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Towards nonlinear thermohydrodynamic simulations via the Onsager-Regularized Lattice Boltzmann Method

Anirudh Jonnalagadda, Amit Agrawal, Atul Sharma, Walter Rocchia, Sauro Succi

TL;DR

This work develops and analyzes the Onsager-Regularized LBM as a local, assumption-free framework to mitigate spurious isotropy errors on standard lattices. Through a Chapman-Enskog analysis, it demonstrates how NS-level thermohydrodynamics can be recovered for generic equilibria and, in particular, when paired with a guided or polynomial equilibrium, yields significantly higher accuracy than BGK while avoiding nonlocal corrections. The guided-equilibrium implementation on the D2Q9 lattice achieves correction-free isothermal NS dynamics with substantial accuracy gains, and numerical benchmarks show improved dissipation handling and elimination of spurious oscillations compared to existing regularization schemes. Overall, OReg provides a pathway to fully local nonlinear thermohydrodynamic LB simulations on standard lattices, with clear implications for scalable HPC applications and complex-flow modeling.

Abstract

This work presents a generalized, assumption-free, and stencil-independent theoretical analyses of the recently proposed Onsager-Regularized (OReg) lattice Boltzmann (LB) method [Jonnalagadda et al., Phys. Rev. E 104, 015313 (2021)] and demonstrates its ability to mitigate spurious errors associated with the insufficient isotropy of standard first-neighbor lattices without the inclusion of any external correction terms. The hydrodynamic limit recovered by the OReg scheme is derived for two equilibrium distribution functions, namely the so-called thermal guided equilibrium and the popular second order polynomial equilibrium, to show that the OReg scheme yields macroscopic dynamics that are O(u) times more accurate than that of the bare BGK collision model. Specifically, we show that, with the guided equilibrium on the D2Q9 standard lattice, the OReg scheme inherently compensates for the insufficient lattice isotropy of the standard D2Q9 lattice by automatically adjusting the lattice viscosity, yielding O(u4) and O(u2) accurate kinetic models when operated at the reference and arbitrary temperatures respectively. Further, we also show that the OReg scheme presents an O(u3) accurate kinetic model for the D2Q9 lattice when used with the second order polynomial equilibrium formulation. Thereafter, the accuracy of the OReg-guided-equilibrium kinetic model is numerically demonstrated for quasi-one-dimensional simulations of the rotated decaying shear wave and isothermal shocktube problems. The present work lays the theoretical foundation of a generic framework which can enable fully local, correction-free, nonlinear thermohydrodynamic LB simulations on standard lattices, thereby facilitating scalable simulations of physically challenging fluid flows.

Towards nonlinear thermohydrodynamic simulations via the Onsager-Regularized Lattice Boltzmann Method

TL;DR

This work develops and analyzes the Onsager-Regularized LBM as a local, assumption-free framework to mitigate spurious isotropy errors on standard lattices. Through a Chapman-Enskog analysis, it demonstrates how NS-level thermohydrodynamics can be recovered for generic equilibria and, in particular, when paired with a guided or polynomial equilibrium, yields significantly higher accuracy than BGK while avoiding nonlocal corrections. The guided-equilibrium implementation on the D2Q9 lattice achieves correction-free isothermal NS dynamics with substantial accuracy gains, and numerical benchmarks show improved dissipation handling and elimination of spurious oscillations compared to existing regularization schemes. Overall, OReg provides a pathway to fully local nonlinear thermohydrodynamic LB simulations on standard lattices, with clear implications for scalable HPC applications and complex-flow modeling.

Abstract

This work presents a generalized, assumption-free, and stencil-independent theoretical analyses of the recently proposed Onsager-Regularized (OReg) lattice Boltzmann (LB) method [Jonnalagadda et al., Phys. Rev. E 104, 015313 (2021)] and demonstrates its ability to mitigate spurious errors associated with the insufficient isotropy of standard first-neighbor lattices without the inclusion of any external correction terms. The hydrodynamic limit recovered by the OReg scheme is derived for two equilibrium distribution functions, namely the so-called thermal guided equilibrium and the popular second order polynomial equilibrium, to show that the OReg scheme yields macroscopic dynamics that are O(u) times more accurate than that of the bare BGK collision model. Specifically, we show that, with the guided equilibrium on the D2Q9 standard lattice, the OReg scheme inherently compensates for the insufficient lattice isotropy of the standard D2Q9 lattice by automatically adjusting the lattice viscosity, yielding O(u4) and O(u2) accurate kinetic models when operated at the reference and arbitrary temperatures respectively. Further, we also show that the OReg scheme presents an O(u3) accurate kinetic model for the D2Q9 lattice when used with the second order polynomial equilibrium formulation. Thereafter, the accuracy of the OReg-guided-equilibrium kinetic model is numerically demonstrated for quasi-one-dimensional simulations of the rotated decaying shear wave and isothermal shocktube problems. The present work lays the theoretical foundation of a generic framework which can enable fully local, correction-free, nonlinear thermohydrodynamic LB simulations on standard lattices, thereby facilitating scalable simulations of physically challenging fluid flows.

Paper Structure

This paper contains 9 sections, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Onsager-Regularized LBM
  3. Hydrodynamic limit using the Chapman-Enskog multiscale expansion
  4. General outcomes
  5. Onsager-Regularized macrodynamics for a generic equilibrium distribution function
  6. Onsager-Regularized macrodynamics using the Guided Equilibrium
  7. Onsager-Regularized macrodynamics using the second order polynomial equilibrium
  8. Numerical Benchmarks
  9. Conclusion

Figures (4)

  • Figure 1: Velocity space representations in two- and three-dimensions using the standard D2Q9 and D3Q27 lattices respectively.
  • Figure 2: Comparison of the numerically computed and physically imposed fluid viscosities, $\widetilde{\nu}$ and $\nu$, for a decaying shear wave at different Mach numbers obtained with the OReg, Lattice-BGK and Projected Regularized (PR) LB schemes using the guided equilibrium on the D2Q9 lattice. For the axis-aligned shear wave (panel (a)) all schemes correctly model the viscous dissipation as demonstrated by $\widetilde{\nu}/\nu$ = 1 while for the $\pi/4$ rotated wave (panel (b)) only the OReg scheme recovers the imposed viscous dissipation rate.
  • Figure 3: Isothermal shocktube results obtained using the guided equilibrium on the D2Q9 lattice using different LB schemes at $\theta = 0.35$ and $\nu = 10^{-5}$. Panel (A) shows the OReg (black) and the Lattice-BGK (blue) schemes, while panel (B) shows the first order Essentially Entropic LB (brown) and the projected regularized (green) schemes. The round black symbols correspond to the analytical solution.
  • Figure 4: Isothermal shocktube results obtained using the guided equilibrium on the D2Q9 lattice using different LB schemes at $\theta = 2/5$ and $\nu = 10^{-9}$. The curves have the same meaning as in Fig. \ref{['fig:isothermal-shocktube-case1']}.