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The simulation of rarefied gas flow based on the Boltzmann equation is challenging, especially when the gas mixtures have disparate molecular masses. In this paper, a computationally tractable kinetic model is proposed for monatomic gas mixtures, to mimic the Boltzmann collision operator as closely as possible. The intra- and inter-collisions are modelled separately using relaxation approximations, to correctly recover the relaxation time scales that could span several orders of magnitude. The proposed kinetic model preserves the accuracy of the Boltzmann equation in the continuum regime by recovering four critical transport properties of a gas mixture: the shear viscosity, the thermal conductivity, the coefficients of diffusion and the thermal diffusion. While in the rarefied flow regimes, the kinetic model is found to be accurate when comparing its solutions with those from the direct simulation Monte Carlo method in several representative cases (e.g. one-dimensional normal shock wave, Fourier flow and Couette flow, two-dimensional supersonic flow passing a cylinder and nozzle flow into a vacuum), for binary mixtures with a wide range of mass ratios, species concentrations and different intermolecular potentials. Pronounced separations in species properties have been observed, and the flow characteristics of gas mixtures in shock waves are found to change as the molecular mass ratio increases from 10 to 1000.

Simulating complex gas flows from turbulent to rarefied regimes is a long-standing challenge, since turbulence and rarefied flow represent contrasting extremes of computational aerodynamics. We propose a multiscale method to bridge this gap. Our method builds upon the general synthetic iterative scheme for the mesoscopic Boltzmann equation, and integrates the $k$–$\\omega$ model in the macroscopic synthetic equation to address turbulent effects. Asymptotic analysis and numerical simulations show that the macroscopic–mesoscopic coupling adaptively selects the turbulence model and the laminar Boltzmann equation. The multiscale method is then applied to opposing jet problems in hypersonic flight surrounding by rarefied gas flows, showing that the turbulence could cause significant effects on the surface heat flux, which cannot be captured by the turbulent model nor the laminar Boltzmann solution alone. This study provides a viable framework for advancing understanding of the interaction between turbulent and rarefied gas flows.

Based on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearized Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager–Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearized Boltzmann equation. Then, a number of two-dimensional microflows in the transition and free-molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearized Boltzmann equation. For thermally driven flows in the free-molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearized Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearized Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

The lattice Boltzmann method (LBM), which was originally designed for near-incompressible Navier–Stokes flows, has been extended to rarefied gas flows with high-order quadrature in recent years. Although the ability of the high-order LBM to capture rarefaction effects has been demonstrated by many authors, its accuracy and efficiency are often undermined by numerical dissipation introduced by the off-lattice abscissas in Gauss–Hermite quadrature. Here, using the spontaneous Rayleigh–Brillouin scattering problem as the benchmark, we assess the accuracy and efficiency of the high-order LBM with on-lattice quadrature rules up to 39th order. The numerical error comprises two parts, one due to the rarefaction effect and the other due to temporal-spatial discretization, and we find that the former depends not only on the number of discrete velocities, but also on their distribution in velocity space. With a quadrature of 29th order, the error between the LBM and the discrete velocity method is found to be below 1 % up to $Kn=2.0$. Compared with a finite-volume Bhatnagar–Gross–Krook solver using Gauss–Hermite quadrature, the on-lattice LBM has a numerical dissipation several orders of magnitude lower, and achieves the same accuracy with fewer discrete velocities.

The apparent gas permeability of a porous medium is an important parameter in the prediction of unconventional gas production, which was first investigated systematically by Klinkenberg in 1941 and found to increase with the reciprocal mean gas pressure (or equivalently, the Knudsen number). Although the underlying rarefaction effects are well known, the reason that the correction factor in Klinkenberg’s famous equation decreases when the Knudsen number increases has not been fully understood. Most of the studies idealize the porous medium as a bundle of straight cylindrical tubes; however, according to the gas kinetic theory, this only results in an increase of the correction factor with the Knudsen number, which clearly contradicts Klinkenberg’s experimental observations. Here, by solving the Bhatnagar–Gross–Krook equation in simplified (but not simple) porous media, we identify, for the first time, two key factors that can explain Klinkenberg’s experimental results: the tortuous flow path and the non-unitary tangential momentum accommodation coefficient for the gas–surface interaction. Moreover, we find that Klinkenberg’s results can only be observed when the ratio between the apparent and intrinsic permeabilities is ${\\lesssim}30$ ; at large ratios (or Knudsen numbers) the correction factor increases with the Knudsen number. Our numerical results could also serve as benchmarking cases to assess the accuracy of macroscopic models and/or numerical schemes for the modelling/simulation of rarefied gas flows in complex geometries over a wide range of gas rarefaction. Specifically, we point out that the Navier–Stokes equations with the first-order velocity-slip boundary condition are often misused to predict the apparent gas permeability of the porous medium; that is, any nonlinear dependence of the apparent gas permeability with the Knudsen number, predicted from the Navier–Stokes equations, is not reliable. Worse still, for some types of gas–surface interactions, even the ‘filtered’ linear dependence of the apparent gas permeability with the Knudsen number is of no practical use since, compared to the numerical solution of the Bhatnagar–Gross–Krook equation, it is only accurate when the ratio between the apparent and intrinsic permeabilities is ${\\lesssim}1.5$ .

