Three-Dimensional Lattice Boltzmann Flux Solver and Its Applications to Incompressible Isothermal and Thermal Flows

A three-dimensional (3D) lattice Boltzmann flux solver (LBFS) is presented in this paper for the simulation of both isothermal and thermal flows. The present solver combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. It applies the finite volume method (FVM) to solve the N-S equations. Different from the conventional N-S solvers, its viscous and inviscid fluxes at the cell interface are evaluated simultaneously by local reconstruction of LBE solution. As compared to the conventional LBE solvers, which apply the lattice Boltzmann method (LBM) globally in the whole computational domain, it only applies LBM locally at each cell interface, and flow variables at cell centers are given from the solution of N-S equations. Since LBM is only applied locally in the 3D LBFS, the drawbacks of the conventional LBM, such as limitation to uniform mesh, tie-up of mesh spacing and time step, tedious implementation of boundary conditions, are completely removed. The accuracy, efficiency and stability of the proposed solver are examined in detail by simulating plane Poiseuille flow, lid-driven cavity flow and natural convection. Numerical results show that the LBFS has a second order of accuracy in space. The efficiency of the LBFS is lower than LBM on the same grids. However, the LBFS needs very less non-uniform grids to get grid-independence results and its efficiency can be greatly improved and even much higher than LBM. In addition, the LBFS is more stable and robust.     

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

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

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

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

Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes

A genuine finite volume method based on the lattice Boltzmann equation (LBE) for nearly incompressible flows is developed. The proposed finite volume lattice Boltzmann method (FV-LBM) is grid-transparent, i.e., it requires no knowledge of cell topology, thus it can be implemented on arbitrary unstructured meshes for effective and efficient treatment of complex geometries. Due to the linear advection term in the LBE, it is easy to construct multi-dimensional schemes. In addition, inviscid and viscous fluxes are computed in one step in the LBE, as opposed to in two separate steps for the traditional finite-volume discretization of the Navier-Stokes equations. Because of its conservation constraints, the collision term of the kinetic equation can be treated implicitly without linearization or any other approximation, thus the computational efficiency is enhanced. The collision with multiple-relaxation-time (MRT) model is used in the LBE. The developed FV-LBM is of second-order convergence. The proposed FV-LBM is validated with three test cases in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the Blasius boundary layer; and (c) the steady flow past a cylinder at the Reynolds numbers Re=10, 20, and 40. The results verify the designed accuracy and efficacy of the proposed FV-LBM.     

Semiclassical Lattice Boltzmann Simulations of Rarefied Circular Pipe Flows

Computations of microscopic circular pipe flow in a rarefied quantum gas are presented using a semiclassical axisymmetric lattice Boltzmann method. The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al. [Phys. Rev. E., 64 (2001), 011208] is adopted by adding forcing terms into the resulting microdynamic evolution equation. The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form. The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion. The velocity profiles and the mass flow rates of pipe flows with several Knudsen numbers covering different flow regimes are presented. It is found the Knudsen minimum can be captured in all three statistics studied. The results also indicate distinct characteristics of the effects of quantum statistics.     

Continuum Formulation for Non-Equilibrium Shock Structure Calculation

Are extensions to continuum formulations for solving fluid dynamic problems in the transition-to-rarefied regimes viable alternatives to particle methods? It is well known that for increasingly rarefied flow fields, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies are attributed primarily to the linear approximations of the stress and heat flux terms in the Navier-Stokes equations. The inclusion of higher-order terms, such as Burnett or high-order moment equations, could improve the predictive capabilities of such continuum formulations, but there has been limited success in the shock structure calculations, especially for the high Mach number case. Here, after reformulating the viscosity and heat conduction coefficients appropriate for the rarefied flow regime, we will show that the Navier-Stokes-type continuum formulation may still be properly used. The equations with generalization of the dissipative coefficients based on the closed solution of the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using the gas-kinetic numerical scheme. This paper concentrates on the non-equilibrium shock structure calculations for both monatomic and diatomic gases. The Landau-Teller-Jeans relaxation model for the rotational energy is used to evaluate the quantitative difference between the translational and rotational temperatures inside the shock layer. Variations of shear stress, heat flux, temperatures, and densities in the internal structure of the shock waves are compared with, (a) existing theoretical solutions of the Boltzmann solution, (b) existing numerical predictions of the direct simulation Monte Carlo (DSMC) method, and (c) available experimental measurements. The present continuum formulation for calculating the shock structures for monatomic and diatomic gases in the Mach number range of 1.2 to 12.9 is found to be satisfactory.     

