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Viscous stress approximations in diffuse interface methods for two-phase flow based on mechanical jump conditions
... In contrast to a sharp theory, the dividing surfaces between different fluids are replaced by a diffuse transition region, also called diffuse interface (cf. [60] for a detailed discussion). Therefore, phase variables ϕ α (x, t ) ∈ [0, 1] are introduced for each of the N considered phases. ...
The present work addresses the simulation of pore emptying during the drying of battery electrodes. For this purpose, a model based on the multiphase-field (MPF) method is used, since it is an established approach for modeling and simulating multiphysical problems. A model based on phase fields is introduced that takes into account fluid flow, capillary effects, and wetting behavior, all of which play an important role in drying. In addition, the MPF method makes it possible to track the movement of the liquid-air interface without computationally expensive adaptive mesh generation. The presented model is used to investigate pore emptying in real hard carbon microstructures. For this purpose, the microstructures of real dried electrodes are used as input for the simulations. The simulations performed here demonstrate the importance of considering the resolved microstructural information compared to models that rely only on statistical geometry parameters such as pore size distributions. The influence of various parameters such as different microstructures, fluid viscosity, and the contact angle on pore emptying are investigated. In addition, this work establishes a correlation between the capillary number and the breakthrough time of the solvent as well as the height difference of the solvent front at the time of breakthrough. The results indicate that the drying process can be optimized by doping the particle surface, which changes the contact angle between the fluids and the particles.
Published by the American Physical Society 2025
An implementation of the crystal plasticity theory in the context of the multiphase-field method provides a numerically efficient tracking of evolving grain boundaries, modeled as diffuse interfaces. In literature, several approaches exist for the implementation of the plastic material behavior within the diffuse interface, based on interpolation, homogenization, or the mechanical jump conditions. Among these, only the jump condition approach exhibits an intrinsic relationship to the sharp interface (SI) theory. Therefore, in the work at hand, the implementation of the crystal plasticity theory within the jump condition approach, referred to as phase-specific plastic fields approach (PSPFA), is discussed in detail. The PSPFA is compared to the interpolation approach, referred to as common plastic fields approach (CPFA), using three-dimensional benchmark simulations of a bicrystal set-up. The comparison reveals that the PSPFA and SI coincide convincingly regarding the accumulated plastic slip and the Mises stress. In contrast, a significant deviation of CPFA and SI is observed both quantitatively and qualitatively, not only within the diffuse interface region, but throughout the complete simulation domain. A variation of the interface width illustrates that this observation can be transferred to the normal components of the total strain, even for smaller interface widths. Consequently, a quantitative estimate of the plastic material behavior, which is crucial for the prediction of the dynamic behavior of grain boundaries, is only provided by the PSPFA. The application of the crystal plasticity in the context of PSPFA to more complex microstructures is illustrated with respect to a periodic honeycomb-structure and an octotuple.
Metal foams constitute a promising and emerging material class in the context of lightweight construction. There exists a variety of different foam topologies, on which resulting mechanical properties depend. To maximize the potential of foams in material use under mechanical load, the present work addresses the question how different geometrical parameters influence the material behaviour. Therefore, an algorithm for digital generation and design of open pore foam structures is presented, that allows to regulate the geometry precisely. A method for retrieving effective mechanical properties from numerical simulations of compression tests in the elastic regime is introduced. Additionally, the representativeness of foam volumes considered for simulations is investigated. This yields a fully digital workflow, which enables the investigation of geometry influence on mechanical properties. This approach is used to conduct simulation studies on generated foam structures with a systematic variation of geometrical parameters. Herein, a range of effective Young's moduli varying by up to a factor of 1.3 for different foam structures at the same porosity is found. This shows a significant impact of the foam geometry on the elastic properties of metal foams. The presented methodology yields insights, which can guide design and optimization of materials for specific applications.
