Diffuse Interface Method Research Articles

Self-similar diffuse boundary method for phase boundary driven flow

Interactions between an evolving solid and inviscid flow can result in substantial computational complexity, particularly in circumstances involving varied boundary conditions between the solid and fluid phases. Examples of such interactions include melting, sublimation, and deflagration, all of which exhibit bidirectional coupling, mass/heat transfer, and topological change of the solid–fluid interface. The diffuse interface method is a powerful technique that has been used to describe a wide range of solid-phase interface-driven phenomena. The implicit treatment of the interface eliminates the need for cumbersome interface tracking, and advances in adaptive mesh refinement have provided a way to sufficiently resolve diffuse interfaces without excessive computational cost. However, the general scale-invariant coupling of these techniques to flow solvers has been relatively unexplored. In this work, a robust method is presented for treating diffuse solid–fluid interfaces with arbitrary boundary conditions. Source terms defined over the diffuse region mimic boundary conditions at the solid–fluid interface, and it is demonstrated that the diffuse length scale has no adverse effects. To show the efficacy of the method, a one-dimensional implementation is introduced and tested for three types of boundaries: mass flux through the boundary, a moving boundary, and passive interaction of the boundary with an incident acoustic wave. Two-dimensional results are presented as well these demonstrate expected behavior in all cases. Convergence analysis is also performed and compared against the sharp-interface solution, and linear convergence is observed. This method lays the groundwork for the extension to viscous flow and the solution of problems involving time-varying mass-flux boundaries.

Large-Eddy Simulation of a Water Jet Exhausting into Quiescent Air

This study focuses on a water injection system, typical of those used to inject water in the motor jet plume of rocket engines to reduce noise, and particularly on the destabilization of the liquid jet that leads to a spray system. For that purpose, a large-eddy simulation is conducted for an experimental water jet of the literature. It is a sprinkler whose Reynolds number is and nozzle exit diameter is , similar to injectors used for jet noise reduction studies at a reduced scale. A diffuse interface method is used to calculate the dense liquid phase, and the dispersed phase containing droplets is calculated with an Eulerian solver. Transfers between the dense and the dispersed phases are modeled in the coupling procedure, including the atomization and impingement processes. The jet mean radius and initial instability frequency are consistent with the experimental data, while the position of the jet transition varies with the fluctuation levels at the nozzle exit. The droplets are created in the shear layer and accumulate on the jet center when only the atomization process is considered. When also implemented, the impingement phenomenon, however, is shown to absorb the droplets before they reach the center.

Width effect on contact angle hysteresis in a patterned heterogeneous microchannel

The width effect on contact angle hysteresis in a microchannel with patterned heterogeneous surfaces is systematically investigated. In the model, identical defects periodically appear on the background surface. To this end, a droplet's evaporation and condensation processes inside the microchannel are studied by theoretical analysis and numerical simulation based on a diffuse-interface lattice Boltzmann method. The microchannel width effect on the system's equilibrium properties is studied. The results demonstrate that the number of equilibrium configurations increases linearly with the microchannel width ( $b$ ), and has a quadratic relationship with the cosine of the reference contact angle and the heterogeneity strength ( $\varepsilon$ ). The average most stable contact angle is independent of $b$ and is always equal to the contact angle predicted by the Cassie–Baxter equation. For contact angle hysteresis ( $H$ ), when the microchannels are narrow and wide, there are individual-effect-dominated hysteresis (IDH) and collective-effect-dominated hysteresis (CDH), respectively. The IDH and CDH are hysteresis modes corresponding to the jumping behaviour of contact lines affected by individual defects and two neighbouring defects, respectively. Based on the graphical force balance approach, we establish a scaling law to quantify the connection between $H$ , $b$ and $\varepsilon$ . Specifically, in the IDH mode, $H\sim b \varepsilon ^2$ , while in the CDH mode, $H$ increases linearly with $\varepsilon$ but nonlinearly with $b$ .

An Interface-Corrected Diffuse Interface Model for Incompressible Multiphase Flows with Large Density Ratios

An interface-corrected diffuse interface method is presented in this work for the simulation of incompressible multiphase flows with large density ratios. In this method, an interface correction term together with a mass correction term is introduced into the diffuse-interface Cahn–Hilliard model to maintain both mass conservation and interface shapes between binary fluids simultaneously. The interface correction term is obtained by connecting the signed distance functions in the Hamilton–Jacobian equation with the order parameter of the Cahn–Hilliard model. In addition, an improved multiphase lattice Boltzmann flux solver is introduced, in which the fluxes are obtained by considering the contributions of the particle distribution functions before and after the streaming process through a local switch function. The proposed method is validated by simulating multiphase flows, such as the Laplace law, the evolution of a square bubble, the merging of two bubbles, Rayleigh–Taylor instability, and a droplet impacting on a film with a density ratio of 1000. Numerical results show that the presented method can not only reduce the interface diffusion but also has good control over the interface thickness and mass conservation. The improved numerical method has great potential for use in practical applications involving multiphase flows.

