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Modelling anisotropic Cahn-Hilliard equation with the lattice Boltzmann method
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Abstract
The anisotropic Cahn-Hilliard equation is often used to model the formation of faceted pyramids on nanoscale crystal surfaces. In comparison to the isotropic Cahn-Hilliard model, the nonlinear terms associated with strong anisotropic coefficients present challenges for developing an effective numerical scheme. In this work, we propose a multiple-relaxation-time lattice Boltzmann method to solve the anisotropic Cahn-Hilliard equation. To this end, we reformulate the original equation into a nonlinear convection-diffusion equation with source terms. Then the modified equilibrium distribution function and source terms are incorporated into the computations. Through Chapman-Enskog analysis, it successfully recovers the macroscopic governing equation. To validate the proposed approach, we perform numerical simulations, including cases like droplet deformation and spinodal decomposition. These results consistent with available works, confirming the effectiveness of the proposed approach. Furthermore, the simulations demonstrate that the model adheres to the energy dissipation law, further highlighting the effectiveness of the developed lattice Boltzmann method.
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In this paper, we present a brief overview of the phase-field-based lattice Boltzmann method (LBM) that is a distinct and efficient numerical algorithm for multiphase flow problems. We first give an introduction to the mathematical theory of phase-field models for multiphase flows, and then present some recent progress on the LBM for the phase-field models which are composed of the classic Navier-Stokes equations and the Cahn-Hilliard or Allen-Cahn equation. Finally, some applications of the phase-field-based LBM are also discussed.
We develop in this paper efficient and robust numerical methods for solving anisotropic Cahn-Hilliard systems. We construct energy stable schemes for the time discretization of the highly nonlinear anisotropic Cahn-Hilliard systems by using a stabilization technique. At each time step, these schemes lead to a sequence of linear coupled elliptic equations with constant coefficients that can be efficiently solved by using a spectral-Galerkin method. We present numerical results that are consistent with earlier work on this topic, and also carry out various simulations, such as the linear bi-Laplacian regularization and the nonlinear Willmore regularization, to demonstrate the efficiency and robustness of the new schemes.
Discretization of interaction force is an important issue in the lattice Boltzmann equation (LBE) method for two-phase systems. A variety of discretization schemes for the interaction force have been developed from different viewpoints. Although these schemes are totally identical mathematically, they may have quite different behaviors in actual applications. In this paper we investigate the mass conservation properties of four discretization forcing schemes that have been adopted in a variety of two-phase LBE models. It is found that LBE with the mixed scheme does not conserve the total mass, and the violation is related to grid resolution, fluid velocity, and density variation.
We present a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn–Hilliard-typ equations. Such problems arise during the growth and coarsening of thin films. When the anisotropic surface energy is sufficientl strong, sharp corners form and unregularized anisotropic Cahn–Hilliard equations become ill-posed. Our models contain a high-orde Willmore regularization, where the square of the mean curvature is added to the energy, to remove the ill-posedness. The regularize equations are sixth order in space. A key feature of our approach is the development of a new formulation in which the interfac thickness is independent of crystallographic orientation. Using the method of matched asymptotic expansions, we show the convergenc of our phase-field model to the general sharp-interface model. We present two- and three-dimensional numerical results usin an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the dynamics of the new phase-fiel model and the sharp-interface model. The computed equilibrium shapes using the new model also match a recently developed analytica sharp-interface theory that describes the rounding of the sharp corners by the Willmore regularization.
In this paper, the lattice Boltzmann equation is directly derived from the Boltzmann equation. It is shown that the lattice Boltzmann equation is a special discretized form of the Boltzmann equation. Various approximations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail. A general procedure to derive the lattice Boltzmann model from the continuous Boltzmann equation is demonstrated explicitly. The lattice Boltzmann models derived include the two-dimensional 6-bit, 7-bit, and 9-bit, and three-dimensional 27-bit models.
Modeling and numerical approximation of two-phase incompressible flows with different densities and viscosities are considered. A physically consistent phase-field model that admits an energy law is proposed, and several energy stable, efficient, and accurate time discretization schemes for the coupled nonlinear phase-field model are constructed and analyzed. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios.
