Li-Shi Luo

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• PhD in Physics, Georgia Institute of Technology
• Distinguished Endowed Professor (full) at Old Dominion University

This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of 15-velocity and 19-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re = 500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of t...

An analysis for the acoustic and thermal properties of the energy-conserving lattice Boltzmann models was presented. To eliminate the spurious mode coupling, a hybrid thermal lattice Boltzmann equation (HTLBE), in which the mass and momentum conservation equations were solved by using the multiple-relaxation-time model, was proposed. The simulation...

Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space derivatives. Several previous works have been devoted to analyzing the accuracy of these models with special emph...

Matter, conceptually classified into fluids and solids, can be completely described by the microscopic physics of its constituent atoms or molecules. However, for most engineering applications, a macroscopic or continuum description has usually been sufficient due to the large disparity between the spatial and temporal scales relevant to these appl...

The Stokes eigenmodes on two-dimensional regular polygons of N apexes, 3≤N≤40, are studied numerically using two different solvers: the lattice Boltzmann equation and the Legendre-Galerkin spectral element method. In particular, the lowest 55 eigenmodes on regular N-polygons have been computed and investigated for the following properties including...

This review summarizes the rigorous mathematical theory behind the lattice Boltzmann equation (LBE). Relevant properties of the Boltzmann equation and a derivation of the LBE from the Boltzmann equation are presented. A summary of some important LBE models is provided. Focus is given to results from the numerical analysis of the LBE as a solver for...

We present an interface conforming method for simulating two-dimensional and axisymmetric multiphase flows. In the proposed method, the interface is composed of straight segments which are part of mesh and move with the flow. This interface representation is an integral part of an Arbitrary Lagrangian-Eulerian (ALE) method on an moving adaptive uns...

This work combines the lattice Boltzmann equation (LBE) and the overset method to simulate moving boundary problems in Navier-Stokes flows in two dimensions (2D). The transformation of the velocity moments of the distribution functions between a moving frame of reference and the one at rest is analyzed. The flow past a cylinder moving with a prescr...

A numerical scheme is developed for the evaluation of Abramowitz functions Jn in the right half of the complex plane. For n=−1,…,2, the scheme utilizes series expansions for |z|<1, asymptotic expansions for |z|>R with R determined by the required precision, and least squares Laurent polynomial approximations on each sub-region in the intermediate r...

A numerical scheme is developed for the evaluation of Abramowitz functions $J_n$ in the right half of the complex plane. For $n=-1,\, \ldots,\, 2$, the scheme utilizes series expansions for $|z|<1$ and asymptotic expansions for $|z|>R$ with $R$ determined by the required precision, and modified Laurent series expansions which are precomputed via a...

In this work, a comparative study for two simulation methods is conducted for interfacial flows in two dimensions: an arbitrary Lagrangian Eulerian (ALE) finite element method (FEM) on interface-conforming meshes and a phase field lattice Boltzmann method (LBM) on Cartesian meshes with quadtree adaptive mesh refinement (AMR). The methods are valida...

Using an arbitrary Lagrangian-Eulerian method on an adaptive moving unstructured mesh, we carry out numerical simulations for a rising bubble interacting with a solid wall. Driven by the buoyancy force, the axisymmetric bubble rises in a viscous liquid toward a horizontal wall, with impact on and possible bounce from the wall. First, our simulation...

To compute the non-oscillating mutual interaction for a system with N points, the fast multipole method (FMM) has an efficiency that scales linearly with the number of points. Specifically, for Coulomb interaction, FMM can be constructed using either the spherical harmonic functions or the totally symmetric Cartesian tensors. In this paper, we will...

An exponential time-integrator scheme of second-order accuracy based on the predictor-corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability...

