A corrected open boundary framework for lattice Boltzmann immiscible pseudopotential models

The pseudopotential lattice Boltzmann (LB) method has been widely used for simulating multiphase flow due to its concise concept and computational simplicity. In this paper, based on the weighted orthogonal transformation matrix, a three-dimensional (3D) weighted multiple-relaxation-time pseudopotential lattice Boltzmann method (WRMT-LBM) is developed, in which the standard lattice stencil D3Q19 is adopted. Compared with the classical multiple-relaxation-time pseudopotential lattice Boltzmann method (CMRT-LBM) based on the orthogonal transformation matrix, the expressions of the equilibrium density distribution function and discrete force term in moment space are simplified in the present model, which contributes to simplifying the program implementation and improving the computational efficiency. Moreover, an additional discrete source term in moment space compatible with the proposed model is introduced to achieve tunable surface tension. A series of numerical tests are then implemented to investigate the performance of the proposed model. Compared with the CMRT-LBM, the results of the present model can achieve lower spurious velocity and higher computational efficiency while keeping comparable accuracy. Furthermore, using the present model, three benchmark cases, including droplet oscillation, droplet impacting on wall and droplet impact on thin film, are performed to investigate the performance of this model. The numerical results are in good agreement with the analytical solutions or the empirical correlations in the literature, which demonstrates that the present model can simulate the multiphase flow with large density ratio.

The pseudopotential lattice Boltzmann (LB) method has significant advantages in the simulation of liquid-vapor two-phase flow and phase-transition behavior of cryogenic fluids. However, prior research on multiphase flows employing the LB method primarily utilizes the cubic Peng-Robinson equation of state, which has limited accuracy in predicting thermodynamic properties of fluids, in particular for cryogens. This study presents a thermodynamic perspective on the performance of different equations of state in LB simulations, including high-precision equations of state such as those in Helmholtz energy form and the modified Benedict-Webb-Rubin equation of state. The performance of various equations of state within the entire two-phase region, ranging from the triple-point temperature to the critical-point temperature, is comparatively evaluated. It is found that the two-phase region characteristics of equations of state, which are often overlooked in traditional thermodynamics, significantly impact simulation stability. The feasibility and superiority of high-precision equations of state in simulating cryogenic fluids are further substantiated through analysis in three key aspects: liquid-vapor coexistence density, surface tension, and spurious velocity.

In single-component two-phase flow simulations by the lattice Boltzmann method (LBM), one key topic of interest is the equation of state (EOS), which has been studied more in the pseudo-potential approach but much less in the free energy approach. Here, we conducted fair assessment of several popular non-ideal gas EOS in free energy LBM. The inability to independently tune the surface tension and interface thickness was identified for existing free energy formulation using most EOS. A simple modification was proposed and incorporated into three free energy LBM models, including the pressure tensor model, Lee–Fischer's model, and Guo's well-balanced model. Good compatibility with all of them was demonstrated by numerical simulations. In addition, a piecewise-linear (PWL) EOS proposed in pseudo-potential LBM was found to be also feasible in free energy LBM. Extensive numerical experiments on diverse problems (some containing contact lines) revealed that the influence of EOS was rather limited when key parameters are nearly the same (easily achievable with the proposed modification). The capability of the Lee–Fischer and well-balanced models to reduce the spurious currents down to machine accuracy (for static drop tests) was found to be independent of the EOS (including the customized PWL EOS). Overall, the pressure tensor LBM is associated with larger spurious currents and deviations in the density and chemical potential, thus less accurate than the other two. However, it is less sensitive to the relaxation parameter and grid number, thus more robust.

Study on C–S and P–R EOS in pseudo-potential lattice Boltzmann model for two-phase flows

