Multi-relaxation Time Research Articles

An automatic approach for the stability analysis of multi-relaxation-time lattice Boltzmann models

This paper is concerned with the weighted L2-stability of the lattice Boltzmann method (LBM) for the incompressible Navier-Stokes equations. Such stability is formulated for the linearized LBM and can lead to convergence proof of the nonlinear problem. A key step in this stability analysis is to find an appropriate decomposition for the Jacobian matrix of the collision term. For multi-relaxation-time (MRT) models, it is very difficult to analytically obtain such a decomposition and how to overcome this difficulty for MRT models is still a challenge. In this paper, we propose an automatic approach to prove the weighted L2-stability of the LBM with general collision models. Instead of exploring the decomposition for each specific model, we partially fix the decomposition and turn to show the symmetry and semi-negative definiteness of the other part of the decomposition, which can be automatically examined by a simple computer code. We apply our automatic approach to ten MRT models and stability conditions are generated by the code immediately. In particular, for the popular D2Q9 orthogonal MRT model, the stability condition we obtained has one less constraint than that given in previous work. As verifications, we also provide some analytical proofs for the stability of orthogonal and weighted-orthogonal MRT models. The proofs are much more complicated than the automatic approach, not to mention that those for nonorthogonal models are unavailable. These demonstrate the efficiency and superiority of the proposed automatic approach. With this approach, proving the weighted L2-stability and thereby the convergence of general MRT models becomes straightforward.

Thermal management optimization of natural convection in a triangular chamber: Role of heating positions and ternary hybrid nanofluid

In this paper, we used the multi-relaxation time lattice Boltzmann method to investigate natural convection in a triangular-shaped cavity filled with a tri-hybrid nanofluid. The cavity is partially heated by a chip of fixed size (l=L/2), the position of which varies on the left and bottom walls in order to find the optimal positions. The inclined side is maintained at a cool temperature, while the other parts are adiabatic. A detailed analysis is carried out on the impact of four essential parameters on the optimization of heat transfer: the Rayleigh number, ranging between Ra = 103 and Ra = 106; the partial heating position, showing the cavity in six different configurations; the fluid type, including pure water, nanofluid, hybrid nanofluid, and tri-hybrid nanofluid; and finally, the volume concentration of the nanoparticles for three values, ϕ = 0%, 3%, and 6%. Results are presented in the form of isotherms, streamlines, temperature and velocity profiles, and the mean Nusselt number values. As the results show, the position of the partial heater plays a crucial role, influencing natural convection heat transfer significantly in certain positions at all values of the Rayleigh number. The type of fluid has a remarkable impact on the amplification of natural convection at large values of the Rayleigh number, where the buoyancy force becomes strong. Notably, the use of tri-hybrid nanofluid shows a clear improvement in natural convection heat transfer. Furthermore, a substantial increase in thermal transmittance is observed with an increasing nanoparticle volume fraction. The validation results agree well with both numerical results and experimental data published in the literature.

Lattice Boltzmann simulation of water droplet impact and freezing on inclined supercooled surfaces with different roughnesses and wettabilities

Water droplet's impact and subsequent freezing on inclined supercooled surfaces, having micro-pillar structures with three different types of wettabilities (hydrophilic, neutral, or hydrophobic) are investigated numerically by a three-dimensional, multi-relaxation time, pseudo-potential liquid-vapor phase-change lattice Boltzmann method. Effects of volume expansion of water at 0 °C as well as the advancing contact angle of the droplet on real rough surfaces are considered in this model. Temporal variations of droplet morphological changes after impact and subsequent freezing on the supercooled inclined surfaces under various wall temperatures are illustrated. Simulation results indicate that after impacting on the supercooled rough inclined surfaces at a given Weber number, a water droplet can form fully penetrated elliptical ice morphologies and stretched stream-like ice morphologies attached on the supercooled surface, or partially and fully rebounding from the supercooled surface depending on the wettability of the surface, the surface inclination, and the degree of supercooling. It is shown that elliptical ice patterns are formed on hydrophilic substrates where droplets do not recoil after spreading, while stream-like ice patterns are mostly formed on neutral substrates where droplets are gradually stretched due to the gravity because droplets are attached on the inclined surface. On the other hand, partial and full rebounds of droplets occur on hydrophobic surfaces having high inclination angle where droplets undergo strong recoiling and rebounding after spreading on rough surfaces. At low degrees of supercooling (i.e., higher substrate temperatures), partial rebounds occur because a portion of the water droplet freezes on the substrate during contact with the supercooled substrate; otherwise, fully rebounding occurs. These results elucidate the effects of wettability, roughness, temperature and inclined angle of the wall on ice morphologies. Additionally, effects of Stefan number (substrate temperature) on the four different ice morphologies of a water droplet (D0 = 100 and Pr = 13.5) after its impact on supercooled inclined rough surfaces at a Weber number of We = 112 (Reynolds number of Re = 164.9) are illustrated.