Stokes-Cahn-Hilliard formulation in sliding bi-periodic frames for the simulation of two-phase flows

A direct numerical simulation technique is proposed in this study for droplet emulsions in simple shear flow in Newtonian-Newtonian liquid systems. We combine the sliding bi-periodic frame with the diffuse interface method in the finite element framework to treat the multi-drop problem without wall effects. We employ the standard velocity-pressure formulation in a creeping regime and the diffuse interface method with the Galerkin weak formulation. Sliding bi-periodic frame constraints are implemented by Lagrangian multiplier. The results are presented for the morphological development of a single drop, two drops, and multiple drops in a sliding bi-periodic frame. To preserve the identical solution for a given relative configuration of droplets (with different droplet locations) within the context of the bi-periodic frame, a phase-translation method is proposed. The method corrects the inconsistency in the time integration, which yields discrepancy in flow solution particularly for problems with thin interfaces. By applying this procedure, accurate solutions for a thin interface are obtained regardless of location of the identical relative configuration of droplets. The effect of phase-field translation is also analyzed for the coalescence of two drops. For the first time, the sliding bi-periodic frame has been implemented with the combination of the standard finite element method and the diffuse interface method, which can be easily extended to more complex rheological liquids to solve industrially important problems.

Regimes of evaporation and mixing behaviors of nanodroplets at transcritical conditions

The objective of this paper is to examine the fundamental mechanisms responsible for the transition between subcritical evaporation and supercritical dense-fluid-mixing in the absence of convection effects, specifically focusing on the liquid–vapor interfacial dynamics. To isolate the dynamics of this transition process, we characterize the different physical behaviors exhibited by an n-dodecane nanoscale droplet placed in different nitrogen ambient conditions across the fuel’s critical point. We employ a continuum-based interface-resolving diffuse-interface method to explore the underlying phase-exchange mechanisms that bring about such distinct dynamics. Following the comparison against molecular dynamics simulations and experiments of evaporating droplets and experimental data for vapor–liquid equilibria, a parametric study at various ambient conditions and droplet sizing is performed to identify four regimes of evaporation/mixing behaviors: sub- and supercritical droplet evaporation, and sub- and supercritical dense-fluid-mixing. It is shown that the distinction in the phase-exchange mechanisms in these four regimes are brought about by the different thermodynamic phases the droplet center can exhibit during the evaporation/mixing process: subcritical liquid, supercritical liquid-like, subcritical gaseous, and supercritical gas-like, respectively. It is shown that the subcritical dense-fluid-mixing behavior is a direct result of nanoconfinement of the liquid–vapor interfacial structure and thus is not present for large droplet sizes. The present study also shows that the supercritical phase-exchange dynamics can follow two different pathways: supercritical droplet-like evaporation and supercritical dense-fluid-mixing. Furthermore, promoting the early transition to supercritical dense-fluid-mixing can significantly expedite the phase-exchange process through the disintegration of the liquid-like droplet core.

Large eddy simulations of reacting and non-reacting transcritical fuel sprays using multiphase thermodynamics

We present a novel framework for high-fidelity simulations of inert and reacting sprays at transcritical conditions with highly accurate and computationally efficient models for complex real-gas effects in high-pressure environments, especially for the hybrid subcritical/supercritical mode of evaporation during the mixing of fuel and oxidizer. The high-pressure jet disintegration is modeled using a diffuse interface method with multiphase thermodynamics, which combines multi-component real-fluid volumetric and caloric state equations with vapor–liquid equilibrium calculations for the computation of thermodynamic properties of mixtures at transcritical pressures. Combustion source terms are evaluated using a finite-rate chemistry model, including real-gas effects based on the fugacity of the species in the mixture. The adaptive local deconvolution method is used as a physically consistent turbulence model for large eddy simulation (LES). The proposed method represents multiphase turbulent fluid flows at transcritical pressures without relying on any semi-empirical breakup and evaporation models. All multiphase thermodynamic model equations are presented for general cubic state equations coupled with a rapid phase-equilibrium calculation method that is formulated in a reduced space based on the molar specific volume function. LES results show a very good agreement with available experimental data for the reacting and non-reacting engine combustion network benchmark spray A at transcritical operating conditions.

