Ellipsoidal Statistical Bhatnagar-Gross-Krook Model Research Articles

Extension of the Shakhov Bhatnagar–Gross–Krook model for nonequilibrium gas flows

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 with the other BGK models, the proposed S-BGK+ model exhibits better performance for various flow regimes.

Particle-based fluid dynamics: Comparison of different Bhatnagar-Gross-Krook models and the direct simulation Monte Carlo method for hypersonic flows

The Bhatnagar-Gross-Krook (BGK) model and its extensions (ellipsoidal statistical BGK, Shakhov BGK, and unified BGK) are used in particle-based fluid dynamics and compared with the Direct Simulation Monte Carlo (DSMC) method. To this end, different methods are investigated that allow efficient sampling of the Shakhov and the unified target distribution functions. As a consequence, particle simulations based on the Shakhov BGK and unified BGK models are possible in an efficient way. Furthermore, different energy conservation schemes are tested for the BGK models. It is shown that the choice of the energy conservation scheme has a major impact on the quality of the results for each model. The models are verified with a Couette flow problem at different Knudsen numbers and wall velocities. Furthermore, the models are compared with the DSMC results of a hypersonic flow around a 70° blunted cone. It is shown that the unified BGK model is able to reproduce rarefied gas phenomena. Furthermore, it is shown that the difference in the reproduction of the shock structure is not significant between the ellipsoidal statistical BGK and Shakhov BGK models for the flow around a 70° blunted cone and significantly depends on the energy conservation method. The choice of the energy conservation method is especially crucial for the Shakhov model whereas the ellipsoidal statistical BGK model is much more robust concerning the energy conservation scheme. Additionally, a computational time study is performed to show the efficiency of BGK-based simulations for low Knudsen number flows compared with DSMC.

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

We investigate the accuracy of the ellipsoidal-statistical Bhatnagar–Gross–Krook (ES-BGK) kinetic model for planar force-driven Poiseuille flows. Our numerical simulations are conducted using the deterministic discrete velocity method, for Knudsen numbers (Kn) ranging from 0.05 to 10. While we provide numerically accurate data, our aim is to assess the accuracy of the ES-BGK model for these flows. By comparing with data from the direct simulation Monte Carlo (DSMC) method and the Boltzmann equation, the ES-BGK model is found to be able to predict accurate velocity and temperature profiles in the slip flow regime (0.01<Kn⩽0.1), for both low-speed and high-speed flows. In the transition flow regime (0.1<Kn⩽10), however, the model does not quantitatively capture the viscous heating effect.

Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows

AbstractA thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar–Gross–Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small-Mach-number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately $0. 5$. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased.

Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls

The ellipsoidal-statistical Bhatnagar–Gross–Krook (ES-BGK) kinetic model is investigated for steady gas-phase transport of heat, tangential momentum, and mass between parallel walls (i.e., Fourier, Couette, and Fickian flows). This investigation extends the original study of Cercignani and Tironi, who first applied the ES-BGK model to heat transport (i.e., Fourier flow) shortly after this model was proposed by Holway. The ES-BGK model is implemented in a molecular-gas-dynamics code so that results from this model can be compared directly to results from the full Boltzmann collision term, as computed by the same code with the direct simulation Monte Carlo (DSMC) algorithm of Bird. A gas of monatomic molecules is considered. These molecules collide in a pairwise fashion according to either the Maxwell or the hard-sphere interaction and reflect from the walls according to the Cercignani–Lampis–Lord model with unity accommodation coefficients. Simulations are performed at pressures from near-free-molecular to near-continuum. Unlike the BGK model, the ES-BGK model produces heat-flux and shear-stress values that both agree closely with the DSMC values at all pressures. However, for both interactions, the ES-BGK model produces molecular-velocity-distribution functions that are qualitatively similar to those determined for the Maxwell interaction from Chapman–Enskog theory for small wall temperature differences and moment-hierarchy theory for large wall temperature differences. Moreover, the ES-BGK model does not produce accurate values of the mass self-diffusion coefficient for either interaction. Nevertheless, given its reasonable accuracy for heat and tangential-momentum transport, its sound theoretical foundation (it obeys the H-theorem), and its available extension to polyatomic molecules, the ES-BGK model may be a useful method for simulating certain classes of single-species noncontinuum gas flows, as Cercignani suggested.

Numerical comparison of Bhatnagar–Gross–Krook models with proper Prandtl number

While the standard Bhatnagar–Gross–Krook (BGK) model leads to the wrong Prandtl number, the BGK model with velocity dependent collision frequency as well as the ellipsoidal statistical BGK (ES-BGK) model can be adjusted to give its proper value of 2/3. In this paper, the BGK model with velocity dependent collision frequency is considered in some detail. The corresponding thermal conductivity and viscosity are computed from the Chapman–Enskog method, and several velocity-dependent collision frequencies are introduced which all give the proper Prandtl number. The models are tested for Couette flow, and the results are compared to solutions obtained with the ES-BGK model, and the direct simulation Monte Carlo method. The simulations rely on a numerical scheme that ensures positivity of solutions, conservation of moments, and dissipation of entropy. The advantages and disadvantages of the various BGK models are discussed.