The aim of this work is to propose a new strategy to control the mixing in two-dimensional jets. As is known, theoretical and experimental works have shown that the dynamic of heated jets which develop a self-excited global mode have better mixing properties than homogeneous jets. With this idea in mind, in this paper, we put forward an analysis of the vorticity field dynamic of an heated jet which focus on the baroclinic source term presents in the vorticity equation of variable density flows. We show how this analysis is able to define a new control strategy in homogeneous jets, based on a reorganization of the vorticity field through a controlled vorticity generation mechanism which will force the jet to develop a global mode as hot jet do naturally. This mechanism is implemented through the sinusoidal rotation of two small cylinders well-placed in the near field *Research Engineer, Fluid Mechanics Department of ENSICA, Toulouse, France. Member AIAA. tsocrates student, University Politechnico de Torino, Turin, Italie. t Professor, Director of the Fluid Mechanics Department of ENSICA, Toulouse, France. t Copyright @ 1999 by the American Institute of Aeronautics and Astronautics, Inc. A11 rights reserved. jects used to control or to manipulate flow properties through well founded control strategy. This is a desirable objective which demands : to have a better understanding of the spatial and temporal development of instabilities, by tracking the vorticity dynamic for example, and to develop numerical codes able to simulate these kinds of flows. With theses two pre-requisites, we will show that it is possible to derive an effective control strategy. The linear stability theory have shown that velocity profiles of inhomogeneous jets may be convectively unstable or absolutely unstable, as a function of the density ratio S = pj/pw. This density ratio should be sufficiently low to have an absolutely unstable velocity profile. Moreover, theoretical works and experiments have shown that when the region of absolute instability is long enough, a self-excited global mode is present, enhancing the mixing properties of that flow (Chomaz et a1.2 (1988), Monkewitz et a1.3 (1990), Kyle et a1.4 (1993)). Using these results, in this paper, we will propose a novel control strategy for inhomogeneous jets and begin to evaluate its potential efficiency through a parametric study of the control model. The first part is devoted to the numerical methodology used for our simulations. It emphasized main critical points which are related to the choice and (c)l999 American Institute of Aeronautics & Astionairtics loca.tion of the inlet and outlet boundary conditions because of the convective nature of the simulated flows and because of the vorticity generation mechanism. The second part presents numerical results. It show the spatio-temporal evolution of the vorticity field to enhance our understanding of the forcing mechanism. Next, we define quantities to illustrate global change on the mixing of the jet nea.r field r& gion. Results are discussed and perspectives will be put forward. Numerical methodology The flow simtilations consists of a spatially and temporally developing air jet emerging into quiescent ambiant air. The time-dependent compressible hyperviscous equations (Passot et a1.5 (1988)), with characteristic based boundary conditions, are timemarched with a 5-stage, second order, Runge-Kutta scheme, in conservative variables with a finite volume method (Jameson et a1.6 (1981)). The spatial discretization is based on second order centered differences. The relevance of this hyperviscous model to the study has been detailed in Chapin et ale7 (1996), Chapin et aL8 (1998)). Grid & boundary conditions As is known, a critical point to do this kind of simulations is boundary conditions. Two contradictory properties shduld be verified by bounda.ry conditions. First of all, they should be able to specify the mean flow field characteristics of the simulated flow without sny drift in time.’ But this is not enough when simula.ted flows a.re unsteady for any reasons (intrinsic instability development and/or unsteady forcing). Open bounda.ry conditions should also be able to let coherent structures and waves generated by the flow to propa.gate out,side of the computational domain without. any spurious reflexions on these artificial boundaries. This last, point is particularly important when we try to simulate a convectively unsta,ble flow which is very sen&ve to ba.ckground perturbations a.s wa.s previously shown for example by (Grinstein et a.l.g (1991), Astruc” (1993)). Hence, we have done a prelimina,ry study which have shown the ina.bilit,y oi HedstrGml 1 (1979) and Thompson12 (1987) nonreflecting boundary conditions, applied to the outlet boundary, to completely verify the two necessary criteria, especially when high amplitude coherent vortical $tr&tu&s reached the outlet boundary of the calculation domain. So, it has been necessary to put the outlet boundary sufficiently far from the inlet boundary to be able to analyse the flow before that perturbations going through the outlet frontier have time to modify the dynamic of the flow field in the region of interest through acoustic feedback. To do so, we have chosen a 40-D length calculation domain (with D the jet diameter). The convection velocity of the vertical structures being in this case U, = Uj/2 = 25 m/s, and the height of the jet D=15.6 mm. Hence, the time that a vertical structure, hypothetically forming on the inlet boundary, takes to reach the outlet is 24.4 ms, while the simulated time was chosen to be 23 ms. Moreover, mesh refinement is used in the potential core region to increase resolution. For precision, a typical large scale vertical structure in the finely resolved region.is a.pproximately described with 40x40 gridpoints. The boundary conditions have been chosen in the following way : slip conditions at the lateral boundaries, non-reflecting HedstrGm at the outlet and reservoir condition at the inlet. In the next section, we explain how the control strategy is defined. Numerical experiments

