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

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.

Thermodynamic nonequilibrium effects in bubble coalescence: A discrete Boltzmann study

The thermodynamic nonequilibrium (TNE) effects in a coalescence process of two initially static bubbles under thermal conditions are investigated by a discrete Boltzmann model. The spatial distributions of the typical nonequilibrium quantity, i.e., nonorganized momentum fluxes (NOMFs), during evolutions are investigated in detail. The density-weighted statistical method is used to highlight the relationship between the TNE effects and the morphological and kinetics characteristics of bubble coalescence. The results show that the xx component and yy component of NOMFs are antisymmetrical; the xy component changes from an antisymmetric internal and external double quadrupole structure to an outer octupole structure during the coalescence process. Moreover, the evolution of the averaged xx component of NOMFs provides two characteristic instants, which divide the nonequilibrium process into three stages. The first instant, when the averaged xx component of the NOMFs reaches its first local minimum, corresponds to the moment when the mean coalescence speed gets the maximum, and at this time the ratio of minor and major axes is about 1/2. The second instant, when the averaged xx component of the NOMFs gets its second local maximum, corresponds to the moment when the ratio of minor and major axes becomes 1 for the first time. It is interesting to find that the three quantities, TNE intensity, acceleration of coalescence, and the slope of boundary length, show a high degree of correlation and attain their maxima simultaneously. The surface tension and the heat conduction accelerate the process of bubble coalescence, while the viscosity delays it. Both the surface tension and the viscosity enhance the global nonequilibrium intensity, whereas the heat conduction restrains it. These TNE features and findings present some insights into the kinetics of bubble coalescence.

Particles on demand for flows with strong discontinuities

Particles-on-demand formulation of kinetic theory [B. Dorschner, F. Bösch and I. V. Karlin, Phys. Rev. Lett. 121, 130602 (2018)0031-900710.1103/PhysRevLett.121.130602] is used to simulate a variety of compressible flows with strong discontinuities in density, pressure, and velocity. Two modifications are applied to the original formulation of the particles-on-demand method. First, a regularization by Grad's projection of particles populations is combined with the reference frame transformations in order to enhance stability and accuracy. Second, a finite-volume scheme is implemented which allows tight control of mass, momentum, and energy conservation. The proposed model is validated with an array of challenging one- and two-dimensional benchmarks of compressible flows, including hypersonic and near-vacuum situations, Richtmyer-Meshkov instability, double Mach reflection, and astrophysical jet. Excellent performance of the modified particles-on-demand method is demonstrated beyond the limitations of other lattice Boltzmann-like approaches to compressible flows.

Discrete Boltzmann modeling of Rayleigh-Taylor instability: Effects of interfacial tension, viscosity, and heat conductivity

The two-dimensional Rayleigh-Taylor instability (RTI) in compressible flow with intermolecular interactions is probed via the discrete Boltzmann method. The effects of interfacial tension, viscosity, and heat conduction are investigated. It is found that the influences of interfacial tension on the perturbation amplitude, bubble velocity, and two kinds of entropy production rates all show differences at different stages of RTI evolution. It inhibits the RTI evolution at the bubble acceleration stage, while at the asymptotic velocity stage, it first promotes and then inhibits the RTI evolution. Viscosity and heat conduction inhibit the RTI evolution. Viscosity shows a suppressive effect on the entropy generation rate related to heat flow at the early stage but a first promotive and then suppressive effect on the entropy generation rate related to heat flow at a later stage. Heat conduction shows a promotive effect on the entropy generation rate related to heat flow at an early stage. Still, it offers a first promotive and then suppressive effect on the entropy generation rate related to heat flow at a later stage. By introducing the morphological boundary length, we find that the stage of exponential growth of the interface length with time corresponds to the bubble acceleration stage. The first maximum point of the interface length change rate and the first maximum point of the change rate of the entropy generation rate related to viscous stress can be used as a criterion for RTI to enter the asymptotic velocity stage.

Growth mechanism of interfacial fluid-mixing width induced by successive nonlinear wave interactions