Lattice Boltzmann simulation on particle suspensions containing porous particles in a narrow channel

The suspension of porous particles in fluids occurs widely in various natural and industrial processes. However, the sedimentation behavior of porous particles is not extensively understood as the solid impermeable counterparts. In this work, the drafting–kissing–tumbling (DKT) phenomenon in a narrow channel containing porous particles is investigated by the multi-relaxation-time (MRT) lattice Boltzmann method (LBM). The initial particle spacing Lp* (1.5∼6) and Darcy number Da (8×10−6∼6×10−2) are examined on the sedimentation process of two particles under three initial arrangements, i.e., the trailing particle is porous (case 1), the leading particle is porous (case 2), and both the particles are porous (case 3). The results show that the presence of porous particles can enhance the interactions between two particles, and increasing the penetrability reduces the particle drag force to accelerate sedimentation. The drafting time is insensitive to Da at small Lp*, and it decreases with Da at large Lp* in cases 1 and 3 while it changes to increase with Da in case 2. A phase diagram with respect to Da and Lp* is further extracted to identify three sedimentation modes of particle pairs. It is found that the transition between the one-off DKT and repeated DKT modes is not affected by Lp* in cases 2 and 3, while the critical condition for the non-DKT and one-off DKT modes depends strongly on Da and Lp* in case 2.

Longitudinal transducer based membrane resonant ejector for high-viscosity and high-frequency micro-droplet jetting

Aiming at the requirement for high-precision on-demand droplet jetting of high-viscosity industrial inks, a longitudinal-transducer-based membrane resonant droplet ejector (LT-MRDE) is designed and fabricated in this work. The jetting and observing system of LT-MRDE is set up to quantitatively explore the variations of droplet volume and velocity with driving voltage. Moreover, the three-dimensional (3D) multi-relaxation-time (MRT) lattice Boltzmann (LB) model coupled immersed boundary (IB) is employed to elucidate the effects of driving voltage and liquid viscosity on droplet characteristics. Experimental results demonstrate that the LT-MRDE could stably eject the silicone oil with a viscosity of 100 mPa·s from ∼50 μm vibrating orifice. Furthermore, both the volume and velocity of droplet exhibit a proportionality to voltage and an inverse proportionality to viscosity. The optimal droplet volume and velocity of 100 mPa·s silicon oil are 67.27 pL and 9.07 m/s with the driving parameters of 75 Vpp/14.754 kHz. With high jetting frequency and controllable droplet volume, LT-MRDE provides a new alternative for on-demand precision jetting of high-viscosity industrial inks.

Impact dynamics of droplets on circular bodies: Exploring the influence of wettability, viscosity, and body dimensions using the lattice Boltzmann method

This study used the multi-relaxation time pseudopotential lattice Boltzmann method to examine the impact dynamics of droplets on circular bodies, focusing on the influence of the surface wettability, the viscosity of droplets by choosing three distinct Reynolds numbers (Re = 100, 300, and 500), and the body diameters. Initially, the study examined impact behavior under non-wetting and wetting conditions, revealing distinct behaviors characterized by dimensional stretch lengths in horizontal and vertical directions. Furthermore, the study evaluated the impact of viscosity by varying the Reynolds number, providing a better understanding of droplet behavior on the solid body. In addition, the effect of changing the diameter of the circular body was examined. This research underlines the importance of surface wettability in the dynamics of contact with the droplets impinging on it. In particular, higher wettability correlates with a monotonic increase in viscosity by acting on the Reynolds number. In addition, the droplet profile responds to changes in Reynolds number, albeit with relatively limited deformation, even in the case of intense interaction with the wetting surface. These results highlight the complex interplay between wettability, droplet dynamics, viscosity, body dimensions, and surface interactions during impact processes.