For reduction of the cost of a polymer electrolyte fuel cell (PEFC), it is desired to increase the maximum current density. Under high current density operation, it is important to control water drainage for smooth supply of oxygen. A gas diffusion layer (GDL) which is placed between separator and CL plays an important role for the water transport. Therefore, the improvement of GDL design is important to increase the maximum current density. Recently, there is a report which shows that better cell performance was obtained using the GDL with designed hydrophobic and hydrophilic lines (1). However, the water transport in the GDL is still unclear because the detail observation of liquid water behavior in GDL is difficult due to the quite complex structure and the very small pores (pore diameter: approximately 10 mm). Therefore, there is some room for improving the GDL structure. The objective of this study is to investigate the water transport in GDL with designed wettability pattern and to develop the ideal GDL. To evaluate the water transport in the GDL, a lattice Boltzmann method (LBM) is applied. The LBM is a simulation method to analyze the fluid behavior by tracking movements of large number of particle ensembles. The particle density is expressed by distribution functions, and the distribution functions are calculated by a simple law of collision and transition ensuring the continuity equation and the Navier-Stokes equations for incompressible fluids. Due to the simplicity of the algorithm, it has an advantage of adaptability for complex boundary geometries. Therefore, the LBM is suitable model to simulate the water behavior in the GDL. For high speed calculation, the LBM treating two-phases as having the same density is applied in this study although the densities of air and water are very different. The applicability of this LBM model for the simulation in GDL has been confirmed in the previous study (2). The density of the two-phase in the simulation is set identical to water, ρ = 978kg/m3. The interfacial tension between the two-phase is σ = 6.40×10-2N/m, and liquid and gas viscosities are μL = 4.04×10-4Pa∙s and μG = 2.30×10-5Pa∙s respectively. Figure 1 shows the simulation domain and boundaries. The lower half of simulation domain is GDL layer, on which rib and channel space are placed. Size of GDL is 240μm × 240μm × 100μm and the porosity of GDL is 75%. Sizes of rib and channel are both 120μm × 240μm × 160μm. The bottom of the simulation domain is assumed the micro-porous layer (MPL) and it is treated as a solid wall. Commonly, much water stays in the GDL region under rib. Therefore, to focus on the water behavior in GDL under rib, MPL crack as inlet boundary is set at MPL under rib. Top of the channel is outlet boundary. Four side boundaries are solid walls. The contact angle of GDL, MPL and side walls is set 130º. The contact angle of rib is set 50º. Figure 2 shows simulation results of water transport in GDL without designed wettability pattern. In the simulation, the water is transported randomly and water spreads in the x-y plane of GDL. Finally, the water is drained to channel from pores which is slightly away from the rib. Figure 3 shows simulation results of water transport in GDL with designed wettability pattern. The pattern is set in the center of GDL structure and the size is 50μm × 240μm × 100μm. The contact angle of designed wettability pattern is 50º. Here, the GDL structure is same as Fig. 2. In the simulation, the water is accumulated in the designed wettability pattern of GDL and the smooth water transport from the GDL region under rib to the GDL region under channel is achieved. It is known that water tends to retain in the GDL under ribs and smooth removal of the water is important to avoid flooding. In Ref. (1), it seems that the cell performance using the GDL with designed wettability pattern was improved by the smooth water transport from GDL under rib to GDL under channel. In the final full paper, the effect of the structural characteristics of the GDL such as the contact angle of the pattern and the pore distributions will be discussed. Figure 1

A methodology combining the lattice Boltzmann method (LBM) and discrete element method (DEM) is proposed to simulate gas–liquid–solid multiphase flows and interphase interactions. Specifically, the phase-field LBM is employed to simulate the fluid flow, incorporating the virtual density boundary method. This method effectively enables the realization of wetting phenomena with relatively small spurious velocities, in comparison with the pseudopotential LBM. The DEM is utilized to simulate the motion of multiple solid particles. As for the interaction between fluid and solid, leveraging the features of LBM, the calculation of fluid forces acting on solids can be achieved by going through all fluid nodes surrounding the solid walls, enabling a straightforward and efficient calculation process. The numerical stability and accuracy of the hybrid LBM-DEM are demonstrated via benchmark cases. It is then applied successfully to simulate the upward migration of leaked gas bubbles through a deformable porous medium composed of solid particles.