Comparison of Simulations of Convective Flows

We show that a single particle distribution for the "energy-conserving" D2Q13 lattice Boltzmann scheme can simulate coupled effects involving advection and diffusion of velocity and temperature. We consider various test cases: non-linear waves with periodic boundary conditions, a test case with buoyancy, propagation of transverse waves, Couette and Poiseuille flows. We test various boundary conditions and propose to mix bounce-back and anti-bounce-back numerical boundary conditions to take into account velocity and temperature Dirichlet conditions. We present also first results for the de Vahl Davis heated cavity. Our results are compared with the coupled D2Q9-D2Q5 lattice Boltzmann approach for the Boussinesq system and with an elementary finite differences solver for the compressible Navier-Stokes equations. Our main experimental result is the loss of symmetry in the de Vahl Davis cavity computed with the single D2Q13 lattice Boltzmann model without the Boussinesq hypothesis. This result is confirmed by a direct Navier Stokes simulation with finite differences.     

Lattice Boltzmann Finite Volume Formulation with Improved Stability

The most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especially when there is a need for high resolutions near the body or the walls. Among the recent advances in lattice Boltzmann research to handle complex geometries, a particularly remarkable option is represented by changing the solution procedure from the original "stream and collide" to a finite volume technique. However, most of the presented schemes have stability problems. This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme. To enhance stability, upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice. The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers: boundary layer, mixing layer and plane jet flows. The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data. Results in self-similar coordinates are also investigated and show that the time-averaged statistics for velocity and vorticity express self-similarity at low Reynolds numbers. Furthermore, the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent. The agreement of the numerical results with previous results is satisfactory.     

Numerical Simulation of Rarefied Gas Flows with Specified Heat Flux Boundary Conditions

We investigate unidirectional rarefied flows confined between two infinite parallel plates with specified heat flux boundary conditions. Both Couette and force-driven Poiseuille flows are considered. The flow behaviors are analyzed numerically by solving the Shakhov model of the Boltzmann equation. We find that a zero-heat-flux wall can significantly influence the flow behavior, including the velocity slip and temperature jump at the wall, especially for high-speed flows. The predicted bimodal-like temperature profile for force-driven flows cannot even be qualitatively captured by the Navier-Stokes-Fourier equations.     

On the Five-Moment Maximum Entropy System of One-Dimensional Boltzmann Equation

The maximum entropy moment system extends the Euler equation to nonequilibrium gas flows by considering higher order moments such as the heat flux. This paper presents a systematic study of the maximum entropy moment system of Boltzmann equation. We consider a hypothetical one-dimensional gas and study a five-moment model. A numerical algorithm for solving the optimization problem is developed to produce the maximum entropy distribution function from known moments, and the asymptotic behaviour of the system around the singular region known as the Junk’s line, as well as that near the boundary of the realizability domain is analyzed. Furthermore, we study the properties of the system numerically, including the behaviour of the system around the Maxwellian and within the interior of the realizability domain, and properties of its characteristic fields. Our studies show the higher order entropy-based moment models to differ significantly from the Euler equations. Much of this difference comes from the singularity near the Junk’s line, which would be removed if a truncation of the velocity domain is employed.     

Elements of the Lattice Boltzmann Method I: Linear Advection Equation

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

Dimension-Reduced Hyperbolic Moment Method for the Boltzmann Equation with BGK-Type Collision

We develop the dimension-reduced hyperbolic moment method for the Boltzmann equation, to improve solution efficiency using a numerical regularized moment method for problems with low-dimensional macroscopic variables and high-dimensional microscopic variables. In the present work, we deduce the globally hyperbolic moment equations for the dimension-reduced Boltzmann equation based on the Hermite expansion and a globally hyperbolic regularization. The numbers of Maxwell boundary condition required for well-posedness are studied. The numerical scheme is then developed and an improved projection algorithm between two different Hermite expansion spaces is developed. By solving several benchmark problems, we validate the method developed and demonstrate the significant efficiency improvement by dimension-reduction.     