The capability of modeling and simulating complex interfacial dynamics of multiphase flows has been recognized as one of the main advantages of the lattice Boltzmann equation (LBE). A basic feature of two-phase LBE models, i.e., the force balance condition at the discrete lattice level of LBE, is investigated in this work. An explicit force-balance formulation is derived for a flat interface by analyzing the two-dimensional nine-velocity (D2Q9) two-phase LBE model without invoking the Chapman-Enskog expansion. The result suggests that generally the balance between the interaction force and the pressure does not hold exactly on the discrete lattice due to numerical errors. It is also shown that such force imbalance can lead to some artificial velocities in the vicinity of phase interface.
The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyperviscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters that optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some two-dimensional LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long-wavelength limit (wave vector k=0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.
Consider faceting of a crystal surface caused by strongly anisotropic surface tension, driven by surface diffusion and accompanied by deposition (etching) due to fluxes normal to the surface. Nonlinear evolution equations describing the faceting of 1+1 and 2+1 crystal surfaces are studied analytically, by means of matched asymptotic expansions for small growth rates, and numerically otherwise. Stationary shapes and dynamics of faceted pyramidal structures are found as functions of the growth rate. In the 1+1 case it is shown that a solitary hill as well as periodic hill-and-valley solutions are unique, while solutions in the form of a solitary valley form a one-parameter family. It is found that with the increase of the growth rate, the faceting dynamics exhibits transitions from the power-law coarsening to the formation of pyramidal structures with a fixed average size and finally to spatiotemporally chaotic surfaces resembling the kinetic roughening.
In this paper, a lattice Boltzmann for quasi-incompressible two-phase flows is proposed based on the Cahn-Hilliard phase-field theory, which can be viewed as an improved model of a previous one [Yang and Guo, Phys. Rev. E 93, 043303 (2016)]. The model is composed of two LBE's, one for the Cahn-Hilliard equation (CHE) with a singular mobility, and the other for the quasi-incompressible Navier-Stokes equations (qINSE). Particularly, the LBE for the CHE uses an equilibrium distribution function containing a free parameter associated with the gradient of chemical potential, such that the variable (and even zero) mobility can be handled. In addition, the LBE for the qINSE uses an equilibrium distribution function containing another free parameter associated with the local shear rate, such that the large viscosity ratio problems can be handled. Several tests are first carried out to test the capability of the proposed LBE for the CHE in capturing phase interface, and the results demonstrate that the proposed model outperforms the original LBE model in terms of accuracy and stability. Furthermore, by coupling the hydrodynamic equations, the tests of double-stationary droplets and droplets falling problems indicate that the proposed model can reduce numerical dissipation and produce physically acceptable results at large time scales. The results of droplets falling and phase separation of binary fluid problems show that the present model can handle two-phase flows with large viscosity ratio up to the magnitude of 104.
Solid particle erosion is a mechanical process of destroying a wall surface material due to the impacts of solid particles entrained with a fluid. It is a frequent phenomenon encountered within various industries such as chemical processes, oil and gas, and hydraulic transportation. Erosion problem has led to enormous consequences such as oil spills caused by equipment failure of oil transmission pipelines, chokes valves, pipe fittings etc; resulting in considerable economic loss as well as safety and environmental concerns. In this study, a 3-D simulations are performed using CFD code ANSYS FLUENT to predict sand erosion rates under different engine-oil viscosity conditions for multiphase liquid, in a 90-degree standard (R/D = 1.5) elbow pipe. The CFD utilizes Eulerian-Lagrangian method to model the multiphase flow oil-water-sand in elbow. The realizable k -ε model is adopted for the fluid turbulence effects. The velocity and pressure distributions are analysed as contours for the fluid flow. In order to understand the dynamics of the erosion process, the motion of the solid particles are also investigated based on Stokes number as well as the effect of secondary flows. The results indicated that erosion rates decrease with the increase in oil viscosity. Additionally, erosion mainly occurs in two locations; at the extrados near the bend exit and also on the side walls of the downstream straight pipe. The unusual distribution of erosion on the side walls occurred as a result of the effect of secondary flows due to centrifugal force. The numerical results are in qualitative good agreement with the experimental data available in the literature for elbows in order to validate the presented modelling approach.
