Effects of Inclined Interface Angle on Compressible Rayleigh-Taylor Instability: A Numerical Study Based on the Discrete Boltzmann Method.

Rayleigh-Taylor (RT) instability is a basic fluid interface instability that widely exists in nature and in the engineering field. To investigate the impact of the initial inclined interface on compressible RT instability, the two-component discrete Boltzmann method is employed. Both the thermodynamic non-equilibrium (TNE) and hydrodynamic non-equilibrium (HNE) effects are studied. It can be found that the global average density gradient in the horizontal direction, the non-organized energy fluxes, the global average non-equilibrium intensity and the proportion of the non-equilibrium region first increase and then reduce with time. However, the global average density gradient in the vertical direction and the non-organized moment fluxes first descend, then rise, and finally descend. Furthermore, the global average density gradient, the typical TNE intensity and the proportion of non-equilibrium region increase with increasing angle of the initial inclined interface. Physically, there are three competitive mechanisms: (1) As the perturbed interface elongates, the contact area between the two fluids expands, which results in an increasing gradient of macroscopic physical quantities and leads to a strengthening of the TNE effects. (2) Under the influence of viscosity, the perturbation pressure waves on both sides of the material interface decrease with time, which makes the gradient of the macroscopic physical quantity decrease, resulting in a weakening of the TNE strength. (3) Due to dissipation and/or mutual penetration of the two fluids, the gradient of macroscopic physical quantities gradually diminishes, resulting in a decrease in the intensity of the TNE.

Multiple-relaxation-time discrete Boltzmann modeling of multicomponent mixture with nonequilibrium effects.

A multiple-relaxation-time discrete Boltzmann model (DBM) is proposed for multicomponent mixtures, where compressible, hydrodynamic, and thermodynamic nonequilibrium effects are taken into account. It allows the specific heat ratio and the Prandtl number to be adjustable, and is suitable for both low and high speed fluid flows. From the physical side, besides being consistent with the multicomponent Navier-Stokes equations, Fick's law, and Stefan-Maxwell diffusion equation in the hydrodynamic limit, the DBM provides more kinetic information about the nonequilibrium effects. The physical capability of DBM to describe the nonequilibrium flows, beyond the Navier-Stokes representation, enables the study of the entropy production mechanism in complex flows, especially in multicomponent mixtures. Moreover, the current kinetic model is employed to investigate nonequilibrium behaviors of the compressible Kelvin-Helmholtz instability (KHI). The entropy of mixing, the mixing area, the mixing width, the kinetic and internal energies, and the maximum and minimum temperatures are investigated during the dynamic KHI process. It is found that the mixing degree and fluid flow are similar in the KHI process for cases with various thermal conductivity and initial temperature configurations, while the maximum and minimum temperatures show different trends in cases with or without initial temperature gradients. Physically, both heat conduction and temperature exert slight influences on the formation and evolution of the KHI morphological structure.

Knudsen Number Effects on Two-Dimensional Rayleigh-Taylor Instability in Compressible Fluid: Based on a Discrete Boltzmann Method.

Based on the framework of our previous work [H.L. Lai et al., Phys. Rev. E, 94, 023106 (2016)], we continue to study the effects of Knudsen number on two-dimensional Rayleigh-Taylor (RT) instability in compressible fluid via the discrete Boltzmann method. It is found that the Knudsen number effects strongly inhibit the RT instability but always enormously strengthen both the global hydrodynamic non-equilibrium (HNE) and thermodynamic non-equilibrium (TNE) effects. Moreover, when Knudsen number increases, the Kelvin-Helmholtz instability induced by the development of the RT instability is difficult to sufficiently develop in the later stage. Different from the traditional computational fluid dynamics, the discrete Boltzmann method further presents a wealth of non-equilibrium information. Specifically, the two-dimensional TNE quantities demonstrate that, far from the disturbance interface, the value of TNE strength is basically zero; the TNE effects are mainly concentrated on both sides of the interface, which is closely related to the gradient of macroscopic quantities. The global TNE first decreases then increases with evolution. The relevant physical mechanisms are analyzed and discussed.

