Sharp Interface Methods Research Articles

Comparison of Sharp and Diffuse Interface Methods for Radially Symmetric Compressible Multi-Medium Fluid Flows

Comparison of Sharp and Diffuse Interface Methods for Radially Symmetric Compressible Multi-Medium Fluid Flows

Sharp Interface Methods for Simulation and Analysis of Free Surface Flows with Singularities: Breakup and Coalescence

A common feature of many free surface flows—drop/bubble breakup or coalescence and film/sheet rupture—is the occurrence of hydrodynamic singularities. Accurately computing such flows with continuum mechanical, multidimensional free surface flow algorithms is a challenging task given these problems’ multiscale nature, which necessitates capturing dynamics occurring over disparate length scales across 5–6 orders of magnitude. In drop breakup, the thinning of fluid threads that form and eventually pinch-off must be simulated until the thread's radius is about 10 nm. When two drops approach one another, the thickness of the fluid film separating them must fall below 10 nm before coalescence is said to have occurred. If the initial drop radii are 1 mm, simulations must remain faithful to the physics as thread radius or film thickness falls from 10−3 m to below 10−8 m. Here we review significant findings in interfacial flows with hydrodynamic singularities spearheaded by sharp interface algorithms. These multidimensional algorithms can achieve resolution that to date has only been possible with the use of simple 1D evolution equations.

The immersed boundary method for confined flows: Numerical diffusion and simulation accuracy of a boundary retraction scheme

Diffuse-interface immersed boundary methods (IBM) have been successfully applied to numerous complex fluid–structure interaction problems because of their simple and efficient implementation. Despite their ability to suppress numerical oscillations significantly compared to sharp-interface methods, the diffuse interface is likely to reduce simulation accuracy of the flow field around the solid boundary. The present work investigates the diffusive effects of IBM and the mitigation method for surface-confined particulate flows by comparing results to sharp-interface methods. It is found that increasingly-confined geometries accentuate interface diffusion effects and decrease simulation accuracy. To minimise the diffusive effects of IBM, a boundary retraction scheme is used and its effectiveness is examined, in particular for particles in close contact where the diffuse interfaces overlap. It is shown that this simple implementation is capable of alleviating interface diffusion errors, thus increasing accuracy while limiting computational costs. With an optimal boundary retraction scheme, IBM can successfully capture fluid–structure interactions at different degrees of confinement, comparable to sharp-interface methods.

Smoothed boundary method for simulating incompressible flow in complex geometries

Simulating flow through porous media with explicit considerations of complex microstructures is very challenging using conventional sharp-interface methods because of the difficulties in generating meshes conformal to complex geometries. In this work, a diffuse interface embedded boundary method known as the Smoothed Boundary Method (SBM) is utilized to facilitate simulations of fluid dynamics involving complex geometries. In diffuse-interface methods, the geometry is described by a domain parameter. The SBM allows the straightforward reformulation of the time-dependent Navier–Stokes equations in terms of this domain parameter, using only differentiation identities. Thus, enforcing the appropriate boundary conditions at the irregular embedded boundary is greatly simplified. Adaptive mesh refinement is used to increase the accuracy of the diffuse interface method by allowing a thinner interfacial thickness to be used in the domain parameter. The SBM-formulated Navier–Stokes equations are solved with the Finite Difference Method on refined mesh systems. Sharp-interface Finite Element Method simulations using the commercial software COMSOL on body-conforming meshes are also provided for comparison. Favorable agreement between the two methods is observed. Since it is no longer necessary for the mesh to conform to the complex geometry, the grid system for the SBM simulations can be generated rapidly and without additional manual interventions, making the entire simulation process more expedient.