Sensitivity analysis of natural convection in a porous cavity filled with nanofluid and equipped with horizontal fins using various optimization methods and MRT-LB

In the present study, natural convection heat transfer is investigated in a porous cavity filled with Cu/water nanofluid and equipped with horizontal fins. Optimization and sensitivity analysis of the fin’s geometry, porous medium and nanofluid properties to maximize heat transfer rate is the aim of this work. To achieve this purpose, a design space is created by input parameters which include length, number of fins, distance between fins, porosity, Darcy number and volumetric fraction of the nanoparticles. Several tools have been used to implement optimization methods including the Taguchi method (TM) for design points generation, sensitivity analysis of design variables by using signal-to-noise ratio (SNR) and analysis of variance (ANOVA), response surface method (RSM) for interpolation and regression by using nonparametric regression, and genetic algorithm (GA) for finding optimum design point. The double multi-relaxation time lattice Boltzmann method (MRT-LBM) is used to analyze and simulate the flow field and heat transfer in each design point. The results show that the optimal configuration leads to an average Nusselt number of 5.56. This optimal configuration is at the length of fins L/2, the number of fins 2, the distance between fins L/12, porosity 0.8, Darcy number 0.1, and the volumetric fraction of the nanoparticles 0.02. By using the SNR results, the Darcy number and the number of fins have the most and the least effect in maximizing the average Nusselt number, respectively. The ANOVA results and global sensitivity analysis (GSA) findings further validated this conclusion.

Study on the flow structure in curved open channels with suspended vegetation using multi relaxation time lattice Boltzmann method

Study on the flow structure in curved open channels with suspended vegetation using multi relaxation time lattice Boltzmann method

Development of a cascaded lattice Boltzmann model for two‐layer shallow water flows

SummaryMany environmental phenomena, such as flows in rivers or in coastal region can be characterised by means of the ‘shallow approach’. A multi‐layer scheme allows to extend it to density layered shallow water flows (e.g., gravity currents). Although a variety of models allowing numerical investigation of single and multi‐layer shallow water flows, based on continuum and particle approaches, have been widely discussed, there are still some computational aspects that need further investigation. Focusing on the Lattice Boltzmann models (LBM), available multi‐layer models generally use the standard linear collision operator (CO). In this work we adopt a multi relaxation time (MRT) cascaded collision operator to develop a two‐layered liquid Lattice‐Boltzmann model (CaLB‐2). Specifically, the model solves the shallow water equations, taking into account two separate sets of particle distribution function (PDF), one for each layer, solved separately. Layers are connected through coupling terms, defined as external forces that model the mutual actions between the two layers. The model is validated through comparisons with experimental and numerical results from test cases available in literature. First results are very promising, highlighting a good correspondence between simulation results and literature benchmarks.

Lattice Boltzmann method for particulate multiphase flow system

This study proposes a numerical model for particulate three-phase flow in microchannels based on multiphase lattice Boltzmann method (LBM). The model combines the color-gradient method to track the immiscible fluid-fluid interface and the two-fluid model (TFM) to describe particle-particle and particle-fluid interactions, which can efficiently simulate transport and displacement processes involving large amounts of particles. A mixture-rheology TFM algorithm is proposed by introducing a mixture phase with rheology properties obtained from experiments instead of the conventional TFM particle phase with artificial viscosity models. Multi-relaxation-time (MRT) collision operator and GPU computing are adopted to enhance the numerical stability and efficiency. Various theoretical benchmarks for particle transport and two-phase flow are performed respectively to verify the accuracy of the proposed model. Exceptional consistency between results from particulate three-phase flow simulation and microfluidic experiments further confirms the reliability of our model, especially in capturing the inertial lagging and accumulation phenomena under multiphase and porous flow conditions. The proposed numerical framework will benefit our understanding of multiphase displacement with microgels in microchannels with complex geometries.

