Lattice Boltzmann Equation Method Research Articles

Decay of Taylor–Green flow type initial conditions in a two-dimensional domain

Geophysical flows are often two-dimensional (2D) in nature. The decay dynamics of the canonical, symmetric initial conditions defined by the Taylor–Green (TG) flow is simulated in a 2D domain. The standard lattice Boltzmann equation method with the Bhatnagar–Gross–Krook collision model is employed. The vortex dynamics is investigated for bounded and unbounded flows with various sets of Gaussian vortices and varying Reynolds numbers. The formation of thin filaments and the generation of small-scale structures near the no-slip walls are found. To characterize the decay properties, the integral quantities, namely, kinetic energy, E(t), enstrophy, Ω(t) , and palinstrophy, P(t), are tracked as a function of the eddy-turnover-time. The initial vortex population in the domain dictates the decay rate of these integral quantities. The energy spectra, E(k), are computed to investigate the effect of domain boundaries and vortex populations on energy distribution among different length scales for Re=50000 . The evolution of TG flow in a bounded domain shows an exponent close to −3, which may be attributed to the generation of thin filaments and small-scale structures in vortex-wall interactions. Exponent changes with an increase in the vortex populations.

Well-balanced kinetic schemes for two-phase flows

At equilibrium state of a two-phase fluid system, the chemical potential is constant and the velocity vanishes. However, such equilibrium state usually cannot be captured by numerical methods due to discretization errors. Consequently, inconsistent thermodynamic interfacial properties due to non-constant chemical potential, and spurious velocities due to discrete force imbalance, are frequently encountered in numerical simulations. Therefore, numerical schemes with well-balanced properties, which can capture the equilibrium state at discrete level, are preferred for simulating two-phase flows. In this paper, we report the progress of developing well-balanced mesoscopic numerical methods based on kinetic theory, focusing on the lattice Boltzmann equation (LBE) method and the discrete unified gas-kinetic scheme (DUGKS). First, the discrete balance property of a free-energy LBE for single-component two-phase flows is analyzed to identify the structure of force imbalance. Then, a well-balanced LBE (WB-LBE) model which has the same algorithm structure as the standard one is reviewed. The WB-LBE is theoretically shown to be able to achieve the discrete equilibrium state, and the well-balance properties are confirmed numerically. Some extensions to other two-phase LBE models are discussed with enhanced numerical stability. The well-balanced DUGKS, which can use non-uniform meshes and exhibits better numerical stability for large density ratio systems is also reported.

Hybrid Lattice Boltzmann Equation Method in Problems of Coupled Heat Transfer

Hybrid Lattice Boltzmann Equation Method in Problems of Coupled Heat Transfer

Competition of localized thermal buoyancy and Lorentz forces in an electrolyte enclosed in a cavity

We study the flow of an electrolyte inside a slender square cavity produced by two competing localized forces, the thermal buoyancy force produced by the heating of a central part of the horizontal bottom wall and a vertical downward Lorentz force. This force is produced by a constant horizontal electric current between the slender vertical walls and a couple of circular magnets with opposing polarization placed on the center of both square vertical walls. The flow inside the cavity is considered two dimensional and simulated using the lattice Boltzmann equation method. We find the map of the flow patterns and their transitions for a Grashof number Gr in the range 0≤Gr≤4×103 and a Chandrasekhar number Ch in 0≤Ch≤1×107. For Gr≤2×103, the plane Gr-Ch can be divided in three regions with sharp changes in the flow patterns and the average Nusselt number on the top wall and the average asymmetry. One region is dominated by the buoyancy force, another by the Lorentz force, and there is an intermediate one, where neither of these forces dominates. For larger Gr, there are no sharp transitions.

Three-dimensional single framework multicomponent lattice Boltzmann equation method for vesicle hydrodynamics

We develop a three-dimensional immersed boundary chromodynamic multicomponent lattice Boltzmann method capable of simulating vesicles, such as erythrocytes. The presented method is encapsulated in a single framework, where the application of the immersed boundary force in the automatically adaptive interfacial region results in correct vesicle behavior. We also set down a methodology for computing the principal curvatures of a surface in a three-dimensional, physical space which is defined solely in terms of its surface normal vectors. The benefits of such a model are its transparent methodology, stability at high levels of deformation, automatic-adaptive interface, and potential for the simulation of many erythrocytes. We demonstrate the utility of the model by examining the steady-state properties, as well as dynamical behavior within shear flow. The stability of the method is highlighted through its handling of high deformations, as well as interaction with another vesicle.