Fractional phase field models have been reported to suitably describe the anomalous two-phase transport in heterogeneous porous media, evolution of structural damage, and image inpainting process. It is commonly different to derive their analytical solutions, and the numerical solution to these fractional models is an attractive work. As one of the popular fractional phase-field models, in this paper we propose a fresh lattice Boltzmann (LB) method for the fractional Cahn-Hilliard equation. To this end, we first transform the fractional Cahn-Hilliard equation into the standard one based on the Caputo derivative. Then the modified equilibrium distribution function and proper source term are incorporated into the LB method in order to recover the targeting equation. Several numerical experiments, including the circular disk, quadrate interface, droplet coalescence and spinodal decomposition, are carried out to validate the present LB method. It is shown that the numerical results at different fractional orders agree well with the analytical solution or some available results. Besides, it is found that increasing the fractional order promotes a faster evolution of phase interface in accordance with its physical definition, and also the system energy predicted by the present LB method conforms to the energy dissipation law.
In this paper, we consider numerical approximations for solving the anisotropic Cahn–Hilliard model. We combine the Scalar Auxiliary Variable (SAV) approach with the stabilization technique to arrive at a novel Stabilized-SAV approach, where three linear stabilization terms, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient, are added to enhance the stability while keeping the required accuracy. The schemes are very easy-to-implement and fast in the sense that all nonlinear terms are treated in a semi-explicit way, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. We further prove the unconditional energy stabilities rigorously and present numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy.
Theory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the model most suitable for the problems they are interested in. The book is targeted at graduate students and researchers who plan to investigate multiphase flows using LBMs. Throughout the text most of the popular multiphase LBMs are analyzed both theoretically and through numerical simulation. The authors present many of the mathematical derivations of the models in greater detail than is currently found in the existing literature. The approach to understanding and classifying the various models is principally based on simulation compared against analytical and observational results and discovery of undesirable terms in the derived macroscopic equations and sometimes their correction. A repository of FORTRAN codes for multiphase LBM models is also provided.
Over the past few decades, tremendous progress has been made in the
development of particle-based discrete simulation methods versus the
conventional continuum-based methods. In particular, the lattice Boltzmann (LB)
method has evolved from a theoretical novelty to a ubiquitous, versatile and
powerful computational methodology for both fundamental research and
engineering applications. It is a kinetic-based mesoscopic approach that
bridges the microscales and macroscales, which offers distinctive advantages in
simulation fidelity and computational efficiency. Applications of the LB method
have been found in a wide range of disciplines including physics, chemistry,
materials, biomedicine and various branches of engineering. The present work
provides a comprehensive review of the LB method for thermofluids and energy
applications, focusing on multiphase flows, thermal flows and thermal
multiphase flows with phase change. The review first covers the theoretical
framework of the LB method, revealing the existing inconsistencies and defects
as well as common features of multiphase and thermal LB models. Recent
developments in improving the thermodynamic and hydrodynamic consistency,
reducing the spurious currents, enhancing the numerical stability, etc., are
highlighted. These efforts have put the LB method on a firmer theoretical
foundation with enhanced LB models that can achieve larger liquid-gas density
ratio, higher Reynolds number and flexible surface tension. Examples of
applications are provided in fuel cells and batteries, droplet collision,
boiling heat transfer and evaporation, and energy storage. Finally, further
developments and future prospect of the LB method are outlined for thermofluids
and energy applications.
In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.
A level set method for capturing the interface between two fluids is
combined with a variable density projection method to allow for
computation of two-phase flow where the interface can merge/break and
the flow can have a high Reynolds number. A distance function
formulation of the level set method enables one to compute flows with
large density ratios (1000/1) and flows that are surface tension driven;
with no emotional involvement. Recent work has improved the accuracy of
the distance function formulation and the accuracy of the advection
scheme. We compute flows involving air bubbles and water drops, to name
a few. We validate our code against experiments and theory.
