." arXiv preprint arXiv:2301.05985 (2023). doi: https://doi.org/10.48550/arXiv.2301.05985. Copyright the authors 2024. CC BY