The pseudopotential lattice Boltzmann method is a prominent approach in computational fluid dynamics, valued for its physical intuitiveness and computational tractability. However, existing solid–fluid interaction schemes for modeling wettability in multi-component immiscible fluid systems face persistent challenges, notably spurious currents and unphysical mass-transfer layers. This study proposes an improved solid–fluid interaction scheme that preserves implementation simplicity while effectively mitigating these numerical artifacts. Furthermore, leveraging the superior isotropy of eighth-order discrete schemes, we develop a novel boundary treatment methodology for second-layer lattice data reconstruction at complex interfaces. To validate the universality of the proposed improved scheme, four benchmark scenarios are simulated: static contact angle measurement on a cylindrical surface, droplet dynamics through confined geometries, immiscible displacement processes, and co-current flow in microchannels. Results demonstrate that the improved scheme accurately captures diverse complex immiscible multiphase flows.

SummaryMultiphase lattice Boltzmann methods are known to generate spurious or parasitic currents at the fluid–fluid interfaces. This nonphysical phenomenon has to be avoided, or at least controlled, in order to achieve reliable solutions. In this article, a method to control these fictitious velocities via lattice refinement is proposed, which is based on interface thickness control for which both the spurious currents and the physical fluid–fluid interface thickness vanishes as the spatial resolution increases. It has been found that a proper interface thickness adjustment is required as the lattice refinement is applied, or an increase in spurious currents, instead of a reduction, can occur. By combining the new method with an appropriate multiphase flow initialization, the overall stability for high density O(1000) and viscosity O(100) ratios is greatly improved. Although this research has been conducted with a Rothman and Keller type lattice Boltzmann model, it is believed that other types of multiphase lattice Boltzmann models could benefit from the basic ideas underlying this research. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical wave simulation using a modified lattice Boltzmann scheme

Application of LBM-SGS model to flows in a pumping-station forebay

Numerical modeling of free surface flow in hydraulic structures using Smoothed Particle Hydrodynamics

In this work, we find that two-dimensional characteristics of hydrodynamic phonon transports in two-dimensional dielectric materials lead to the anomalous length dependency and even divergence of thermal conductivities. For suspended monolayer graphene the analytical solution of the two-dimensional Guyer-Krumhansl equations shows that the thermal conductivity increases with the increase of its length from 1 micron to 9 micron. When a specific temperature distribution at the inlet boundary is given, and the positive partial derivative of the heat flux along the length direction at the inlet and outlet boundaries is prescribed, analytical results show that the thermal conductivity of suspended monolayer graphene at the room temperature tends to infinity with the increase of its length. This result can be used to artificially regulate the thermal conductivity of suspended single-layer graphene.

Thermodynamic inconsistencies, particularly the prominent spurious currents near three-phase contact lines, present a major obstacle in pseudopotential lattice Boltzmann (LB) simulations of contact angles. Previous solutions have predominantly focused on equations of state (EOSs) and forcing schemes (FSs). In this work, a hierarchical force computation architecture (HFCA) is established through a comprehensive assessment of EOSs, interparticle interaction force terms (IFTs), and FSs, aiming to suppress spurious currents at three-phase contact lines in lattice Boltzmann simulations. The results demonstrate that the Peng-Robinson EOS shows excellent numerical stability at high density ratios, yet it leads to elevated spurious currents when used alone. More significantly, Gong’s term, when incorporated into the HFCA, ensures universal stability, full compatibility with all EOSs and FSs, and systematically reduces the maximum spurious currents through hierarchical coordinated optimization. Additionally, the relaxation time τ is identified as a sensitive system-level parameter whose optimal value depends on the configuration established by the higher layers of the HFCA. By leveraging the synergistic effects within the HFCA, specifically by optimizing τ in a configured framework, spurious currents can be significantly reduced. Evidently, this approach is more effective than optimizations based solely on schemes and EOSs. Based on these findings, specific combinations of τ, forcing terms, schemes, and EOSs are recommended to minimize spurious currents in LB simulations.

The pseudopotential method has grown as a powerful tool for multiphase fluid flow simulations within the lattice Boltzmann method framework. We consider that due to its simplicity and computational efficiency, the pseudopotential method could be explored also inside the framework of more traditional Computational Fluid Dynamic methods such as Finite Difference, Finite Volume or Finite Element methods. Following this idea, in this work we start from the macroscopic equations resulting from the pseudopotential lattice Boltzmann method and discretize it by a simple Finite Difference scheme. This pseudopotential based finite difference method is then tested in different benchmark problems such as a planar interface, a smooth droplet oscillation, and a single droplet evaporation. Excellent results were obtained in all tests. One of the advantages of the proposed method is that mesh refinement is straightforward, and converged solutions can be used as a tool of validation for the lattice Boltzmann method. Results indicate that the pseudopotential method is suitable to be used with standard discretization methods such as Finite Difference and that in future works more robust discretizations can be used to further enhance the pseudopotential method application.