When the Lattice Boltzmann Method (LBM) is used for simulating continuum fluid flow, the discrete mass distribution must satisfy imposed constraints for density and momentum along the boundaries of the lattice. These constraints uniquely determine the three‐dimensional (3‐D) mass distribution for boundary nodes only when the number of external (inward‐pointing) lattice links does not exceed four. We propose supplementary rules for computing the boundary distribution where the number of external links does exceed four, which is the case for all except simple rectangular lattices. Results obtained with 3‐D body‐centered‐cubic lattices are presented for Poiseuille flow, porous‐plate Couette flow, pipe flow, and rectangular duct flow. The accuracy of the two‐dimensional (2‐D) Poiseuille and Couette flows persists even when the mean free path between collisions is large, but that of the 3‐D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3‐D low‐Reynolds‐number flow.
This article presents a comprehensive review of numerical methods and models for interface resolving simulations of multiphase flows in microfluidics and micro process engineering. The focus of the paper is on continuum methods where it covers the three common approaches in the sharp interface limit, namely the volume-of-fluid method with interface reconstruction, the level set method and the front tracking method, as well as methods with finite interface thickness such as color-function based methods and the phase-field method. Variants of the mesoscopic lattice Boltzmann method for two-fluid flows are also discussed, as well as various hybrid approaches. The mathematical foundation of each method is given and its specific advantages and limitations are highlighted. For continuum methods, the coupling of the interface evolution equation with the single-field Navier–Stokes equations and related issues are discussed. Methods and models for surface tension forces, contact lines, heat and mass transfer and phase change are presented. In the second part of this article applications of the methods in microfluidics and micro process engineering are reviewed, including flow hydrodynamics (separated and segmented flow, bubble and drop formation, breakup and coalescence), heat and mass transfer (with and without chemical reactions), mixing and dispersion, Marangoni flows and surfactants, and boiling.
We present a stable numerical scheme for modelling multiphase flow in porous media, where the characteristic size of the flow domain is of the order of microns to millimetres. The numerical method is developed for efficient modelling of multiphase flow in porous media with complex interface motion and irregular solid boundaries. The Navier–Stokes equations are discretised using a finite volume approach, while the volume-of-fluid method is used to capture the location of interfaces. Capillary forces are computed using a semi-sharp surface force model, in which the transition area for capillary pressure is effectively limited to one grid block. This new formulation along with two new filtering methods, developed for correcting capillary forces, permits simulations at very low capillary numbers and avoids non-physical velocities. Capillary forces are implemented using a semi-implicit formulation, which allows larger time step sizes at low capillary numbers. We verify the accuracy and stability of the numerical method on several test cases, which indicate the potential of the method to predict multiphase flow processes.
In this paper, we consider the vector-valued Allen–Cahn equations which model phase separation in NN-component systems. The considerations of solving numerically the vector-valued Allen–Cahn equations are as follows: (1) the use of a small time step is appropriate to obtain a stable solution and (2) a sufficient number of phase-field variables is required to capture the correct dynamics. However, stability restrictions on the time step and a large number of phase-field variables cause huge computational costs and make the calculation very inefficient. To overcome this problem, we present an efficient and accurate numerical algorithm which is based on an operator splitting technique and is solved by a fast solver such as a linear geometric multigrid method. The algorithm allows us to convert the vector-valued Allen–Cahn equations with NN components into a system of N−1N−1 binary Allen–Cahn equations and drastically reduces the required computational time and memory. We demonstrate the efficiency and accuracy of the algorithm with various numerical experiments. Furthermore, using our algorithm, we can simulate the growth of multiple crystals with different orientation angles and fold numbers on a single domain. Finally, the efficiency of our algorithm is validated with an example that includes the growth of multiple crystals with consideration of randomness effects.