An exponential time-integrator scheme of second-order accuracy based on the predictor-corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability...

http://dx.doi.org/10.1016/j.compfluid.2017.08.022

We present a lattice Boltzmann method (LBM) with a weighted multiple-relaxation-time (WMRT) collision model and an adaptive mesh refinement (AMR) algorithm for direct numerical simulation of two-phase flows in three dimensions. The proposed WMRT model enhances the numerical stability of the LBM for immiscible fluids at high density ratios, particul...

The fast multipole method (FMM) using traceless totally symmetric Cartesian tensor to calculate the Coulomb interaction between charged particles will be presented. The Cartesian tensor based FMM can be generalized to treat other non-oscillating interactions with the help of the differential algebra or the truncated power series algebra. Issues on...

NASA Technical Report CR-2002-211659 ICASE Report 2002-19 Also published in Lecture Notes in Computational Science and Engineering Vol. 21, edited by M. Breuer, F. Durst, and C. Zenger, pp. 123-130 (2002). Springer Berlin

This work studies the stability of a class of the globally hyperbolic moment system (GHMS) with the single relaxation-time collision model in the sense of hyperbolic relaxation systems. We prove the equilibrium stability of the GHMS in both one- and multi-dimensional space. For a five-moment system in one dimension, we prove its linear instability...

This work proposes a fully implicit lattice Boltzmann (LB) scheme based on finite-volume (FV) discretization on arbitrary unstructured meshes. The linear system derived from the finite-volume lattice Boltzmann equation (LBE) is solved by the block lower-upper (BLU) symmetric-Gauss-Seidel (SGS) algorithm. The proposed implicit FV-LB scheme is effici...

Multiple numerical approaches have been developed to simulate porous media fluid flow and solute transport at the pore scale. These include 1) methods that explicitly model the three-dimensional geometry of pore spaces and 2) methods that conceptualize the pore space as a topologically consistent set of stylized pore bodies and pore throats. In pre...

A genuine finite volume method based on the lattice Boltzmann equation (LBE) for near 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...

The integral equation for the flow velocity u(x;k) in the steady Couette flow derived from the linearized Bhatnagar-Gross-Krook-Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the...

We propose using the Maxwell iteration to derive the hydrodynamic equations from the lattice Boltzmann equation (LBE) with an external forcing term. The proposed methodology differs from existing approaches in several aspects. First, it need not explicitly introduce multiple-timescales or the Knudsen number, both of which are required in the Chapma...

We numerically study the Stokes eigen-modes in two dimensions on isosceles triangles with apex angle π/2, and 2π/3 by using two spectral solvers, i.e., a Lagrangian collocation method with a weak formulation for the primitive variables and a Legendre–Galerkin method for the stream-function. We compute the first 6,400 Stokes eigen-modes. With 72 col...

An efficient parallel algorithm for the time dependent incompressible Navier-Stokes equations is developed in this paper. The time discretization is based on a direction splitting method which only requires solving a sequence of one-dimensional Poisson type equations at each time step. Then, a spectral-element method is used to approximate these on...

The thermal lattice Boltzmann equation (TLBE) with multiple-relaxation-times (MRT) collision model is used to simulate the steady thermal convective flows in the two-dimensional square cavity with differentially heated vertical walls at high Rayleigh numbers. The MRT-TLBE consists of two sets of distribution functions, i.e., a D2Q9 model for the ma...

MRT lattice Boltzmann model for thermo-hydrodynamics in 2D Square cavity with differentially heated vertical walls Rayleigh–Bénard convection Boundary conditions for thermal flows a b s t r a c t In this paper we study the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model for incompressible thermo-hydrodynamics wi...

Based on the theory of asymptotic analysis, we prove that the viscous stress tensor computed with the lattice Boltzmann equation (LBE) in a two-dimensional domain is indeed second-order accurate in space. We only consider simple bounce-back boundary conditions which can be reduced to the periodic boundary conditions by using the method of image. Wh...

This Reply addresses two issues raised in the Comment [Phys. Rev. E 84, 068701 (2011)] by Karlin, Succi, and Chikatamarla (KSC): (1) A lattice Boltzmann (LB) model, which is claimed to have an $H$ theorem, is not qualified to be called an entropic lattice Boltzmann equation (ELBE); and (2) the real ELBE with a variable relaxation time performs exce...

