Standard Lattice Boltzmann Equation 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.

Achieving thermodynamic consistency in a class of free-energy multiphase lattice Boltzmann models.

The free-energy lattice Boltzmann (LB) model is one of the major multiphase models in the LB community. The present study is focused on a class of free-energy LB models in which the divergence of thermodynamic pressure tensor or its equivalent form expressed by the chemical potential is incorporated into the LB equation via a forcing term. Although this class of free-energy LB models may be thermodynamically consistent at the continuum level, it suffers from thermodynamic inconsistency at the discrete lattice level owing to numerical errors [Guo et al., Phys. Rev. E 83, 036707 (2010)10.1103/PhysRevE.83.036707]. The numerical error term mainly includes two parts: one comes from the discrete gradient operator and the other can be identified in a high-order Chapman-Enskog analysis. In this paper, we propose an improved scheme to eliminate the thermodynamic inconsistency of the aforementioned class of free-energy LB models. The improved scheme is constructed by modifying the equation of state of the standard LB equation, through which the discretization of ∇(ρc_{s}^{2}) is no longer involved in the force calculation and then the numerical errors can be significantly reduced. Numerical simulations are subsequently performed to validate the proposed scheme. The numerical results show that the improved scheme is capable of eliminating the thermodynamic inconsistency and can significantly reduce the spurious currents in comparison with the standard forcing-based free-energy LB model.

An improved immersed moving boundary for hydrodynamic force calculation in lattice Boltzmann method

SummaryParticles suspension is considerably prevalent in petroleum industry and chemical engineering. The efficient and accurate simulation of such a process is always a challenge for both the traditional computational fluid dynamics and lattice Boltzmann method. Immersed moving boundary (IMB) method is promising to resolve this issue by introducing a particle‐fluid interaction term in the standard lattice Boltzmann equation, which allows for the smooth hydrodynamic force calculation even for a large grid size relative to the solid particle. Although the IMB method was proved good for stationary particles, the deviation of hydrodynamic force on moving particles exists. In this work, we reveal the physical origin of this problem first and figure out that the internal fluid effect on the hydrodynamic force calculation is not counted in the previous IMB. An improved immersed moving boundary method is therefore proposed by considering the internal fluid correction, which is easy to implement with the little extra computation cost. A 2D single elliptical particle and a 3D sphere sedimentation in Newtonian fluid is simulated directly for the validation of the corrected model by excellent agreements with the standard data.

A New Formulation for Prediction of Permeability of Nano Porous Structures using Lattice Botzmann Method

The standard lattice Boltzmann equation, LBE, is inadequate for simulating gas flows in nano scale flows with Knudsen numbers higher than 0.1. In the present study, rarefied gas flow in nano porous structures is simulated using the modified Lattice Boltzmann Method, LBM, which is able to cover wide range of flow regimes. The present study, reports the effects of the Knudsen number and porosity on the flow rate and permeability in slip and transitional flow regimes. For the first time, the Knudsen’s minimum effect in micro/nano porous was observed. A new correlation between the permeability, the porosity and the Knudsen number is then proposed which is able to predict the permeability of in-line and staggered nano porous structures in slip and transitional regimes.

Variable property-based lattice Boltzmann flux solver for thermal flows in the low Mach number limit

A variable property-based lattice Boltzmann flux solver (VPLBFS) is proposed in this paper for thermal flows with partial or total variation in fluid properties in the low Mach number limit. In the solver, one particle distribution function is introduced for pressure and momentum, and another for fluid temperature. The fluid properties are assumed to be functions of only the local fluid temperature. The macroscopic variables at cell centers are determined from the solution of macroscopic governing equations by the finite volume method. The fluxes at cell interfaces are evaluated by the local construction of the solution for the standard lattice Boltzmann equation. The additional terms due to fluid property variations and body forces are regarded as source terms and treated by the finite volume discretization. These features make its application on non-uniform grids with a fixed time interval more flexible in comparison with conventional lattice Boltzmann models. The VPLBFS is validated by several numerical examples, including fluid flows between two parallel plates, with one plate to be isothermally heated from a certain moment, the natural convection in a square cavity with a large temperature difference and the natural convection of supercritical carbon dioxide. Numerical results show the reliability of the VPLBFS for thermal flows with partial or total variation in fluid properties.