The equilibrium shape of a crystal is the shape that minimizes its anisotropic interfacial free energy subject to the constraint of constant volume. This shape can be determined geometrically by using the Wulff construction; it can have missing orientations, sharp edges and corners, and its faces can be rounded or flat (facets). In two dimensions, when the surface free energy, gamma, is a function of a single angle theta, the analytical criterion for the onset of missing orientations is that gamma+gamma changes sign from positive to negative. In three dimensions, for which gamma depends on two angles, theta and varphi, no such analytical criterion is known. A geometrical criterion for missing orientations can be based on the Herring sphere construction. By means of inversion through the origin, an equivalent criterion becomes whether any portion of a polar plot of the reciprocal of gamma (1/gamma-plot) lies outside a tangent plane. The onset of missing orientations occurs when the 1/gamma-plot changes from convex to concave. An equivalent analytical criterion is obtained by showing that the normal to the 1/gamma-plot is proportional to the xi vector of Hoffman and Cahn and then using an invariant representation of the Gaussian curvature of the 1/gamma-plot to test its convexity. Results are illustrated for cubic symmetry. As the magnitude of anisotropy is increased, the equilibrium shape loses orientations and tends to an octahedron (positive anisotropy) or a cube (negative anisotropy). A similar formalism can be used to find an analytical criterion for missing orientations on growth shapes of crystals that are growing under the control of anisotropic interface kinetics.
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.
A lattice Boltzmann equation (LBE) method for incompressible binary fluids is proposed to model the contact line dynamics on partially wetting surfaces. Intermolecular interactions between a wall and fluids are represented by the inclusion of the cubic wall energy in the expression of the total free energy. The proposed boundary conditions eliminate the parasitic currents in the vicinity of the contact line. The LBE method is applied to micron-scale drop impact on dry surfaces, which is commonly encountered in drop-on-demand inkjet applications. For comparison with the existing experimental results [H. Dong, W.W. Carr, D.G. Bucknall, J.F. Morris, Temporally-resolved inkjet drop impaction on surfaces, AIChE J. 53 (2007) 2606–2617], computations are performed in the range of equilibrium contact angles from 31° to 107° for a fixed density ratio of 842, viscosity ratio of 51, Ohnesorge number (Oh) of 0.015, and two Weber numbers (We) of 13 and 103.
We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hy- drodynamic equations with variable source terms based on equivalent equilibrium functions. A special parametrization of the free relaxation parameter is derived. It controls, in addition to the non-dimensional hydrodynamic numbers, any TRT macro- scopic steady solution and governs the spatial discretization of transient flows. In this framework, the multi-reflection approach (16,18) is generalized and extended for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary condi- tions. We propose second and third-order accurate boundary schemes and adapt them for corners. The boundary schemes are analyzed for exactness of the parametrization, uniqueness of their steady solutions, support of staggered invariants and for the ef- fective accuracy in case of time dependent boundary conditions and transient flow. When the boundary scheme obeys the parametrization properly, the derived perme- ability values become independent of the selected viscosity for any porous structure and can be computed efficiently. The linear interpolations (5,46) are improved with respect to this property.
The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite ele-ment solutions are not common because primal variational formulations of fourth-order operators are well defined and integrable only if the finite element basis func-tions are piecewise smooth and globally C 1 -continuous. There are a very limited number of two-dimensional finite elements possessing C 1 -continuity applicable to complex geometries, but none in three-dimensions. We propose Isogeometric Anal-ysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accu-racy, robustness, two-and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of C 1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two-and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitiz-ing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.
This brief review deals with the characterization of multiphase flows needed in reaction engineering and describes the techniques and methods available to accomplish it. It emphasizes the fact that most multiphase systems of industrial interest (e.g. stirred tanks, slurry bubble columns, gas–solid or liquid–solid risers and fluidized beds, pneumatic transport reactors, etc.) operate in churn turbulent flow at large volume fraction of the dispersed phase and are opaque. This inevitably leads to the use of the Euler–Euler, interpenetrating fluid model in describing their fluid dynamics, which raises issues of closure for the interfacial momentum terms and turbulence. Hence, computational fluid dynamic (CFD) models based on the Euler–Euler approach with selected closures require experimental validation. Due to the non-transparency of the systems involved, non-conventional experimental techniques are required.The paper briefly describes two such techniques which have been successfully implemented at the Chemical Reaction Engineering Laboratory for characterization of opaque, multiphase flows. One is the gamma ray computed tomography (CT), the other is the computer automated radioactive particle tracking (CARPT). Their use in characterizing diverse systems such as liquid–solid risers and bubble columns is illustrated as well as the utility of CARPT-CT data in validation of the Euler–Euler CFD models. The goal is to enable the reader to grasp the advantages and limitations of the described experimental techniques and computational fluid dynamic models.