Diffuse-interface lattice Boltzmann modeling of charged particle transport in Poiseuille flow.

In this paper, we developed a coupled diffuse-interface lattice Boltzmann method (DI-LBM) to study the transport of a charged particle in the Poiseuille flow, which is governed by the Navier-Stokes equationsfor fluid field and the Poisson-Boltzmann equationfor electric potential field. We first validated the present DI-LBM through some classical benchmark problems, and then investigated the effect of electric field on the lateral migration of the particle in the Poiseuille flow. The numerical results show that the electric field has a significant influence on the particle migration. When an electric field in the vertical direction is applied to the charged particle initially located above the centerline of the channel, the equilibrium position of the particle would drop suddenly as the electric field is larger than a critical value. This is caused by the wall repulsion due to lubrication, the inertial lift related to shear slip, the lift owing to particle rotation, the lift due to the curvature of the undisturbed velocity profile, and the electric force. On the other hand, when an electric field in the horizontal direction is adopted, the equilibrium position of the particle would move toward the centerline of the channel with the increase of the electric field.

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.

A diffuse interface method for solid-phase modeling of regression behavior in solid composite propellants

Solid composite propellants (SCPs) are ubiquitous in the field of propulsion. In order to design and control solid rocket motors, it is critical to understand and accurately predict SCP regression. Regression of the burn surface is a complex process resulting from thermo-chemical-mechanical interactions, often exhibiting extreme morphological changes and topological transitions. Diffuse interface methods, such as phase field (PF), are well-suited for modeling processes of this type, and offer some distinct numerical advantages over their sharp-interface counterparts. They also provide a convenient framework for incorporating multiple multiphysical dynamics. In this work, we present a phase-field method for modeling the regression of SCPs with varying species and geometry. We construct the model from a thermodynamic perspective, leaving the base formulation general. A diffuse-species-interface field is employed as a mechanism for capturing complex burn chemistry in a reduced-order fashion, making it possible to model regression from the solid phase only. The computational implementation, which uses block-structured adaptive mesh refinement and temporal substepping for increased performance, is briefly discussed. The model is then applied to four test cases: (i) pure AP monopropellant, (ii) AP PBAN sandwich, (iii) AP HTPB sandwich, and (iv) spherical AP particles packed in HTPB matrix in two and three dimensions. In all cases, reasonable quantitative agreement is observed, even when the model is applied predictively (i.e., no parameter adjustment), as in the case of (iv). The validation of the proposed PF model demonstrates its efficacy as a numerical design tool for future SCP investigation.

A kinetic energy–and entropy-preserving scheme for compressible two-phase flows

Accurate numerical modeling of compressible flows, particularly in the turbulent regime, requires a method that is non-dissipative and stable at high Reynolds (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.In this study, we adopt the recently developed conservative diffuse-interface method (Jain et al., 2020 [30]) for the simulation of compressible two-phase flows. This method discretely conserves the mass of each phase, momentum, and total energy of the system and consistently reduces to the single-phase Navier-Stokes system when the properties of the two phases are identical. We here propose discrete consistency conditions between the numerical fluxes, such that any set of numerical fluxes that satisfy these conditions would not spuriously contribute to the kinetic energy and entropy of the system. We also present a set of numerical fluxes—which satisfies these consistency conditions—that results in an exact conservation of kinetic energy and approximate conservation of entropy in the absence of pressure work, viscosity, thermal diffusion effects, and time-discretization errors. Since the model consistently reduces to the single-phase Navier-Stokes system when the properties of the two phases are identical, the proposed consistency conditions and numerical fluxes are also applicable for a broader class of single-phase flows.To this end, we present coarse-grid numerical simulations of compressible single-phase and two-phase turbulent flows at infinite Re, to illustrate the stability of the proposed method in canonical test cases, such as an isotropic turbulence and Taylor-Green vortex flows. A higher-resolution simulation of a droplet-laden compressible decaying isotropic turbulence is also presented, and the effect of the presence of droplets on the flow is analyzed.