Complex Electrode Microstructure Simulations using a Smoothed Boundary Method with Adaptive Mesh Refinement

Lithium-ion batteries have gained significant attention in the research community due to their increasing utilization in various applications for energy storage. However, the dependence of macroscopic battery performance on microscopic electrode structures is not fully understood, thus hindering a systematic, comprehensive manner of optimizing electrode performance via the microstructures. Computer simulation can serve as a powerful tool to close the knowledge gaps in our understanding of microstructure phenomena. In this work, we present a smoothed boundary method (SBM) electrochemical simulation framework with adaptive mesh refinement (AMR). This method allows the use of mesh non-conforming to the domain of interest when solving the governing equations. Therefore, the arduous tasks of generating meshes conforming to the highly complex electrode microstructures required in the conventional sharp-interface methods can be circumvented. The accuracy of the SBM approach can be significantly enhanced with AMR. The material properties of Li x Ni1/3Mn1/3Co1/3O2 from literature data are parameterized to be the input of the simulations. One-dimensional and three-dimensional simulations are utilized to study the error behavior and demonstrate this framework’s capabilities. This simulation framework can be easily adapted and extended to study a wide variety of electrode microstructure phenomena in other electrochemical systems.

A dual resolution phase-field solver for wetting of viscoelastic droplets.

We present a new and efficient phase‐field solver for viscoelastic fluids with moving contact line based on a dual‐resolution strategy. The interface between two immiscible fluids is tracked by using the Cahn‐Hilliard phase‐field model, and the viscoelasticity incorporated into the phase‐field framework. The main challenge of this approach is to have enough resolution at the interface to approach the sharp‐interface methods. The method presented here addresses this problem by solving the phase field variable on a mesh twice as fine as that used for the velocities, pressure, and polymer‐stress constitutive equations. The method is based on second‐order finite differences for the discretization of the fully coupled Navier–Stokes, polymeric constitutive, and Cahn–Hilliard equations, and it is implemented in a 2D pencil‐like domain decomposition to benefit from existing highly scalable parallel algorithms. An FFT‐based solver is used for the Helmholtz and Poisson equations with different global sizes. A splitting method is used to impose the dynamic contact angle boundary conditions in the case of large density and viscosity ratios. The implementation is validated against experimental data and previous numerical studies in 2D and 3D. The results indicate that the dual‐resolution approach produces nearly identical results while saving computational time for both Newtonian and viscoelastic flows in 3D.

A Direct Ghost Fluid Method for Modeling Explosive Gas and Water Flows

This work presents a Direct Ghost Fluid Method (DGFM) as part of a two-fluid numerical framework suitable to model explosive gas and water flows resulting from underwater explosion (UNDEX). Due to the presence of explosive gas and water with shock waves in the modeling domain, classic Eulerian methods with inherent diffusion may not be effective. Numerical diffusion occurs due to nonphysical diffused density at material interfaces, which creates spurious pressure oscillations and significantly degrades the quality of the numerical results. To eliminate or minimize numerical diffusion, sharp interface methods having no mixed elements may be used in multifluid flow computations. The Direct Ghost Fluid Method (DGFM) described in this paper uses direct extrapolation of density (vice pressure) and tangential velocity from real to ghost fluid. The spurious pressure oscillations near the material interface are therefore minimized. One-, two-, and three-dimensional computational fluid dynamics (CFD) solvers that have DGFM as an essential part in their framework to model UNDEX interface conditions are developed, explored, and applied to the simulation of a series of benchmark problems. Excellent agreement is obtained among the simulations, the analytical solutions, and the experiments.

Development of a high-order continuous Galerkin sharp-interface immersed boundary method and its application to incompressible flow problems

The sharp-interface immersed boundary method is a strategy to impose boundary conditions on complex geometries while simplifying the meshing process. This method can offer a high-order of accuracy. Most sharp-interface methods have been developed in the context of the finite volume or the finite difference methods. In this paper, we introduce, verify and validate a novel high-order sharp-interface immersed boundary method in the context of the finite element method. We apply this method to the incompressible Navier–Stokes equations using a pressure-stabilizing/Petrov–Galerkin (PSPG) stabilization. We verify that we obtain a high order of convergence using a Taylor–Couette flow. We validate the results obtained for the drag, lift, and Strouhal number of the flow behind a cylinder at Re=200. We investigate the flow around a sphere at Re=100 and compare the drag force and the characteristics of the recirculating zones with experimental and numerical results obtained in the literature. Finally, the sharp-interface method is used to study a packing of 10 spheres at Re=50, and the results are compared to those obtained with a conformal mesh. It is shown that the sharp-interface immersed boundary preserves the high-order of the finite element scheme and accurately predicts the steady-state and transient flow around particles including the evaluation of the particle-fluid forces.