In this paper, a numerical study of nonlinear flow phenomena in two-dimensional symmetric channels using the lattice-Boltzmann equation method is presented. The results are compared with both experimental results and other numerical results using some traditional methods. Comparisons are found to be quantitatively accurate.

In this Comment we reveal the falsehood of the claim that the lattice Bhatnagar-Gross-Krook (BGK) model "is capable of modeling shear-driven, pressure-driven, and mixed shear-pressure-driven rarified [sic] flows and heat transfer up to Kn=1 in the transitional regime" made in a recent paper [Ghazanfarian and Abbassi, Phys. Rev. E 82, 026307 (2010)]...

Although the lattice-gas automata (LGA) or lattice-gas cellular automata (LGCA) and the lattice Boltzmann equation (LBE) have a rather short history extending only over a decade or so, they have attracted much attention among physicists in various disciplines. The reason is that the methods of LGA and LBE have demonstrated their great potentials to...

Under strong shock loading, the pressure distribution in porous material is very complicated. We simulate such a system using a mesoscopic particle method. Morphological analysis is used to characterize the high pressure regimes defined by P>=P"t"h, ...

We conduct a comparative study to evaluate several lattice Boltzmann (LB) models for solving the near incompressible Navier-Stokes equations, including the lattice Boltzmann equation with the multiple-relaxation-time (MRT), the two-relaxation-time (TRT), the single-relaxation-time (SRT) collision models, and the entropic lattice Boltzmann equation...

This article provides a concise survey of the lattice Boltzmann equation: its mathematical theory and its capabilities for applications in computational fluid dynamics (CFD). The lattice Boltzmann method stems from the Boltzmann equation, and thus differentiates from any conventional method for CFD based on direct discretizations of the Navier–Stok...

We numerically study a vortex ring impacting a flat wall with an angle of incidence θ ≥ 0°) in three dimensions by using the lattice Boltzmann equation. The hydrodynamic behaviour of the ring–wall interacting flow is investigated by systematically varying the angle of incidence θ in the range of 0° ≤ θ ≤ 40° and the Reynolds number in the range of...

The lattice Boltzmann space/time discretisation, as usually derived from integration along characteristics, is shown to correspond to a Strang splitting between decoupled streaming and collision steps. Strang splitting offers a second-order accurate ...

We conduct a detailed comparison of the lattice Boltzmann equation (LBE) and the pseudo-spectral (PS) methods for direct numerical simulations (DNS) of the decaying homogeneous isotropic turbulence in a three-dimensional periodic cube. We use a mesh size of N(3) = 128(3) and the Taylor micro-scale Reynolds number 24.35 <= Re(lambda) <= 72.37, and c...

We study the effects of the rotational-translational energy exchange on the compressible decaying homogeneous isotropic turbulence (DHIT) in three dimensions through direct numerical simulations. We use the gas-kinetic scheme coupled with multitemperature nonequilibrium based on the Jeans-Landau-Teller model. We investigate the effects of the relax...

We study the effects of the rotational-translational energy exchange on the compressible decaying homogeneous isotropic turbulence (DHIT) in three dimensions through direct numerical simulations. We use the gas-kinetic scheme coupled with multi-temperature non-equilibrium based on the Jeans-Landau-Teller model. We investigate the effects of the rel...

Understanding and predicting of transition and turbulence under non-thermodynamical-equilibrium (NTE) conditions are important for hypersonic flight and other industrial applications. In NTE turbulence, the Kolmogorov paradigm, which forms the basis of most equilibrium turbulence models, may be invalid. Furthermore, under the NTE conditions, multip...

One undesirable feature of LBE methods as diffuse interface methods is the existence of parasitic currents. Recently, Lee and Fischer have shown that if the potential form of the intermolecular force is used, the parasitic currents can be eliminated. In their study, the LBGK collision model is used. As we know that multiple-relaxation-time (MRT) co...

