Effective Force Stabilising Technique for the Immersed Boundary Method

The immersed boundary method has emerged as an efficient approach forthe simulation of finite-sized solid particles in complex fluid flows. However, one ofthe well known shortcomings of the method is the limited support for the simulationof light particles, i.e. particles with a density lower than that of the surrounding fluid,both in terms of accuracy and numerical stability.Although a broad literature exists, with several authors reporting different approachesfor improving the stability of the method, most of these attempts introduce extra complexities and are very costly from a computational point of view.In this work, we introduce an effective force stabilizing technique, allowing to extendthe stability range of the method by filtering spurious oscillations arising when dealingwith light-particles, pushing down the particle-to-fluid density ratio as low as 0.04.We thoroughly validate the method comparing with both experimental and numericaldata available in literature.     

A Characteristic Boundary Condition for Multispeed Lattice Boltzmann Methods

We present the development of a non-reflecting boundary condition, basedon the Local One-Dimensional Inviscid (LODI) approach, for Lattice Boltzmann Models working with multi-speed stencils.We test and evaluate the LODI implementation with numerical benchmarks, showing significant accuracy gains with respect to the results produced by a simple zero-gradient condition. We also implement a simplified approach, which allows handlingthe unknown distribution functions spanning several layers of nodes in a unified way,still preserving a comparable level of accuracy with respect to the standard formulation.     

Burnett Order Stress and Spatially-Dependent Boundary Conditions for the Lattice Boltzmann Method

Stress boundary conditions for the lattice Boltzmann equation that are consistent to Burnett order are proposed and imposed using a moment-based method.The accuracy of the method with complicated spatially-dependent boundary conditions for stress and velocity is investigated using the regularized lid-driven cavity flow.The complete set of boundary conditions, which involve gradients evaluated at theboundaries, are implemented locally. A recently-derived collision operator with modified equilibria and velocity-dependent collision rates to reduce the defect in Galileaninvariance is also investigated. Numerical results are in excellent agreement with existing benchmark data and exhibit second-order convergence. The lattice Boltzmannstress field is studied and shown to depart significantly from the Newtonian viscousstress when the ratio of Mach to Reynolds numbers is not negligibly small.     

Phase-Field-Based Axisymmetric Lattice Boltzmann Method for Two-Phase Electro-Hydrodynamic Flows

In this work, a novel and simple phase-field-based lattice Boltzmann (LB) method is proposed for the axisymmetric two-phase electro-hydrodynamic flows. The present LB method is composed of three LB models, which are used to solve the axisymmetric Allen-Cahn equation for the phase field, the axisymmetric Poisson equation for the electric potential, and the axisymmetric Navier-Stokes equations for the flow field. Compared with the previous LB models for the axisymmetric Poisson equation, which can be viewed as the solvers to the convection-diffusion equation, the present model is a genuine solver to the axisymmetric Poisson equation. To test the capacity of the LB method, the deformation of a single leaky or perfect dielectric drop under a uniform electric field is considered, and the effects of electric strength, conductivity ratio, and permittivity ratio are investigated in detail. It is found that the present numerical results are in good agreement with some available theoretical, numerical and/or experimental data.     

A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

In this work, we proposed a diffuse-interface model for the dendritic growth with thermosolutal convection. In this model, the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface, which is described by the order parameter, and the diffuse-interface based governing equations for the dendritic growth are presented. To solve the model for the dendritic growth with thermosolutal convection, we also developed a diffuse-interface multi-relaxation-time lattice Boltzmann (LB) method. In this method, the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction, and the treatment on the complex fluid-solid interface can be avoided. In addition, four LB models are designed for the phase-field equation, concentration equation, temperature equation and the Navier-Stokes equations in a unified framework. Finally, we performed some simulations of the dendritic growth to test the present diffuse-interface LB method, and found that the numerical results are in good agreements with some previous works.     

