Towards nonlinear thermohydrodynamic simulations via the Onsager-regularized lattice-Boltzmann method

This work presents a generalized, assumption-free, and stencil-independent theoretical analyses of the recently proposed Onsager-regularized (OReg) lattice Boltzmann (LB) method [Jonnalagadda , ] 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi mathvariant="script">O</a:mi><a:mo>(</a:mo><a:mi>u</a:mi><a:mo>)</a:mo></a:mrow></a:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi mathvariant="script">O</c:mi><c:mo>(</c:mo><c:msup><c:mi>u</c:mi><c:mn>4</c:mn></c:msup><c:mo>)</c:mo></c:mrow></c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi mathvariant="script">O</e:mi><e:mo>(</e:mo><e:msup><e:mi>u</e:mi><e:mn>2</e:mn></e:msup><e:mo>)</e:mo></e:mrow></e:math> accurate kinetic models when operated at the reference and arbitrary temperatures, respectively. Further, we also show that the OReg scheme presents an <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi mathvariant="script">O</g:mi><g:mo>(</g:mo><g:msup><g:mi>u</g:mi><g:mn>3</g:mn></g:msup><g:mo>)</g:mo></g:mrow></g:math> 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.