[Paper Review] Entropic multi-relaxation lattice Boltzmann scheme for turbulent flows

Fabian Bösch, S. S. Chikatamarla+1 more|arXiv (Cornell University)|Jul 9, 2015

Lattice Boltzmann Simulation Studies4 citations

TL;DR

This paper presents a three-dimensional entropic multi-relaxation lattice Boltzmann scheme (KBC model) on the D3Q27 lattice that achieves high numerical stability and second-order convergence in turbulent flows, even at high Reynolds numbers and under-resolved conditions. The method uses entropy-based stabilization to dynamically adjust relaxation parameters, ensuring thermodynamic consistency and enabling accurate simulation of low-order statistics and Kolmogorov scaling without explicit turbulence models.

ABSTRACT

We present three dimensional realizations of the model introduced recently by (Karlin, Bösch, Chikatamarla, Phys. Rev. E 2014) and review the role of the entropic stabilizer. The presented models achieve outstanding numerical stability in presence of turbulent high Reynolds number flows. We report accurate results for low order moments for homogeneous isotropic decaying turbulence and second order grid convergence for most assessed statistical quantities. The explicit and efficient nature of the scheme renders it a very promising candidate for both engineering and scientific purposes in the vicinity of highly turbulent flows.

Motivation & Objective

• To develop a stable, parameter-free lattice Boltzmann scheme for simulating high-Reynolds-number turbulent flows.
• To address the numerical instability of conventional LBGK and RLB schemes in under-resolved turbulent regimes.
• To demonstrate that entropic stabilization enables accurate capture of low-order statistical quantities and energy spectrum scaling.
• To validate the scheme's performance in both symmetric (Kida flow) and random initial condition (decaying turbulence) scenarios.
• To establish the method as a viable alternative for engineering and scientific applications requiring high accuracy and stability in turbulent flows.

Proposed method

• The KBC model employs an entropic stabilizer that dynamically adjusts relaxation parameters to satisfy the second law of thermodynamics, ensuring numerical stability.
• The scheme uses a multi-relaxation approach with eight distinct variations, each tailored to stabilize different flow regimes.
• The D3Q27 lattice is used for three-dimensional simulations, providing sufficient velocity directions to capture hydrodynamic and non-hydrodynamic moments.
• The equilibrium distribution is derived as the entropy maximizer subject to conserved moments (density and momentum), with weights $W_i$ specific to the lattice.
• The method avoids explicit turbulence models and maintains a constant kinematic viscosity, ensuring consistency with Navier-Stokes dynamics.
• The algorithm is explicitly computed and fully explicit, enabling high computational efficiency suitable for large-scale simulations.

Experimental results

Research questions

• RQ1 Can the entropic multi-relaxation lattice Boltzmann scheme maintain numerical stability in under-resolved, high-Reynolds-number turbulent flows?
• RQ2 Does the KBC model accurately reproduce Kolmogorov scaling in the energy spectrum for decaying homogeneous isotropic turbulence?
• RQ3 How does the scheme's accuracy and convergence rate compare to LBGK and RLB in terms of second-order convergence for statistical quantities?
• RQ4 To what extent can the KBC model capture low-order statistics such as kinetic energy, enstrophy, and dissipation rate without explicit turbulence modeling?
• RQ5 Is the scheme robust under random, asymmetric initial conditions typical of real turbulent flows?

Key findings

• The KBC scheme exhibits second-order convergence in most statistical quantities, including energy spectra, enstrophy, and dissipation rate, across multiple grid resolutions.
• The method achieves numerical stability at Reynolds numbers up to $\mathrm{Re}_\lambda \approx 600$ and under-resolved conditions with Kolmogorov length scale $\eta \approx 10$ lattice units.
• The energy spectrum shows a well-developed inertial subrange with Kolmogorov scaling $E(k) \sim k^{-5/3}$, even in coarse-grid simulations.
• Velocity correlation functions decay rapidly beyond $r/N > 0.2$, indicating isotropic and homogeneous turbulent behavior.
• Low-order statistics such as kinetic energy, enstrophy, and dissipation rate are accurately captured despite under-resolution, matching results from resolved simulations.
• The scheme outperforms LBGK and RLB in stability, remaining unconditionally stable across all tested turbulent regimes without requiring tuning or explicit turbulence models.