Mean-field homogenization is an established and computationally efficient method estimating the effective linear elastic behavior of composites. In view of short-fiber reinforced materials, it is important to homogenize consistently during process simulation. This paper aims to comprehensively reflect and expand the basics of mean-field homogenization of anisotropic linear viscous properties and to show the parallelism to the anisotropic linear elastic properties. In particular, the Hill–Mandel condition, which is generally independent of a specific material behavior, is revisited in the context of boundary conditions for viscous suspensions. This study is limited to isothermal conditions, linear viscous and incompressible fiber suspensions and to linear elastic solid composites, both of which consisting of isotropic phases with phase-wise constant properties. In the context of homogenization of viscous properties, the fibers are considered as rigid bodies. Based on a chosen fiber orientation state, different mean-field models are compared with each other, all of which are formulated with respect to orientation averaging. Within a consistent mean-field modeling for both fluid suspensions and solid composites, all considered methods can be recommended to be applied for fiber volume fractions up to 10%. With respect to larger, industrial-relevant, fiber volume fractions up to 20%, the (two-step) Mori–Tanaka model and the lower Hashin–Shtrikman bound are well suited.
In this work, we propose a novel phase-field model for the simulation of two-phase flows that is accurate, conservative, bounded, and robust. The proposed model conserves the mass of each of the phases, and results in bounded transport of the volume fraction. We present results from the canonical test cases of a drop advection and a drop in a shear flow, showing significant improvement in the accuracy over the commonly used conservative phase-field method.
Moreover, the proposed model imposes a lesser restrictive Courant-Friedrichs-Lewy condition, and hence, is less expensive compared to other conservative phase-field models. We also propose improvements on computation of surface tension forces and show that the proposed improvement significantly reduces the magnitude of spurious velocities at the interface.
We also derive a consistent and conservative momentum transport equation for the proposed phase-field model and show that the proposed model when coupled with the consistent momentum transport equation results in discrete conservation of kinetic energy, which is a sufficient condition for the numerical stability of incompressible flows, in the absence of dissipative mechanisms. To illustrate the robustness of the method in simulating high-density ratio turbulent two-phase flows, we present the numerical simulations of high-density ratio droplet-laden isotropic turbulence at finite and infinite Reynolds numbers.
A distributed Lagrange multiplier/fictitious domain method in a phase‐field formulation for the simulation of rigid bodies in incompressible fluid flow is presented. The phase‐field method yields an implicit representation of geometries and thus rigid body particulate flows within arbitrary geometries can be simulated based on a fixed Cartesian grid. Therefore, a phase‐field based collision model is introduced in order to address contact of particles with arbitrary solid structures as boundaries. Additionally, grain growth within the boundary geometry can be considered leading to changes in its shape during the simulation. The method is validated on benchmark problems and a convergence study is performed. Multiple numerical experiments are carried out in order to show the methods’ capability to simulate problems with differently shaped rigid bodies and particulate flows involving complex boundary geometries like foam structures.
We report on two- and three-dimensional numerical simulations of Rayleigh–Taylor instabilities in immiscible fluids. A diffuse-interface model that combines the Cahn–Hilliard equation, governing the evolution of the volume fraction of one fluid, and the Navier–Stokes equations, governing the bulk velocity and pressure, is used. The study is limited to low Atwood numbers owing to the use of the Boussinesq approximation. The code is based on a pseudo-spectral method. A linear analysis is first performed in a two-dimensional case of Rayleigh–Taylor instability to confirm that the model very well captures this phenomenon in the case of inviscid or viscid fluids. One key aspect of this work is that the influence of the thermodynamic parameters related to the Cahn–Hilliard equation (interface thickness and mobility) is quantitively studied. Three-dimensional results of Rayleigh–Taylor instabilities in viscous fluids are then presented to show the possibilities of this modeling. We observe the effect of the viscosity and the wavelength of an initial single-mode perturbation on the mass transport during the nonlinear regime.
The presence of grain boundaries significantly influences the overall mechanical behavior of materials with an underlying crystalline microstructure. They act as an obstacle against the movement of dislocations and, thus, significantly contribute to size effects as for example the Hall–Petch effect. Hence, the thermodynamically consistent modeling of the behavior of the plastic slip at a grain boundary is of utmost interest. To this end, balance equations at a grain boundary are derived from an extended energy balance by means of invariance considerations. The grain boundary is considered as a material singular surface with own internal and kinetic energy as well as energy supply. Consequently, the balances at the grain boundary depend on its mean curvature. The framework presented is applied to a small strain slip gradient crystal plasticity theory, regarding single slip. Accounting for the derived balance equations, thermodynamically consistent flow rules for the plastic slip at the grain boundary are obtained by exploitation of the Coleman–Noll procedure. In this context, a classification of flow rules for the plastic slip at the grain boundary is provided. Finally, the distribution of the plastic slip is presented regarding a three-phase laminate material with two elastoplastic phases and one elastic phase. The two elastoplastic phases represent the grains of a bicrystal. A grain boundary effect is discussed which is based on a variation of the internal length scale in one of the two elastoplastic phases.