Numerical investigation of a plunging flat-plate airfoil using a diffuse interface immersed boundary method

This study investigates the capabilities of a diffuse interface immersed boundary method in the simulation of fluid flow over an oscillating flat-plate airfoil at moderately high Reynolds numbers where interacting unsteady vortical flows control the flow behavior. The recently developed and modified Smoothed Profile Method (SPM) is adopted as a diffuse interface immersed boundary solver to model the fluid-structure interaction (FSI) problem. While in general diffuse interface methods are simpler to implement relative to sharp-interface methods (SIMs), they have been thought to lose accuracy in the simulation of moderate Reynolds number flows. The simulation results show that the lift coefficient and vorticity contours from SPM match well with experiments. Moreover, our detailed analysis of the vorticity fluxes at the airfoil leading edge and the comparison with experiments also show that SPM can be used for moderate Reynolds number external aerodynamics problems applicable to micro-air vehicles.

A flux-enriched Godunov method for multi-material problems with interface slide and void opening

This work outlines a new three-dimensional diffuse interface finite volume method for the simulation of multiple solid and fluid components featuring large deformations, sliding and void opening. This is achieved by extending an existing reduced-equation diffuse interface method by means of a number of novel flux-modifiers and interface seeding routines that enable the application of different material boundary conditions. The method allows for slip boundary conditions across solid interfaces, material-void interaction, and interface separation. The method is designed to be straightforward to implement, inexpensive and highly parallelisable. This makes it suitable for use in large, multi-dimensional simulations that feature many complex materials and physical processes interacting over multiple levels of adaptive mesh refinement. Furthermore, the method allows for the generation of new interfaces in a conservative fashion and therefore naturally facilitates the simulation of high-strain rate fracture. Hence, the model is augmented to include ductile damage to allow for validation of the method against demanding physical experiments. The method is shown to give excellent agreement with both experiment and existing Eulerian interface tracking algorithms that employ sharp interface methods.

Adaptive interface thickness based mobility—Phase-field method for incompressible fluids

The ubiquity of two-phase systems has rendered them a subject of prime importance especially from numerical perspectives. Among several methods described in the literature, which are generally classified as sharp or diffuse interface methods, phase-field (diffuse interface) method has been at the paramount of several recent investigations owing to its several advantages, primarily to handle complex topological changes. Though several advancements have been made in the subject, one of the important challenges with this approach using Cahn-Hilliard equation lies in the determination of an appropriate value of mobility. Despite certain propositions in the literature in terms of non-dimensional numbers (Peclet and Cahn numbers), ambiguity in the velocity scale to be chosen for evaluating mobility poses a challenge for their straightforward extension to real systems. In addition it renders the system to be dependent on numerical parameters. In the current work, we address this problem using a new approach to calculate the mobility parameter in terms of local equilibrium interface thickness which in itself is evaluated from the phase-field parameter. Consequently, this helps to make the model independent of numerical parameters making it more reliable. We test this approach for several canonical and complex cases, such as the rise of lighter bubble in a heavier medium, bubble coalescence, Rayleigh-Taylor instability. The results are compared with those in literature or obtained using level set method. An excellent agreement is observed illustrating the potential of this approach to make phase-field model more prominent.

An accelerated conservative sharp-interface method for multiphase flows simulations

In this paper, we develop an accelerated conservative sharp-interface method for multiphase flows simulations. Traditional multiphase simulation methods use the minimum time step of all fluids obtained according to the CFL conditions to evolve the fluid states, which limits the computational efficiency, as the sound speed of one fluid may be much larger than of others. To address this issue, based on the original conservative sharp-interface methods, the present method is proposed by solving the governing equations of each individual fluid with the corresponding time step. Without violating the numerical stability requirement, the states of one fluid containing larger time-scale features will be evolved with a larger time step. The interaction step between two fluids is solved for synchronization, which is handled by interpolating the intermediate states of the fluid with larger time steps. In addition, an interfacial flux correction is applied to maintain the conservative property. The present algorithm has be coupled with a wavelet-based adaptive multi-resolution (MR) algorithm to achieve additional computational efficiency. A number of numerical tests indicate that the accuracy of the results obtained by the present method is comparable to of the original costly method, with a significant speedup. The speedup increases with the dimensions and resolution, which indicates its applications in 3D high resolution simulations of multiphase flows.

