Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method

By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improved greatly, especially on simulating high Reynolds number （Re） flow. As a discovery, the super-stability analysed by Lallemand and Luo is verified and the complex structure of the cavity flow is also exhibited in our numerical simulation when Re is high enough. To the best knowledge of the authors, the maximum of Re which has been investigated by direct numerical simulation is only around 50 000 in the literature; however, this paper can readily extend the maximum to 1000 000 with the above combination.

Multi-relaxation-time lattice Boltzmann front tracking method for two-phase flow with surface tension

In this paper, an improved incompressible multi-relaxation-time lattice Boltzmann-front tracking approach is proposed to simulate two-phase flow with a sharp interface, where the surface tension is implemented. The lattice Boltzmann method is used to simulate the incompressible flow with a stationary Eulerian grid, an additional moving Lagrangian grid is adopted to track explicitly the motion of the interface, and an indicator function is introduced to update the fluid properties accurately. The interface is represented by using a four-order Lagrange polynomial through fitting a set of discrete marker points, and then the surface tension is directly computed by using the normal vector and curvature of the interface. Two benchmark problems, including Laplace＇s law for a stationary bubble and the dispersion relation of the capillary wave between two fluids are conducted for validation. Excellent agreement is obtained between the numerical simulations and the theoretical results in the two cases.

Three-dimensional multi-relaxation-time lattice Boltzmann front-tracking method for two-phase flow

We developed a three-dimensional multi-relaxation-time lattice Boltzmann method for incompressible and immiscible two-phase flow by coupling with a front-tracking technique. The flow field was simulated by using an Eulerian grid, an adaptive unstructured triangular Lagrangian grid was applied to track explicitly the motion of the two-fluid interface, and an indicator function was introduced to update accurately the fluid properties. The surface tension was computed directly on a triangular Lagrangian grid, and then the surface tension was distributed to the background Eulerian grid. Three benchmarks of two-phase flow, including the Laplace law for a stationary drop, the oscillation of a three-dimensional ellipsoidal drop, and the drop deformation in a shear flow, were simulated to validate the present model.

Multi-relaxation-time lattice Boltzmann simulations of lid driven flows using graphics processing unit

Large eddy simulation （LES） using the Smagorinsky eddy viscosity model is added to the two-dimensional nine velocity components （D2Q9） lattice Boltzmann equation （LBE） with multi-relaxation-time （MRT） to simulate incompressible turbulent cavity flows with the Reynolds numbers up to 1 × 10^7. To improve the computation efficiency of LBM on the numerical simulations of turbulent flows, the massively parallel computing power from a graphic processing unit （GPU） with a computing unified device architecture （CUDA） is introduced into the MRT-LBE-LES model. The model performs well, compared with the results from others, with an increase of 76 times in computation efficiency. It appears that the higher the Reynolds numbers is, the smaller the Smagorinsky constant should be, if the lattice number is fixed. Also, for a selected high Reynolds number and a selected proper Smagorinsky constant, there is a minimum requirement for the lattice number so that the Smagorinsky eddy viscosity will not be excessively large.

Multi-relaxation-time lattice Boltzmann simulation of slide damping in micro-scale shear-driven rarefied gas flow

To investigate the slide film damping in the micro-scale shear-driven rarefied gas flows, an effective multi-relaxation-time lattice Boltzmann method(MRT-LBM) is proposed. Through the Knudsen boundary layer model, the effects of wall and rarefaction are considered in the correction of relaxation time. The results of gas velocity distributions are compared among the MRT, Monte Carlo model(DSMC) and high-order LBM, and the effects of the tangential momentum accommodation coefficient on the gas velocity distributions are also compared between the MRT and the high-order LBM. It is indicated that the amendatory MRT-LBM can unlock the dilemma of simulation of micro-scale non-equilibrium. Finally, the effects of the Knudsen number, the Stokes number, and the gap between the plates on the damping are researched. The results show that by decreasing the Knudsen number or increasing the Stokes number, the slide film damping increases in the transition regime;however, as the size of the gap increases, the slide film damping decreases substantially.

INCOMPRESSIBLE MULTI-RELAXATION-TIME LATTICE BOLTZMANN MODEL IN 3-D SPACE

Multi-Relaxation-Time Lattice Boltzmann Method （MRT LBM） is of better numerical stability and has attracted more and more research interests. The previous MRT LBM included artificial compressible effects. To overcome the disadvantage, an incompressible MRT LBM has been proposed in two dimensions recently. In this article, we present incompressible MRT LBMs in 3-D space, with example of nineteen-velocity. The equilibria in momentum space are derived from an earlier incompressible Lattice Bhatnagar-Gross-Krook （LBGK） model proposed by Guo et al.. Through the Chapman-Enskog （C-E） expansion, the incompressible Navier-Stokes （N-S） equations can be recovered without artificial compressible effects. Simulations of a lid-driven cavity flow in three dimensions with Re = 1 000, 2 000 and 3 200 are performed. The simulation results agree with the existing data and clearly demonstrate better numerical stability of the presented model over the incompressible LBGK model.