We develop a conservative, second-order accurate fully implicit discretization of the Navier–Stokes (NS) and Cahn–Hilliard (CH) system that has an associated discrete energy functional. This system provides a diffuse-interface description of binary fluid flows with compressible or incompressible flow components [R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998) 2617]. In this work, we focus on the case of flows containing two immiscible, incompressible and density-matched components. The scheme, however, has a straightforward extension to multi-component systems. To efficiently solve the discrete system at the implicit time-level, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an applied external shear, the evolution of the flow is nontrivial and the flow morphology repeats itself in time as multiple pinchoff and reconnection events occur. Eventually, the periodic motion ceases and the system relaxes to a global equilibrium. The equilibria we observe appears has a similar structure in all cases although the dynamics of the evolution is quite different. We view the work presented in this paper as preparatory for a detailed investigation of liquid–liquid interfaces with surface tension where the interfaces separate two immiscible fluids [On the pinchoff of liquid–liquid jets with surface tension, in preparation]. To this end, we also include a simulation of the pinchoff of a liquid thread under the Rayleigh instability at finite Reynolds number.
Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn–Hilliard/Navier–Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn–Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn–Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier–Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank–Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.
We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn–Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn–Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening.
Phase-field models provide a way to model fluid interfaces as having finite thickness. This can allow the computation of interface movement and deformation on fixed grids. This paper applies phase-field modeling to the computation of two-phase incompressible Navier–Stokes flows. The Navier–Stokes equations are modified by the addition of the continuum forcing −Cφ, where C is the composition variable and φ is C's chemical potential. The equation for interface advection is replaced by a continuum advective-diffusion equation, with diffusion driven by C's chemical potential gradients. The paper discusses how solutions to these equations approach those of the original sharp-interface Navier–Stokes equations as the interface thickness ϵ and the diffusivity both go to zero. The basic flow-physics of phase-field interfaces is discussed. Straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion. On the other hand, too large a diffusion will overly damp the flow. These two constraints result in an upper bound for the diffusivity of O(ϵ) and a lower bound of O(ϵ2). Within these two bounds, the phase-field Navier–Stokes equations appear to generate an O(ϵ) error relative to the exact sharp-interface equations. An O(h2/ϵ2) numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing. An O(h4/ϵ4) compact scheme is introduced that takes advantage of the asymptotic, comparatively smooth, behavior of the chemical potential. For O(ϵ) accurate phase-field models the optimum path to convergence for this scheme appears to be ϵ∝h4/5. The asymptotic rate of convergence corresponding to this is O(h4/5) but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear. Extensive analysis and computations show that this scheme is very effective and accurate. It allows the accurate calculation of two-phase flows with interfaces only two cells wide. Computational results are given for linear capillary waves and for Rayleigh–Taylor instabilities. The first set of computations is compared to exact solutions of the diffuse-interface equations and of the original sharp-interface equations. The Rayleigh–Taylor computations test the ability of the method to compute highly deforming flows. These flows include near-singular phenomena such as interface coalescences and breakups, contact line movement, and the formation and breakup of thin wall-films. Grid-refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed, wavelength, and the general physical characteristics of the instability and mass transfer rates.
In this paper, we propose a new lattice Boltzmann scheme for simulation of multiphase flow in the nearly incompressible limit. The new scheme simulates fluid flows based on distribution functions. The interfacial dynamics, such as phase segregation and surface tension, are modeled by incorporating molecular interactions. The lattice Boltzmann equations are derived from the continuous Boltzmann equation with appropriate approximations suitable for incompressible flow. The numerical stability is improved by reducing the effect of numerical errors in calculation of molecular interactions. An index function is used to track interfaces between different phases. Simulations of the two-dimensional Rayleigh–Taylor instability yield satisfactory results. The interface thickness is maintained at 3–4 grid spacings throughout simulations without artificial reconstruction steps.