Smoothed Boundary Method Simulations of Complex Electrode Microstructures Using Adaptive Mesh

Rechargeable battery electrodes comprise of highly complex microstructures and convoluted electrolyte courses. These configurations dictate the macrohomogeneous properties and electrochemical performances of the electrodes. In addition to the geometric complexity, multiple concurrent physical mechanisms take place in the electrochemical processes of charge-discharge cycles. Understanding these detailed processes occurring in the complex electrode microstructure is the key to improve the design of electrodes. In this work, we present a framework to implement detailed electrochemical simulations in complex electrode microstructures. The framework employs the Smoothed Boundary Method(SBM)[1] which allows simple and fast mesh generation for complex geometries. In SBM, a continuous domain parameter is utilized to distinguish the phases of electrode particles and electrolyte, wherein the particle-electrolyte interface is defined by the region where the domain parameter transitions from zero to one in a continuous manner. With the electrochemical governing equations reformulated using the domain parameter, conformal mesh to the complex geometries is circumvented in the simulations. The SBM is implemented with Octree-based adaptive mesh refinement (AMR)[2], in which fine cubic meshes are generated near the particle-electrolyte interfaces while coarse cubic meshes are used in the bulk regions of particles and electrolyte. As a result, the diffuse interface in the SBM is effectively decreased and the accuracy of the electrochemical simulations is comparable to those of conventional sharp-interface methods. Intercalation electrode materials, namely NMC for the cathode and graphite for the anode, are used in the simulations to reveal the detailed electrochemical processes in the complex electrode microstructures and as a demonstration of the presented simulation framework. Material parameters in the simulations are directly obtained from experimental data to study the electrochemical phenomena occurring in the microstructures. The results can be used to extract macrohomogeneous electrode properties in the Porous Electrode Modeling for device-level simulations. Thus, it is shown that the presented simulation framework can serve as a useful tool for examining the effects of electrode microstructures and intrinsic material properties on the electrodes’ performance and as well as for electrode design.

A critical comparison of smooth and sharp interface methods for phase transition

In this study, the Local Front Reconstruction Method (LFRM) is extended to allow for the direct numerical simulation of flows with phase transition. The LFRM is a hybrid front tracking method without connectivity, which can easily handle complex topological changes. The expansion due to phase change is incorporated as a non-zero divergence condition at the interface. The energy equation is treated with two different approaches: smooth interface approach and sharp interface approach. The smooth interface approach uses a one fluid formulation to solve the energy equation with an interfacial source term accounting for phase change. This interfacial source term enforces the saturation temperature at the interface. However, in the sharp interface approach, the thermal properties are not volume-averaged near the interface and the saturation temperature is imposed as a boundary condition at the interface. A detailed mathematical formulation and numerical implementation pertaining to both approaches is presented. Both implementations are verified using 1D and 3D test cases and produce a good match with analytical solutions. A comparison of results highlights certain advantages of the sharp interface approach over the smooth interface approach such as better accuracy and convergence rate, reduced fluctuations in the velocity field and a physically bounded temperature field near the interface. Finally, both approaches are validated with a 3D simulation of the rise and growth of a vapor bubble in a superheated liquid under gravity, where a good agreement with experimental data is observed for the bubble growth rate.