Interactions Between Shock Waves and Liquid Droplet Clusters: Interfacial Physics

Abstract This study quantitatively investigates the behaviors of single and multiple liquid cylinders placed in the path of a traveling normal shock wave using high-fidelity numerical simulations. The research is motivated by next-generation liquid-fueled scramjet and rotating detonation engines (RDE) where the liquid fuel interacts with shock waves and undergoes deformation, fragmentation, atomization, and vaporization before it mixes with the air and subsequently burns—the focus of this study is on the deformation and interfacial physics. The mathematical formulation to investigate this multiphase problem is based on a modified five-equation Kapila model that incorporates pressure-relaxation, viscous, and surface tension effects. A diffuse interface method is used to capture the liquid–gas interface. Two configurations are studied in this effort: (1) a single column of diameter 22 mm exposed to a shock wave traveling at Mach 2.4 and (2) a two identical cylinder system with diameters of 4.8 mm and 30 mm apart, and exposed to a shock wave moving a Mach number of 1.47. The computational results show excellent agreement with high-speed images and droplet deformation measured in the experiments. For both cases, it is found that the shock and the flow field in its wake leads to the flattening of the cylinder, followed by the formation of instability waves that are amplified by the baroclinic torque and the continuous reflections of the waves transmitted inside the liquid interior, eventually leading to ligament stripping. Based on the spatiotemporal evolution of the liquid and gaseous flowfields, time evolution of shock strength and parent droplet's mass and translation distance are also discussed.

On the maximal spreading of drops impacting onto a no-slip substrate

We numerically study the impact of a liquid drop onto no-slip rigid substrates with different wettabilities using a diffuse interface method, aiming to obtain a universal model for the maximal spreading of the impacting drop at moderate Weber numbers. We find that the wettability plays an important role in the maximal spreading and that the ratio of the surface energy to the initial kinetic energy of the drop at the maximal spreading, η, follows η∼We−1/2 at high fixed Reynolds numbers, where We is the Weber number. Taking account of the wettability effect, we obtain a scaling law at high Reynolds numbers from an analysis of energy transformation. This scaling law is compatible with the one derived from the momentum balance at the high impact velocity by Clanet et al. [“Maximal deformation of an impacting drop,” J. Fluid Mech. 517, 199–208 (2004)]. Moreover, we attribute it to the presence of a viscous–capillary regime, in which the viscous dissipation of the kinetic energy from the substrate is as significant as the kinetic energy transformed into the surface energy. Accordingly, we identify a new impact parameter, which makes all the numerical results of maximum drop deformation (from the viscous regime to the viscous–capillary regime with Reynolds number up to 104) collapse onto a single curve. Finally, we propose a universal model, the predictions of which are shown to agree well with numerical results for a wide range of Weber and Reynolds numbers.

Numerical simulation of meniscus evaporation in microchannel based on diffuse interface method

Numerical simulation of meniscus evaporation in microchannel based on diffuse interface method

A unified diffuse interface method for the interaction of rigid bodies with elastoplastic solids and multi-phase mixtures

This work outlines a new multi-physics-compatible immersed rigid body method for Eulerian finite-volume simulations. To achieve this, rigid bodies are represented as a diffuse scalar field, and an interface seeding method is employed to mediate the interface boundary conditions. The method is based on an existing multi-material diffuse interface method that is capable of handling an arbitrary mixture of fluids and elastoplastic solids. The underlying method is general and can be extended to a range of different applications including high-strain rate deformation in elastoplastic solids and reactive fluid mixtures. As such, the new method presented here is thoroughly tested through a variety of problems, including fluid–rigid body interaction, elastoplastic–rigid body interaction, and detonation–structure interaction. Comparison is drawn between both experimental work and previous numerical results, with excellent agreement in both cases. The new method is straightforward to implement, highly local, and parallelizable. This allows the method to be employed in three dimensions with multiple levels of adaptive mesh refinement using complex immersed geometries. The rigid body field can be static or dynamic, with the tangent of hyperbola interface capturing reconstruction method being used to keep the interface sharp in the dynamic case.

Buoyancy-driven bubbles in a constricted vertical capillary

We numerically study the dynamics of buoyancy-driven bubbles in a constricted vertical capillary in which a throat with an arc shape is present. To investigate at what conditions and how the bubble would be entrapped at the capillary throat, a diffuse-interface immersed-boundary method is used in numerical simulations. Axisymmetric simulations are performed for various bubble and throat sizes, represented by the diameter ratio of the throat to the bubble, η (0.55≤η≤1.35), the Bond number (0.1≤Bo≤15), and the Reynolds number (78.5≤Re≤3367). We find that small bubbles have insignificant deformation and, thus, cannot pass through a throat with η<1, while relatively large bubbles encounter noticeable interface oscillations at their lower part when approaching the throat. In particular, the interface oscillations are composed of a standing wave arising from buoyancy and a capillary wave propagating radially. A phase diagram is presented regarding the eventual bubble morphology: pass-through and entrapment. For the critical diameter ratio ηc at the onset of bubble entrapment, we proposed two scaling laws based on the analysis of the deformability and oscillation of the bubble, i.e., ηc≈1.1 for Bo < 1 and ηc∼Bo−1/4 for Bo > 1. These theoretical predictions are in good agreement with our numerical results.