We conduct a comparative study to evaluate several lattice Boltzmann (LB) models for solving the near incompressible Navier-Stokes equations, including the lattice Boltzmann equation with the multiple-relaxation-time (MRT), the two-relaxation-time (TRT), the single-relaxation-time (SRT) collision models, and the entropic lattice Boltzmann equation (ELBE). The lid-driven square cavity flow in two dimensions is used as a benchmark test. Our results demonstrate that the ELBE does not improve the numerical stability of the SRT or the lattice Bhatnagar-Gross-Krook (LBGK) model. Our results also show that the MRT and TRT LB models are superior to the ELBE and LBGK models in terms of accuracy, stability, and computational efficiency and that the ELBE scheme is the most inferior among the LB models tested in this study, thus is unfit for carrying out numerical simulations in practice. Our study suggests that, to optimize the accuracy, stability, and efficiency in the MRT model, it requires at least three independently adjustable relaxation rates: one for the shear viscosity ν (or the Reynolds number Re), one for the bulk viscosity ζ, and one to satisfy the criterion imposed by the Dirichlet boundary conditions which are realized by the bounce-back-type boundary conditions.
A new design and construction methodology for integration of complicated chemical processing on a microchip was proposed. This methodology, continuous-flow chemical processing (CFCP), is based on a combination of microunit operations (MUOs) and a multiphase flow network. Chemical operations in microchannels, such as mixing, reaction, and extraction, were classified into several MUOs. The complete procedure for Co(II) wet analysis, including a chelating reaction, solvent extraction, and purification was decomposed into MUOs and reconstructed as CFCP on a microchip. Chemical reaction and molecular transport were realized in and between continuous liquid flows in a multiphase flow network, such as aqueous/aqueous, aqueous/organic, and aqueous/organic/aqueous flows. When the determination of Co(II) in an admixture of Cu(II) was carried out using this methodology, the determination limit (2sigma) was obtained as 18 nM, and the absolute amount of Co chelates detected was 0.13 zmol, that is, 78 chelates. The sample analysis time was faster than that of a conventional processing system. Moreover, troublesome operations such as phase separation and acid and alkali washing, all necessary for the conventional system, were simplified. The CFCP methodology proposed here can be applied to various on-chip applications.
Dynamic models for facet formation often employ a regularization of the surface energy based on a corner energy term. Here we consider the effect of this regularization on the equilibrium shape of a solid particle in two dimensions. Using matched asymptotic expansions we determine the explicit solution for the corner shape in the presence of the regularization. Our results show that for a class of surface energy anisotropy models the regularized solution approaches the classic sharp-corner results as the regularization approaches zero. The results validate the use of the regularization in numerical calculations for the equilibrium problem. Finally, a byproduct of the analysis is an exact solution for the equilibrium shape of a semi-infinite wedge in the presence of the regularization.
In this paper, a two-phase lattice Boltzmann (LB) model, developed for simulating fluid flows on a Cartesian grid at high liquid-to-gas density ratios, is adapted to an axisymmetric coordinate system. This is achieved by incorporating additional source terms in the planar evolution equations for the density and pressure distribution functions such that the axisymmetric mass and momentum conservation equations are recovered in the macroscopic limit. Appropriate numerical treatment of the terms is performed to obtain stable computations at high density ratio for this axisymmetric model. The particle collision is modeled by employing multiple relaxation times to attain stability at low viscosity. The model is evaluated by verifying the Laplace-Young relation for a liquid drop, comparing computed frequency of oscillations of an initially ellipsoidal drop with analytical values and comparing the behavior of a spherical drop impinging on a wet wall with prior results. The time evolution of the radial distance of the tip of the corona, formed when the drop impinges, agrees well with prior data.
Research progress of capillary flow in microchannels and its engineering application
- L I Yating
- W Zhongdong
- D Yanpeng
- Z Chunying
- M Youguang
- F Taotao
Yating LI, Zhongdong W, Yanpeng D, Chunying Z, Youguang M, Taotao F. Research progress of capillary flow in microchannels and its
engineering application. CIESC Journal, 2024;75(1):159.
Multiple-relaxation-time lattice Boltzmann models in three dimensions
- D Humières
d'Humières D. Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philos. Trans. R. Soc. Lond. A 2002;360(1792):437-451.