Jianping Meng

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar-Gross-Krook (BGK) equation. Therefore, when the same Gauss-Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss-Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss-Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss-Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier-Stokes level and rarefied gas flows at the linearized Boltzmann model equation level.

Lattice ellipsoidal statistical BGK model for thermal non- equilibrium flows

A 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 Monte Carlo 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 that have been 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.

Multiscale lattice Boltzmann approach to modeling gas flows

For multiscale gas flows, the kinetic-continuum hybrid method is usually used to balance the computational accuracy and efficiency. However, the kinetic-continuum coupling is not straightforward since the coupled methods are based on different theoretical frameworks. In particular, it is not easy to recover the nonequilibrium information required by the kinetic method, which is lost by the continuum model at the coupling interface. Therefore, we present a multiscale lattice Boltzmann (LB) method that deploys high-order LB models in highly rarefied flow regions and low-order ones in less rarefied regions. Since this multiscale approach is based on the same theoretical framework, the coupling precess becomes simple. The nonequilibrium information will not be lost at the interface as low-order LB models can also retain this information. The simulation results confirm that the present method can achieve modeling accuracy with reduced computational cost.

Diffuse reflection boundary condition for high-order lattice Boltzmann models with streaming-collision mechanism

For lattice Boltzmann (LB) models, their particle feature is ensured by the streaming-collision mechanism. However, this mechanism is often discarded for high-order LB models due to multi-speed lattices. Here, we propose a new way of implementing the diffuse reflection boundary condition that can maintain the characteristic streaming-collision mechanism for high-order models. It is then tested on four lattice models, namely D2Q16, D2Q17, D2Q37 and D3Q121 for isothermal and thermal Couette flows and lid-driven cavity flows. Our implementation is shown to be able to achieve second-order accuracy globally.

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

Kinetic modelling of the quantum gases in the normal phase

Using the maximum entropy principle, a kinetic model equation is proposed to simplify the intricate collision term in the semi-classical Boltzmann equation for dilute quantum gases in the normal phase. The kinetic model equation keeps the main properties of the Boltzmann equation, including conservation of mass, momentum and energy, the entropy dissipation property, and rotational invariance. It also produces the correct Prandtl numbers for the Fermi gases. To validate the proposed model, the kinetic model equation is numerically solved in the hydrodynamic and kinetic flow regimes using the asymptotic preserving scheme. The results agree well with those of the quantum Euler and Boltzmann equations.

Analysis of non-physical slip velocity in lattice Boltzmann simulations using the bounce-back scheme

Abstract In this work, we investigate the non-physical velocity-slip at a solid wall associated with lattice Boltzmann simulations using the bounce-back boundary scheme. By comparing the analytical solutions for the force-driven Poiseuille flow problem of two lattice models involving four and nine discrete velocities respectively, we found that the bounce-back scheme fails to define the behavior of certain discrete velocities and gives room for other factors to generate the commonly observed non-physical velocity-slip. To eliminate this deficiency, it is necessary to supplement the missing definition. For a lid-driven cavity flow, this is shown to be very simple although further study is necessary for more complex geometries.

A kinetic switching criterion for hybrid modelling of multiscale gas flows

In some important engineering applications, gas flows are often found to be hydrodynamic in one part of device and highly rarefied in the others. To solve this kind of multi-scale flow problems efficiently and accurately, hybrid methods coupling hydrodynamic and kinetic methods are attractive. The successful implementation of hybrid methods relies on the accurate assessment of the level of non-equilibrium (rarefaction) in the local flowfield. Currently available criteria, such as Knudsen and Mach numbers, are based on macroscopic parameters and have been shown to be restrictive. Here, we propose a new kinetic criterion that utilises the fundamental molecular distribution function to assess the local flow field. Through numerical evaluation we show that our criterion provides a reasonable assessment and, in particular, it behaves consistently for both high-speed and low-speed flows, which is not the case for the other criteria based on macroscopic parameters. As our criterion fully utilises the accurate information provided by the molecular distribution function, it is particularly suitable for recently developed multi-scale kinetic methods.