In this paper, we develop a novel phase-field model for fluid-structure interaction (FSI), that is capable to handle very large deformations as well as topology changes like contact of the solid to the domain boundary. The model is based on a fully Eulerian description of the velocity field in both, the fluid and the elastic domain. Viscous and elastic stresses in the Navier-Stokes equations are restricted to the corresponding domains by multiplication with their characteristic functions. To obtain the elastic stress, an additional Oldroyd-B - like equation is solved. Thermodynamically consistent forces are derived by energy variation. The convergence of the derived equations to the traditional sharp interface formulation of fluid-structure interaction is shown by matched asymptotic analysis. The model is evaluated in a challenging benchmark scenario of an elastic body traversing a fluid channel. A comparison to reference values from Arbitrary Lagrangian Eulerian (ALE) simulations shows very good agreement. We highlight some distinct advantages of the new model, like the avoidance of re-triangulations and the stable inclusion of surface tension. Further, we demonstrate how simple it is to include contact dynamics into the model, by simulating a ball bouncing off a wall. We extend this scenario to include adhesion of the ball, which to our knowledge, cannot be simulated with any other FSI model. While we have restricted simulations to fluid-structure interaction, the model is capable to simulate any combination of viscous fluids, visco-elastic fluids and elastic solids.
Computational models based on the phase-field method have become an essential tool in material science and physics in order to investigate materials with complex microstructures. The models typically operate on a mesoscopic length scale resolving structural changes of the material and provide valuable information about the evolution of microstructures and mechanical property relations. For many interesting and important phenomena, such as martensitic phase transformation, mechanical driving forces play an important role in the evolution of microstructures. In order to investigate such physical processes, an accurate calculation of the stresses and the strain energy in the transition region is indispensable. We recall a multiphase-field elasticity model based on the force balance and the Hadamard jump condition at the interface. We show the quantitative characteristics of the model by comparing the stresses, strains and configurational forces with theoretical predictions in two-phase cases and with results from sharp interface calculations in a multiphase case. As an application, we choose the martensitic phase transformation process in multigrain systems and demonstrate the influence of the local homogenization scheme within the transition regions on the resulting microstructures.
Numerical simulations based on phase-field methods are indispensable in order to investigate interesting and important phenomena in the evolution of microstructures. Microscopic phase transitions are highly affected by mechanical driving forces and therefore the accurate calculation of the stresses in the transition region is essential. We present a method for stress calculations within the phase-field framework, which satisfies the mechanical jump conditions corresponding to sharp interfaces, although the sharp interface is represented as a volumetric region using the phase-field approach. This model is formulated for finite deformations, is independent of constitutive laws, and allows using any type of phase inherent inelastic strains.
Background
The commonly used flow models for Fiber Reinforced Polymers (FRPs) often neglect the flow-induced anisotropy of the suspension, but with increasing fiber volume fraction, this plays an important role. There exist already some models which count on this effect. They are, however, phenomenological and need a fitted model parameter. In this paper, a micromechanically-based constitutive law is proposed which considers the flow-induced anisotropic viscosity of the fiber suspension. Methods
The introduced viscosity tensor can handle arbitrary anisotropy of the fluid-fiber suspension which depends on the actual fiber orientation distribution. Assuming incompressible material behaviour, a homogenization method for unidirectional structures in contribution with orientation averaging is used to determine the effective viscosity tensor. The motion of rigid ellipsoidal fibers induced by the flow of the matrix material is described based on Jeffery’s equation. The reorientation of the fibers is modeled in two ways: by describing them with fiber orientation vectors, and by fiber orientation tensors. A numerical implementation of the introduced model is applied to representative flow modes. ResultsThe predicted effective stress values depending on the actual fiber orientation distribution through the anisotropic viscosity are analyzed in transient and stationary flow cases. In the case of the assumed incompressibility, they show similar effective viscous material behaviour as the results obtained by the use of the Dinh-Armstrong constitutive law. Conclusions
The introduced model is a possible way to describe the flow-induced anisotropic viscosity of a fluid-fiber suspension based on the mean field theory.
