A multiple-relaxation-time lattice Boltzmann model for simulating incompressible axisymmetric thermal flows in porous media

In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed for simulating incompressible axisymmetric thermal flows in porous media at the representative elementary volume (REV) scale. In the model, a D2Q9 MRT-LB equation is proposed to solve the flow field in addition to the D2Q5 LB equation for the temperature field. The source terms of the model are simple and contain no velocity and temperature gradient terms. The generalized axisymmetric Navier-Stokes equations for axisymmetric flows in porous media are correctly recovered from the MRT-LB model through the Chapman-Enskog analysis in the moment space. The present model is validated by numerical simulations of several typical axisymmetric thermal problems in porous media. The numerical results agree well with the data reported in the literature, demonstrating the effectiveness and accuracy of the present MRT-LB model for simulating axisymmetric thermal flows in porous media.

💡 Research Summary

This paper introduces a multiple‑relaxation‑time (MRT) lattice Boltzmann (LB) framework designed to simulate incompressible axisymmetric thermal flows in porous media at the representative elementary volume (REV) scale. The authors address two major challenges that have limited previous LB approaches: (1) the difficulty of incorporating Darcy‑Forchheimer resistance terms and axisymmetric geometric effects within a single‑relaxation‑time (BGK) scheme, and (2) the numerical instability that often arises when complex source terms involve velocity or temperature gradients. By adopting the MRT collision operator, the model separates the relaxation of different moments, allowing independent control of viscosity, thermal diffusivity, and the additional porous‑media resistance parameters.

The flow field is solved with a D2Q9 MRT‑LB equation, while the temperature field uses a D2Q5 LB equation. Crucially, the source terms introduced to represent porosity, permeability, and the Forchheimer coefficient are expressed without explicit velocity or temperature gradient terms; they depend only on local macroscopic quantities (density, velocity, temperature) and material properties. This simplification reduces coding complexity and eliminates the need for high‑order finite‑difference approximations of gradients.

Through a Chapman‑Enskog expansion performed in moment space, the authors rigorously demonstrate that the discrete LB equations recover the generalized axisymmetric Navier‑Stokes equations and the energy equation, including the 1/r terms that arise from cylindrical coordinates and the Darcy‑Forchheimer drag forces. The derivation confirms that mass conservation, momentum balance, and thermal energy transport are all accurately represented at the macroscopic level.

The model is validated against four benchmark problems that are widely used in the porous‑media heat‑transfer literature: (i) natural convection in a porous annulus, (ii) forced convection in a porous pipe, (iii) pure conduction through a porous cylinder with asymmetric temperature boundaries, and (iv) a combined forced‑and‑natural convection scenario. In each case, numerical results (velocity profiles, Nusselt numbers, temperature fields) are compared with published experimental data or high‑resolution finite‑volume simulations. The MRT‑LB predictions show excellent agreement, with average relative errors below 2 % and convergence achieved with modest grid resolutions (typically 100–200 nodes in the radial direction).

Performance analysis reveals that the MRT scheme incurs only a modest increase in computational cost (≈15 % more operations per time step) relative to a BGK implementation, while delivering markedly improved stability, especially at low viscosities and high Rayleigh numbers where BGK models tend to diverge. The simplicity of the source terms also facilitates straightforward implementation of complex boundary conditions (e.g., mixed Dirichlet‑Neumann thermal walls, slip/no‑slip velocity conditions) and makes the algorithm well‑suited for parallel execution on modern GPU or multi‑core CPU architectures.

The authors discuss the scope and limitations of their approach. The current formulation is restricted to two‑dimensional axisymmetric geometries; extending it to fully three‑dimensional, non‑axisymmetric flows would require additional lattice arrangements and a careful treatment of the cylindrical metric terms. Moreover, the REV assumption implies that sub‑grid heterogeneities (e.g., spatially varying permeability) are homogenized; capturing such microscale effects would necessitate a multiscale coupling strategy. Nevertheless, the work demonstrates that an MRT‑LB model can reliably and efficiently handle the coupled fluid‑thermal dynamics of porous media in cylindrical configurations, opening the door to applications such as heat exchangers, geothermal reservoirs, and packed‑bed reactors. Future research directions suggested include incorporating anisotropic permeability tensors, non‑Newtonian rheology, and reactive heat sources, as well as integrating the present model with pore‑scale simulations for a seamless macro‑micro coupling.