Lattice-Gas-Automaton Modeling of Income Distribution.

A simple and effective lattice-gas-automaton (LGA) economic model is proposed for the income distribution. It consists of four stages: random propagation, economic transaction, income tax, and charity. Two types of discrete models are introduced: two-dimensional four-neighbor model (D2N4) and D2N8. For the former, an agent either remains motionless or travels to one of its four neighboring empty sites randomly. For the latter, the agent may travel to one of its nearest four sites or the four diagonal sites. Afterwards, an economic transaction takes place randomly when two agents are located in the nearest (plus the diagonal) neighboring sites for the D2N4 (D2N8). During the exchange, the Matthew effect could be taken into account in the way that the rich own a higher probability of earning money than the poor. Moreover, two kinds of income tax models are incorporated. One is the detailed taxable income brackets and rates, and the other is a simplified tax model based on a fitting power function. Meanwhile, charity is considered with the assumption that a richer agent donates a part of his income to charity with a certain probability. Finally, the LGA economic model is validated by using two kinds of benchmarks. One is the income distributions of individual agents and two-earner families in a free market. The other is the shares of total income in the USA and UK, respectively. Besides, impacts of the Matthew effect, income tax and charity upon the redistribution of income are investigated. It is confirmed that the model has the potential to offer valuable references for formulating financial laws and regulations.

Hydrodynamic and Thermodynamic Nonequilibrium Effects around Shock Waves: Based on a Discrete Boltzmann Method.

A shock wave that is characterized by sharp physical gradients always draws the medium out of equilibrium. In this work, both hydrodynamic and thermodynamic nonequilibrium effects around the shock wave are investigated using a discrete Boltzmann model. Via Chapman-Enskog analysis, the local equilibrium and nonequilibrium velocity distribution functions in one-, two-, and three-dimensional velocity space are recovered across the shock wave. Besides, the absolute and relative deviation degrees are defined in order to describe the departure of the fluid system from the equilibrium state. The local and global nonequilibrium effects, nonorganized energy, and nonorganized energy flux are also investigated. Moreover, the impacts of the relaxation frequency, Mach number, thermal conductivity, viscosity, and the specific heat ratio on the nonequilibrium behaviours around shock waves are studied. This work is helpful for a deeper understanding of the fine structures of shock wave and nonequilibrium statistical mechanics.

Discrete Boltzmann modeling of unsteady reactive flows with nonequilibrium effects.

A multiple-relaxation-time discrete Boltzmann model (DBM) is developed for compressible thermal reactive flows. A unified Boltzmann equation set is solved for hydrodynamic and thermodynamic quantities as well as higher order kinetic moments. The collision, reaction, and force terms are uniformly calculated with a matrix inversion method, which is physically accurate, numerically efficient, and convenient for coding. Via the Chapman-Enskog analysis, the DBM is demonstrated to recover reactive Navier-Stokes (NS) equations in the hydrodynamic limit. Both specific heat ratio and Prandtl number are adjustable. Moreover, it provides quantification of hydrodynamic and thermodynamic nonequilibrium effects beyond the NS equations. The capability of the DBM is demonstrated through simulations of chemical reactions in the free falling process, sound wave, thermal Couette flow, and steady and unsteady detonation cases. Moreover, nonequilibrium effects on the predicted physical quantities in unsteady combustion are quantified via the DBM. It is demonstrated that nonequilibrium effects suppress detonation instability and dissipate small oscillations of fluid flows.

Mesoscopic Simulation of the Two-Component System of Coupled Sine-Gordon Equations with Lattice Boltzmann Method.