Elements of the Lattice Boltzmann Method I: Linear Advection Equation

This paper opens a series of papers aimed at finalizing the development of the lattice Boltzmann method for complex hydrodynamic systems. The lattice Boltzmann method is introduced at the elementary level of the linear advection equation. Details are provided on lifting the target macroscopic equations to a kinetic equation, and, after that, to the fully discrete lattice Boltzmann scheme. The over-relaxation method is put forward as a cornerstone of the second-order temporal discretization, and its enhancement with the use of the entropy estimate is explained in detail. A new asymptotic expansion of the entropy estimate is derived, and implemented in the sample code. It is shown that the lattice Boltzmann method provides a computationally efficient way of numerically solving the advection equation with a controlled amount of numerical dissipation, while retaining positivity.     

Hybrid Diffuse and Sharp Interface Immersed Boundary Methods for Particulate Flows in the Presence of Complex Boundaries

A coupling framework that leverages the advantages of the diffuse and sharp interface immersed boundary (IB) methods is presented for handling the interaction among particles and particles with the static complex geometries of the environment. In the proposed coupling approach, the curvilinear IB method is employed to represent the static complex geometries, a variant of the direct forcing IB method is proposed for simulating particles, and the discrete element method is employed for particle-particle and particle-wall collisions. The proposed approach is validated using several classical benchmark problems, which include flow around a sphere, sedimentation of a sphere, collision of two sedimenting spheres, and collision between a particle and a flat wall, with the present predictions showing an overall good agreement with the results reported in the literature. The capability of the proposed framework is further demonstrated by simulating the interaction between multiple particles and a wall-mounted cylinder, and the particle-laden turbulent flow over periodic hills. The proposed method provides an efficient way to simulate particle-laden turbulent flows in environments with complex boundaries.     

A Hybrid Immersed Boundary-Lattice Boltzmann Method for Simulation of Viscoelastic Fluid Flows Interaction with Complex Boundaries

In this study, a numerical technique based on the Lattice Boltzmann method is presented to model viscoelastic fluid interaction with complex boundaries which are commonly seen in biological systems and industrial practices. In order to accomplish numerical simulation of viscoelastic fluid flows, the Newtonian part of the momentum equations is solved by the Lattice Boltzmann Method (LBM) and the divergence of the elastic tensor, which is solved by the finite difference method, is added as a force term to the governing equations. The fluid-structure interaction forces are implemented through the Immersed Boundary Method (IBM). The numerical approach is validated for Newtonian and viscoelastic fluid flows in a straight channel, a four-roll mill geometry as well as flow over a stationary and rotating circular cylinder. Then, a numerical simulation of Oldroyd-B fluid flow around a confined elliptical cylinder with different aspect ratios is carried out for the first time. Finally, the present numerical approach is used to simulate a biological problem which is the mucociliary transport process of human respiratory system. The present numerical results are compared with appropriate analytical, numerical and experimental results obtained from the literature.     

Consistent Forcing Scheme in the Simplified Lattice Boltzmann Method for Incompressible Flows

Considering the fact that the lattice discrete effects are neglected while introducing a body force into the simplified lattice Boltzmann method (SLBM), we propose a consistent forcing scheme in SLBM for incompressible flows with external forces. The lattice discrete effects are considered at the level of distribution functions in the present forcing scheme. Consequently, it is more accurate compared with the original forcing scheme used in SLBM. Through Taylor series expansion and Chapman-Enskog (CE) expansion analysis, the present forcing scheme can be proven to recover the macroscopic Navier-Stokes (N-S) equations. Then, the macroscopic equations are resolved through a fractional step technique. Furthermore, the material derivative term is discretized by the central difference method. To verify the results of the present scheme, we simulate with multiple forms of external force interactions including the space- and time-dependent body forces. Hence, the present forcing scheme overcomes the disadvantages of the original forcing scheme and the body force can be accurately imposed in the present scheme even when a coarse mesh is applied while the original scheme fails. Excellent agreements between the analytical solutions and our numerical results can be observed.     

Numerical Simulations of Particle Sedimentation Using the Immersed Boundary Method

We study the settling of solid particles in a viscous incompressible fluid contained within a two-dimensional channel, where the mass density of the particles is greater than that of the fluid. The fluid-structure interaction problem is simulated numerically using the immersed boundary method, where the added mass is incorporated using a Boussinesq approximation. Simulations are performed with a single circular particle, and also with two particles in various initial configurations. The terminal particle settling velocity and drag coefficient correspond closely with other theoretical, experimental and numerical results, and the particle trajectories reproduce the expected behavior qualitatively. In particular, simulations of a pair of interacting particles similar drafting-kissing-tumbling dynamics to that observed in other experimental and numerical studies.     

