Preconditioned lattice Boltzmann method for steady flows: A noncascaded central-moments-based approach

We present a concise yet effective central-moments-based lattice Boltzmann method with an accelerated convergence to the steady state through preconditioning. It is demonstrated that the proposed scheme reduces to a slight modification of the unaccelerated one, as the preconditioning affects only the equilibrium state. Different from previous efforts carried out within the lattice Boltzmann community, the present scheme is built on an original model. In fact, the corresponding collision operator loses the pyramidal orchestrated nature that is typical of the cascaded scheme, hence we coin the name “noncascaded.” Our model is very general, characterized by highly intelligible formulations, simple to implement, and it can be derived for any lattice velocity space.

Figure 1

Poiseuille flow: Results from a convergence analysis with $\gamma =0.01$ (magenta), 0.05 (blue), 0.1 (green), 0.5 (red), and 1 (black). The dashed line has slope equal to $-2$, corresponding to the optimal convergence rate of the LBE. To interpret the figure in a print gray-scale version, notice that the value of $\gamma$ increases moving closer to the dashed line.

Figure 2

Lid-driven cavity: Normalized profiles of the horizontal component of velocity in the vertical midsection (left column) and vertical component of the velocity in the horizontal midsection (right column) at $\text{Re}=100$ (top row), 400 (central row), and 1000 (bottom row) for $\gamma =0.01$ (magenta), 0.05 (blue), 0.1 (green), 0.5 (red), and 1 (black). Except for findings corresponding to the lowest value of the preconditioning factor, all the curves are in a very good agreement with reference benchmark values provided in [45] (circles). To interpret the figure in a print gray-scale version, notice that the value of $\gamma$ increases moving closer to the circles.

Figure 3

Lid-driven cavity: Time evolution of the residual error $\epsilon$ for different values of $\gamma$, i.e., 0.01 (magenta), 0.05 (blue), 0.1 (green), 0.5 (red), and 1 (black), at (a) $\text{Re}=100$, (b) $\text{Re}=400$, and (c) $\text{Re}=1000$. The remarkable speedup conferred by the preconditioning factor can be appreciated. To interpret the figure in a print gray-scale version, notice that the value of $\gamma$ increases from left to right.

Figure 4

Lid-driven cavity: Number of time iterations $t_{\max }$ required to achieve the steady state at $\text{Re}=100$ (red squares), 400 (blue circles), and 1000 (green triangles) in log-log scale. The black dashed line has slope equal to 1, highlighting that $\epsilon$ scale linearly with $t_{\max }$, i.e., $t_{\max }\sim \gamma$.

Figure 5

Lid-driven cavity with nonunitary aspect ratio: Contour map of the magnitude of the velocity field $u$ normalized with respect to $u_{lid}$ at $\text{Re}=2000$ for different values of $\gamma$, i.e., (a) 0.05, (b) 0.1, (c) 0.5, and (d) 1.

Figure 6

Taylor-Green vortex: Results from a convergence analysis carried out by the cascaded (C) noncascaded (N-C) methods.