Parameter-free thermal lattice Boltzmann boundary treatment for rarefied gas microflow: maintaining accuracy beyond the first-order slip regime (Kn around 0.1 and above) at large Eckert and Prandtl numbers

Tian, Zou, Liu, Guo, Liu and Zheng (Physica A 385, 2007) introduced a thermal lattice Boltzmann scheme for gaseous microflow whose parameter-free wall treatment combines first-order Maxwell velocity-slip and temperature-jump conditions with non-equilibrium extrapolation of the wall distribution functions; it reproduces analytical slip-regime solutions at Knudsen numbers up to 0.08, but deviates near $Kn\approx 0.1$, with deviations growing at larger Eckert and Prandtl numbers, and it was formulated and tested only in two-dimensional planar channels. The open question is whether a parameter-free thermal lattice Boltzmann closure (for example higher-order slip/jump conditions or a refined non-equilibrium-extrapolation-type kinetic closure) remain quantitatively accurate, against direct simulation Monte Carlo or independently converged Boltzmann-model references, for thermally coupled microflow with viscous heat dissipation at Knudsen numbers around 0.1 and above, at Eckert and Prandtl numbers beyond the source validation envelope. Meng et al. (2013) document the viscous-heating-in-the-transition-regime accuracy limitation even for high-order Hermite lattices, and the boundary-treatment literature to date remains confined to the slip regime.

Background

Gaseous flow and heat transfer in microchannels govern the thermal design of micro-electro-mechanical systems (MEMS). When the molecular mean free path $\lambda$ becomes comparable to a characteristic device dimension $H$, the flow is rarefied; the Knudsen number $Kn=\lambda/H$ classifies regimes, with the slip regime roughly $10^{-3}\lesssim Kn\lesssim 0.1$ (continuum equations with velocity-slip and temperature-jump wall corrections still adequate) and the transition regime roughly $0.1\lesssim Kn\lesssim 10$ (continuum closures fail and kinetic descriptions are required). Directly solving the Boltzmann equation or running particle methods such as molecular dynamics and direct simulation Monte Carlo (DSMC, a stochastic particle solver of the Boltzmann equation) is often too costly for engineering-scale microdevices, which motivates mesoscopic solvers.

The lattice Boltzmann method (LBM) evolves discrete particle-velocity distribution functions on a lattice through local collisions and free streaming, and recovers the Navier-Stokes-Fourier equations in the hydrodynamic limit via Chapman-Enskog analysis (a systematic expansion of the kinetic equation in powers of $Kn$). Thermal (non-isothermal) LBM variants either evolve a second distribution for the internal-energy/temperature field (double-distribution models) or use single-distribution kinetic collision models with an adjustable Prandtl number, such as the ellipsoidal statistical Bhatnagar-Gross-Krook model (ES-BGK, a relaxation-time kinetic model that corrects the BGK model's fixed Prandtl number) or the Shakhov model. Here $Pr=\nu/\alpha$ is the ratio of kinematic viscosity to thermal diffusivity, and the Eckert number $Ec=U_0^2/(c_v\,\Delta T)$, with $U_0$ a characteristic flow speed, $c_v$ the specific heat at constant volume, and $\Delta T$ a wall-imposed temperature difference, measures the importance of viscous heat dissipation relative to the imposed thermal driving: large $Ec$ means viscous heating strongly distorts the temperature field, especially near walls.

At a gas-solid interface the kinetic boundary condition generates a velocity slip and a temperature jump. To first order in $Kn$ the Maxwell and Smoluchowski conditions read

$u_{\mathrm{slip}}=\frac{2-\sigma_v}{\sigma_v}\,\lambda\left.\frac{\partial u}{\partial n}\right|_{w},\qquad T_{\mathrm{jump}}=\frac{2-\sigma_T}{\sigma_T}\,\frac{2\gamma}{\gamma+1}\,\frac{\lambda}{Pr}\left.\frac{\partial T}{\partial n}\right|_{w}$