Three dimensional vortex ring impacting a wall at different angles of incidence has been numerically investigated using the lattice Boltzmann model. The detailed flow behavior, vortex evolution, and pressure distribution on wall have been studied systematically with the Reynolds number of 100 < Re < 1000, and the impact angle of the range of 0 < th...

We apply the gas-kinetic scheme (GKS) for the direct numerical simulations (DNSs) of compressible decaying homogeneous isotropic turbulence (DHIT). We intend to study the accuracy, stability, and efficiency of the gas-kinetic scheme for DNS of compressible homogeneous turbulence depending on both flow conditions and numerics. In particular, we stud...

We prove for generic steady solutions of the Lattice Boltzmann (LB) models that the variation of the numerical errors is set by specific combinations (called ''magic numbers'') of the relaxation rates associated with the symmetric and anti-symmetric ...

We prove for generic steady solutions of the Lattice Boltzmann (LB) models that the variation of the numerical errors is set by specific combinations (called ''magic numbers'') of the relaxation rates associated with the symmetric and anti-symmetric ...

We apply the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model to simulate laminar flows in two-dimensions (2D). In order to simulate flows in an unbounded domain with the LBE method, we need to address two issues: stretched non-uniform mesh and in-flow and out-flow boundary conditions. We use the interpolated gri...

We present the lattice Boltzmann equation (LBE) with multiple relaxation times (MRT) to simulate pressure-driven gaseous flow in a long microchannel. We obtain analytic solutions of the MRT-LBE with various boundary conditions for the incompressible Poiseuille flow with its walls aligned with a lattice axis. The analytical solutions are used to rea...

We compare the lattice Boltzmann method (LBM) and Pseudo-Spectral (PS) method for direct numerical simulation of decaying homogeneous isotropic turbulence. In this study we use the generalized lattice Boltzmann equation (GLBE) with multiple-relaxation-time (MRT) collision model, which overcomes all the apparent defects in the popular lattice BGK eq...

Recently Reis and Phillips [Phys. Rev. E 77, 026702 (2008)] proposed a perturbative method to solve the dispersion equation derived from the linearized lattice Boltzmann equation. We will demonstrate that the method proposed by Reis and Phillips is a reinvention of an existing method. We would also like to refute a number of claims made by Reis and...

We conduct a comparison of the lattice Boltzmann (LB) and the pseudo-spectral (PS) methods for direct numerical simulations (DNS) of the decaying turbulence in a three dimensional periodic cube. We use a mesh size of 128^3 and the Taylor micro-scale Reynolds number 24.35

We apply the gas-kinetic scheme to direct numerical simulation of decaying compressible turbulence. We compute the kinetic energy K(t), dissipation rate ε(t), probability density functions (PDFs) of the two-point longitudinal velocity difference, shocklet strength, and local Mach number. Our results reveal the following features of decaying compres...

A very efficient implementation of a lattice Boltzmann (LB) kernel in 3D on a graphical processing unit using the compute unified device architecture interface developed by nVIDIA is presented. By exploiting the explicit parallelism offered by the graphics ...

Implicit time-integration techniques are envisioned to be the methods of choice for direct numerical simulations (DNS) for flows at high Reynolds numbers. Therefore, the computational efficiency of implicit flow solvers becomes critically important. The textbook multigrid efficiency (TME), which is the optimal efficiency of a multigrid method, is a...

We compare the lattice Boltzmann equation (LBE) and the gas-kinetic scheme (GKS) applied to 2D incompressible laminar flows. Although both methods are derived from the Boltzmann equation thus share a common kinetic origin, numerically they are rather different. The LBE is a finite difference method, while the GKS is a finite-volume one. In addition...

We propose the derivation of acoustic-type isotropic partial differential equations that are equivalent to linear lattice Boltzmann schemes with a density scalar field and a momentum vector field as conserved moments. The corresponding linear equivalent ...