High-order methods for diffuse-interface models in compressible multi-medium flows: A review

The diffuse interface models, part of the family of the front capturing methods, provide an efficient and robust framework for the simulation of multi-species flows. They allow the integration of additional physical phenomena of increasing complexity while ensuring discrete conservation of mass, momentum, and energy. The main drawback brought by the adoption of these models consists of the interface smearing, increasing with the simulation time, therefore, requiring a counteraction through the introduction of sharpening terms and a careful selection of the discretization level. In recent years, the diffuse interface models have been solved using several numerical frameworks including finite volume, discontinuous Galerkin, and hybrid lattice Boltzmann method, in conjunction with shock and contact wave capturing schemes. The present review aims to present the recent advancements of high-order accuracy schemes with the capability of solving discontinuities without the introduction of numerical instabilities and to put them in perspective for the solution of multi-species flows with the diffuse interface method.

A diffuse interface method for simulation‐based screening of heat transfer processes with complex geometries

AbstractFrequently, the simulation‐based design of physicochemical processes requires screening of large numbers of alternative designs with complex geometries. These geometries may result in conformal meshes which introduce stability issues, significant computational complexity, and require user‐interaction for their creation. In this work, a method for simulation of heat transfer using the diffuse interface method to capture a complex geometry is presented as an alternative to conformal meshing, with analysis and comparisons given. The methods presented include automated non‐iterative generation of phase fields from CAD geometries and an extension of the diffuse interface method for mixed boundary conditions. Simple measures of diffuse interface quality are presented and used to predict performance. The method is applied to a realistic three‐dimensional heat transfer problem (LED heat sink) and compared to the traditional conformal mesh approach. It is found to enable reasonable accuracy at an order‐of‐magnitude reduction in simulation time or comparable accuracy for equivalent simulation times.

Dynamics of a gas bubble penetrating through porous media

Porous media have been widely used for liquid-gas separation, benefiting from the strong capillary force generated from the micro/nanoscale pores. Understanding the flow characteristics in pore scale is significant for the design of porous structure. In this study, a numerical model was established to investigate the dynamics of a bubble penetrating through porous media at the pore scale. The two-phase interface was captured using the diffuse interface method. The influence of pore shape, width, and height on the bubble deformation, velocity, and critical pressure was investigated. For the same pore size, the largest bubble centroid velocity and the highest critical pressure exist in the quadrilateral pores compared rather than in the circular or triangular pores. As the pore width decreases, both the average velocity of the bubble centroid and the critical pressure increase. However, the critical pressure is independent of the pore height. As the pore height increases, the average velocity of the bubble centroid increases. A new correlation of the critical pressure for bubble penetration has been proposed, which is a function of the shape factor, the pore width, and the bubble diameter. The findings of this work can contribute to improving the design of porous media for two-phase separation.

Sharp Interface Capturing in Compressible Multi-Material Flows with a Diffuse Interface Method

Compressible multi-materialflows are encountered in a wide range of natural phenomena and industrial applications, such as supernova explosions in space, high speed flows in jet and rocket propulsion, underwater explosions, and vapor explosions in post accidental situations in nuclear reactors. In the numerical simulations of these flows, interfaces play a crucial role. A poor numerical resolution of the interfaces could make it difficult to account for the physics, such as material separation, location of the shocks and contact discontinuities, and transfer of the mass, momentum and heat between different materials/phases. Owing to such importance, sharp interface capturing remains an active area of research in the field of computational physics. To address this problem in this paper we focus on the Interface Capturing (IC) strategy, and thus we make use of a newly developed Diffuse Interface Method (DIM) called Multidimensional Limiting Process-Upper Bound (MLP-UB). Our analysis shows that this method is easy to implement, can deal with any number of material interfaces, and produces sharp, shape-preserving interfaces, along with their accurate interaction with the shocks. Numerical experiments show good results even with the use of coarse meshes.