Discrete unified gas kinetic scheme with a force term for incompressible fluid flows

The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which combines the advantages of both lattice Boltzmann equation (LBE) and unified gas kinetic scheme (UGKS), including the simplified flux evaluation scheme, flexible mesh adaption and the asymptotic preserving properties. However, similar to standard LBE, the DUGKS can also be considered as a compressible scheme, and the compressible effect may bring some undesirable errors when it is used to investigate incompressible fluid flows. To eliminate the compressible effect, in this work a new DUGKS with a force term is developed through modifying the equilibrium distribution function. And simultaneously, the non-equilibrium extrapolation (NEE) scheme is also introduced to treat the velocity and pressure boundary conditions. To illustrate the capacity of the present DUGKS, we first performed some numerical simulations of two-dimensional steady and unsteady flows, and conducted a comparison between the present DUGKS and the original one. The results indicate that the present DUGKS can reduce the compressible effect efficiently, and the NEE scheme is also consistent with the second-order accuracy of DUGKS. We then extended the present DUGKS to study the three-dimensional lid-driven flows (LDF) in cubic and deep cavities, and found that the present results are in good agreement with available benchmark results, which indicates the present DUGKS is also accurate and efficient in the study of three-dimensional problems. At last, the structures of vortex in the cubic and deep cavities are also considered, and the symmetric affiliated vortices aside the secondary vortex at Re≥600 can be observed in the deep LDF.

Interface-capturing lattice Boltzmann equation model for two-phase flows.

In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.

Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio

A multiphase lattice Boltzmann flux solver (MLBFS) is proposed in this paper for incompressible multiphase flows with low- and large-density-ratios. In the solver, the flow variables at cell centers are given from the solution of macroscopic governing differential equations (Navier–Stokes equations recovered by multiphase lattice Boltzmann (LB) model) by the finite volume method. At each cell interface, the viscous and inviscid fluxes are evaluated simultaneously by local reconstruction of solution for the standard lattice Boltzmann equation (LBE). The forcing terms in the governing equations are directly treated by the finite volume discretization. The phase interfaces are captured by solving the phase-field Cahn–Hilliard equation with a fifth order upwind scheme. Unlike the conventional multiphase LB models, which restrict their applications on uniform grids with fixed time step, the MLBFS has the capability and advantage to simulate multiphase flows on non-uniform grids. The proposed solver is validated by several benchmark problems, such as two-phase co-current flow, Taylor–Couette flow in an annulus, Rayleigh–Taylor instability, and droplet splashing on a thin film at density ratio of 1000 with Reynolds numbers ranging from 20 to 1000. Numerical results show the reliability of the proposed solver for multiphase flows with high density ratio and high Reynolds number.

An Orthorhombic Lattice Boltzmann Model for Pore-Scale Simulation of Fluid Flow in Porous Media

One application of the lattice Boltzmann equation (LBE) models is in combination with tomography to simulate pore-scale flow and transport processes in porous media. Most LBE models in the literature are based on cubic lattice, and if the voxels in a tomography image are not cubic or cannot be divided into cubes due to computational limitations, these models will lose most of their advantages. How to deal with such images is, hence, an interest in use of the LBE model to simulate pore-scale processes. In this paper, we present an orthorhombic LBE model based on the single-relaxation time approach with the relaxation parameter varying with lattice directions. The equilibrium distribution functions in the standard LBE model were modified to correct the anisotropy induced by the non-cubic lattice, and the calculations of the fluid density and momentum were also redefined in order to maintain the conservation of mass and momentum during the collision. We tested the model against analytical solution for fluid flow in a tube, and against the standard cubic-based LBE model for fluid flow in a duct with an island inside. The model was then applied to simulate fluid flow in a 3D image in attempts to analyse the errors if the voxels in the image are not cubic but are assumed to be cubic.