We propose a consistent lattice Boltzmann equation (LBE) with baroclinic coupling between species and mixture dynamics to model the active scalar dynamics in multi-species mixtures. The proposed LBE model is directly derived from the linearized Boltzmann equations for mixtures and it has the following two distinctive features. First, it uses the mu...

The lattice Boltzmann equation replaces continuous particle velocity space by a finite set; the velocity distribution function then varies over a finite-dimensional vector space instead of over an infinite-dimensional function space. The number of linearly independent moments of the distribution function in a lattice Boltzmann model cannot exceed t...

We compare the lattice Boltzmann equation (LBE) and the gas-kinetic scheme (GKS) for direct numerical simulation of decaying homogeneous isotropic turbulence. Although both methods are derived from the Boltzmann equation thus share a common kinetic origin, numerically they are rather different. The LBE is a finite difference method, while the GKS i...

The gas-kinetic scheme (GKS) is a finite-volume method in which the fluxes are constructed from the single particle velocity distribution function f . The distribution function f is obtained from the linearised Boltzmann equation and is retained only to the Navier-Stokes order in terms of the Chapman-Enskog expansion. Higher order non-equilibrium e...

The Interpolated Bounce-Back (IBB) method and Immersed-Boundary (IB) method are compared for fluid-solid boundary conditions in the Lattice Boltzmann Equation (LBE) in terms of their numerical accuracy and computational efficiency. We carry out simulations for the flow past a circular cylinder asymmetrically placed in the channel in two dimensions...

In recent years, quite a few particle-resolved simulation methods have emerged for treating moving solid particles in a viscous fluid. A common advantageous feature shared by these methods is the use of a simple fixed mesh. The no-slip boundary condition on the surface of a particle is handled locally by a consistent coupling or interaction scheme....

We propose to combine the lattice Boltzmann equation (LBE) and the front-tracking (FT) method to simulate interfacial dynamics with surface tension in two dimensions (2D). In the proposed LBE-FT method, the flow is modeled by the LBE on a fixed Cartesian mesh, whereas interfaces are explicitly tracked by a set of markers that are advected by the fl...

We simulate a two-dimensional incompressible flow around a rotating circular cylinder near a plane wall at the Reynolds number Re=200 by using the lattice Boltzmann equation with multiple relaxation times. We investigate the flow pattern in the parameter space of the rotational rate gamma:=omega a/U and the normalized gap h:=H/D, where omega is the...

We present a unified approach for both continuum and near-continuum flows based on the Boltzmann equation and kinetic theory. We use the gas-kinetic scheme developed from the linearized Boltzmann equation for the continuum flows and a modified gas-kinetic scheme with a variable relaxation time for the near-continuum flows. The gas-kinetic schemes a...

We study the asymptotic behavior, as the degree approaches infinity, of the Christoffel function at a fixed point z corresponding to a weight function of the type exp(−|z|λ) on the set |arg z|=π/2+α. The method generalizes that of Rakhmanov and also Mhaskar and Saff.

We propose a simple and effective iterative procedure to generate consistent initial conditions for the lattice Boltzmann equation (LBE) for incompressible flows with a given initial velocity field u0. Using the Chapman-Enskog analysis we show that not only the proposed procedure effectively solves the Poisson equation for the pressure field p0 cor...

In this paper we consider the application of multiple-relaxation-time (MRT) lattice Boltzmann equation (LBE) for large-eddy simulation (LES) of turbulent flows. The implementation is discussed in the context of 19-velocity (D3Q19) MRT-LBE model in conjunction with the Smagorinsky subgrid closure model. The MRT-LBE-LES is then tested in the turbulen...

A two dimensional incompressible flow past a rotating circular cylinder near a plane wall at Re 200 is investigated by using the lattice Boltzmann equation (LBE). The effects of the gap between the cylinder and the wall, and tangential speed of the cylinder on the frequency of vortex shedding, and the lift and drag forces on the cylinder are quanti...

