Discrete effects on boundary conditions of the lattice Boltzmann method for convection-diffusion equations with curved geometries
Introduction
In the past three decades, the lattice Boltzmann method (LBM) been successfully advanced from simulating complex fluid flows [1] to heat and mass transfer processes, which are usually governed by convection-diffusion equations (CDEs) [2]. In addition to the lattice Boltzmann (LB) models for CDEs, the boundary treatment implemented at boundary nodes (the lattice nodes nearest to the physical boundary) is another important problem. As have been revealed in flow simulations [[3], [4], [5], [6], [7], [8]], the discrete effect on the boundary condition also must be minimized to derive reliable results for CDEs [[9], [10], [11], [12], [13]]. Since the local information at the current node is only contained, the single-node boundary technique would be more expected to match the local computation of LBM, especially for the case of complex boundaries and narrow spaces in porous media [14]. With the two-relaxation-time (TRT) model, Ginzburg [9,10] analyzed the single-node anti-bounce-back (ABB) boundary condition for CDEs. On the basis of the multiple-relaxation-time (MRT) model, Dubois et al. [11,15] used the Taylor expansion to investigate the ABB boundary scheme for the Dirichlet boundary condition. Their results both show that the accuracy of the boundary condition is affected by the combination of two relaxation rates. Zhang et al. [12] theoretically studied their halfway bounce-back boundary (HBB) scheme within the Bhatnagar-Gross-Krook (BGK) model for CDEs. For the diffusion in Couette flow with wall injection, their mathematical derivations reveal a nonzero numerical slip related with the relaxation time and the square of lattice spacing. Later, the discrete effect of the HBB condition was specially analyzed by Cui et al. [13] using the MRT model for CDEs. They further provided the formulae to eliminate the numerical slip by the free relaxation parameter s2 in the MRT model. However, in all these works regarding the halfway boundary scheme, the derivation results are theoretically valid only for the case of aligned straight walls with respect to the underlying lattice [9,15].
For the case of curved boundaries, the location of boundary nodes shall change locally from the physical wall. One straightforward strategy for this issue is to approximate the curved surface by some halfway points between the boundary and solid nodes [12,13]. However, such implementation will lose its accuracy under coarse grid resolutions. To improve the accuracy of numerical simulations for CDEs, various LB curved boundary conditions have been proposed to account for the curved geometry [9,[16], [17], [18], [19]]. Noteworthily, it is revealed in Refs. [9, 15, 20] that besides the relaxation time and the lattice spacing, the numerical slip (or the accuracy) of curved boundary conditions is related with the location of physical wall between lattice nodes. Therefore, when one attempts to remove the discrete effects from curved boundary conditions, the corresponding relaxation parameter would be varied at different boundary nodes with different lattice directions. This will make the collision operator anisotropic in the LBM. Therefore, it is inspiring us to resolve this important problem. In the present paper, a single-node boundary condition [21], which contains a free parameter besides the distance ratio, is analyzed detailedly within the MRT model for the diffusion problems with a parabolic distribution. Based on the mathematical derivations, a strategy to obtain uniform relaxation parameters is then provided to eliminate the numerical slip. Some numerical examples are then performed with the case of inclined and curved boundaries. The numerical results demonstrate the availability of our theoretical analysis for diffusion and thermal convective flow problems.
Section snippets
Lattice Boltzmann model for convection-diffusion equations
Without loss of generality, the convection-diffusion equation considered here are in two-dimensional case. The CDE with a source term is correspondingly written aswhere ϕ is the scalar variable usually encompassing the temperature or concentration, D is the diffusion coefficient, u = (u, v) is the convective velocity, and R is the source term. In the present work, the LB equation (LBE) with the MRT collision operator is employed to solve the CDE [13].
Discrete effect on curved boundary conditions
We now theoretically investigate the discrete effect of boundary conditions for CDEs with curved boundaries. Note that the boundary condition with a curved wall can be treated separately for each velocity direction. As illustrated in Fig. 1(a), the curved boundary is intersected with a lattice link from xr to xf along the lattice direction ei = ci/c. In the context of LBM, the unknown distribution function fi(xf, t + δt) is specified according to the boundary condition after a time step δt of
Numerical tests
Two benchmark problems with inclined or curved boundaries, i.e., the diffusion in an inclined channel, and the diffusion between two concentric cylinders, which have analytical solutions, are first simulated to validate the above theoretical analysis. Subsequently, the availability of our theoretical results are further examined for the thermal convective flows between two cylinders. In the simulations, the relaxation time τ0 = 1.0 is assigned for the conserved variable ϕ, τ1 is determined by
Conclusions
The discrete effects of boundary conditions were analyzed within the framework of MRT model for CDEs with curved boundaries. For previous curved boundary conditions which only contain the distance ratio γ, the mathematical derivations show that the relaxation time τ2 to eliminate the numerical slip should be changeable with the distance ratio. By virtue of a free parameter l besides γ, a theoretical strategy, based on a curved single-node boundary scheme [21], is provided to achieve the uniform
Declaration of Competing Interest
All authors have participated in conception and design, or analysis and interpretation of the data; drafting the article or revising it critically for important intellectual content; and approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript. All the authors have
Acknowledgements
This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 51776068, 51906044 and 12072127), and the Fundamental Research Funds for the Central Universities (No. 2018MS060). LW would like to thank Profs. Wen-An Yong, Zhaoli Guo and Dr. Weifeng Zhao for helpful discussions and advices.
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LaBCof: Lattice Boltzmann boundary condition framework
2023, Computer Physics CommunicationsCitation Excerpt :Dubois et al. [20] proposed a general bounce-back BC with the Taylor expansion method to increase the accuracy of the original bounce-back. But now there are many studies that model the curve boundaries [21–40] in order to make them more accurate. After reviewing the above studies, it appears that the implementation of these BCs depends on the lattice topology and orientation of the boundaries to which they should be applied.