A flux bounce-back scheme for the filtered Spectral Element Lattice Boltzmann Method

TL;DR

This work develops a spectral element lattice Boltzmann method (SELBM) with a flux bounce-back boundary condition to enable accurate single-phase flows on unstructured meshes. By replacing perfect-shift streaming with an Eulerian advection formulation and employing a continuous Galerkin spatial discretization alongside SSPRK3 time integration, the method achieves high-order accuracy with stability enhanced by an explicit non-dissipative filter applied to distribution functions. The approach is validated through a sequence of benchmarks—double shear layer, unsteady Couette flow, planar Poiseuille flow, Taylor-Green vortex, and fully turbulent pipe flow—showing strong agreement with Nek5000 and analytical solutions, exponential convergence with polynomial order, and robust handling of complex boundaries. The results demonstrate the method’s potential for accurate, high-fidelity simulations on unstructured meshes, including isotropic turbulence and wall-bounded flows, with practical implications for engineering and scientific computing where mesh flexibility is essential.

Abstract

We develop a spectral element lattice Boltzmann method (SELBM) with the flux bounce-back (FBB) scheme, to enable accurate simulations of single-phase fluid dynamics in unstructured mesh. We adopt an Eulerian description of the streaming process in place of the perfect shift in the regular LBM. The spectral element method is used to spatially discretize the convective term, while the strong stability-preserving Runge-Kutta (SSPRK) method is used for time integration. To increase stability, we investigate the use of an explicit filter, particularly in the context of the sensitive double shear layer problem. The results indicate that by using the high-order polynomial, we can effectively eliminate the small vortices around the neck region. We introduce the flux bounce-back scheme to enable the current scheme to handle complex boundaries. The proposed scheme and flux boundary method are validated through benchmark simulations, including the unsteady Couette flow and the planar Poiseuille flow. Further validation is provided through the Taylor-Green vortex problem, demonstrating the accuracy and convergence of the scheme for isotropic turbulence. Finally, we consider a fully developed turbulent flow within a cylindrical pipe and correctly predict the turbulent boundary layer profile.

Figures (9)

• Figure 1: Comparison of the double shear layer simulation conducted by NekLBM, Nek5000, and regular LBM. (a) Vortex contours $\Delta \omega=5$ for $N_e=32$, $N=4$ by NekLBM(top), Nek5000 (middle), and regular LBM with $128^2$ number of nodes. (b) Vortex contours $\Delta \omega=5$ for $N_e=16$, $N=8$ by NekLBM(top), Nek5000 (middle), and regular LBM with $256^2$ number of nodes.
• Figure 2: (a) Comparison between simulation results (triangular markers) and analytical solutions (dashed line) of the unsteady Couette flow during $T/t_0=[0.5-64]$ for $Re=10$, $Ma=0.1$. (b) Spatial convergence for the unsteady Couette flow with different $\delta t$ for $Re=2000$, $Ma=0.05$.
• Figure 3: Comparison between simulation results (black dashed line) and analytical solutions (gray) of (a)velocity and (b) stress for the 2D planar Poiseuille flow at $T/t_0=64$ with $Ma=0.1/\sqrt{3}$, $Re=100$.
• Figure 4: Convergence test of the dimenionless slip velocity $u_{slip}^*=u_x(y=0)/u_{max}$ with relaxation time $\tau=[0.1,0.4]$, and kinematic viscosity $\nu=[0.0025,0.01]$.
• Figure 5: Evolution of (a) Kinetic energy and (b) energy dissipation during $T/t_0=[0,20]$ of the Taylor Green vortex simulation for the same element number $N_e=16$ on each direction, and different polynomial order $N=8$, $N=12$.