We quantitatively evaluate the capability and accuracy of the lattice Boltzmann equation (LBE) for modeling flow through porous media. In particular, we conduct a comparative study of the LBE models with the multiple-relaxation-time (MRT) and the Bhatnagar– Gross–Krook (BGK) single-relaxation-time (SRT) collision operators. We also investigate seve...

In this article we analyze the lattice Boltzmann equation (LBE) by using the asymptotic expansion technique. We first relate the LBE to the finite discrete-velocity model (FDVM) of the Boltzmann equation with the diffusive scaling. The analysis of this model directly leads to the incompressible Navier–Stokes equations, as opposed to the compressibl...

The objective of the paper is to assess the effectiveness of the lattice Boltzmann equation (LBE) as a computational tool for performing direct numerical simulations (DNS) and large-eddy simulations (LES) of turbulent flows. Decaying homogeneous isotropic turbulence (HIT) in inertial and rotating frames is considered for this investigation. We perf...

In this paper, we provide a set of sufficient conditions under which a lattice Boltzmann model does not admit an H theorem. By verifying the conditions, we prove that a number of existing lattice Boltzmann models does not admit an H theorem. These models include D2Q6, D2Q9 and D3Q15 athermal models, and D2Q16 and D3Q40 thermal (energy-conserving) m...

For a simple discrete model of Boltzmann equation, we study the derivatives of H-Boltzmann function, and prove that all derivatives of odd order are negative, instead all derivatives of even order are postive. These result is a first and small generalisation of the classical H-Boltzmann theorem.

Decaying homogeneous isotropic turbulence in inertial and rotating reference frames is investigated to evaluate the capability of the lattice Boltzmann method in turbulence. In the inertial frame case, the decay exponents of kinetic energy and dissipation and the low wave-number scaling of the spectrum are studied. The results are in agreement with...

We evaluated lattice Boltzmann equation (LBE) methods for modeling flow through porous media. We compared a three-dimensional, 19-velocity, multiple-relaxation-time (MRT) LBE model with a popular single-relaxation-time; Bhatnagar-Gross-Krook (BGK) LBE model. It can be shown that the latter (BGK-LBE) model is a special case of the former (MRT-LBE) m...

A Comment on the Letter by Baoming Li and Daniel Y. Kwok, Phys. Rev. Lett. 90, 124502 (2003). The authors of the Letter offer a Reply.

The Lattice Boltzmann equation (LBE) was used to perform direct numerical simulations (DNS) and large-eddy simulations (LES) of benchmark problems in turbulence, scalar mixing, and reaction. The most important contribution was the development of LBE theory for binary mixing, which can be extended in a straight-forward manner to multi-scalar mixing....

Homogeneous isotropic turbulence subject to linearly increasing forcing is investigated as a unit problem for statistically unsteady turbulence. The transient spectral dynamics is analysed using a closure theory. A long time asymptotic state is found with k-7/3 corrections to the energy spectrum as proposed by Yoshizawa. Although the cancellation o...

The method of lattice Boltzmann equation (LBE) is a kinetic-based approach for fluid flow computations. This method has been successfully applied to the multi-phase and multi-component flows. To extend the application of LBE to high Reynolds number incompressible flows, some critical issues need to be addressed, noticeably flexible spatial resoluti...

We prove that no H theorem exists for the athermal lattice Boltzmann equation with polynomial equilibria satisfying the conservation laws exactly and explicitly. The proof is demonstrated by using the seven-velocity model in a triangular lattice in two dimensions, and can be readily extended to other lattice Boltzmann models in two and three dimens...

A two-fluid lattice Boltzmann model for binary mixtures is developed. The model is derived formally from kinetic theory by discretizing two-fluid Boltzmann equations in which mutual collisions and self-collisions are treated independently. In the resulting lattice Boltzmann model, viscosity and diffusion coefficients can be varied independently by...