Reduction-consistent Cahn–Hilliard theory based lattice Boltzmann equation method for N immiscible incompressible fluids

When some fluid components are absent from N (N≥ 2) immiscible fluids, the reduction-consistent property should be guaranteed. In phase-field theory, the evolution of fluid–fluid interface in N immiscible fluids can be captured by a reduction-consistent Cahn–Hilliard equation (CHE), which has a variable dependent mobility. However, it is difficult for lattice Boltzmann equation (LBE) method to solve this kind of CHE with variable mobility. To eliminate this issue, in this paper, a reduction-consistent LBE is proposed for N immiscible fluids. In the model, the reduction-consistent formulation of fluid–fluid interface force is reformulated into a chemical potential form, which can be implemented by a force term in LBE, while a source term treatment is used to achieve the reduction-consistent property for CHE. Numerical simulations of spreading of a liquid lens, spinodal decomposition, and dynamic interaction of drops are carried out to validate present LBE, and the results show the accuracy and capability of present phase-field based LBE for N (N≥2) immiscible fluids.

Reduction-consistent multiple-relaxation-time lattice Boltzmann equation method for wall bounded N immiscible incompressible fluids

A multiple-relaxation-time (MRT) lattice Boltzmann equation (LBE) method is developed for N-phase (N≥ 2) flow with moving contact lines. In the model, MRT LBE is applied to solve the incompressible Navier-Stokes equation, and another MRT LBE is developed for fluid-fluid interface capturing, where the MRT LBE has the reduction-consistent property and can guarantee the mass conservation of each phase. The wettability of solid in the presence of N (N≥ 2) immiscible fluids can be achieved by a reduction-consistent wettability boundary condition, which can treat different contact angle between the fluids and the solid. A series of benchmark tests such as spreading of droplets and compound droplet on a solid wall, and compound droplet impact on a dry solid wall are conducted to validate the MRT LBEs, and it is shown that the predictions of LBE agree well with the theory/other numerical results.

Well-balanced lattice Boltzmann model for two-phase systems

The standard two-phase lattice Boltzmann equation (LBE) method usually fails to capture the physical equilibrium state (zero velocity and constant chemical potential). Consequently, spurious velocities and inconsistent thermodynamic density properties are frequently encountered in LBE simulations. In this work, based on a rigorous analysis of the discrete balance equation of LBE, the structure of force imbalance due to discretization errors is identified. Then, a well-balanced LBE model is proposed, which can achieve the discrete equilibrium state. The well-balance properties of the model are confirmed by simulating a flat interface problem and a droplet system.

Reduction-consistent axisymmetric lattice Boltzmann equation method for [formula omitted]-phase fluids

We develop a reduction-consistent axisymmetric lattice Boltzmann equation method (LBE) for hydrodynamics and interface capturing among N (N≥ 2) immiscible fluids with large density and viscosity contrasts. In present work, the chemical potential form of interface force among N immiscible fluids is incorporated to the axisymmetric hydrodynamic equations, and the reduction-consistent axisymmetric conservative Allen–Cahn equation (ACACE) is developed for interface capturing of N (N≥ 2) immiscible fluids, then we propose a simple ACACE LBE for interface capturing, which has reduction-consistent property, and the axisymmetric hydrodynamic equations are solved by another LBE. Some axisymmetric benchmark multiphase problems are carried out to validate present LBE including stationary compound droplet, break up of compound liquid thread and compound droplet splashing on a thin liquid film, numerical results show that the predictions by present LBE agree well with the theoretical solutions/other numerical and available experimental results.

Reduction-consistent phase-field lattice Boltzmann equation for N immiscible incompressible fluids

In this paper, we develop a reduction-consistent conservative phase-field method for interface-capturing among N (N≥ 2) immiscible fluids, which is governed by conservative Allen-Cahn equation (CACE); here the reduction-consistent property is that if only M (1≤M≤N-1) immiscible fluids are present in a N-phase system, the governing equations for N immiscible fluids must reduce to the corresponding M immiscible fluids system. Then we propose a reduction-consistent lattice Boltzmann equation (LBE) method for solving N immiscible incompressible fluids with high density and viscosity contrasts. Some numerical simulations are carried out to validate the present LBE such as stationary droplets, spreading of a liquid lens, and spinodal decomposition together with the reduction-consistent property, and the numerical results predicted by present LBE are in good agreement with the analytical solutions/other numerical results, which also demonstrate the reduction-consistent property by present LBE.

Modeling the effects of slip on dipole–wall collision problems using a lattice Boltzmann equation method

We study the physics of flow due to the interaction between a viscous dipole and boundaries that permit slip. This includes partial and free slip, and interactions near corners. The problem is investigated by using a two relaxation time lattice Boltzmann equation with moment-based boundary conditions. Navier-slip conditions, which involve gradients of the velocity, are formulated and applied locally. The implementation of free-slip conditions with the moment-based approach is discussed. Collision angles of 0°, 30°, and 45° are investigated. Stable simulations are shown for Reynolds numbers between 625 and 10 000 and various slip lengths. Vorticity generation on the wall is shown to be affected by slip length, angle of incidence, and Reynolds number. An increase in wall slippage causes a reduction in the number of higher-order dipoles created. This leads to a decrease in the magnitude of the enstrophy peaks and reduces the dissipation of energy. The dissipation of the energy and its relation to the enstrophy are also investigated theoretically, confirming quantitatively how the presence of slip modifies this relation.