In this paper, a new lattice Boltzmann model for the two-component system of coupled sine-Gordon equations is presented by using the coupled mesoscopic Boltzmann equations. Via the Chapman-Enskog multiscale expansion, the macroscopical governing evolution system can be recovered correctly by selecting suitable discrete equilibrium distribution functions and the amending functions. The mesoscopic model has been validated by several related issues where analytic solutions are available. The experimental results show that the numerical results are consistent with the analytic solutions. From the mesoscopic point of view, the present approach provides a new way for studying the complex nonlinear partial differential equations arising in natural nonlinear phenomena of engineering and science.

A multi-component discrete Boltzmann model for nonequilibrium reactive flows.

We propose a multi-component discrete Boltzmann model (DBM) for premixed, nonpremixed, or partially premixed nonequilibrium reactive flows. This model is suitable for both subsonic and supersonic flows with or without chemical reaction and/or external force. A two-dimensional sixteen-velocity model is constructed for the DBM. In the hydrodynamic limit, the DBM recovers the modified Navier-Stokes equations for reacting species in a force field. Compared to standard lattice Boltzmann models, the DBM presents not only more accurate hydrodynamic quantities, but also detailed nonequilibrium effects that are essential yet long-neglected by traditional fluid dynamics. Apart from nonequilibrium terms (viscous stress and heat flux) in conventional models, specific hydrodynamic and thermodynamic nonequilibrium quantities (high order kinetic moments and their departure from equilibrium) are dynamically obtained from the DBM in a straightforward way. Due to its generality, the developed methodology is applicable to a wide range of phenomena across many energy technologies, emissions reduction, environmental protection, mining accident prevention, chemical and process industry.

Discrete Boltzmann modeling of Rayleigh-Taylor instability in two-component compressible flows.

A discrete Boltzmann model (DBM) is proposed to probe the Rayleigh-Taylor instability (RTI) in two-component compressible flows. Each species has a flexible specific-heat ratio and is described by one discrete Boltzmann equation (DBE). Independent discrete velocities are adopted for the two DBEs. The collision and force terms in the DBE account for the molecular collision and external force, respectively. Two types of force terms are exploited. In addition to recovering the modified Navier-Stokes equations in the hydrodynamic limit, the DBM has the capability of capturing detailed nonequilibrium effects. Furthermore, we use the DBM to investigate the dynamic process of the RTI. The invariants of tensors for nonequilibrium effects are presented and studied. For low Reynolds numbers, both global nonequilibrium manifestations and the growth rate of the entropy of mixing show three stages (i.e., the reducing, increasing, and then decreasing trends) in the evolution of the RTI. On the other hand, the early reducing tendency is suppressed and even eliminated for high Reynolds numbers. Relevant physical mechanisms are analyzed and discussed.

Multiple-relaxation-time lattice Boltzmann kinetic model for combustion.

To probe both the hydrodynamic nonequilibrium (HNE) and thermodynamic nonequilibrium (TNE) in the combustion process, a two-dimensional multiple-relaxation-time (MRT) version of lattice Boltzmann kinetic model (LBKM) for combustion phenomena is presented. The chemical energy released in the progress of combustion is dynamically coupled into the system by adding a chemical term to the LB kinetic equation. Aside from describing the evolutions of the conserved quantities, the density, momentum, and energy, which are what the Navier-Stokes model describes, the MRT-LBKM presents also a coarse-grained description on the evolutions of some nonconserved quantities. The current model works for both subsonic and supersonic flows with or without chemical reaction. In this model, both the specific-heat ratio and the Prandtl number are flexible, the TNE effects are naturally presented in each simulation step. The model is verified and validated via well-known benchmark tests. As an initial application, various nonequilibrium behaviors, including the complex interplays between various HNEs, between various TNEs, and between the HNE and TNE, around the detonation wave in the unsteady and steady one-dimensional detonation processes are preliminarily probed. It is found that the system viscosity (or heat conductivity) decreases the local TNE, but increases the global TNE around the detonation wave, that even locally, the system viscosity (or heat conductivity) results in two kinds of competing trends, to increase and to decrease the TNE effects. The physical reason is that the viscosity (or heat conductivity) takes part in both the thermodynamic and hydrodynamic responses.

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.