Li-Shi Luo

Multiple-relaxation-time lattice Boltzmann models in three dimensions

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 the multiple–relaxation–time lattice Boltzmann equation over the popular lattice Bhatnagar–Gross–Krook equation.

Viscous flow computations with the method of lattice Boltzmann equation

Abstract 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 resolution, boundary treatments for curved solid wall, dispersion and mode of relaxation, and turbulence model. Recent developments in these aspects are highlighted in this paper. These efforts include the study of force evaluation methods, the development of multi-block methods which provide a means to satisfy different resolution requirement in the near wall region and the far field and reduce the memory requirement and computational time, the progress in constructing the second-order boundary condition for curved solid wall, and the analyses of the single-relaxation-time and multiple-relaxation-time models in LBE. These efforts have lead to successful applications of the LBE method to the simulation of incompressible laminar flows and demonstrated the potential of applying the LBE method to higher Reynolds flows. The progress in developing thermal and compressible LBE models and the applications of LBE method in multi-phase flows, multi-component flows, particulate suspensions, turbulent flow, and micro-flows are reviewed.

Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model

In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as Σαfαeα=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids.

Lattice Boltzmann method for moving boundaries

We propose a lattice Boltzmann method to treat moving boundary problems for solid objects moving in a fluid. The method is based on the simple bounce-back boundary scheme and interpolations. The proposed method is tested in two flows past an impulsively started cylinder moving in a channel in two dimensions: (a) the flow past an impulsively started cylinder moving in a transient Couette flow; and (b) the flow past an impulsively started cylinder moving in a channel flow at rest. We obtain satisfactory results and also verify the Galilean invariance of the lattice Boltzmann method.

LARGE-EDDY SIMULATIONS WITH A MULTIPLE-RELAXATION-TIME LBE MODEL

We include Smagorinskys algebraic eddy viscosity approach into the multiple-relaxation-time (MRT) lattice Boltzmann equation (LBE) for large-eddy simulations (LES) of turbulent flows. The main advantage of the MRT-LBE model over the popular lattice BGK model is a significant improvement of numerical stability which leads to a substantial reduction of oscillations in the pressure field, especially for turbulent flow simulations near the numerical stability limit. The MRT-LBE model for LES is validated with a benchmark case of a surface mounted cube in a channel at Re = 40 000. Our preliminary results agree well with experimental data.

Lattice Boltzmann modeling of microchannel flow in slip flow regime

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 realize the Dirichlet boundary conditions in the LBE. We use the first-order slip boundary conditions at the walls and consistent pressure boundary conditions at both ends of the long microchannel. We validate the LBE results using the compressible Navier-Stokes (NS) equations with a first-order slip velocity, the information-preservation direct simulation Monte Carlo (IP-DSMC) and DSMC methods. As expected, the LBE results agree very well with IP-DSMC and DSMC results in the slip velocity regime, but deviate significantly from IP-DSMC and DSMC results in the transition-flow regime in part due to the inadequacy of the slip velocity model, while still agreeing very well with the slip NS results. Possible extensions of the LBE for transition flows are discussed.

Lattice Boltzmann simulations of thermal convective flows in two dimensions

In this paper we study the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model for incompressible thermo-hydrodynamics with the Boussinesq approximation. We use the MRT thermal LBE (TLBE) to simulate the following two flows in two dimensions: the square cavity with differentially heated vertical walls and the Rayleigh-Benard convection in a rectangle heated from below. For the square cavity, the flow parameters in this study are the Rayleigh number Ra=10^3-10^6, and the Prandtl number Pr=0.71; and for the Rayleigh-Benard convection in a rectangle, Ra=2@?10^3, 10^4 and 5@?10^4, and Pr=0.71 and 7.0.

HYBRID FINITEDIFFERENCE THERMAL LATTICE BOLTZMANN EQUATION

We analyze the acoustic and thermal properties of athermal and thermal lattice Boltzmann equation (LBE) in 2D and show that the numerical instability in the thermal lattice Boltzmann equation (TLBE) is related to the algebraic coupling among different modes of the linearized evolution operator. We propose a hybrid finite-difference (FD) thermal lattice Boltzmann equation (TLBE). The hybrid FD-TLBE scheme is far superior over the existing thermal LBE schemes in terms of numerical stability. We point out that the lattice BGK equation is incompatible with the multiple relaxation time model.

A comparative study of the LBE and GKS methods for 2D near incompressible laminar flows

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, the LBE is valid for near incompressible flows with low-Mach number restriction Ma<0.3, while the GKS is valid for fully compressible flows. 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 equation. We use both the LBE and GKS methods to simulate the flow past a square block symmetrically placed in a 2D channel with the Reynolds number Re between 10 and 300. The LBE and GKS results are validated against the well-resolved results obtained using finite-volume method. Our results show that both the LBE and GKS yield quantitatively similar results for laminar flow simulations, and agree well with existing ones, provided that sufficient grid resolution is given. For 2D problems, the LBE is about 10 and 3 times faster than the GKS for steady and unsteady flow calculations, respectively, while the GKS uses less memory. We also observe that the GKS method is much more robust and stable for under-resolved cases due to its upwinding nature and interpolations used in calculating fluxes.

A comparative study of immersed-boundary and interpolated bounce-back methods in LBE

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 with the Reynolds number Re = 20 and 100, corresponding to steady and unsteady flows, respectively. The results obtained by the LBE method are compared with the existing data. We observe that the LBE with either the IBB or IB methods for the no-slip boundary conditions exhibits a second-order rate of convergence. While the computational cost for both methods are comparable, the interpolated bounce-back method is more accurate than the immersed-boundary method with the same mesh size. Consequently the IBB method is more efficient computationally, while the IB method is easier to implement.