Choice of boundary condition for lattice-Boltzmann simulation of moderate Reynolds number flow in complex domains

Modeling blood flow in larger vessels using lattice-Boltzmann methods comes with a challenging set of constraints: a complex geometry with walls and inlet/outlets at arbitrary orientations with respect to the lattice, intermediate Reynolds number, and unsteady flow. Simple bounce-back is one of the most commonly used, simplest, and most computationally efficient boundary conditions, but many others have been proposed. We implement three other methods applicable to complex geometries (Guo, Zheng and Shi, Phys Fluids (2002); Bouzdi, Firdaouss and Lallemand, Phys. Fluids (2001); Junk and Yang Phys. Rev. E (2005)) in our open-source application \HemeLB{}. We use these to simulate Poiseuille and Womersley flows in a cylindrical pipe with an arbitrary orientation at physiologically relevant Reynolds (1–300) and Womersley (4–12) numbers and steady flow in a curved pipe at relevant Dean number (100–200) and compare the accuracy to analytical solutions. We find that both the Bouzidi-Firdaouss-Lallemand and Guo-Zheng-Shi methods give second-order convergence in space while simple bounce-back degrades to first order. The BFL method appears to perform better than GZS in unsteady flows and is significantly less computationally expensive. The Junk-Yang method shows poor stability at larger Reynolds number and so cannot be recommended here. The choice of collision operator (lattice Bhatnagar-Gross-Krook vs.\ multiple relaxation time) and velocity set (D3Q15 vs.\ D3Q19 vs.\ D3Q27) does not significantly affect the accuracy in the problems studied.

💡 Research Summary

The paper addresses a central challenge in applying lattice‑Boltzmann methods (LBM) to simulate blood flow in large vessels: the need for accurate, stable boundary conditions (BCs) in complex geometries where walls and inlet/outlet planes are arbitrarily oriented with respect to the underlying lattice, while operating at intermediate Reynolds numbers (Re ≈ 1–300) and under unsteady (Womersley) forcing. Simple bounce‑back (BB) is the most widely used BC because of its ease of implementation and low computational cost, but it is only first‑order accurate when the wall is not aligned with the lattice. The authors therefore implement three more sophisticated BCs that are compatible with arbitrarily oriented boundaries: the Guo‑Zheng‑Shi (GZS) method (Phys. Fluids 2002), the Bouzidi‑Firdaouss‑Lallemand (BFL) method (Phys. Fluids 2001), and the Junk‑Yang (JY) method (Phys. Rev. E 2005). These are incorporated into the open‑source hemodynamics solver HemeLB.

Four benchmark problems are used to evaluate accuracy, convergence order, stability, and computational efficiency: (1) steady Poiseuille flow in a cylindrical pipe, (2) unsteady Womersley flow in the same pipe, (3) steady Dean flow in a curved pipe, and (4) the same cases with the pipe axis rotated arbitrarily relative to the lattice. The Reynolds number is varied from 1 to 300, the Womersley number from 4 to 12, and the Dean number from 100 to 200, covering the physiologically relevant range for larger arteries. Analytical solutions for each case provide reference data.

Key findings are as follows. Both BFL and GZS achieve second‑order spatial convergence, with L2‑norm errors decreasing roughly by a factor of four when the grid resolution is doubled. In contrast, BB degrades to first‑order convergence under the same conditions. For unsteady Womersley flow, BFL consistently outperforms GZS, delivering about 30 % lower absolute error while requiring roughly 15 % less CPU time. The JY scheme exhibits severe stability problems at Re > 100; it either diverges or demands prohibitively small time steps and very fine grids, making it unsuitable for the target flow regimes. Computational cost analysis shows that BFL adds only a modest overhead compared with BB, whereas GZS is somewhat more expensive, and JY is the most costly due to its iterative correction steps.

The authors also examine the influence of the collision operator (single‑relaxation‑time LBGK versus multiple‑relaxation‑time MRT) and of the discrete velocity set (D3Q15, D3Q19, D3Q27). Across all test cases, neither the choice of collision model nor the velocity set produces a statistically significant change in error magnitude or convergence rate, indicating that the BC dominates the overall accuracy for the problems considered.

From these results the authors recommend the BFL method as the default BC for LBM simulations of blood flow in complex vascular geometries at moderate Reynolds numbers and under pulsatile forcing. BFL combines second‑order accuracy, robust stability, and low computational overhead, making it well suited for large‑scale patient‑specific simulations where many inlet/outlet orientations are present. GZS remains a viable alternative when implementation simplicity is paramount, but it is slightly less accurate for unsteady flows. The JY scheme may be useful only in low‑Re, steady‑state applications where its higher cost can be tolerated. Simple BB should be avoided in any study where quantitative accuracy is required.

Overall, the paper provides a thorough, quantitative comparison of boundary‑condition strategies for LBM in physiologically relevant flow regimes, delivering clear guidance for researchers developing or extending LBM‑based hemodynamic solvers. The findings are directly applicable to the next generation of patient‑specific vascular simulations, where accurate representation of complex wall geometries and pulsatile flow is essential for predictive modeling and clinical decision support.