Lattice Boltzmann modeling for conduction heat transfer based on the Qadyan method

Technical Summary: Lattice Boltzmann Modeling for Conduction Heat Transfer Based on the Qadyan Method

Problem Statement
The paper addresses the simulation of conductive heat transfer using the Lattice Boltzmann Method (LBM). While LBM is effective for fluid dynamics, its application to heat conduction faces limitations regarding grid uniformity and numerical stability. Standard LBM schemes, particularly those based on the Bhatnagar–Gross–Krook (BGK) approximation, are often restricted to uniform grids with equal spatial steps. Furthermore, standard spatial discretization methods, such as second-order finite difference schemes, can induce numerical oscillations in the solution. The study seeks to evaluate alternative spatial discretization strategies to enhance accuracy and stability without introducing additional complexity to the standard LBM framework or resorting to non-uniform grids.

Methodology
The research implements and compares three distinct approaches for solving the lattice Boltzmann equation (LBE) for heat conduction:

1. Standard LBM: Utilizes conventional lattice models (D2Q9, D2Q8, and D2Q5) with standard collision-streaming processes. The study notes that for purely diffusive problems, models with fewer velocity sets (like D2Q8 and D2Q5) are sufficient, as heat diffusion lacks directional preference.
2. Finite Difference Lattice Boltzmann Method (FDLBM): This approach discretizes the convective term of the LBE using finite difference schemes. Specifically, it employs a first-order forward difference for time derivatives and a first-order upwind scheme for spatial derivatives to avoid oscillations, though this limits spatial accuracy to the first order near boundaries.
3. The Qadyan Method: A novel approach proposed by Ghadyani et al. that combines semi-discrete schemes with the Differential Quadrature Method (DQM). In this method, the spatial derivatives in the LBE are discretized using DQM weighting coefficients. To replicate the upwind effect inherent in advection-diffusion problems, the method modifies the weighting coefficients to assign greater weight to upstream nodes than downstream nodes. This allows the solution of the LBE as a system of ordinary differential equations without being tied to a specific discrete velocity model type.

The study validates these methods using one-dimensional transient heat conduction in a slab with both Dirichlet (constant temperature) and Neumann (insulated/zero flux) boundary conditions. Analytical solutions derived from the heat equation serve as benchmarks. Error norms ($L_1$, $L_2$, and $L_\infty$) are calculated to assess accuracy, and a 2D plate diffusion problem is solved to demonstrate the method's applicability to multi-dimensional cases.

Key Contributions and Results

• Accuracy vs. Stability: The results indicate that while the traditional LBM offers high accuracy, the proposed Qadyan method provides a robust alternative that improves accuracy over the FDLBM approach, particularly for Neumann boundary conditions. For Dirichlet conditions, the standard D2Q5 LBM showed the lowest error norms, but the Qadyan method significantly outperformed the FDLBM formulations.
• Grid Independence: The Qadyan method successfully widens the accuracy and stability range of solutions, increasing accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids.
• Computational Cost: A critical finding is the trade-off between accuracy and efficiency. The Qadyan method entails a higher computational cost and longer simulation time compared to both standard LBM and FDLBM due to the increased number of calculation operations required for the DQM weighting coefficients.
• Parameter Sensitivity: The study establishes that for standard LBM and FDLBM, the reference values for lattice velocity are dependent on the type of boundary condition (temperature vs. flux). In contrast, the Qadyan method utilizes a consistent lattice velocity value ($c=0.015$) regardless of the boundary condition type, simplifying parameter selection.
• 2D Validation: In a two-dimensional square plate simulation, the Qadyan method produced isotherm contours and centerline temperature distributions that showed good agreement with finite difference method results, demonstrating its capability for multi-dimensional heat transfer problems.

Significance and Claims
The paper claims that the Qadyan method is a suitable approach for a wide range of conductive heat transfer problems. Its primary significance lies in its ability to increase numerical accuracy without introducing the complexity of non-uniform grids or abandoning the LBM framework. The authors assert that the method is robust and straightforward to implement. However, the paper maintains a modest stance regarding efficiency, acknowledging that the method is computationally more expensive than traditional approaches. The study concludes that while the Qadyan method requires longer computational time, its improved accuracy and stability for specific boundary conditions justify its application, offering a competitive alternative to standard LBM formulations for conductive heat transfer simulations.