In this work, we present our numerical results of the application of
Galerkin-based Isogeometric Analysis (IGA) to incompressible
Navier--Stokes--Cahn--Hilliard (NSCH) equations in
velocity--pressure--phase field--chemical potential formulation.
For the approximation of the velocity and pressure fields, LBB compatible
non-uniform rational B-spline spaces are used
which can be regarded as smooth generalizations of Taylor--Hood pairs of
finite element spaces.
The one-step -scheme is used for the discretization in time.
The static and rising bubble, in addition to the nonlinear Rayleigh-Taylor
instability flow problems, are considered in two dimensions as model problems
in order to investigate the numerical properties of the scheme.
In this paper, we present a detailed analysis of the computation of the viscous terms for the simulation of incompressible two-phase flows in the framework of Level Set/Ghost Fluid Method when viscosity is discontinuous across the interface. Two pioneering papers on the topic, Kang et al. [10] and Sussman et al. [26], proposed two different approaches to deal with viscous terms. However, a definitive assessment of their respective efficiency is currently not available. In this paper, we demonstrate from theoretical arguments and confirm from numerical simulations that these two approaches are equivalent from a continuous point of view and we compare their accuracies in relevant test-cases. We also propose a new intermediate method which uses the properties of the two previous methods. This new method enables a simple implementation for an implicit temporal discretization of the viscous terms. In addition, the efficiency of the Delta Function method [24] is also assessed and compared to the three previous ones, which allow us to propose a general overview of the accuracy of all available methods. The selected test-cases involve configurations wherein viscosity plays a major role and for which either theoretical results or experimental data are available as reference solutions: simulations of spherical rising bubbles, shape-oscillating bubbles and deformed rising bubbles at low Reynolds numbers.
In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenom-ena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immis-cible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.
We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds for n=2,3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier-Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier-Stokes-Voight model and the Navier-Stokes model) for the fluid velocity suitably coupled with a convective Allen-Cahn equation for the (phase) order parameter. We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parametrizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability and regularity results, and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α->0 limit in α-models. Then, we also show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory converges to a single equilibrium by means of a LS inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier-Stokes equations and magnetohydrodynamics models which improve and complement the results of Holst et. al. [J. Nonlinear Science 20, 2010, 523-567]. Finally, our analysis is applied to certain regularized Ericksen-Leslie (RSEL) models for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds.
Phase-field models are gaining importance as application-oriented tools
for alloy and process optimization. In this context, numerical
performance plays an essential role, but is often difficult to reconcile
with accuracy. The numerical solution of the phase-field equation based
on finite differences inherently implies a discretization error which
may strongly bias the simulation results. A direct correlation between
the accuracy of the finite-difference solution and the number of
numerical cells resolving the diffuse interface profile has been
observed. This presents a problem, because high numbers of interface
cells are unfavorable for computational performance. To overcome the
problem, an improved numerical solution of the phase-field equation is
proposed, which allows obtaining highly accurate results with three or
four interface cells only. The solution takes advantage of the a priori
knowledge of the phase-field profile, determined by the choice of the
potential function in the free energy functional. This knowledge enables
an almost exact quantification and compensation of the error evoked by
the discrete nature of the grid spacing and interface width. Benchmark
simulations demonstrate the significant gain in both accuracy and
numerical efficiency
The Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model.
A theory of fracture is presented that is based upon an extension of continuum mechanics to the nanoscale through the incorporation of long-range intermolecular forces which correct bulk material descriptions near interfaces. The surface energy on crack surfaces, which is given in terms of the long-range intermolecular forces, plays an important role in an expression for the stress distribution near the crack tip. It is observed through numerical simulation that the incorporation of these long-range intermolecular forces removes the square-root stress singularity predicted by classical linear elastic fracture mechanics.
The motion of a free rising skirt bubble is studied using direct numerical simulation covering the range of Eötvös (Bond) and Morton numbers where they can be observed. We investigate the skirt bubble motion (terminal velocity and drag), the flow field structure (inside the skirt film, around the bubble, and inside the gas film that forms the skirt) as well as the geometrical characteristics of the skirt (both length and thickness). A unified relation for the terminal velocity (and drag force) is provided for all bubble shapes (dimple, skirt, and spherical cap bubbles) when considering Eötvös numbers (based on bubble diameter) larger than 100. The velocity field around and inside the bubble is investigated resulting in the clarification of the wake structure with a major difference with the flow deduced from experimental observations during the 1970s. The numerical simulations reveal that only one recirculation is observed inside the bubble while two vortices are highlighted in the wake. We confirm that a parallel Poiseuille flow develops inside the skirt film resulting in an intense vorticity field. Original modeling is improved with a Reynolds correction that makes possible the prediction of the film thickness in very good agreement with the only experimental value available in the literature. Our simulations combined with experimental observations suggest the existence of a maximum value for the skirt length.
