Aiguo Xu

Multiple-relaxation-time lattice Boltzmann approach to compressible flows with flexible specific-heat ratio and Prandtl number

A new multiple-relaxation-time lattice Boltzmann scheme for compressible flows with arbitrary specific-heat ratio and Prandtl number is presented. In the new scheme, which is based on a two-dimensional 16-discrete-velocity model, the kinetic moment space and the corresponding transformation matrix are constructed according to the seven-moment relations associated with the local-equilibrium distribution function. In the continuum limit, the model recovers the compressible Navier-Stokes equations with flexible specific-heat ratio and Prandtl number. Numerical experiments show that compressible flows with strong shocks can be simulated by the present model up to Mach numbers Ma~5.

Lattice BGK kinetic model for high-speed compressible flows: Hydrodynamic and nonequilibrium behaviors

We present a simple and general approach to formulate the lattice BGK model for high-speed compressible flows. The main point consists of two parts: an appropriate discrete equilibrium distribution function (DEDF) feq and a discrete velocity model with flexible velocity size. The DEDF is obtained by feqxa0=xa0C−1M, where M is a set of moments of the Maxwellian distribution function, and C is the matrix connecting the DEDF and the moments. The numerical components of C are determined by the discrete velocity model. The calculation of C−1 is based on the analytic solution which is a function of the parameter controlling the sizes of discrete velocity. The choice of the discrete velocity model has a high flexibility. The specific-heat ratio of the system can be flexible. The approach works for the one-, two- and three-dimensional model constructions. As an example, we compose a new lattice BGK kinetic model which works not only for recovering the Navier-Stokes equations in the continuum limit but also for measuring the departure of the system from its thermodynamic equilibrium. Via adjusting the sizes of the discrete velocities, the stably simulated Mach number can be significantly increased up to 30 or even higher. The model is verified and validated by well-known benchmark tests. Some macroscopic behaviors of the system due to the deviation from thermodynamic equilibrium around the shock wave interfaces are shown.

Polar-coordinate lattice Boltzmann modeling of compressible flows.

We present a polar coordinate lattice Boltzmann kinetic model for compressible flows. A method to recover the continuum distribution function from the discrete distribution function is indicated. Within the model, a hybrid scheme being similar to, but different from, the operator splitting is proposed. The temporal evolution is calculated analytically, and the convection term is solved via a modified Warming-Beam (MWB) scheme. Within the MWB scheme a suitable switch function is introduced. The current model works not only for subsonic flows but also for supersonic flows. It is validated and verified via the following well-known benchmark tests: (i) the rotational flow, (ii) the stable shock tube problem, (iii) the Richtmyer-Meshkov (RM) instability, and (iv) the Kelvin-Helmholtz instability. As an original application, we studied the nonequilibrium characteristics of the system around three kinds of interfaces, the shock wave, the rarefaction wave, and the material interface, for two specific cases. In one of the two cases, the material interface is initially perturbed, and consequently the RM instability occurs. It is found that the macroscopic effects due to deviating from thermodynamic equilibrium around the material interface differ significantly from those around the mechanical interfaces. The initial perturbation at the material interface enhances the coupling of molecular motions in different degrees of freedom. The amplitude of deviation from thermodynamic equilibrium around the shock wave is much higher than those around the rarefaction wave and material interface. By comparing each component of the high-order moments and its value in equilibrium, we can draw qualitatively the main behavior of the actual distribution function. These results deepen our understanding of the mechanical and material interfaces from a more fundamental level, which is indicative for constructing macroscopic models and other kinds of kinetic models.

Lattice Boltzmann study of thermal phase separation: Effects of heat conduction, viscosity and Prandtl number

We investigate the effects of heat conduction, viscosity, and Prandtl number on thermal liquid-vapor separation via a lattice Boltzmann model for van der Waals fluids. The set of Minkowski measures on the density field enables to divide exactly the stages of the spinodal decomposition (SD) and domain growth. The duration tSD of the SD stage decreases with increasing the heat conductivity κT but increases with increasing the viscosity η. The two relations can be fitted by tSD=a+b/κT and tSD=c+dη+(eη)3, respectively, where a, b, c, d and e are fitting parameters. For fixed Prandtl number Pr, when η is less than a critical value ηc, tSD shows an inverse power-law relationship with η. However, when η>ηc, tSD for Pr>1 shows qualitatively different behavior. From the evolution of the Peclet number Pe, the separation procedure can also be divided into two stages. During the first stage, the convection effects become more dominant with time over those of the diffusivity, while they are reverse in the second stage.

Nonequilibrium thermohydrodynamic effects on the Rayleigh-Taylor instability in compressible flows

The effects of compressibility on Rayleigh-Taylor instability (RTI) are investigated by inspecting the interplay between thermodynamic and hydrodynamic nonequilibrium phenomena (TNE, HNE, respectively) via a discrete Boltzmann model. Two effective approaches are presented, one tracking the evolution of the local TNE effects and the other focusing on the evolution of the mean temperature of the fluid, to track the complex interfaces separating the bubble and the spike regions of the flow. It is found that both the compressibility effects and the global TNE intensity show opposite trends in the initial and the later stages of the RTI. Compressibility delays the initial stage of RTI and accelerates the later stage. Meanwhile, the TNE characteristics are generally enhanced by the compressibility, especially in the later stage. The global or mean thermodynamic nonequilibrium indicators provide physical criteria to discriminate between the two stages of the RTI.

Lattice BBGKY scheme for two-phase flows: one-dimensional case

A novel lattice Boltzmann model for two-phase fluids is presented. We begin with the two-body BBGKY equation, and perform a coordinate transformation to split it into a Boltzmann equation for the one-body distribution, coupled to a kinetic equation for the correlation function. The coupling is accomplished by a self-consistent force. The resulting lattice Boltzmann model for nonideal fluids is grounded in the physics of the two-body distribution function. The discrete velocity model is described in detail, and numerical results are given for phase separation in one dimension.