Exploring the Applicability of the Lattice-Boltzmann Method for Two-Dimensional Turbulence Simulation

TL;DR

This work assesses the applicability of the Lattice-Boltzmann method (LBM) to simulate two-dimensional turbulence generated by flows around randomly placed disk obstacles. Using a D2Q9 BGK LBM with Smagorinsky-like eddy viscosity and standard boundary conditions, the authors analyze kinetic energy and enstrophy spectra to compare with Kraichnan's 2D turbulence theory, reporting an inverse energy cascade and spectral slopes near $\gamma_E \approx -3$ and $\gamma_Z \approx -1$ under certain conditions. The study demonstrates that LBM can reproduce key 2D turbulence features at low computational cost, provides open-source Python code for educational use, and suggests extensions for deeper exploration of spectral behavior and vortex dynamics in constrained geometries. Overall, the approach offers a practical, accessible platform for teaching and exploring 2D turbulent flows in academic settings.

Abstract

The Lattice-Boltzmann method is a mesoscopic approach for solving hydrodynamic problems involving both laminar and turbulent fluids. Although the suitability for the former cases is supported by a myriad of studies, turbulent flows always give rise to additional challenges that need to be addressed properly. In this paper, we estimate the accuracy of the simulation results obtained via a custom implementation of a Lattice-Boltzmann solver for a two-dimensional turbulent flow. To this end, a two-dimensional flow field filled with randomly located rigid disks was simulated, and the von Karman vortex street generated after the wake of such obstacles was studied. To ensure reproducibility, the implementation underlying these results is provided as supplementary material.

Figures (6)

• Figure 1: Schematic of the D2Q9 velocity set. A fictitious particle located at the central node can move in eight possible directions, either though the diagonal links or the main axis links, or stay stationary at its location. The square marked with gray lines has a length of $2\Delta x$. Rest velocity $0$ located at central node not shown. See Table \ref{['tab:vel_set']} for more details.
• Figure 2: Representation of the simulation setup. $N$ disks are located randomly within the gray area (three shown here as a visual example). No-slip (bounce-back) boundary conditions are imposed at the channel lateral walls and on the surface of the immersed disks, For the inlet and outlet boundaries we enforce velocity and pressure (respectively) boundary conditions using the Zou/He scheme. The inlet velocity profile is simplified using standard arrows.
• Figure 3: Instantaneous visualization of a 2D flow passing through an array of disks for different disks sizes and Reynolds number: $R=5$, $Re=200$ (top); $R=30$, $Re=200$ (middle); and $R=30$, $Re=800$ (bottom). Number of disks $N=16$ for all three examples.
• Figure 4: Spatio-temporal kinetic energy and enstrophy mean values as a function of the disk radius for several number of disks. Reynolds number $Re=200$.
• Figure 5: Power spectral density (PSD) for the kinetic energy and enstrophy (left panel), and scaling exponents $\gamma$ for $E_k$ and $Z_k$ (right panel). Blue solid lines and dots correspond to enstrophy and red dashed lines and squares to energy, respectively. Set of parameters: $Re=200$, $R=20$ and $N=16$.