Zhenhua Chai

A Multiple-Relaxation-Time Lattice Boltzmann Model for General Nonlinear Anisotropic Convection---Diffusion Equations

In this paper, based on the previous work (Shi and Guo in Phys Rev E 79:016701, 2009), we develop a multiple-relaxation-time (MRT) lattice Boltzmann model for general nonlinear anisotropic convection–diffusion equation (NACDE), and show that the NACDE can be recovered correctly from the present model through the Chapman–Enskog analysis. We then test the MRT model through some classic CDEs, and find that the numerical results are in good agreement with analytical solutions or some available results. Besides, the numerical results also show that similar to the single-relaxation-time lattice Boltzmann model or so-called BGK model, the present MRT model also has a second-order convergence rate in space. Finally, we also perform a comparative study on the accuracy and stability of the MRT model and BGK model by using two examples. In terms of the accuracy, both the analysis and numerical results show that a numerical slip on the boundary would be caused in the BGK model, and cannot be eliminated unless the relaxation parameter is fixed to be a special value, while the numerical slip in the MRT model can be overcome once the relaxation parameters satisfy some constrains. The results in terms of stability also demonstrate that the MRT model could be more stable than the BGK model through tuning the free relaxation parameters.

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 R e ź 600 can be observed in the deep LDF.

Regularized lattice Boltzmann model for a class of convection-diffusion equations.

In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations.

Lattice Boltzmann model for a class of convectiondiffusion equations with variable coefficients

In this work, a lattice Boltzmann model for a class of n-dimensional convectiondiffusion equations with variable coefficients is proposed through introducing an auxiliary distribution function. The model can exactly recover the convectiondiffusion equation without any assumptions. A detailed numerical study on several types of convectiondiffusion equations is performed to validate the present model, and the results show that the accuracy of the present model is better than previous models.

A Coupled Lattice Boltzmann Method to Solve Nernst---Planck Model for Simulating Electro-osmotic Flows

In this paper, we focus on the nonlinear coupling mechanism of the Nernst–Planck model and propose a coupled lattice Boltzmann method (LBM) to solve it. In this method, a new LBM for the Nernst–Planck equation is developed, a multi-relaxation-time (MRT)-LBM for flow field and an LBM for the Poisson equation are used. And then, we discuss the choice of the model and found that the MRT-LBM is much more stable and accurate than the LBGK model. A reasonable iterative sequence and evolution number for each LBM are proposed by considering the properties of the coupled LBM. The accuracy and stability of the presented coupled LBM are also discussed through simulating electro-osmotic flows (EOF) in micro-channels. Furthermore, to test the applicability of it, the EOF with non-uniform surface potential in micro-channels based on the Nernst–Planck model is simulated. And we investigate the effects of non-uniform surface potential on the pattern of the EOF at different external applied electric fields. Finally, a comparison of the difference between the Nernst–Planck model and the Poisson–Boltzmann model is presented.

Coupled lattice Boltzmann method for generalized Keller-Segel chemotaxis model

In this work, an efficient and stable lattice Boltzmann method (LBM) for generalized Keller-Segel (K-S) chemotaxis model is proposed. Through the Chapman-Enskog analysis, the proposed LBM can correctly recover to the K-S model. The stability of the proposed LBM has been improved through adding correction terms in the evolution equations. Moreover, a local computational scheme for the gradient operator, which is included in the evolution equation, is developed, making the proposed LBM be implemented locally. Hence, both 2D and 3D problems with arbitrary geometries can be processed easily. In the numerical experiments, several representative chemotaxis problems are studied, including the blow up problem in square and circle domains, two-species chemotaxis blow up problem, chemotactic bacteria pattern formation in semi-solid medium in circle domain, 3D pattern formation in liquid medium, and the tumor invasion into surrounding healthy tissue. The numerical results demonstrate the high efficiency, stability and robustness of the proposed LBM. Furthermore, the capability of the proposed LBM in handling both 2D and 3D problems with complex domain is also illustrated.