The interaction of immersed rigid bodies with two‐phase flow is of high interest in many applications. A model for the coupling of a Hohenberg‐Halperin type model for two‐phase flow and a fictitious domain method for consideration of rigid bodies is introduced leading to a full multi‐phase‐field method to address the overall problem. A normalised phase variable is used alongside a method for application of wetting boundary conditions over a diffuse fluid‐solid interface. This enables the representation of capillary effects and different wetting behaviour based on Young's law. A number of simulations is conducted in order to validate the model and highlight its ability to handle a variety of setups for two‐phase particulate flow. This includes dynamic wetting situations, the motion of multiple particles within the two‐phase flow and the interaction with arbitrarily shaped solid structures inside the domain.
Grand-potential based multiphase-field model is extended to include surface diffusion. Diffusion is elevated in the interface through a scalar degenerate term. In contrast to the classical Cahn-Hilliard-based formulations, the present model circumvents the related difficulties in restricting diffusion solely to the interface by combining two second-order equations, an Allen-Cahn-type equation for the phase field supplemented with an obstacle-type potential and a conservative diffusion equation for the chemical potential or composition evolution. The sharp interface limiting behavior of the model is deduced by means of asymptotic analysis. A combination of surface diffusion and finite attachment kinetics is retrieved as the governing law. Infinite attachment kinetics can be achieved through a minor modification of the model, and with a slight change in the interpretation, the same model handles the cases of pure substances and alloys. Relations between model parameters and physical properties are obtained which allow one to quantitatively interpret simulation results. An extensive study of thermal grooving is conducted to validate the model based on existing theories. The results show good agreement with the theoretical sharp-interface solutions. The obviation of fourth-order derivatives and the usage of the obstacle potential make the model computationally cost-effective.
A given symmetric and compatible linear transformation field D determines, through Cesaro representation theorem, a vector field v whose symmetric gradient ∇sv equals D, up to an additive vector field vh satisfying ∇svh=0. If D represents an infinitesimal strain field, then vh is an infinitesimal rigid-body displacement. If D represents a stretching field, then vh is a rigid-body velocity which characterizes a rate of translation and rotation of the whole (fluid or solid) body.
If D is compatible in regions separated by a smooth singular surface across which D suffers a jump, then in our main Theorem 2.3 we show that the associated vector field v must satisfy the Hadamard rank 1 compatibility condition across the surface, i.e., the jump ⟦∇v⟧ is rank 1 of the form d⊗n, where n is a unit normal to the surface, if and only if ⟦∇v⟧ is constant across the singular surface. Importantly, no a priori stated restriction on the skewsymmetric part of ∇v is needed. We, then, record two corollaries that characterize the possible jump behavior at a corner line where two smooth singular surfaces meet, discussing how our results apply in two example elementary flow problems in classical fluid dynamics.
We present a novel partitioned iterative formulation for modeling of fluid-structure interaction in two-phase flows. The variational formulation consists of a stable and robust integration of three blocks of differential equations, viz., incompressible viscous fluid, a rigid or flexible structure and two-phase indicator field. The fluid-fluid interface between the two phases, which may have high density and viscosity ratios, is evolved by solving the conservative phase-field Allen-Cahn equation in the arbitrary Lagrangian-Eulerian coordinates. While the Navier-Stokes equations are solved by a stabilized Petrov-Galerkin method, the conservative Allen-Chan phase-field equation is discretized by the positivity preserving variational scheme. Fully decoupled implicit solvers for the two-phase fluid and the structure are integrated by the nonlinear iterative force correction in a staggered partitioned manner. We assess the accuracy and stability of the new phase-field/ALE variational formulation for two- and three-dimensional problems involving the dynamical interaction of rigid bodies with free-surface. We consider the decay test problems of increasing complexity, namely free translational heave decay of a circular cylinder and free rotation of a rectangular barge. Through numerical experiments, we show that the proposed formulation is stable and robust for high density ratios across fluid-fluid interface and for low structure-to-fluid mass ratio with strong added-mass effects. Using three-dimensional unstructured meshes, we demonstrate the second-order temporal accuracy of the coupled phase-field/ALE method. Finally, we demonstrate the three-dimensional phase-field FSI formulation for a practical problem of internal two-phase flow in a flexible circular pipe subjected to vortex-induced vibrations due to external fluid flow.