A machine learning based-method to generate random circle-packed porous media with the desired porosity and permeability

The generation of digital porous media facilitates the fabrication of artificial porous media and the analysis of their properties. The past random digital porous medium generation methods are unable to generate a porous medium with a specified permeability. In this work, a new method is proposed to generate a random circle-packed digital porous medium with a specified porosity and permeability. Firstly, the process of generating the random circle-packed digital porous medium with a specified porosity is detailed. Secondly, the permeability of the digital porous medium is calculated by the multi-relaxation time lattice Boltzmann method. A total of 3,000 digital porous medium samples are generated, and their microstructure data and permeabilities are prepared for the training of a convolutional neural network (CNN) model, which is then applied to effectively predict the permeability of a digital porous medium. Finally, our method is elaborated and the choice of target permeability in this method is discussed. Our approach has the potential to be applied to the generation of other types of porous media.

Direct numerical simulations of turbulent channel flow with wall transpiration using lattice Boltzmann method

Abstract Direct numerical simulations of turbulent channel flow with wall transpiration are conducted using the multi-relaxation time lattice Boltzmann method for the first time. Flows with two friction Reynolds numbers, $Re_\tau$ = 150 and 250, are investigated, and the explored transpiration rates based on the friction velocity range from 0.05 to 0.26. A successive block mesh-refinement strategy is adopted to resolve the near-wall structure, where the grid sizes are $\Delta ^+ \sim$1, 2, and 4. Under such resolution, the size of the maximum grid density employed is $775 \times 10^6$. At the low transpiration rate ($V^+=0.05$), the turbulence level at the suction side is damped relative to the less altered values along the injection side. However, at the high level of wall transpiration, i.e. $V^+=0.26$, turbulence levels are severely retarded on both the injection and suction sides. The predicted mean and turbulence quantities show good agreements with the available direct numerical simulation data, indicating the effectiveness of lattice Boltzmann method combined with mesh refinement in reproducing the correct physics of turbulent channel flows with different levels of wall transpiration.

Liquid water transport phenomena in the porous transport layer of proton exchange membrane fuel cell based on lattice Boltzmann simulation

The porous transport layer (PTL) of proton exchange membrane fuel cell (PEMFC) plays an important role in water management. The liquid water transport behavior with real density ratio and viscosity ratio is discussed by establishing a multi-relaxation time (MRT) multiphase flow lattice Boltzmann model. The effects of temperature, wettability and microporous layer (MPL) thickness on the water transport process and water distribution are included. As the temperature increased from 278.15 K to 353.15 K, the water saturation in the PTL decreased from 0.501 to 0.219. The breaking phenomenon of water clusters are captured at temperatures of 328.15 K and 353.15 K. The hydrophobic structure has less water content under the ribs and the whole, which is beneficial to the reaction gas transport and the performance improvement of PEMFC. Adding MPL with a thickness of 25 µm can significantly improve water management. However, excessively thick MPL will drive water to flow under the ribs and reduce the permeability of the structure. Appropriate MPL thickness design is a feasible solution to improve water management.

An Eulerian based geometry conforming grid-block dynamic mesh refinement for the lattice Boltzmann method

An algorithm for a three-dimensional fine-grid block encompassing a moving solid body using a multi-relaxation time model of the lattice Boltzmann method is proposed and developed. In this numerical framework, a geometry-conforming fine block shifts a coarse lattice unit as the object translates by same amount. The effective utilization of higher-order spatial interpolation has been demonstrated in place of the commonly used low-order temporal interpolation in traditional grid refinement techniques within the lattice Boltzmann method. The successful application of this method has been showcased through three distinct cases: the settling of a solid sphere in a fluid tank under the influence of gravity, hovering motion of an elliptic airfoil, and the “clap and fling” motion of an insect wing. In this regard, various interpolation schemes based on the location of nodes in the overlapping zones of fine and coarse block are discussed. In addition, two cases were evaluated, case 1 where distributions are interpolated and case 2 where macroscopic variables are directly interpolated. It was observed that both interpolations gave same computational accuracy for low Reynolds number [∼O(102)]. However, as Re is increased [∼O(103)], direct interpolation of macroscopic variables proved erroneous and resulted in a large deviation in fluid forces and is not recommended. The geometry-conforming dynamic mesh refinement results in a substantial decrease in computation time, approximately 90%, along with a memory reduction of about 80% compared to the fully refined counterpart.