Lattice Boltzmann method for filtering and contour detection of the natural images

In this paper, the lattice Boltzmann method (LBM) is extended to study the filtering and contour detection of natural images, and a new lattice Boltzmann model is proposed for more complicated image processing model, like the Ambrosio and Tortorelli (A-T) model that contains two coupled nonlinear partial differential equations. The numerical results of image filtering and contour detection show that the noises in the image can be removed greatly, and simultaneously, important contours of the image are protected well. To improve the computational efficiency, we implement the developed lattice Boltzmann model on Graphic Processing Unit (GPU), and find that, compared to the CPU based algorithm, the GPU based LBM can gain more than 25 x speedup, which is very important in the further lattice Boltzmann study of large-scale image processing problems. And finally, these numerical results also show that the LBM is a feasible and efficient approach for filtering and contour detection of the natural images.

A lattice Boltzmann study of the asymmetry effect on the hemodynamics in stented fusiform aneurysms

In this paper, the effect of asymmetric bulges on the hemodynamics in stented fusiform aneurysms under pulsatile Newtonian flow condition has been studied numerically by the lattice Boltzmann method. In order to guarantee the efficiency and accuracy of the method, a domain decomposition technique is also considered. Numerical results show that the flow structures are significantly affected by the asymmetric bulges. For non-stented fusiform aneurysms, the maximum wall shear stress (WSS) and maximum wall pressure are found to occur near the distal end of the aneurysms at peak systole, and the magnitude of peak WSS and wall pressure usually increase along with the increase of the maximum height of the dilated region. In addition, the implantation of a stent can reduce the magnitude of the maximum wall pressure and WSS at the distal end, and a low-porosity stent gives better performance than a high-porosity stent in terms of the reduction. In particular, for stented fusiform aneurysms, the effect of asymmetry on wall pressure is found insignificant. Further, a comparison between pulsatile solution and steady-state solution at peak systole is also presented, and the results show that the difference of WSS near the proximal neck for two conditions is not apparent, while the location of the maximum wall pressure obtained from steady-state condition moves toward downstream in contrast to pulsatile condition, and the maximum WSS at the distal end is underestimated by the condition of steady-state.

The computation of strain rate tensor in multiple-relaxation-time lattice Boltzmann model

Multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is an important class of LB model with lots of advantages over traditional single-relaxation-time (SRT) LB model. In addition, the computation of strain rate tensor is crucial in MRT-LB simulations of some complex flows. Up to now, there are only two formulas to compute the strain rate tensor in the MRT LB model. One is to compute the strain rate tensor by using non-equilibrium parts of macroscopic moments (Yu formula). The other is to compute the strain rate tensor by using non-equilibrium parts of density distribution functions (Chai formula). The mathematical expressions of these two formulas are so different that we do not know which formula to choose for computing the strain rate tensor in MRT LB model. In this paper, we study the relationship of these two formulas. It is found that Yu formula can be deduced from Chai formula in a particular procedure. However, these two formulas have their own advantages and disadvantages. Yu formula is more efficient in the computation aspect while Chai formula can be applied to more lattice patterns of MRT LB models. It is also found that deducing Yu formula for a particular lattice pattern from Chai formula is more convenient than the way proposed by Yu et al.

A lattice Boltzmann based local feedback control approach for spiral wave

Abstract Suppression of spiral wave attracts more and more attention in nonlinear systems. In this paper, a spiral wave local feedback control approach based on the FitzHugh–Nagumo (FHN) model is studied with lattice Boltzmann method. Numerical simulations are performed to investigate the effects of initial conditions for the spiral wave formation, model parameters, size and position of the feedback control region, and feedback control parameters on the behavior of spiral wave. The results show that there are three characteristics of spiral wave elimination. The first is that initial conditions of the spiral wave formation have little influence on feedback control of spiral wave. Secondly, the model parameters are related to the time needed for the elimination of spiral wave, for example, the larger the mutual time scales, the faster the elimination of spiral wave. Finally, through selecting the size and position of the feedback region, spiral wave can be effectively removed with weak feedback signal.