Restoring the conservativity of characteristic-based segregated models: Application to the hybrid lattice Boltzmann method

Abstract

A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum, and energy equations are solved by different methods. It is especially designed here for developing new numerical discretizations of the total energy equation and adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is equivalently written under a finite-volume formulation involving fluxes, which are further included in the total energy equation as numerical corrections. The viscous heat production is implicitly considered thanks to the knowledge of the LBM momentum flux. Three models are subsequently derived: a first-order upwind, a Lax–Wendroff, and a third-order Godunov-type schemes. They are assessed on standard academic test cases: a Couette flow, entropy spot and vortex convections, a Sod shock tube, several two-dimensional Riemann problems, and a shock–vortex interaction. Three key features are then exhibited: (1) the models are conservative by construction, recovering correct jump relations across shock waves; (2) the stability and accuracy of entropy modes can be explicitly controlled; and (3) the low dissipation of the LBM for isentropic phenomena is preserved.

A hybrid lattice Boltzmann method for gaseous detonations

“…The aim of this section is to detail the construction of stable conservative schemes for the hybrid LBM in presence of multi-species reactive flows. It is an extension of a previous work dedicated to the non-reactive case [50].…”

“…However, the present work aims to initiate from a generalized form of the hybrid system based on the entropy equation, as it offers a distinct advantage over other equations: the evolution equations for entropy and mass fraction assume a characteristic form, manifesting as straightforward advection equations for quantities s and Y k at velocity u. As stated in previous work [50,53,54], this characteristic nature yields two key benefits: 1) the characteristic equations remain linearly decoupled with each other, affording enhanced stability and accuracy control for the entire system, and 2) a substantial literature exists on numerical discretizations of advection equations [57]. This is why coupling the LBM with a non-conservative entropy equation has been the favored approach for compressible non-reactive flows [42,[45][46][47]55].…”

“…Its primary strengths stem from a simple numerical scheme, achieved by splitting the Boltzmann equation with discrete velocities into two parts: a local aforementioned benefits of dealing with a characteristic equation such as the entropy one may be lost when working with a total energy formulation. Recently, a solution to this predicament was proposed by Wissocq et al [50] for non-reactive flows. The idea is to construct FV total energy schemes maintaining all the linear properties of a nonconservative entropy-based hybrid system.…”

Lattice-Boltzmann modeling of the quiet and unstable PRECCINSTA burner modes

“…Given (i) the excellent dissipation properties of LBM for acoustic propagation [26], including for hybrid methods [12,13] for which vortical and acoustic mode propagations are convected by the LBM scheme [27][28][29] while species/entropy modes are convected with a specifically designed scheme [30,31] ; and (ii) the success encountered in simulating burners with complex geometries [23,25] for a reasonable cost, the next logical step is to investigate and develop LBM able to model thermo-acoustic instabilities.…”

Exponential distribution functions for positivity-preserving lattice Boltzmann schemes: Application to 2D compressible flow simulations

“…[38][39][40][41][42] Nonetheless, more recent approaches are also able to correctly enforce energy conservation across shockwaves while remaining stable. 43,44 Third, one can stick to pure LB approaches based on the collideand-stream algorithm and work on the collision model to increase stability instead. Several works were proposed on that topic with various degrees of success.…”