Boosting of CO2 sensitivity in Zn-doped SnO2 thin film: Analysis based on structural, optical and dielectric characterization

Zn-incorporated SnO2 thin films are deposited by spray pyrolysis technique from an ethanolic solution of SnCl4·5H2O at 310 °C. The structural analysis shows that both undoped and Zn-doped films are tetragonal rutile structures with crystallite sizes ranging from 44 to 35 nm. Morphological analysis shows that the particles are spherical in shape and uniformly distributed. The UV–Vis absorbance spectra of the samples are recorded and the band gap energies are estimated. The CO2 sensing properties are investigated in detail. The dielectric constant is found to vary with frequency that is attributed to the multi-relaxation time constant of the energy states responsible for the conduction mechanism. The AC conductivity has been found to obey a power law, indicating a multi-hopping conduction mechanism in the samples. These results are discussed in relation to the sensing characteristics of the sample.

Comprehensive performance analysis and structural improvement of latent heat thermal energy storage (LHTES) unit using a novel parallel enthalpy-based lattice Boltzmann model

Latent heat thermal energy storage (LHTES) utilizing phase change material (PCM) is one of the critical enablers in developing sustainable and low-carbon energy systems. To fill the knowledge gap, this paper presents an enthalpy-based solid–liquid model through the lattice Boltzmann method (LBM) with a multi-relaxation-time (MRT) approach, aiming to simulate convective phase change in LHTES units with and without porous media in Cartesian or axisymmetric domains. To improve accuracy and efficiency, the model integrates a differential scanning calorimetry (DSC) correlated equation for enthalpy modeling, couples with a 1D heat-transfer-fluid (HTF) model for boundary treatment of HTF side, and employs a parallel LBM scheme for efficient parametric studies. The validation demonstrates the model’s success in predicting PCM phase change, with errors below 10%. A comprehensive numerical analysis is then conducted to quantitatively evaluate the effect of convection on PCM melting. Novel metrics, such as acceleration rates (ac) of PCM melting and threshold Rayleigh numbers (Radc) at various aspect ratios, are introduced. Furthermore, PCM melting in the porous cylindrical unit is explored. Findings reveal up to 86% acceleration in melting compared to pure PCM at 80% energy storage, and the porous media with porosity above 0.9 is recommended for thermal enhancement. Moreover, this paper analyzes the negative effect of uneven temperature distributions caused by convection on LHTES unit efficiency. A modified LHTES unit geometry is proposed to offset this negative effect, and the study demonstrates successful mitigation of uneven temperature distributions, achieving up to 57 % acceleration in PCM melting.

Representative elementary volumes for various characteristics of two-phase flows in porous media: A statistical approach

This article presents a systematic study of the influence of two-phase flowpatterns on the representative elementary volume for various displacement characteristics. This paper considers both drainage and imbibition regimes and related patterns such as compact displacement, viscous, and capillary fingering. As two-phase flow characteristics, the displacement efficiency and interfacial contact areas between the invaded fluid and the displaced fluid, as well as between the invaded fluid and solid particles, are considered. Not limited to displacement patterns, we study the effect of capillary number on the minimum REV size. The findings of this paper are based on numerical simulations performed in two-dimensional artificially created porous structures using the Monte-Carlo movement algorithm. The main conclusions are validated on 3D X-ray computed tomography images of natural sandstones. Two-phase immiscible displacement in a porous medium is modeled using the lattice Boltzmann equations with the multi-relaxation time collision operator in combination with the color-gradient method describing interfacial phenomena and wetting effects. Minimum REV sizes have been identified using statistical data collected from a large number of numerical simulations in two-dimensional samples. Coefficient of variation and average value of the investigated property have been used to identify REV. The results show that REV for two-phase flow in porous media depends on the displacement pattern. Minimum REV sizes for viscous fingering and compact displacement are close. The accuracy of predicting the displacement efficiency for viscous fingering and compact displacement modes is much higher than for capillary fingering. Minimum REV sizes for interfacial contacts are lower than for displacement efficiency.