Simulation of Power-Law Fluid Flows in Two-Dimensional Square Cavity Using Multi-Relaxation-Time Lattice Boltzmann Method

In this paper,the power-law fluid flows in a two-dimensional square cavity are investigated in detail with multi-relaxation-time lattice Boltzmann method(MRTLBM).The influence of the Reynolds number(Re)and the power-law index(n)on the vortex strength,vortex position and velocity distribution are extensively studied.In our numerical simulations,Re is varied from 100 to 10000,and n is ranged from 0.25 to 1.75,covering both cases of shear-thinning and shear-thickening.Compared with the Newtonian fluid,numerical results show that the flow structure and number of vortex of power-law fluid are not only dependent on the Reynolds number,but also related to power-law index.

A multiple-relaxation-time lattice Boltzmann method for high-speed compressible flows

This paper presents a coupling compressible model of the lattice Boltzmann method. In this model, the multiplerelaxation-time lattice Boltzmann scheme is used for the evolution of density distribution functions, whereas the modified single-relaxation-time （SRT） lattice Boltzmann scheme is applied for the evolution of potential energy distribution functions. The governing equations are discretized with the third-order Monotone Upwind Schemes for scalar conservation laws finite volume scheme. The choice of relaxation coefficients is discussed simply. Through the numerical simulations, it is found that compressible flows with strong shocks can be well simulated by present model. The numerical results agree well with the reference results and are better than that of the SRT version.

Study on the flow structure in curved open channels with suspended vegetation using multi relaxation time lattice Boltzmann method

Suspended vegetation in rivers,lakes,reservoirs and canals can change flow structure,which will in turn affect the sediment transport and cause the variation of water ecological environment.In order to study the characteristics of bend flow through suspended vegetation,three-dimensional numerical simulations are carried out by using the multi-relaxation-time lattice Boltzmann method(MRT-LBM).The drag force induced by vegetation is added in the velocity correction in the equilibrium distribution and a hybrid format combined bounce and specular reflection scheme is applied in the solid-fluid boundaries.After the validation of this model,six cases are designed to conduct the numerical simulations according to the root depth and the arrangement of vegetation.The simulated results show that the suspended vegetation can redistribute the flow structure in curved open channels.After the arrangement of suspended vegetation,the main flow moves to the side without vegetation,and the distribution of velocity tends to be balanced when vegetation is arranged on the entire cross section,the range of circulating current is reduced from the whole cross section to the local position without vegetation,however,the circulating current can still exist in the curve where the suspended vegetation enters less than half of the water depth.In addition,it can also be concluded that the suspended vegetation can affect the lateral gradient of flow velocity,and the bed shear stress in the curved channel.

Gas Flow Through Square Arrays of Circular Cylinders with Klinkenberg Effect:A Lattice Boltzmann Study

It is well known that,as non-continuum gas flows through microscale porous media,the gas permeability derived from Darcy law is larger than the absolute permeability,which is caused by the so-called Klinkenberg effect or slippage effect.In this paper,an effective definition of Knudsen number for gas flows through square arrays of circular cylinders and a local boundary condition for non-continuum gas flows are first proposed,and then the multi-relaxation-time lattice Boltzmann equation including discrete effects on boundary condition is used to investigate Klinkenberg effect on gas flow through circular cylinders in square arrays.Numerical results show that the celebrated Klinkenberg equation is only correct for low Knudsen number,and secondorder correction to Klinkenberg equation is necessary with the increase of Knudsen number.Finally,the present numerical results are also compared to some available results,and in general an agreement between them is observed.

Lattice Boltzmann model for shallow water in curvilinear coordinate grid

In this study, a multi-relaxation time lattice Boltzmann model for shallow water in a curvilinear coordinate grid is developed using the generalized form of the interpolation supplemented lattice Boltzmann method. The Taylor-Colette flow tests show that the proposed model enjoys a second order accuracy in space. The proposed model is applied to three types of meandering channels with 180°, 90°and 60° consecutive bends. The numerical results demonstrate that the simulated results agree well with previous computational and experimental data. In addition, the model can achieve the acceptable accuracy in terms of the water depth and the depth-averaged velocities for shallow water flows in curved and meandering channels over a wide range of bend angles.