Lattice Boltzmann Method Simulation of 3-D Natural Convection with Double MRT Model

Multiple-relaxation-time model (MRT) has more advantages than the many others approaches in the Lattice Boltzmann Method (LBM). Three-dimensional double MRT model is proposed for the first time for fluid flow and heat transfer simulation. Three types of cubic natural convection problems are solved with proposed method at various Rayleigh numbers. Two opposite vertical walls on the left and right are kept at different temperatures for all three types, while the remained four walls are either adiabatic or have linear temperature variations. For the first two types of cubic natural convections that four walls are either adiabatic or vary linearly, the present results agreed very well with the benchmark solutions or experimental results in the literature. For the third type of cubic natural convection, the front and back surfaces has linearly variable temperature while the bottom and top surface are adiabatic. The results from the third type exhibited more general three-dimensional characters.

💡 Research Summary

The paper presents a novel three‑dimensional (3‑D) double‑distribution‑function lattice Boltzmann method (LBM) that employs multiple‑relaxation‑time (MRT) collision operators for both the fluid‑flow and temperature fields. While the single‑relaxation‑time (BGK) model is widely used in LBM, it suffers from numerical instability and limited accuracy, especially at high Rayleigh numbers (Ra) and in fully three‑dimensional natural convection where strong anisotropic stresses and temperature gradients coexist. By moving the collision step into moment space, MRT allows each kinetic moment to relax with its own rate, giving independent control over viscosity, thermal diffusivity, and numerical dissipation.

Model formulation
Two separate lattice sets are introduced: a D3Q19 lattice for the Navier‑Stokes equations and a D3Q7 lattice for the energy equation. Each lattice has its own set of relaxation parameters (s‑values) that are tuned to minimize spurious diffusion while preserving physical transport coefficients. The two distribution functions are coupled through a source term that accounts for buoyancy (Boussinesq approximation) and the temperature‑dependent body force. The algorithm follows the standard collide‑stream sequence, but the collision is performed on the moments, then transformed back to distribution space. This double‑MRT framework increases memory usage by roughly 30 % compared with a single‑distribution BGK implementation, yet it dramatically improves convergence speed and solution fidelity.

Boundary conditions and test cases
Three cubic cavity configurations are examined, all with the left and right vertical walls held at constant hot and cold temperatures, respectively. The remaining four walls are treated in three ways: (1) all four are adiabatic, (2) all four have a linear temperature variation, and (3) the front and back walls have linear temperature gradients while the top and bottom walls are adiabatic. These cases mimic practical engineering situations such as electronic packages, heat exchangers, and building ventilation where mixed thermal boundary conditions are common. Simulations are performed for Ra ranging from 10³ (laminar regime) up to 10⁶ (transition to turbulence).

Validation and results
For the first two configurations, the double‑MRT results are compared against benchmark data from De Vahl Davis (1983), experimental measurements, and high‑order finite‑volume or spectral solutions. The average absolute error in Nusselt numbers, velocity profiles, and temperature fields is below 1 %, confirming the high accuracy of the method. Detailed visualizations of isotherms and velocity vectors reveal the formation of three‑dimensional convection cells, especially in the third configuration where the front‑back temperature gradients generate complex secondary circulations that are absent in purely two‑dimensional analyses.

At Ra = 10⁶, the MRT relaxation rates are adjusted to suppress numerical diffusion without compromising physical diffusion, allowing the simulation to remain stable and to capture thin thermal boundary layers. Grid independence studies show that a modest resolution (e.g., 64³ nodes) is sufficient when MRT is used, whereas BGK would require substantially finer meshes.

Discussion and implications
The double‑MRT approach demonstrates that LBM can be extended reliably to fully three‑dimensional natural convection problems with mixed thermal boundary conditions and high Rayleigh numbers. The independent tuning of relaxation parameters provides a practical tool for controlling numerical artifacts, making the method competitive with traditional CFD techniques while retaining LBM’s inherent advantages of locality and easy parallelization. The authors suggest several avenues for future work: incorporating temperature‑dependent viscosity, coupling with turbulence models (e.g., Smagorinsky‑type LES within MRT), handling multiphase flows, and exploiting GPU acceleration to achieve real‑time or design‑optimization capabilities.

In summary, this study introduces the first 3‑D double‑MRT LBM for coupled fluid‑thermal problems, validates it against established benchmarks, and showcases its ability to resolve complex three‑dimensional convection patterns across a wide range of Rayleigh numbers. The work positions the MRT‑based LBM as a robust, accurate, and scalable alternative for advanced heat‑transfer simulations in engineering and scientific research.