where $\sigma_v$ and $\sigma_T$ are the tangential-momentum and thermal-accommodation coefficients (unity for fully diffuse reflection), $\gamma$ is the ratio of specific heats, and derivatives are wall-normal. Implementing these inside the LBM is delicate because the unknowns are the discrete distributions themselves. Prior approaches mixed bounce-back (distribution reversed at the wall) with specular reflection through an adjustable fraction, or used diffuse-reflection schemes; Sofonea and Sekerka (2005) needed an ad hoc factor of 1.15 on the slip velocity to match data. Tian et al. (2007), building on their 2006 conference version, instead imposed the first-order slip and jump conditions directly on the wall-node macroscopic velocity and temperature (with second-order implicit finite-difference wall gradients) and reconstructed the wall distributions as the equilibrium distribution at those wall values plus a non-equilibrium part extrapolated from the adjacent fluid nodes (the non-equilibrium extrapolation idea of Guo and co-workers). This treatment involves no adjustable parameter for $\sigma_v=\sigma_T=1$ (diffuse reflection). In a two-dimensional heated micro-Poiseuille benchmark with viscous dissipation it matched the analytical slip-corrected solutions for $Kn=0.02$, $0.05$, $0.08$ at $Ec=1$, $2$, $5$ with $Pr=0.7$, and at $Pr=3.0$ with $Ec=3.0$; the deviations appear as $Kn$ approaches $0.1$ and grow at larger $Ec$ and $Pr$, motivating the open question posed here. Later boundary-scheme work frequently uses the multiple-relaxation-time (MRT) collision model, in which different kinetic moments relax at independently set rates for better numerical stability and Knudsen-number control.

Two facts from the later literature frame the problem. First, standard low-order lattices (e.g., D2Q9) cannot represent the higher-order velocity moments that enter second-order (Burnett-level) slip corrections or strongly non-equilibrium transition-regime states, so accuracy beyond first-order slip requires higher-order Gauss-Hermite or half-range discrete velocity sets at increased cost (Meng and Zhang 2011; Watari 2016). Second, beyond the slip regime the analytical Maxwell/Smoluchowski benchmark solutions themselves lose validity, so any improved scheme must be validated against DSMC or independently converged solutions of the underlying kinetic model. Parts of the 2007 improvement agenda have since been realized within the slip regime: three-dimensional thermal slip/jump lattice Boltzmann simulations exist (Watari 2009); a parameter-free curved-boundary thermal treatment has been developed for the slip regime (Liu, Mu and Wu 2019); second-order slip conditions have been built into isothermal lattice Boltzmann boundary schemes (Dai et al. 2022); and variable-Prandtl-number kinetic lattice models reach the early transition regime for canonical flows (Meng et al. 2013; Ilyin 2024). None of these demonstrates the parameter-free thermal closure accuracy at $Kn$ of order 0.1 and above combined with viscous heating at large $Ec$ and $Pr$.

Problem Statement

The question inherited from Tian et al. (2007) is whether a thermal lattice Boltzmann scheme whose wall treatment is parameter-free (the accommodation coefficients fixed at unity for diffuse reflection, with no adjustable bounce-back/specular fraction or slip-velocity correction factor) can be made to remain quantitatively accurate in the parameter range where their first-order treatment degrades. Concretely: construct a thermal lattice Boltzmann wall closure - for example by imposing velocity-slip and temperature-jump boundary conditions of higher order in $Kn$ than the first-order Maxwell/Smoluchowski conditions, or by a refined kinetic closure of the non-equilibrium-extrapolation type, without restricting admissible answers to these families - such that, for the inherited class of gaseous microchannel flows with viscous heat dissipation and heated walls (pressure-driven and shear-driven configurations are both admissible, and either one satisfies this flow-coverage requirement), the scheme reproduces reference solutions (analytical slip-regime profiles where they remain valid, and direct simulation Monte Carlo or independently converged solutions of the adopted kinetic model beyond, where the analytical benchmarks lose validity) with quantified error at Knudsen numbers around and above $Kn\approx 0.1$, where the first-order treatment degrades, and at Eckert and Prandtl numbers beyond the source paper's validated envelope ($Ec\le 5$, $Pr\le 3$). The mandatory demonstration window is: at three representative $(Kn, Ec, Pr)$ points beyond the source envelope - at least one with both $Ec>5$ and $Pr>3$ and at least one at $Kn\gtrsim 0.2$ - reproduce the reference wall-normal velocity and temperature profiles with a relative deviation (for example relative $L_2$ or maximum norm) smaller than that of the 2007 first-order treatment, and no larger than that of the 2007 scheme wherever that scheme was validated. An asymptotic (Chapman-Enskog-level) justification of the closure, an explicit delimitation of where accuracy degrades as a function of $Kn$, $Ec$, and $Pr$, and extension to three-dimensional or geometrically complex micro-device configurations (curved walls, cylinders in channels, non-planar devices) are natural further directions but are not required completion conditions; three-dimensional thermal slip/jump lattice Boltzmann capability already exists within the slip regime (Watari 2009). What is sought is an extension of the verified accuracy envelope of parameter-free thermal LBM in the Kn-Ec-Pr space for the inherited flow class, not a claim of universal validity.

References

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