Simulation of Thrombus Formation in Shear Flows Using Lattice Boltzmann Method

This article describes the prediction of index of thrombus formation in shear blood flow by computational fluid dynamics with the Lattice Boltzmann Method (LBM), applying to backward-facing step flow, which is a simple model of shear flow in the rotary blood pumps and complicated geometry of medical fluid devices. Assuming that the blood flow is a multiphase flow composed of blood plasma and activated fibrinogen, the effects of surface tension and adhesion force to the wall were added to the LBM computational model. It was found that the thrombus formation in the backward-facing step flow occurred just after the reattachment point and behind the step. These results corresponded to our observation results of thrombus formation. For the thrombus formation in every case of blood flow to be predicted, effects of threshold level of physical parameters such as shear rate and adhesion force (effective distance from the wall) were estimated. Moreover, it was also found that the predicted adhesion point on the wall agrees with the visualization of thrombus formation by predicting proper thresholds.     

Three-Dimensional Simulation of Balloon Dynamics by the Immersed Boundary Method Coupled to the Multiple-Relaxation-Time Lattice Boltzmann Method

The immersed boundary method (IBM) has been popular in simulating fluid structure interaction (FSI) problems involving flexible structures, and the recent introduction of the lattice Boltzmann method (LBM) into the IBM makes the method more versatile. In order to test the coupling characteristics of the IBM with the multiple-relaxation-time LBM (MRT-LBM), the three-dimensional (3D) balloon dynamics, including inflation, release and breach processes, are simulated. In this paper, some key issues in the coupling scheme, including the discretization of 3D boundary surfaces, the calculation of boundary force density, and the introduction of external force into the LBM, are described. The good volume conservation and pressure retention properties are verified by two 3D cases. Finally, the three FSI processes of a 3D balloon dynamics are simulated. The large boundary deformation and oscillation, obvious elastic wave propagation, sudden stress release at free edge, and recoil phenomena are all observed. It is evident that the coupling scheme of the IBM and MRT-LBM can handle complicated 3D FSI problems involving large deformation and large pressure gradients with very good accuracy and stability.     

Chromodynamic Lattice Boltzmann Method for the Simulation of Drops,Erythrocytes, and Other Vesicles

Recently, we have validated a three-dimensional, single framework multicomponent lattice Boltzmann method, modified to generate vesicles (rather than drops)[“Three-dimensional single framework multicomponent lattice Boltzmann equationmethod for vesicle hydrodynamics,” Phys. Fluids 33, 077110 (2021)]. This approachimplements an immersed boundary force distribution, characterised by bending rigidity, surface tension, preferred curvature and conserved membrane area, in which workwe successfully validated isolated vesicle flows against other methodologies and experiment. Like most immersed boundary algorithms, our method relies on numericalcomputation of high-order spatial derivatives and an intricate body force density. Thenext step is to verify that it has sufficient numerical stability to address the anticipatedapplication of high volume fraction flows of highly deformable objects in intimate interaction. It is this in silico verification – of both the class of fluid object attainable andthe stability of the later in strong, straining and shearing flows which is at issue, here.We extend our method to simulate multiple variously deflated vesicles and multipleliquid droplets still within a single framework, from which our fluid objects emerge asparticular parameterisations. We present data from simulations containing up to fourvesicles (five immiscible fluid species), which threshold verifies that simulations containing unlimited fluid objects are possible [“Modeling the flow of dense suspensionsof deformable particles in three dimensions,” Phys. Rev. E 75, 066707 (2007)]. Thesedata also assure the ability of our immersed boundary forcing to preserve the character and integrity of fluid objects in interactions characterised by large local velocitygradients (intimate squeezing, shearing and elongational straining). Throughout, wetake interfacial or membrane area, $A,$ as a proxy for stability and physical veracity.     

Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGK Model

This paper aims to study the numerical features of a coupling scheme between the immersed boundary (IB) method and the lattice Boltzmann BGK (LBGK) model by four typical test problems: the relaxation of a circular membrane, the shearing flow induced by a moving fiber in the middle of a channel, the shearing flow near a non-slip rigid wall, and the circular Couette flow between two inversely rotating cylinders. The accuracy and robustness of the IB-LBGK coupling scheme, the performances of different discrete Dirac delta functions, the effect of iteration on the coupling scheme, the importance of the external forcing term treatment, the sensitivity of the coupling scheme to flow and boundary parameters, the velocity slip near non-slip rigid wall, and the origination of numerical instabilities are investigated in detail via the four test cases. It is found that the iteration in the coupling cycle can effectively improve stability, the introduction of a second-order forcing term in LBGK model is crucial, the discrete fiber segment length and the orientation of the fiber boundary obviously affect accuracy and stability, and the emergence of both temporal and spatial fluctuations of boundary parameters seems to be the indication of numerical instability. These elaborate results shed light on the nature of the coupling scheme and may benefit those who wish to use or improve the method.     

Numerical Stability Analysis for a Stationary and Translating Droplet at Extremely Low Viscosity Values Using the Lattice Boltzmann Method Color-Gradient Multi-Component Model with Central Moments Formulation

Multicomponent models based on the Lattice Boltzmann Method (LBM) have clear advantages with respect to other approaches, such as good parallel performances and scalability and the automatic resolution of breakup and coalescence events. Multicomponent flow simulations are useful for a wide range of applications, yet many multicomponent models for LBM are limited in their numerical stability and therefore do not allow exploration of physically relevant low viscosity regimes. Here we perform a quantitative study and validations, varying parameters such as viscosity, droplet radius, domain size and acceleration for stationary and translating droplet simulations for the color-gradient method with central moments (CG-CM) formulation, as this method promises increased numerical stability with respect to the non-CM formulation. We focus on numerical stability and on the effect of decreasing grid-spacing, i.e. increasing resolution, in the extremely low viscosity regime for stationary droplet simulations. The effects of small- and large-scale anisotropy, due to grid-spacing and domain-size, respectively, are investigated for a stationary droplet. The effects on numerical stability of applying a uniform acceleration in one direction on the domain, i.e. on both the droplet and the ambient, is explored into the low viscosity regime, to probe the numerical stability of the method under dynamical conditions.     

Natural Convection Heat Transfer in a Porous Cavity with Sinusoidal Temperature Distribution Using Cu/Water Nanofluid: Double MRT Lattice Boltzmann Method

In this study, natural convection flow in a porous cavity with sinusoidal temperature distribution has been analyzed by a new double multi relaxation time (MRT) Lattice Boltzmann method (LBM). We consider a copper/water nanofluid filling a porous cavity. For simulating the temperature and flow fields, D2Q5 and D2Q9 lattices are utilized respectively, and the effects of different Darcy numbers (Da) (0.001-0.1) and various Rayleigh numbers (Ra) ($10^3$-$10^5$) for porosity ($ε$) between 0.4 and 0.9 have been considered. Phase deviation ($θ$) changed from 0 to $π$ and the volume fraction of nanoparticles (Ø) varied from 0 to 6%. The present results show a good agreement with the previous works, thus confirming the reliability the new numerical method proposed in this paper. It is indicated that the heat transfer rate increases at increasing Darcy number, porosity, Rayleigh number, the volume fraction of nanoparticles and phase deviation. However, the most sensitive parameter is the Rayleigh number. The maximum Nusselt deviation is 10%, 32% and 33% for Ra=$10^3$, $10^4$ and $10^5$, respectively, with $ε = 0.4$ to $ε = 0.9$. It can be concluded that the effect of Darcy number on the heat transfer rate increases at increasing Rayleigh number, yielding a maximum enhancement of the average Nusselt number around 12% and 61% for Ra=$10^3$ and Ra=$10^5$, respectively.     

