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.

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In this work, we develop a mesoscopic lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model to solve (2 + 1)-dimensional wave equation with the nonlinear damping and source terms. Through the Chapman-Enskog multiscale expansion, the macroscopic governing evolution equation can be obtained accurately by choosing appropriate local equilibrium distribution functions. We validate the present mesoscopic model by some related issues where the exact solution is known. It turned out that the numerical solution is in very good agreement with exact one, which shows that the present mesoscopic model is pretty valid, and can be used to solve more similar nonlinear wave equations with nonlinear damping and source terms, and predict and enrich the internal mechanism of nonlinearity and complexity in nonlinear dynamic phenomenon.

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.

We use a discrete Boltzmann model (DBM) to simulate the multi-mode Rayleigh-Taylor instability (RTI) in a compressible flow.This DBM is physically equivalent to a Navier-Stokes model supplemented by a coarse-grained model for thermodynamic nonequilibrium behavior.The validity of the model is verified by comparing simulation results of Riemann problems,Sod shock tube,collision between two strong shock waves,and thermal Couette flow with analytical solutions.Grid independence is verified.The DBM is utilized to simulate the nonlinear evolution of the RTI from multi-mode initial perturbation with discontinuous interface.We obtain the basic process of the initial disturbance interface which develops into mushroom graphs.The evolution of the system is relatively slow at the beginning,and the interface moves down on a whole.This is mainly due to the fact that the heat transfer plays a leading role,and the exchange of internal energy occurs near the interface of fluid.The overlying fluid absorbs heat,which causes the volume to expand,and the underlying fluid releases heat,which causes the volume to shrink,consequently the fluid interface moves downward.Meanwhile,due to the effects of viscosity and thermal conduction,the perturbed interface is smoothed.The evolution rate is slow at the initial stage.As the modes couple with each other,the evolution begins to grow faster.As the interface evolves gradually into the gravity dominated stage,the overlying and underlying fluids begin to exchange the gravitational potentials via nonlinear evolution.Lately,the two parts of fluid permeate each other near the interface.The system goes through the nonlinear disturbance and irregular nonlinear stages,then develops into the typical “mushroom” stage.Afterwards,the system evolves into the turbulent mixing stage.Owing to the coupling and development of perturbation modes,and the transformation among the gravitational potential energy,compression energy and kinetic energy,the system first approaches to a transient local thermodynamic equilibrium,then deviates from it and the perturbation grows linearly.After that,at the beginning,the fluid system tends to approach to an equilibrium state,which is caused by the adjustment of the system,and the disturbance of the multi-mode initial interface moves toward a process of the eigenmode stage.Then,the system deviates from the equilibrium state linearly,which is due to the flattening of the system interface and the conversing of the compression energy into internal energy.Moreover, the system tends to approach to the equilibrium state again,and this is because the modes couple and the disturbance interface is further “screened”.The system is in a relatively stable state.Furthermore,the system is farther away from the equilibrium state because of the gravitational potential energy of the fluid system transformation.The compression energy of the system is released further,and the kinetic energy is further increased.After that,the nonequilibrium intensity decreases,and then the system is slowly away from thermodynamic equilibrium.The interface becomes more and more complicated,and the nonequilibrium modes also become more and more abundant.