We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio and using meshes of arbitrary topology. The variational finite element technique relies on the Allen-Cahn phase-field equation for capturing the phase interface on a fixed mesh with a mass conservative and energy-stable discretization. Mass is conserved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the solution of the phase-field equation. The spatial part of the Lagrange multiplier is written as a mid-point approximation to make the scheme energy-stable. This enables us to form a conservative, energy-stable and positivity preserving scheme. The proposed variational technique reduces spurious and unphysical oscillations in the solution while maintaining second-order spatial accuracy. To model a generic two-phase free-surface flow, we couple the Allen-Cahn phase-field equation with the Navier-Stokes equations. Comparison of results between standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen-Cahn two-phase solver. Standard two-phase flow benchmarks such as Laplace-Young law and sloshing tank problem are carried out to assess the convergence and accuracy of the coupled Navier-Stokes and Allen-Cahn solver. Two- and three-dimensional dam break problem are then solved to assess the scheme for the problem with topological changes of the air-water interface on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical problem of wave-structure interaction in offshore engineering using general three-dimensional unstructured meshes.
A computational tool is developed to capture the interaction of solid object with two-phase flow. The full two-dimensional Navier-Stokes equations are solved on a regular structured grid to resolve the flow field. The level set and the immersed boundary methods are used to capture the free surface of a fluid and a solid object, respectively. A two-step projection method along with Multi-Processing (OpenMP) is employed to solve the flow equations. The computational tool is verified based on numerical and experimental data with three scenarios: a cylinder falling into a rectangular domain due to gravity, transient vertical oscillation of a cylinder by releasing above its equilibrium position, and a dam breaking in the presence of a fixed obstacle. In the first two validation simulations, the accuracy of the immersed boundary method is verified. However the accuracy of the level set method while the computational tool can model the high density ratio is confirmed in the dam breaking simulation. The results obtained from the current method are in good agreement with experimental data and other numerical studies. The applicability of the current computational tool for the interaction of a buoy in a water wave tank with two types of waves; symmetrical and asymmetrical waves; has also been studied.
The conservative Allen–Cahn (AC) equation has been studied analytically and numerically. Our mathematical analysis and numerical experiment, however, show that previous numerical methods are not second-order accurate in time and/or do not conserve the initial mass. The aim of this paper is to propose high-order and mass conservative methods for solving the conservative AC equation. In the methods, we discretize the conservative AC equation by using a Fourier spectral method in space and first-, second-, and third-order implicit–explicit Runge–Kutta schemes in time. We show that the methods inherit the mass conservation. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods.
Computational models based on the phase-field method typically operate on a mesoscopic length scale and resolve structural changes of the material and furthermore provide valuable information about microstructure and mechanical property relations. An accurate calculation of the stresses and mechanical energy at the transition region is therefore indispensable. We derive a quantitative phase-field elasticity model based on force balance and Hadamard jump conditions at the interface. Comparing the simulated stress profiles calculated with Voigt/Taylor (Annalen der Physik 274(12):573, 1889), Reuss/Sachs (Z Angew Math Mech 9:49, 1929) and the proposed model with the theoretically predicted stress fields in a plate with a round inclusion under hydrostatic tension, we show the quantitative characteristics of the model. In order to validate the elastic contribution to the driving force for phase transition, we demonstrate the absence of excess energy, calculated by Durga et al. (Model Simul Mater Sci Eng 21(5):055018, 2013), in a one-dimensional equilibrium condition of serial and parallel material chains. To validate the driving force for systems with curved transition regions, we relate simulations to the Gibbs-Thompson equilibrium condition (Johnson and Alexander, J Appl Phys 59(8):2735, 1986).
