Multiblock approach for the passive scalar thermal lattice Boltzmann method

A multiblock approach for the passive scalar thermal lattice Boltzmann method (TLBM) with multiple-relaxation-time collision scheme is proposed based on the Chapman-Enskog analysis. The interaction between blocks is executed in the moment space directly and an external force term is considered. Theoretical analysis shows that all the nonequilibrium parts of the nonconserved moments should be rescaled, while the nonequilibrium parts of the conserved moments can be calculated directly. Moreover, a local scheme based on the pseudoparticles for computing heat flux is proposed with no need to calculate temperature gradient based on the finite-difference scheme. In order to validate the multiblock approach and local scheme for computing heat flux, thermal Couette flow with wall injection is simulated and good results are obtained, which show that the adoption of the multiblock approach does not deteriorate the convergence rate of TLBM and the local scheme for computing heat flux has second-order convergence rate. Further application of the present approach is the simulation of natural convection in a square cavity with the Rayleigh number up to ${10}^9$.

Figure 1

Interface structure between two blocks of different lattice spacings.

Figure 2

Schematic of spatial and temporal interpolations.

Figure 3

Streaming process at node x with the black solid circles being the lattice nodes, the gray solid circles being the virtual points, the blue dashed lines being the postcollision distribution functions ${\overline{f}}_i^T(\mathbf{x},t)$, and the red solid lines being the poststreaming distribution functions $f_i^T(\mathbf{x},t+{\delta }_t)$.

Figure 4

Schematic diagram of the multiblock arrangement for the thermal Couette flow with boundary conditions shown (the gray area is the fine block with $n=4$, and the white area is the coarse block).

Figure 5

The profiles of the horizontal velocity $u_x$ (a), the temperature $T$ (b), and the heat flux $J_x$ (c) and $J_y$ (d) along the $y$ coordinate when $U_0=V_0=0.05$, $\mathrm{Re}=10$, and $Pr=0.5$.

Figure 6

Relative errors of the horizontal velocity $u_x$ (a), the temperature $T$ (b), and the heat flux $J_x$ (c) and $J_y$ (d) versus lattice spacings ${\delta }_{x,c}$ when $\mathrm{Re}=10$ and $Pr=0.5$.

Figure 7

Schematic diagram of the multiblock arrangement for the natural convection in a square cavity with boundary conditions shown (the gray area is the fine block with $n=4$, and the white area is the coarse block).

Figure 8

Streamlines of the natural convection in a square cavity at $\mathrm{Ra}={10}^3$ (a), $\mathrm{Ra}={10}^4$ (b), $\mathrm{Ra}={10}^5$ (c), and $\mathrm{Ra}={10}^6$ (d) (the red solid line is the inner boundary of the fine block).

Figure 9

Isothermals of the natural convection in a square cavity at $\mathrm{Ra}={10}^3$ (a), $\mathrm{Ra}={10}^4$ (b), $\mathrm{Ra}={10}^5$ (c), and $\mathrm{Ra}={10}^6$ (d) (the red solid line is the inner boundary of the fine block).

Figure 10

Streamlines (left) and isothermals (right) of the natural convection in a square cavity at $\mathrm{Ra}={10}^7$ (a), $\mathrm{Ra}={10}^8$ (b), and $\mathrm{Ra}={10}^9$ (c) (the red solid line is the inner boundary of the fine block).