This is a repository copy of Competition of natural convection and thermal creep in a square enclosure.
Bhatnagar-Gross-Krook (BGK) models are widely used to study rarefied gas dynamics. However, as simplified versions of the Boltzmann collision model, their performances are uncertain and need to be carefully investigated in highly nonequilibrium flows. In this study, several common BGK models, such as the Ellipsoidal Statistical BGK (ES-BGK) and Shakhov BGK (S-BGK) models, are theoretically analyzed using their moment equations. Then, numerical comparisons are performed between the Boltzmann collision model and BGK models based on various benchmarks, such as Fourier flow, Couette flow, and shock wave. The prediction performance of the ES-BGK model is better than that of the S-BGK model in Fourier flow, while prediction performance of the S-BGK model is better than that of the ES-BGK model in Couette flow and shock wave. However, with increasing Knudsen number or Mach number, the results of both ES-BGK and S-BGK deviate from the Boltzmann solutions. These phenomena are attributed to the incorrect governing equations of high-order moments of BGK models. To improve the performance of the current BGK models, the S-BGK model is extended by adding more high-order moments into the target distribution function of the original one. Our analytical and numerical results demonstrate that the extended S-BGK (S-BGK+) model provides the same relaxation coefficients as the Boltzmann collision model for the production terms of high-order moment equations. Compared to the other BGK models, the proposed S-BGK+ model exhibits better performance for various flow regimes.
It is extremely expensive to study turbulence using conventional molecular simulation methods such as direct simulation Monte Carlo and molecular dynamics methods, as the molecular scales and the turbulent characteristic scales are significantly separated. To bridge this gap, we employ a particle Fokker-Planck method, namely, the Langevin dynamics simulation method, to study two-dimensional Kolmogorov flow, which is induced by a spatially periodic external force in an unbounded domain. Our simulation results predict that when the Reynolds number (Re) exceeds the critical value, a sequence of bifurcations takes place in the flow as the Reynolds number increases, forming a variety of flow patterns. Correspondingly, the effective diffusion coefficient is enhanced due to convection. Two main regimes of the flow have been observed: the small-scale cellular structure regime (Rec < Re < 8Rec), and the large-scale coherent structure regime (Re > 8Rec). We demonstrate that Langevin dynamics can capture the double kinetic-energy cascade when the large-scale structure is formed in two-dimensional turbulence: the inverse energy cascade has a scaling law of k−4 due to energy condensation in the large-scale structures, while the direct energy cascade has an exponential decay corresponding to the dissipation mechanism. This work provides strong evidence that Langevin dynamics is a promising multiscale tool to study turbulence from molecular motions to large-scale coherent structures.
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