Multiphase flows of N immiscible incompressible fluids: Conservative Allen-Cahn equation and lattice Boltzmann equation method.

In this paper, we develop a conservative phase-field method for interface-capturing among N (N≥2) immiscible fluids, the evolution of the fluid-fluid interface is captured by conservative Allen-Cahn equation (CACE), and the interface force of N immiscible fluids is incorporated to Navier-Stokes equation (NSE) by chemical potential form. Accordingly, we propose a lattice Boltzmann equation (LBE) method for solving N (N≥2) immiscible incompressible NSE and CACE at high density and viscosity contrasts. Numerical simulations including stationary droplets, Rayleigh-Taylor instability, spreading of liquid lenses, and spinodal decompositions are carried out to show the accuracy and capability of present LBE, and the results show that the predictions by use of the present LBE agree well with the analytical solutions and/or other numerical results.

Analysis of force treatment in lattice Boltzmann equation method

In lattice Boltzmann equation (LBE), density distribution function is apparently relaxed to a given state fi(gs)(ρ,u∗) with an external/internal force, here ρ is fluid density and velocity u∗ can be different with physical velocity u. In the literature, it was shown that fi(0) should be fi(gs)(ρ,u∗), however, present work shows that fi(0) is fi(eq)(ρ,u) with different force scheme in single relaxation time LBE. The corresponding error terms are analyzed at Navier–Stokes level, and numerical results confirm our theoretical analysis.

Heat transfer and flow transitions of a thermal plume generated by a heating element on the enclosure bottom wall

Abstract The two-dimensional buoyancy-induced flow above a heating horizontal element in the center of the bottom of a rectangular enclosure filled with air is numerically studied using the thermal lattice Boltzmann equation method with two distribution functions. The heating element has a higher temperature than that of the remaining walls. The width and height of the enclosure are 5 and 10 times the length of the heating element respectively. The flow is space and time symmetric if after a transient, the flow is symmetric with respect to the vertical axis of the enclosure and is time independent. A suitable quantity, here called the asymmetry, is zero when the flow is time and space symmetric, positive and constant when the first symmetry is lost, and positive and time dependent when both symmetries are broken. Flow transitions are studied as the Rayleigh number, R a , changes in a large interval, 1000 ≤ R a ≤ 350 000 . Eight flow transitions were found, the first ones related to the space and time symmetries. When both symmetries are broken, additional flow transitions were found by studying the Fourier power spectrum of the asymmetry, which allows the identification of the fundamental frequency and the appearance of harmonics and other frequencies. The average Nusselt number as a function of R a is also studied and a transition in their relationship is found at the appearance of harmonics, giving way to two different correlations.

Deformation of bubbles in transformer oil at the action of alternating electric field

Abstract The experimental investigations of the deformation of bubbles floating in transformer oil under the action of uniform electric field were performed. The model based on the lattice Boltzmann equation method for the electrohydrodynamic flows accompanying this process was developed. The degree of deformation of bubbles observed in experiments is in agreement with the one obtained in simulations. Comparison with theoretical calculations show that the experimental data on the deformation of bubbles in transformer oil under the action of strong alternating electric field with a magnitude up to 4 kV/mm are reasonably explained by a model with a changeless surface tension.

LBE simulation of impinging stream inside a T-junction mixer

Lattice Boltzmann Equation (LBE) method is utilized to simulate impinging stream (IS) in a T-junction mixer using a TD2G9 model. It aims to investigate the influence of Reynolds number (Re), aspect ratio of outlet diameter to inlet diameter, ratio of opposite inlet velocities, and the thermal boundary conditions on flow, mixing and heat transfer characteristics. In particular, the vortex evolution, velocity distribution, mixing index and Nusselt number (Nu) distribution in the T-junction mixer are explored in details. Four types of vortices and flow regimes are observed. The instantaneous and time-averaged flow and thermal fields, including vortex structure, transition of flow regimes, streamline and the Nusselt number distribution are discussed. Distinct quantitative transitions, even for dramatic change, are observed near the critical Re. At a low or moderate aspect ratio, the symmetric coherent structure is observed in an unstable flow regime. At a larger aspect ratio, the flow in the T-mixer becomes turbulent and asymmetric. The unequal injections velocities of the nozzles impose significant influence on the flow structure, mixing and heat transfer in vertical tube. Using larger difference between the two inlet velocities can result in more obvious change in flow characteristics. Moreover, mixing index is found to be valid in evaluating the mixing degree under a sinusoidal inlet velocity.