Dispersion effects on the freshwater–seawater interface in subsea aquifers

Recent recognition of the widespread occurrence of freshwater beneath the ocean has renewed interest in approaches to understand and predict its extent. The most straightforward methodologies are based on the sharp-interface approximation, which neglects dispersive mechanisms. The understanding of dispersion effects on freshwater extents in coastal aquifers is based almost entirely on onshore aquifer situations. This study explores dispersion in offshore coastal aquifers, in terms of the steady-state freshwater extent, seawater circulation and freshwater discharge, through numerical experimentation. Results show that increasing dispersion causes a seaward shift in the interface toe location, as expected, whereas the interface tip shows a non-monotonic relationship with dispersion that depends on the contrast between aquifer and aquitard hydraulic conductivities. Higher dispersion leads to enhanced seawater recirculation rates and freshwater discharge, as opposed to non-monotonic relationships obtained previously for onshore aquifers. The mixing zone at the toe widens as dispersion increases, similar to onshore cases, whereas the mixing zone at the tip has a surprisingly non-monotonic relationship with dispersion. The dispersion relationships revealed in this study can be explained by counteractions between dispersion, density and advective forces, and refraction across the aquifer-aquitard interface, which in combination produce offshore aquifer behaviour that differs, in some ways, to the manner in which onshore aquifers respond to dispersive processes. Consequently, previous empirical corrections to sharp-interface methods (to account for dispersive effects) applied to onshore coastal aquifers are ineffective in their application to offshore settings.

Dynamics and stationary configurations of heterogeneous foams.

We consider the variational foam model, where the goal is to minimize the total surface area of a collection of bubbles subject to the constraint that the volume of each bubble is prescribed. We apply sharp interface methods to develop an efficient computational method for this problem. In addition to simulating time dynamics, we also report on stationary states of this flow for ≤ 21 bubbles in two dimensions and ≤ 17 bubbles in three dimensions. For small numbers of bubbles, we recover known analytical results, which we briefly discuss. In two dimensions, we also recover previous numerical results, computed using other methods. Particular attention is given to locally optimal foam configurations and heterogeneous foams, where the volumes of the bubbles are not equal. Configurational transitions are reported for the quasi-stationary flow where the volume of one of the bubbles is varied and, for each volume, the stationary state is computed. The results from these numerical experiments are described and accompanied by many figures and videos.

Coupling the fictitious domain and sharp interface methods for the simulation of convective mass transfer around reactive particles: Towards a reactive Sherwood number correlation for dilute systems

We suggest a reactive Sherwood number model for convective mass transfer around reactive particles in a dilute regime. The model is constructed with a simple external-internal coupling and is validated with Particle-Resolved Simulation (PRS). The PRS of reactive particle-fluid systems requires numerical methods able to handle efficiently sharp gradients of concentration and potential discontinuities of gradient concentrations at the fluid-particle interface. To simulate mass transfer from reactive catalyst beads immersed in a fluid flow, we coupled the Sharp Interface Method (SIM) to a Distributed Lagrange Multiplier/Fictious Domain (DLM/FD) two-phase flow solver. We evaluate the accuracy of our numerical method by comparison to analytic solutions and to generic test cases fully resolved by boundary fitted simulations. A previous theoretical model that couples the internal diffusion-reaction problem with the external advection-diffusion mass transfer in the fluid phase is extended to the configuration of three aligned spherical particles representative of a dilute particle-laden flow. Predictions of surface concentration, mass transfer coefficient and chemical effectiveness factor of catalyst particles are validated by DLM-FD/SIM simulations. We show that the model captures properly the effect of an internal first order chemical reaction on the overall respective reactive Sherwood number of each sphere depending on their relative positions. The proposed correlation for the reactive Sherwood number is based on an existing non-reactive Sherwood number correlation. The model can be later used in Euler/Lagrange or Euler/Euler modelling of dilute reactive particle-laden flows.