Interfacial fluid mixing induced by successive waves, such as shock, rarefaction, and compression waves, plays a fundamental role in engineering applications, e.g., inertial confinement fusion, and in natural phenomena, e.g., supernova explosion. These waves bring nonuniform, unsteady external forces into the mixing zone, which leads to a complex mixing process. The growth rate of the mixing width is analyzed by decomposing the turbulent flow field into the averaged field and the fluctuating counterpart. The growth rate is thus divided into three parts: (i) the stretching or compression (S(C)) effect induced by the averaged-velocity difference between two ends of the mixing zone, (ii) the penetration effect induced by the fluctuations which represent the penetration of the two species into each other, and (iii) the diffusive effect, which is induced by the molecular diffusion and is negligible in high-Reynolds-number flows at Schmidt number of order unity. The penetration effect is further divided into the Richtmyer-Meshkov (RM) effect, which is induced by fluctuations that were deposited by earlier wave interactions, and the Rayleigh-Taylor (RT) effect, which is caused by the fluctuations that arise in an overall acceleration of the mixing zone. During the passage of the rarefaction waves, the mixing zone is stretched, while during the passage of the compression waves or shock waves, the mixing zone is compressed. To illustrate these effects, a physical model of RM mixing with reshock is used. By combining the S(C), RM, and RT effects, the entire evolution of mixing width is restructured, which agrees well with numerical simulations for problems with a wide range of density ratios.

Mesoscopic kinetic approach for studying nonequilibrium hydrodynamic and thermodynamic effects of shock wave, contact discontinuity, and rarefaction wave in the unsteady shock tube

This paper presents a detailed description of a molecular velocity distribution-based mesoscopic kinetic approach that enables a better understanding of various nonequilibrium hydrodynamic and thermodynamic effects in shock waves, contact discontinuities, and rarefaction waves. This builds on the mesoscopic kinetic approach in a previous investigation into regular reflection shocks by further addressing the mesoscopic physical meaning of kinetic moments from the view of kinetics and the implications of the magnitude and sign of nonequilibrium kinetic moments. To deepen understanding of nonequilibrium effects, this work focuses on the one-dimensional unsteady shock tube problem, which contains the typical and essential features of the discontinuous flows, and has no interference of two-dimensional flow direction. The approach uses a lattice Boltzmann method to solve the flow field, and describes nonequilibrium effects through the nonequilibrium kinetic moments of molecular velocity distribution functions. The mechanism of nonequilibrium effect in discontinuous flows is further probed. This work develops the mesoscopic kinetic approach and clarifies the mesoscopic physics of shock waves, contact discontinuities, and rarefaction waves.

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.

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.

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.

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.

Phase-field method based on discrete unified gas-kinetic scheme for large-density-ratio two-phase flows

In this paper, a phase-field method under the framework of discrete unified gas-kinetic scheme (DUGKS) for incompressible multiphase fluid flows is proposed. Two kinetic models are constructed to solve the conservative Allen-Cahn equation that accounts for the interface behavior and the incompressible hydrodynamic equations that govern the flow field, respectively. With a truncated equilibrium distribution function as well as a temporal derivative added to the source term, the macroscopic governing equations can be exactly recovered from the kinetic models through the Chapman-Enskog analysis. Calculation of source terms involving high-order derivatives existed in the quasi-incompressible model is simplified. A series of benchmark cases including four interface-capturing tests and four binary flow tests are carried out. Results compared to that of the lattice Boltzmann method (LBM) have been obtained. A convergence rate of second order can be guaranteed in the test of interface diagonal translation. The capability of the present method to track the interface that undergoes a severe deformation has been verified. Stationary bubble and spinodal decomposition problems, both with a density ratio as high as 1000, are conducted and reliable solutions have been provided. The layered Poiseuille flow with a large viscosity ratio is simulated and numerical results agree well with the analytical solutions. Variation of positions of the bubble front and spike tip during the evolution of Rayleigh-Taylor instability has been predicted precisely. However, the detailed depiction of complicated interface patterns appearing during the evolution process is failed, which is mainly caused by the relatively large numerical dissipation of DUGKS compared to that of LBM. A high-order DUGKS is needed to overcome this problem.

Mesoscopic Simulation of the (2 + 1)-Dimensional Wave Equation with Nonlinear Damping and Source Terms Using the Lattice Boltzmann BGK Model

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.

Discrete Boltzmann trans-scale modeling of high-speed compressible flows

We present a general framework for constructing trans-scale discrete Boltzmann models (DBMs) for high-speed compressible flows ranging from continuum to transition regime. This is achieved by designing a higher-order discrete equilibrium distribution function that satisfies additional nonhydrodynamic kinetic moments. To characterize the thermodynamic nonequilibrium (TNE) effects and estimate the condition under which the DBMs at various levels should be used, two measures are presented: (i) the relative TNE strength, describing the relative strength of the (N+1)th order TNE effects to the Nth order one; (ii) the TNE discrepancy between DBM simulation and relevant theoretical analysis. Whether or not the higher-order TNE effects should be taken into account in the modeling and which level of DBM should be adopted is best described by the relative TNE intensity and/or the discrepancy rather than by the value of the Knudsen number. As a model example, a two-dimensional DBM with 26 discrete velocities at Burnett level is formulated, verified, and validated.

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.

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.