The thermal conductivity of a molecular gas consists of the translational and internal parts. Although in continuum flows the total thermal conductivity itself is adequate to describe the heat transfer, in rarefied gas flows they need to be modelled separately, according to the relaxation rates of translational and internal heat fluxes in an homogeneous system. This paper is dedicated to quantifying how these relaxation rates affect rarefied gas dynamics. The kinetic model of Wu et al. (J. Fluid Mech., vol. 763, 2015, pp. 24–50) is adapted to recover the relaxation of heat fluxes, which is validated by the direct simulation Monte Carlo method. Then the model of Wu et al., which has the freedom to adjust the relaxation rates, is used to investigate the rate effects of thermal relaxation in problems such as the normal shock wave, creep flow driven by Maxwell's demon and thermal transpiration. It is found that the relaxation rates of heat flux affect rarefied gas flows significantly, even when the total thermal conductivity is fixed.

Based on an accurate numerical solution of the kinetic equation using well-resolved spatial and velocity grids, the separation of rarefied gas flow in a microchannel with double rectangular bends is investigated over a wide range of Knudsen and Reynolds numbers. Rarefaction effects are found to play different roles in flow separation (vortex formation) at the concave and convex corners. Flow separations near the concave and convex corners are only observed for a Knudsen number up to $0.04$ and $0.01$, respectively. With further increase of the Knudsen number, flow separation disappears. Due to the velocity slip at the solid walls, the concave (convex) vortex is suppressed (enhanced), which leads to the late (early) onset of separation of rarefied gas flows with respect to the Reynolds number. The critical Reynolds numbers for the emergence of concave and convex vortices are found to be as low as $0.32\\times 10^{-3}$ and $30.8$, respectively. The slip velocity near the concave (convex) corner is found to increase (decrease) when the Knudsen number increases. An adverse pressure gradient appears near the concave corner for all the examined Knudsen numbers, while for the convex corner it only occurs when the Knudsen number is less than $0.1$. Due to the secondary flow and adverse pressure gradient near the rectangular bends, the mass flow rate ratio between the bent and straight channels of the same length is a non-monotonic function of the Knudsen number. Our results clarify the diversified and often contradictory observations reported in the literature about flow rate enhancement and vortex formation in bent microchannels.

Two kinetic models are proposed for high-temperature rarefied (or non-equilibrium) gas flows with internal degrees of freedom and radiation. One of the models uses the Boltzmann collision operator to model the translational motion of gas molecules, which has the ability to capture the influence of intermolecular potentials, while the other adopts the relaxation time approximations, which has higher computational efficiency. In our kinetic model equations, not only the transport coefficients such as the shear/bulk viscosity and thermal conductivity but also their underlying relaxation processes are recovered. The non-equilibrium dynamics of gas flow and radiation are tightly coupled, where the transport properties of gas molecules and photons are correlatively dependent. The proposed kinetic models are validated by the direct simulation Monte Carlo method in several non-radiative rarefied gas flows (e.g. the normal shock wave, Fourier flow, Couette flow and the creep flow driven by the Maxwell demon), and the experimental data of planar heat transfer and normal shock waves in nitrogen. Then, the rarefied gas flows with strong radiation are studied based on the kinetic models, not only in the above one-dimensional gas flows, but also in the two-dimensional radiative hypersonic flow passing a cylinder. The characteristics of heat transfer in the tightly coupled fields of gas and radiation are systematically investigated, particularly the influence of the non-equilibrium photon transport and their interactions with gas molecules are revealed. It is found that the radiation makes a profound contribution to the total heat transfer in radiative hypersonic flow at an intermediate photon Knudsen number.

With the aid of direct simulation Monte Carlo (DSMC), we conduct a detailed investigation pertaining to the fluid and thermal characteristics of rarefied gas flow with regard to various arrangements for radiometric pumps featuring vane and ratchet structures. For the same, we consider three categories of radiometric pumps consisting of channels with their bottom or top surfaces periodically patterned with different structures. The structures in the design of the first category are assumed to be on the bottom wall and consist of either a simple vane, a right-angled triangular fin or an isosceles triangular fin. The arrangements on the second category of radiometric pumps consist of an alternating diffuse–specular right-angled fin and an alternating diffuse–specular isosceles fin on the bottom wall. The third category contains either a channel with double isosceles triangular fins on its lowermost surface or a zigzag channel with double isosceles triangular fins on both walls. In the first and the third categories, the surfaces of the channel and its structures are considered as diffuse reflectors. The temperature is kept steady on the horizontal walls of the channel; thus, radiometric flow is created by subjecting the adjacent sides of the vane/ratchet to constant but unequal temperatures. On the other hand, for the second category, radiometric flow is introduced by specifying different top/bottom channel wall temperatures. The DSMC simulations are performed at a Knudsen number based on the vane/ratchet height of approximately one. The dominant mechanism in the radiometric force production is clarified for the examined configurations. Our results demonstrate that, at the investigated Knudsen number, the zigzag channel experiences maximum induced velocity with a parabolic velocity profile, whereas its net radiometric force vanishes. In the case of all other configurations, the flow pattern resembles a Couette flow in the open section of the channel situated above the vane/ratchet. To further enhance the simulations, the predictions of the finite volume discretization of the Boltzmann Bhatnagar–Gross–Krook (BGK)–Shakhov equation for the mass flux dependence on temperature difference and Knudsen number are also reported for typical test cases.

The top gas temperature is an apparent indicator of the distribution state of gas flow, and it is an important parameter to judge the furnace condition. It is related to the top equipment in addition to the charge characteristics and bottom-up generation and distribution. Therefore, the fluid-solid heat transfer model from burden surface -chute-rising ducts is established in this paper. The finite element method is used to calculate the flow and heat transfer of the gas at the top of the blast furnace, and the influence of the chute on the top gas temperature is analyzed. The accuracy of the model is verified by the actual blast furnace top temperature. The results show that the chute has a guiding effect on the gas flow. As the chute rotates, the top gas temperature fluctuates. This research is helpful for actual blast furnace operators to understand the top temperature and make the condition judgment more accurate.