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Coherency misfit stresses and their related anisotropies are known to influence the equilibrium shapes of precipitates. Additionally, mechanical properties of the alloys are also dependent on the shapes of the precipitates. Therefore, in order to investigate the mechanical response of a material which undergoes precipitation during heat treatment, it is important to derive the range of precipitate shapes that evolve. In this regard, several studies have been conducted in the past using sharp interface approaches where the influence of elastic energy anisotropy on the precipitate shapes has been investigated. In this paper, we propose a diffuse interface approach which allows us to minimize grid-anisotropy related issues applicable in sharp-interface methods. In this context, we introduce a novel phase-field method where we minimize the functional consisting of the elastic and surface energy contributions while preserving the precipitate volume. Using this method we reproduce the shape-bifurcation diagrams for the cases of pure dilatational misfit that have been studied previously using sharp interface methods and then extend them to include interfacial energy anisotropy with different anisotropy strengths which has not been a part of previous sharp-interface models. While we restrict ourselves to cubic anisotropies in both elastic and interfacial energies in this study, the model is generic enough to handle any combination of anisotropies in both the bulk and interfacial terms. Further, we have examined the influence of asymmetry in dilatational misfit strains along with interfacial energy anisotropy on precipitate morphologies.

A diffuse interface immersed boundary method for complex moving boundary problems

A diffuse interface immersed boundary method is proposed for problems with arbitrarily moving complex rigid bodies. The solid-fluid interface is replaced by fluid–fluid Lagrangian boundary. A universal equation that combines the momentum conservation and rigid body velocity through the volume fraction is solved in the entire domain. The ever changing volume fraction field is accurately computed locating the object boundary. As the desired boundary conditions are only enforced directly in a volume average sense scope of it spreads over a computing cell. Thus, by reducing size of the computing cell the proposed method approaches the conventional sharp interface methods though it eliminates the spurious force oscillations. Owing to the single universal equation on a fixed Eulerian Cartesian mesh the proposed method retains the simplicity, robustness and versatility of the original Cartesian grid simulation. Convergence study of the transverse oscillation of a cylinder shows second-order convergence with refinement of numerical parameters and significantly diminished force oscillation. The proposed method is parallelized on a mixed shared and distributed memory notions in order to facilitate large scale 2D and 3D simulations. A range of test cases have been carefully chosen to check the quality of prediction of the proposed technique for forced and induced motions of multiple objects. They include in-line, transverse and rotational forced motion of a circular cylinder, motion of a hovering thin wing, vortex induced vibration (VIB) of multiple circular and square cylinder, angular VIB of multiple objects, particle sedimentation, instability driven fluttering of a freely falling plate, settling of a sphere in a quiescent medium, 3D VIB of two cylinders, wake transition due to pitching of a large aspect ratio thin plate and turbulent wake behind a wavy cylinder.

A comparative study of the single-mode Richtmyer–Meshkov instability

In this work, the single-mode Richtmyer–Meshkov instability is studied numerically to find a reasonable nonlinear theoretical model which can be applied to predict the interface evolution from the linear stage to the early nonlinear stage. The cut-cell-based sharp-interface methods $$\hbox {MuSiC}^{+}$$ (Chang et al. in J Comput Phys 242:946–990, 2013) and CCGF (Bai and Deng in Adv Appl Math Mech 9(5):1052–1075, 2017) are applied to generate numerical results for comparisons. Classical Air–SF $$_{6}$$ and Air–Helium conditions are applied in this study, and initial amplitude and Atwood number are varied for comparison. Comparisons to the simulation results from the literature show the applicability of $$\hbox {MuSiC}^{+}$$ and CCGF. Comparisons to the nonlinear theoretical models show that ZS (Zhang and Sohn in Phys Lett A 212:149–155, 1996; Phys Fluids 9:1106–1124, 1997), SEA (Sadot et al. in Phys Rev Lett 80:1654–1657, 1998), and DR (Dimonte and Ramaprabhu in Phys Fluids 22:014104, 2010) models are valid for both spike and bubble growth rates, and MIK (Mikaelian in Phys Rev E 67:026319, 2003) and ZG (Zhang and Guo in J Fluid Mech 786:47–61, 2016) models are valid for bubble growth rate, when the initial perturbation is small and the Atwood number is low, but only the DR model is applicable for both spike and bubble growth rates when the initial perturbation amplitude and the Atwood number are large. A new term of non-dimensional initial perturbation amplitude is presented and multiplied to the DR model to get a unified fitted DR model, which gives consistent results to the simulation ones for small and large initial amplitudes.