Chapman–Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media

The gray lattice Boltzmann equation (GLBE) method has recently been used to simulate fluid flow in porous media. It employs a partial bounce-back of populations (through a fractional coefficient θ, which represents the fraction of populations being reflected by the solid phase) in the evolution equation to account for the linear drag of the medium. Several particular GLBE schemes have been proposed in the literature and these schemes are very easy to implement; but there exists uncertainty about the need for redefining the macroscopic velocity as there has been no systematic analysis to recover the Brinkman equation from the various GLBE schemes. Rigorous Chapman–Enskog analyses are carried out to show that the momentum equation recovered from these schemes can satisfy Brinkman equation to second order in $$ \epsilon $$ only if $$ \theta = {\rm O}\left( \epsilon \right) $$ in which $$ \epsilon $$ is the ratio of the lattice spacing to the characteristic length of physical dimension. The need for redefining macroscopic velocity is shown to be scheme-dependent. When a body force is encountered such as the gravitational force or that caused by a pressure gradient, different forms of forcing redefinitions are required for each GLBE scheme.

Lattice Boltzmann Simulation of the Hydrodynamic Entrance Region of Rectangular Microchannels in the Slip Regime.

Developing a three-dimensional laminar flow in the entrance region of rectangular microchannels has been investigated in this paper. When the hydrodynamic development length is the same magnitude as the microchannel length, entrance effects have to be taken into account, especially in relatively short ducts. Simultaneously, there are a variety of non-continuum or rarefaction effects, such as velocity slip and temperature jump. The available data in the literature appearing on this issue is quite limited, the available study is the semi-theoretical approximate model to predict pressure drop of developing slip flow in rectangular microchannels with different aspect ratios. In this paper, we apply the lattice Boltzmann equation method (LBE) to investigate the developing slip flow through a rectangular microchannel. The effects of the Reynolds number (1 < Re < 1000), channel aspect ratio (0 < ε < 1), and Knudsen number (0.001 < Kn < 0.1) on the dimensionless hydrodynamic entrance length, and the apparent friction factor, and Reynolds number product, are examined in detail. The numerical solution of LBM can recover excellent agreement with the available data in the literature, which proves its accuracy in capturing fundamental fluid characteristics in the slip-flow regime.

Modeling an Ascending Nitrogen Gas Bubble in a Medium Crude Oil by Lattice Boltzmann Method

The study and modeling of oil biphasic systems, liquid-liquid and liquid-gas, focus mainly on the details of the modifications and application of the numerical methods itself. The correspondence between theoretical and experimental results and the information needed to apply a certain numerical method, usually remain in the background. On the other hand, in the particular case of the prediction of minimum miscibility pressure, extremely important parameter in oil exploration, references that show qualitative and numerical data associated with the characterization of the systems are scarce. The above reasons motivated the realization of this work. We used the Lattice Boltzmann Equation method to model a two-dimensional system of the displacement of a nitrogen gas bubble through a medium crude oil, under different pressure conditions keeping the temperature constant. According to experimental data, the bubble is not miscible by the crude, under a pressure range of 5000 psi to 6500 psi; nevertheless, the bubble is miscible in the range of 7000 psi to 7500 psi. Throughout simulations performed under similar conditions, we showed that it can be inferred the critical pressure range of miscibility of a medium crude oil.

Power-law fluid flow in driven enclosures with undulation using MRT-lattice Boltzmann method

In this article, the flow of non-Newtonian fluid (which is represented by the power-law model) in two-dimensional (2D) driven enclosures is studied. The enclosure consists of regular, rectangular shaped undulations on the bottom wall. Multiple-Relaxation time (MRT) collision model for the lattice Boltzmann equation method (LBM) is employed. First, numerical validation is performed by comparing the MRT-LBM results of power-law fluid flow inside the wall-driven square enclosure (no undulation) and flow inside diagonally flipped L-shaped enclosure with the literature. The strain rate profiles for square enclosure without undulations are compared using various equations available for strain rate calculation in the LBM literature. Further, the effect of different values of the non-hydrodynamic relaxation parameters on the flow is examined. Then, for the undulated enclosures, flow features and eddy dynamics are analyzed and discussed for the variations in the power-law index, n, to represent shear-thinning and shear thickening fluids. The effects of various parameters such as Reynolds numbers, wall undulation heights and wavelength of undulations on the power-law fluid flow are analyzed. Also, the variation of viscosity with spatial location for steady-state flow and total kinetic energy within the computational domain are presented for various values of power-law index.