We propose a new computational approach for simulating the coupled interaction between air–water flows and arbitrarily complex floating rigid bodies. The numerical method integrates the fluid–structure interaction (FSI) curvilinear immersed boundary (CURVIB) method of Borazjani et al. (2008) [21] with a level set approach for simulating free surface flows in arbitrarily complex domains. We show that when applying the CURVIB method to simulate two-phase flow FSI problems the approach used to calculate the force imparted on the body is critical for determining the overall accuracy of the method. We develop and demonstrate the accuracy of a new approach for calculating the force, namely the pressure projection boundary condition (PPBC), which is based on projecting the pressure on the surface of the body using the momentum equation along the local normal to the body direction. Extensive numerical tests show that the new approach greatly improves the ability of the method to correctly predict the dynamics of the floating structure motion. To demonstrate the predictive capabilities of the method and its ability to simulate non-linear free surface phenomena, such as breaking waves, we apply it to various two- and three-dimensional problems involving complex rigid bodies interacting with a free surface both with prescribed body motion and coupled FSI. We show that for all cases the proposed method yields results in very good accuracy with benchmark numerical data and available experiments. The simulations also reveal the onset of dynamically rich, energetic coherent structures in the air phase induced by the waves generated as the rigid body interacts with the free surface.
We present a new numerical scheme for solving a conservative Allen–Cahn equation with a space–time dependent Lagrange multiplier. Since the well-known classical Allen–Cahn equation does not have mass conservation property, Rubinstein and Sternberg introduced a nonlocal Allen–Cahn equation with a time dependent Lagrange multiplier to enforce conservation of mass. However, with their model it is difficult to keep small features since they dissolve into the bulk region. One of the reasons for this is that mass conservation is realized by a global correction using the time-dependent Lagrange multiplier. To resolve the problem, we use a space–time dependent Lagrange multiplier to preserve the volume of the system and propose a practically unconditionally stable hybrid scheme to solve the model. The numerical results indicate a potential usefulness of our proposed numerical scheme for accurately calculating geometric features of interfaces.
In this paper we present a full Eulerian model for a dynamic fluid–structure interaction (FSI) problem in terms of phase field approach, and design its full Eulerian finite element discretization and effective iterative method. The present full Eulerian FSI model effectively demonstrates the interaction between fluid flow and solid structure in terms of a uniform system of governing equations defined in a single domain, thus the computational grid is fixed, and the re-meshing and interpolation techniques which are always required by other FSI modeling approaches are no longer needed here. We develop a new stable scheme to discretize the Euler equation of an incompressible hyperelastic structure in Eulerian description, and employ Galerkin/least-square (GLS) stabilization scheme, streamline-upwind/Petrov–Galerkin (SUPG) method, and the second-order backward difference formula (BDF) to solve the derived transient nonlinear system of Navier–Stokes equations and transport equations. Numerical experiment is carried out for a cross spinning around its rotation of axis due to the passing flow field, and the numerical results dramatically show the spinning motion of the cross due to the interaction with the fluid, showing that our model and numerical methods are effective to simulate the dynamic fluid–structure interaction phenomena.
The time-dependent motion for a two-layer Couette flow consisting of fluids of different viscosities is simulated numerically by using an algorithm based on the Volume of Fluid (VOF) method. Interfacial tension is included via a continuous surface force (CSF) algorithm. The algorithm is fine-tuned to handle the motion which is driven by a shear-induced interfacial instability due to the viscosity stratification. The code is validated against linear theory. Two prototypical situations are presented, one at a moderately high Reynolds number and the other at a lower Reynolds number. The initial condition is seeded with the eigenmode of largest growth rate, with amplitudes that are varied from those that capture the linear regime to larger values for nonlinear regimes. Issues of free surface advection and viscosity interpolation are discussed. The onset of nonlinearity occurs at the interface and is quadratic, followed by wave steepening (Research supported by NSF and ONR).
One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description o the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving shar interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivate regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interfac is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularizatio is that it automatically yields a continuous description of surface tension, which can play an important role in topologica transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and th resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into accoun weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–localit introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interfac equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both component are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrat the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show tha when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion eve if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scale quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigat the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we conside a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.
Several methods have been previously used to approximate free boundaries in finite-difference numerical simulations. A simple, but powerful, method is described that is based on the concept of a fractional volume of fluid (VOF). This method is shown to be more flexible and efficient than other methods for treating complicated free boundary configurations. To illustrate the method, a description is given for an incompressible hydrodynamics code, SOLA-VOF, that uses the VOF technique to track free fluid surfaces.
