Generalizing the linearized double-distribution thermal lattice Boltzmann method: coupled shear–thermal noncontinuum flows, stable full-form equilibria, and complex geometries

The lattice Boltzmann (LB) method of Shi, Yap, and Sader [Phys. Rev. E 92, 013307 (2015)] solves the linearized Bhatnagar–Gross–Krook (BGK) Boltzmann equation for noncontinuum gas flow and heat transfer at micro- and nanoscales using two distribution functions (momentum and energy, giving an adjustable Prandtl number) discretized with half-space Gauss–Hermite quadrature and a consistently discretized linearized Maxwell diffusive wall condition. Three scope limitations of that proof-of-concept study are established by its abstract and demonstrations: validation was performed only for thermal Couette flow, where shear and thermal effects decouple; simplified continuum-limit models (SLB-I, SLB-II) were derived by truncating the high-order velocity-polynomial terms of the full-form discrete temperature equilibrium, and no implementation of the full-form equilibrium with characterized numerical stability has been demonstrated; and the quadrature and boundary discretization were demonstrated only for flat parallel walls. No shear–thermally coupled noncontinuum flow (e.g., thermal transpiration) has been simulated with this model class, no stability-characterized full-form implementation exists, and no curved- or three-dimensional-geometry demonstration of its boundary scheme has been given. Moreover, the underlying half-space Gauss–Hermite velocity basis itself loses accuracy at large Knudsen numbers.

Background

When a gas is confined to a device whose characteristic length $L$ is comparable to the molecular mean free path $\lambda$, the ratio $\mathrm{Kn}=\lambda/L$ (the Knudsen number) is not small and the continuum Navier–Stokes–Fourier equations fail; the gas must then be described by the Boltzmann equation, a kinetic equation for the molecular velocity distribution function. In the Bhatnagar–Gross–Krook (BGK) model the complicated collision integral is replaced by relaxation toward local equilibrium, and for low-speed microscale flows the equation can be linearized about a global Maxwellian, which is the setting of the source paper. The lattice Boltzmann (LB) method is a numerical solver of kinetic equations in which velocity space is discretized (classically by Gauss–Hermite quadrature, whose 'algebraic precision' is the highest polynomial degree integrated exactly) and the discrete populations are advanced on a lattice or by finite differences. To represent heat transfer with an arbitrary Prandtl number $\mathrm{Pr}$ (the ratio of momentum to thermal diffusion rates), a double-distribution-function (DDF) formulation is used: separate perturbation functions $h_f$ and $h_T$ carry momentum and energy, each relaxing with its own time ($\tau_f$, $\tau_T$), so that $\mathrm{Pr}=\tau_f/\tau_T$; a Chapman–Enskog expansion (a systematic expansion in $\mathrm{Kn}$) recovers the macroscopic viscous and heat-conducting equations in the continuum limit. Near solid walls with Maxwell's diffusive (fully accommodating) boundary condition, molecules are re-emitted with a wall-temperature Maxwellian, and the wall condition involves half-space velocity integrals; the Knudsen layer, a thin non-equilibrium region of thickness $\sim\lambda$ adjacent to the wall, is then misrepresented by conventional full-space Gauss–Hermite (FGH) quadrature, producing oscillatory, slowly convergent wall quantities. This motivated discretizing the velocity space with half-space Gauss–Hermite (HGH) quadrature — separate quadratures for velocities pointing toward and away from the wall — as developed independently by Ambruş and Sofonea and by Shi, Yap, and Sader. Shi, Yap, and Sader [Phys. Rev. E 92, 013307 (2015)] combined the linearized DDF kinetic model with HGH quadrature into a hierarchy of finite-difference LB algorithms on two-dimensional lattices with 16, 36, and 64 discrete velocities (D2Q16, D2Q36, D2Q64, of algebraic precision 3, 5, and 7 respectively) with a linearized, half-space-consistent discrete diffusive wall condition, and derived continuum-limit simplified models (SLB-I, SLB-II) obtained by truncating high-order velocity-polynomial terms of the discrete temperature equilibrium $h_{T,i}^{eq}$. Benchmarked against accurate direct solutions of the linearized BGK equation for thermal Couette flow, the full-form HGH models clearly outperformed FGH models at moderate-to-large Kn, while the truncated models degraded systematically with increasing Kn. Thermal transpiration (also called thermal creep) — a flow driven by a temperature gradient along a wall, the working principle of Knudsen pumps — is the canonical problem in which shear and thermal fields couple nontrivially in the noncontinuum regime; it is precisely the class of flow the 2015 validation, being restricted to decoupled thermal Couette flow between flat plates, did not address.

Problem Statement

Extend the linearized double-distribution-function thermal lattice Boltzmann framework of Shi, Yap, and Sader (Phys. Rev. E 92, 013307 (2015)) from its proof-of-concept status — validated only for thermal Couette flow between flat parallel plates, with only continuum-limit truncated simplifications (SLB-I/SLB-II) of the discrete temperature equilibrium derived as reduced models — into a validated general-purpose method for noncontinuum micro- and nanoscale gas flow and heat transfer. The single scientific objective is a demonstrated, benchmarked capability of this model class for coupled noncontinuum thermal flows in realistic geometries, with three jointly required components: (1) coupled-flow validation — simulate shear–thermally coupled noncontinuum phenomena, such as thermal transpiration (thermal-creep-driven flow between plates held at differing temperatures along the wall direction) and related flows where the temperature field nontrivially drives or couples with the motion, across the slip and transition regimes, and validate velocity, temperature, and heat-flux profiles against reference solutions of the linearized Boltzmann equation (or equivalent high-accuracy kinetic computations); (2) full-form stability — develop time/space discretizations or stabilization strategies under which the full-form discrete temperature equilibrium $h_{T,i}^{eq}$, with its high-order velocity-polynomial terms intact (not the SLB-I/SLB-II truncations), can be used in stable computations, with the stability properties characterized (analytically or numerically) over the relevant ranges of Knudsen number, Courant number, and quadrature order; and (3) complex-geometry boundary treatment — extend the half-space Gauss–Hermite quadrature and its consistently discretized linearized Maxwell diffusive wall condition to non-planar walls, e.g., curved boundaries or three-dimensional body-fitted coordinates, and demonstrate the resulting scheme against appropriate kinetic benchmarks. Since later work by the same group established that the HGH velocity basis itself loses accuracy at large Knudsen number, a solution may generalize the velocity discretization (e.g., to other Gaussian quadratures) provided the linearized DDF kinetic structure, the adjustable-Prandtl-number capability, and the half-space consistency of the wall treatment are preserved. Completion is expert-reviewable: a solver of this class exists, all three components are demonstrated against kinetic references, and the stabilization is characterized.

References

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