Generalized Onsager-Regularized Lattice Boltzmann Method for error-free Navier-Stokes models on standard lattices

Abstract

This work presents a novel strategy to address Navier-Stokes modelling errors arising on first-nearest neighbour lattice Boltzmann (LB) methods and introduces fully local corrections through Onsager-Regularized (OReg) non-equilibrium populations. The proposed mechanism, which admits partially and completely corrected OReg models, is used to develop representative partially and completely corrected models for the six-moment-constrained guided equilibrium (GEq) representation on the D2Q9 lattice. The former realization only addresses compatibility condition violations and improves the accuracy by two/four orders of magnitude at reference/arbitrary lattice temperatures respectively, while the latter additionally corrects stress tensor modelling errors, resulting in a fully corrected exact model. Numerical benchmarks of the corrected schemes demonstrate improved accuracy and stability in comparison to the Lattice-BGK and uncorrected OReg-GEq schemes thus presenting a promising avenue for OReg based thermohydrodynamic extensions.

Figures (6)

• Figure 1: 2D and 3D Standard lattices(color online).
• Figure 2: Deviation of the numerical viscosity ($\tilde{\nu}$) from the imposed viscosity ($\nu=0.001$) in a decaying rotated shear wave. Results are presented for the Lattice BGK and the uncorrected, partially-corrected and, fully-corrected OReg schemes at different Mach numbers (Ma) and isothermal temperatures. Cases with catastrophic instabilities are represented with $\tilde{\nu} = 0$ and the dotted line represents $\tilde{\nu}=\nu$.
• Figure 3: Isothermal shocktube results for different LB schemes at a mildly elevated isothermal lattice temperatures ($\theta$ = 0.4) and an extremely small lattice viscosity of $\nu = 1 \times 10^{-12}$.
• Figure 4: Isothermal shocktube results for different LB schemes at an elevated isothermal lattice temperatures ($\theta$ = 0.55) and lattice viscosity of $\nu = 1 \times 10^{-9}$. The represented curves have the same meaning as that of \ref{['fig:ist-c1']}. Note that the LBGK yield unphysical results for this case and is therefore not represented.
• Figure 5: Vorticity contours for the double periodic shear layer problem after unit characteristic time at an isothermal lattice temperature of $\theta=0.4$. Simulations are conducted for the LBGK, uncorrected, partially-corrected and completely-corrected OReg schemes (top to bottom) on $N^2$ sized grids with $N$ = 128, 256 and 512 (left to right).