A Fourth-Order Kernel-Free Boundary Integral Method for Interface Problems

This paper presents a fourth-order Cartesian grid based boundary integral method (BIM) for heterogeneous interface problems in two and three dimensional space, where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions. The method reformulates the governing equation with interface conditions into boundary integral equations (BIEs) and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region. Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver. The structure of the coefficient matrix is preserved even with the existence of the interface. In the whole calculation process, analytical expressions of Green’s functions are never determined, formulated or computed. This is the novelty of the proposed kernel-free boundary integral (KFBI) method. Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.     

Comparison of Lattice Boltzmann,Finite Element and Volume of Fluid Multicomponent Methods for Microfluidic Flow Problems and the Jetting of Microdroplets

We show that the lattice Boltzmann method (LBM) based color-gradient model with a central moments formulation (CG-CM) is capable of accurately simulating the droplet-on-demand inkjetting process on a micrometer length scale by comparing it to the Arbitrary Lagrangian Eulerian Finite Element Method (ALE-FEM). A full jetting cycle is simulated using both CG-CM and ALE-FEM and results are quantitatively compared by measuring the ejected ink velocity, volume and contraction rate. We also show that the individual relevant physical phenomena are accurately captured by considering three test-cases; droplet oscillation, ligament contraction and capillary rise. The first two cases test accuracy for a dynamic system where surface tension is the driving force and the third case is designed to test wetting boundary conditions. For the first two cases we also compare the CG-CM and ALE-FEM results to Volume of Fluid (VOF) simulations. Comparison of the three methods shows close agreement when compared to each other and analytical solutions, where available. Finally wedemonstrate that asymmetric jetting is achievable using 3D CG-CM simulations utilizing asymmetric wetting conditions inside the jet-nozzle. This allows for systematic investigation into the physics of asymmetric jetting, e.g. due to jet-nozzle manufacturing imperfections or due to other disturbances.     

Kinetic Slip Boundary Condition for Isothermal Rarefied Gas Flows Through Static Non-Planar Geometries Based on the Regularized Lattice-Boltzmann Method

The simulation of rarefied gas flows through complex porous media is challenging due to the tortuous flow pathways inherent to such structures. The Lattice Boltzmann method (LBM) has been identified as a promising avenue to solve flows through complex geometries due to the simplicity of its scheme and its high parallel computational efficiency. It has been proposed to model the stress-strain relationship with the extended Navier-Stokes equations rather than attempting to directly solve the Boltzmann equation. However, a regularization technique is required to filter out non-resolved higher-order components with a low-order velocity scheme. Although slip boundary conditions (BCs) have been proposed for the non-regularized multiple relaxation time LBM (MRT-LBM) for planar geometries, previous slip BCs have never been verified extensively with the regularization technique. In this work, following an extensive literature review on the imposition of slip BCs for rarefied flows with the LBM, it is proven that earlier values for kinetic parameters developed to impose slip BCs are inaccurate for the regularized MRT-LBM and differ between the D2Q9 and D3Q15 schemes. The error was eliminated for planar flows and good agreement between analytical solutions for arrays of cylinders and spheres was found with a wide range of Knudsen numbers.     

A Quantitative Comparison of Physical Accuracy and Numerical Stability of Lattice Boltzmann Color Gradient and Pseudopotential Multicomponent Models for Microfluidic Applications

The performances of the Color-Gradient (CG) and the Shan-Chen (SC) multicomponent Lattice Boltzmann models are quantitatively compared side-by-side onmultiple physical flow problems where breakup, coalescence and contraction of fluidligaments are important. The flow problems are relevant to microfluidic applications,jetting of microdroplets as seen in inkjet printing, as well as emulsion dynamics. Asignificantly wider range of parameters is shown to be accessible for CG in terms ofdensity-ratio, viscosity-ratio and surface tension values. Numerical stability for a highdensity ratio $mathcal{O}(1000)$ is required for simulating the drop formation process duringinkjet printing which we show here to be achievable using the CG model but not usingthe SC model. Our results show that the CG model is a suitable choice for challengingsimulations of droplet formation, due to a combination of both numerical stability andphysical accuracy. We also present a novel approach to incorporate repulsion forcesbetween interfaces for CG, with possible applications to the study of stabilized emulsions. Specifically, we show that the CG model can produce similar results to a knownmultirange potentials extension of the SC model for modelling a disjoining pressure,opening up its use for the study of dense stabilized emulsions.