Incompressible lattice Boltzmann model for porous flows with large pressure gradient

To drive a fluid in a porous media, a large pressure drop is usually required. Under such cases, the standard lattice Boltzmann equation, which uses the equation of state of ideal gases, may fail to work due to compressibility effect. In this paper, we propose an incompressible lattice Boltzmann model for porous flows, which can overcome this difficulty. Numerical results demonstrate that the model can use a large pressure without the compressibility effect.

Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow

The standard lattice Boltzmann equation (LBE) is inadequate for simulating gas flows with a large Knudsen number. In this paper we propose a generalized lattice Boltzmann equation with effective relaxation times based on a recently developed generalized Navier-Stokes constitution [Guo, Europhys Lett. 80, 24001 (2007)] for nonequilibrium flows. A kinetic boundary condition corresponding to a generalized second-order slip scheme is also designed for the model. The LBE model and the boundary condition are analyzed for a unidirectional flow, and it is found that in order to obtain the generalized Navier-Stokes equations, the relaxation times must be properly chosen and are related to the boundary condition. Numerical results show that the proposed method is able to capture the Knudsen layer phenomenon and can yield improved predictions in comparison with the standard lattice Boltzmann equation.

Lattice Boltzmann model for axisymmetric multiphase flows

A lattice Boltzmann model is presented for axisymmetric multiphase flows. Source terms are added to a two-dimensional standard lattice Boltzmann equation for multiphase flows such that the emergent dynamics can be transformed into the axisymmetric cylindrical coordinate system. The source terms are temporally and spatially dependent and represent the axisymmetric contribution of the order parameter of fluid phases and inertial, viscous, and surface tension forces. A model which is effectively explicit and second order is obtained. This is achieved by taking into account the discrete lattice effects in the Chapman-Enskog multiscale analysis, so that the macroscopic axisymmetric mass and momentum equations for multiphase flows are recovered self-consistently. The model is extended to incorporate reduced compressibility effects. Axisymmetric equilibrium drop formation and oscillations, breakup and formation of satellite droplets from viscous liquid cylindrical jets through Rayleigh capillary instability, and drop collisions are presented. Comparisons of the computed results with available data show satisfactory agreement.

Taylor series expansion- and least square-based Lattice Boltzmann method: an efficient approach for simulation of incompressible viscous flows

The Taylor series expansion- and least-square-based Lattice Boltzmann method (TLLBM) is a flexible Lattice Boltzmann approach capable of simulating incompressible viscous flows with arbitrary geometry. The method is based on the standard Lattice Boltzmann equation (LBE), Taylor series expansion and the least square optimisation. The final formulation is an algebraic form and essentially has no limitation on the mesh structure and lattice model. Successful applications of isothermal and thermal incompressible viscous flows have shown that the TLLBM is an efficient and promising version of LBM. In this paper, we will give the details of TLLBM and present some examples of its application.

A Lattice Boltzmann model for polymeric liquids

A Lattice Boltzmann model for polymeric liquids is developed. In the proposed model, polymers are modelled as dumbbells suspended in a solvent. A distribution function is introduced to describe the configuration of the dumbbells, and is used to evaluate the viscoelastic stresses in flow fields. A discrete kinetic equation for the configurational distribution is derived, and is shown to recover the Oldroyd-B constitutive equation at the continuous level, when coupled with the standard Lattice Boltzmann equation for the flow of the solvent. Numerical simulations of both steady and unsteady flows are performed in order to investigate the validity of the current model. In the simulation of small-amplitude oscillatory shear flows, numerical results are quantitatively compared with the analytical solutions obtained with the Oldroyd-B equation, and good agreements are observed.

A New Differential Lattice Boltzmann Equation and Its Application to Simulate Incompressible Flows on Non-Uniform Grids

A new differential lattice Boltzmann equation (LBE) is presented in this work, which is derived from the standard LBE by using Taylor series expansion only in spatial direction with truncation to the second-order derivatives. The obtained differential equation is not a wave-like equation. When a uniform grid is used, the new differential LBE can be exactly reduced to the standard LBE. The new differential LBE can be applied to solve irregular problems with the help of coordinate transformation. The present scheme inherits the merits of the standard LBE. The 2-D driven cavity flow is chosen as a test case to validate the present method. Favorable results are obtained and indicate that the present scheme has good prospects in practical applications.