Simulation of Acoustic Behavior of Bubbly Liquids with Hybrid Lattice Boltzmann and Homogeneous Equilibrium Models

Homogeneous equilibrium model (HEM) has been widely used in cavitating flow simulations. The major feature of this model is that a single equation of state (EOS) is proposed to describe the thermal behavior of bubbly liquid, where both kinematic and thermal equilibrium are assumed between two phases. In this paper, the HEM was coupled with multi-relaxation-time lattice Boltzmann model (MRT-LBM) and the acoustic behavior was simulated. Two approaches were applied alternatively: adjusting speed of sound (Buick, J. Phys. A, 2006, 39:13807-13815) and setting real gas EOS. Both approaches result in high accuracy in acoustic speed predictions for different void (gas) volume of fractions. It is demonstrated that LBM could be successfully applied as a Navier-Stokes equation solver for industrial applications. However, further dissipation and dispersion analysis shows that Shan-Chen type approaches of LBM are deficient, especially in large wave-number region.     

Lattice BGK Model for Incompressible Axisymmetric Flows

In this paper, a lattice Boltzmann BGK (LBGK) model is proposed for simulating incompressible axisymmetric flows. Unlike other existing axisymmetric lattice Boltzmann models, the present LBGK model can eliminate the compressible effects only with the small Mach number limit. Furthermore, the source terms of the model are simple and contain no velocity gradients. Through the Chapman-Enskog expansion, the macroscopic equations for incompressible axisymmetric flows can be exactly recovered from the present LBGK model. Numerical simulations of the Hagen-Poiseuille flow, the pulsatile Womersley flow, the flow over a sphere, and the swirling flow in a closed cylindrical cavity are performed. The results agree well with the analytic solutions and the existing numerical or experimental data reported in some previous studies.     

A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

The Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.     

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

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

Lattice Boltzmann Methods for Multiphase Flow Simulations across Scales

The simulation of multiphase flows is an outstanding challenge, due to the inherent complexity of the underlying physical phenomena and to the fact that multiphase flows are very diverse in nature, and so are the laws governing their dynamics. In the last two decades, a new class of mesoscopic methods, based on minimal lattice formulation of Boltzmann kinetic equation, has gained significant interest as an efficient alternative to continuum methods based on the discretization of the NS equations for non ideal fluids. In this paper, three different multiphase models based on the lattice Boltzmann method (LBM) are discussed, in order to assess the capability of the method to deal with multiphase flows on a wide spectrum of operating conditions and multiphase phenomena. In particular, the range of application of each method is highlighted and its effectiveness is qualitatively assessed through comparison with numerical and experimental literature data.     

Pressure Distribution of the Gaseous Flow in Microchannel: A Lattice Boltzmann Study

In this paper the pressure distribution of the gaseous flow in a microchannel is studied via a lattice Boltzmann equation (LBE) method. With effective relaxation times and a generalized second order slip boundary condition, the LBE can be used to simulate rarefied gas flows from slip to transition regimes. The Knudsen minimum phenomena of mass flow rate in the pressure driven flow is also investigated. The effects of Knudsen number (rarefaction effect), pressure ratio and aspect ratio (compression effect) on the pressure distribution are analyzed. It is found the rarefaction effect tends to the curvature of the nonlinear pressure distribution, while the compression effect tends to enhance its nonlinearity. The combined effects lead to a local minimum of the pressure deviation. Furthermore, it is also found that the relationship between the pressure deviation and the aspect ratio follows a pow-law.     

Simulation of MHD Flows Using a Hybrid Lattice-Boltzmann Finite-Difference Method

A hybrid lattice-Boltzmann finite-difference method is presented to simulate incompressible, resistive magnetohydrodynamic (MHD) flows. The lattice Boltzmann equation (LBE) with the Lorentz force term is solved to update the flow field while the magnetic induction equation is solved using the finite difference method to calculate the magnetic field. This approach is methodologically intuitive because the governing equations for MHD are solved in their respective original forms. In addition, the extension to 3-D is straightforward. For validation purposes, this approach was applied to simulate the Hartmann flow, the Orszag-Tang vortex system (2-D and 3-D) and the magnetic reconnection driven by doubly periodic coalescence instability. The obtained results agree well with analytical solutions and simulation results available in the literature.