Today matter is universally regarded as composed of molecules. Though molecules cannot be discerned by human senses, they may be defined precisely as the smallest portions of a material to exhibit certain of its distinguishing properties, and much of the behavior of individual molecules is predicted satisfactorily by known physical laws. Molecules in their turn are regarded as composed of atoms; these, of nuclei and electrons; and nuclei themselves as composed of certain elementary particles. The behavior of the elementary particles has been reduced, so far, but to a partial subservience to theory. Whether these elementary particles await analysis into still smaller corpuscles remains for the future.
A lattice Boltzmann model (LBM) is proposed based on the phase-field theory to simulate incompressible binary fluids with density and viscosity contrasts. Unlike many existing diffuse interface models which are limited to density matched binary fluids, the proposed model is capable of dealing with binary fluids with moderate density ratios. A new strategy for projecting the phase field to the viscosity field is proposed on the basis of the continuity of viscosity flux. The new LBM utilizes two lattice Boltzmann equations (LBEs): one for the interface tracking and the other for solving the hydrodynamic properties. The LBE for interface tracking can recover the Chan-Hilliard equation without any additional terms; while the LBE for hydrodynamic properties can recover the exact form of the divergence-free incompressible Navier-Stokes equations avoiding spurious interfacial forces. A series of 2D and 3D benchmark tests have been conducted for validation, which include a rigid-body rotation, stationary and moving droplets, a spinodal decomposition, a buoyancy-driven bubbly flow, a layered Poiseuille flow, and the Rayleigh-Taylor instability. It is shown that the proposed method can track the interface with high accuracy and stability and can significantly and systematically reduce the parasitic current across the interface. Comparisons with momentum-based models indicate that the newly proposed velocity-based model can better satisfy the incompressible condition in the flow fields, and eliminate or reduce the velocity fluctuations in the higher-pressure-gradient region and, therefore, achieve a better numerical stability. In addition, the test of a layered Poiseuille flow demonstrates that the proposed scheme for mixture viscosity performs significantly better than the traditional mixture viscosity methods.
An investigation is made into the moving contact line dynamics of
a Cahn–Hilliard–van der Waals (CHW) diffuse mean-field interface.
The interface separates two incompressible viscous fluids and can evolve
either through convection or through
diffusion driven by chemical potential gradients. The purpose of this paper is to show
how the CHW moving contact line compares to the classical sharp interface contact
line. It therefore discusses the asymptotics of the CHW contact line velocity and
chemical potential fields as the interface thickness ε and the mobility κ both go to
zero. The CHW and classical velocity fields have the same outer behaviour but can
have very different inner behaviours and physics. In the CHW model, wall–liquid
bonds are broken by chemical potential gradients instead of by shear and change of
material at the wall is accomplished by diffusion rather than convection. The result
is, mathematically at least, that the CHW moving contact line can exist even with
no-slip conditions for the velocity. The relevance and realism or lack thereof of this
is considered through the course of the paper.
The connection between the elastic behaviour of an aggregate and a single crystal is considered, with special reference to the theories of Voigt, Reuss, and Huber and Schmid. The elastic limit under various stress systems is also considered, in particular, it is shown that the tensile elastic limit of a face-centred aggregate cannot exceed two-thirds of the stress at which pronounced plastic distortion occurs.
Benchmark configurations for quantitative validation and comparison of incompressible interfacial flow codes, which model two-dimensional bubbles rising in liquid columns, are proposed. The benchmark quantities: circularity, center of mass, and mean rise velocity are defined and measured to monitor convergence toward a reference solution. Comprehensive studies are undertaken by three independent research groups, two representing Eulerian level set finite-element codes and one representing an arbitrary Lagrangian–Eulerian moving grid approach.
The first benchmark test case considers a bubble with small density and viscosity ratios, which undergoes moderate shape deformation. The results from all codes agree very well allowing for target reference values to be established. For the second test case, a bubble with a very low density compared to that of the surrounding fluid, the results for all groups are in good agreement up to the point of break up, after which all three codes predict different bubble shapes. This highlights the need for the research community to invest more effort in obtaining reference solutions to problems involving break up and coalescence.
Other research groups are encouraged to participate in these benchmarks by contacting the authors and submitting their own data. The reference data for the computed benchmark quantities can also be supplied for validation purposes. Copyright