Suitability of body force model for pressure-difference driven flow in porous media

Pore-scale direct numerical simulation is an important tool for studying the flow of porous media nowadays. This work validates the suitability of body force model for simulating pressure-difference-driven porous flows in pore-scale simulations. The multi-relaxation time lattice Boltzmann method (MRT-LBM) with the color-gradient continuum-surface-force (CG-CSF) model is employed to calculate the absolute and relative permeability of randomly generated porous media. Results show that for single-phase flows, the relative errors are below 5% for most homogeneous porous structures while as high as 80% for structures with preferential paths. This study also reveals the pore-scale flow mechanisms and proposes a new heterogeneity parameter quantified by the standard deviation of the pressure at the boundary. For multiphase flows, errors of the body force model are significant at low Ca due to irreversible phase distribution difference appearing at the early stages. The findings of this work could be applied to other transport processes in porous media. It is crucial to consider the structural heterogeneity and proper parameters when employing the body force model in pore-scale simulations.

Three-dimensional Lattice Boltzmann study on structure optimization and heat dissipation performance of pin-fin heat sink integrated with phase change material

Phase change material (PCM) cooling is an excellent approach for reducing the excessive temperature of electronic devices. However, the heat transfer capacity of a heat sink is diminished by the low thermal conductivity of a PCM. Although this issue can be ameliorated by using extended pin-fins and nanoparticles, the latent heat storage capability of a PCM is decreased. To optimize the heat transfer efficiency of PCM heat sink with pin-fins, in this study, the effects of pin-fin shapes (circle, triangle, square, pentagon, hexagon, and octagon), novel arrangements with triangular fins, and the volume fraction of nanoparticles are investigated using the 3D enthalpy-based multi-relaxation time lattice Boltzmann method. The results showed that the PCM heat sink with triangular pin-fins exhibited optimal thermal performance due to the largest heat transmission area between the PCM and pin-fins. After conducting optimization research on triangular pin-fin arrangements, the full melt time was shortened by 14.3 % and the heat dissipation power was increased by up to 15.2 %, when compared with the PCM heat sink with triangular pin-fins. Furthermore, nanoparticles with a volume fraction of <0.1 were beneficial for balancing the elevated heat dissipation capacity and reducing the latent heat duration of the nano-enhanced PCM.

Study of magnetohydrodynamics-based-mixed convection & entropy generation within the rectangular enclosure with two obstacles for Cu-SiO2 multiwalled carbon nanotubes ternary hybrid nanofluids

This work employs the multirelaxation time lattice Boltzmann method (MRT-LBM) to investigate the heat transfer and entropy generation characteristics due to mixed convection in a rectangular enclosure with a discrete heater subjected to an external magnetic field. The enclosure contains Cu-SiO2 multiwalled Carbon nanotubes ternary nanofluid. In particular, the effect of Hartmann number (0 ≤ Ha ≤ 90), Richardson number (0≤ Ri ≤ 10), the volume fraction of nanofluids (0 ≤ ϕ ≤1), and the location of obstacles over the various aspects of thermohydrodynamics and entropy generation characteristics have been elaborated. Three cases are categorized based on the arrangement of the obstacles in the enclosure. Case 1 and Case 3 represent the alienated arrangement of the obstacles where the first obstacle is located below and above the second, respectively, while Case 2 represents the in-tandem arrangement of the obstacles. The results illustrate the reversal of flow velocity at Ha = 90 for all Ri. It is observed that case 2 is best suited for Nusselt number enhancement. It is seen that when Ri = 0.1 and 1 Case 2 and at Ri = 10, Case 1 augments heat transfer for Ha = 0.