Interpolation-supplemented lattice Boltzmann simulation of thermal convection on non-uniform meshes

TL;DR

The paper addresses high-fidelity simulation of buoyancy-driven thermal convection on non-uniform meshes by developing and evaluating an interpolation-supplemented lattice Boltzmann method (ISLBM). ISLBM augments the standard LBM with quadratic interpolation during streaming to enable non-uniform grid refinement near solid boundaries while preserving simplicity and parallel efficiency; it uses a double-distribution-function MRT framework to solve coupled momentum and heat transfer equations. The method is validated in 2-D and 3-D side-heated cavities across wide $Ra$ ranges ($10^6\le Ra\le 10^8$ in 2-D and $10^5\le Ra\le 10^7$ in 3-D) with $Pr=0.71$, showing accurate boundary-layer resolution, close agreement for $Nu$ and $Re$, and nearly third-order convergence for global quantities with ~second-order accuracy for local fields. Performance benchmarks against Nek5000 and OpenFOAM demonstrate substantial speedups on large grids, including strong GPU performance, indicating ISLBM as a robust, scalable path toward high-fidelity simulations and potential extensions to turbulent convection at extreme Rayleigh numbers.

Abstract

We present a systematic evaluation of an interpolation-supplemented lattice Boltzmann method (ISLBM) for simulating buoyancy-driven thermal convection on non-uniform meshes. The ISLBM extends the standard lattice Boltzmann framework by incorporating quadratic interpolation during the streaming step, enabling flexible mesh refinement near solid boundaries while maintaining algorithmic simplicity and parallel scalability. The method is implemented for a two-dimensional side-heated cavity at high Rayleigh numbers $10^6\leq Ra \leq 10^8$, and for a three-dimensional side-heated cavity at $10^5\leq Ra \leq 10^7$, with the Prandtl number fixed at $Pr=0.71$. Benchmark results show that the ISLBM accurately captures thermal and velocity boundary layers, yielding Nusselt and Reynolds numbers in close agreement with high-fidelity reference data. Grid-convergence studies demonstrate nearly third-order accuracy for global quantities and about second-order for local fields. We further assess the computational performance of the in-house LBM solver against two open-source solvers: Nek5000 based on the spectral element method, and OpenFOAM based on the finite volume method. Performance metrics, including million lattice updates per second (MLUPS) and wall-clock time per dimensionless time unit (WCTpDT), indicate that the ISLBM offers one to three orders of magnitude higher efficiency in large-scale simulations. On GPU architectures, the ISLBM retains high computational performance: throughput on non-uniform meshes reaches 60-70% of that on uniform meshes in terms of MLUPS, while the cost in WCTpDT is about three times higher. These results highlight the potential of interpolation-based LBM approaches for high-fidelity simulations of thermal convection on non-uniform meshes, providing a robust foundation for future extensions to turbulent flows.

Figures (14)

• Figure 1: Illustration of the interpolation procedure in the interpolation-supplemented lattice Boltzmann method (ISLBM) on a two-dimensional non-uniform mesh.
• Figure 2: Examples of (a) uniform mesh and (b) non-uniform mesh with half-way bounce-back scheme. Here, l.u. denotes the lattice length-unit huang2015multiphasebook. The first fluid node lies 0.5 l.u. from the wall due to the half-way bounce-back/anti-bounce-back scheme.
• Figure 3: Contours of (a-c) temperature field $T^{*}$ and (d-f) velocity magnitude $\sqrt{u^{*2}+v^{*2}}$ (superposed with streamlines), at (a, d) $Ra=10^6$, (b, e) $Ra=10^7$, and (c, f) $Ra=10^8$.
• Figure 4: Horizontal distributions of vertically averaged (a) temperature $T^{*}$ and (b) vertical velocity $v^{*}$. Panels (c, d) show enlarged views near the hot wall $(x^{*}=0)$, where open circles denote boundary layer locations determined by the 99% criterion method and open squares denote those determined by the slope method.
• Figure 5: (a-c) Contours of the logarithm of velocity divergence magnitude, $\log_{10}(|\nabla\cdot\mathbf{u}|)$. (d-f) Probability density functions (PDFs) of $\log_{10}(|\nabla\cdot\mathbf{u}|)$, obtained over the entire cell, at (a, d) $Ra = 10^6$, (b, e) $Ra = 10^7$ and (c, f) $Ra = 10^8$.