Thermodynamic Nonequilibrium Effects Research Articles

Discrete Boltzmann model with split collision for nonequilibrium reactive flows

Abstract A multi-relaxation-time discrete Boltzmann model (DBM) with split collision is proposed for both subsonic and supersonic compressible reacting flows, where chemical reactions take place among various components. The physical model is based on a unified set of discrete Boltzmann equations that describes the evolution of each chemical species with adjustable acceleration, specific heat ratio, and Prandtl number. On the righ-hand side of discrete Boltzmann equations, the collision, force, and reaction terms denote the change rates of distribution functions due to self- and cross-collisions, external forces, and chemical reactions, respectively. The source terms can be calculated in three ways, among which the matrix inversion method possesses the highest physical accuracy and computational efficiency. Through Chapman-Enskog analysis, it is proved that the DBM is consistent with the reactive Navier-Stokes equations, Fick's law and Stefan-Maxwell diffusion equation in the hydrodynamic limit. Compared with the one-step-relaxation model, the split collision model offers a detailed and precise description of hydrodynamic, thermodynamic, and chemical nonequilibrium effects. Finally, the model is validated by six benchmarks, including multicomponent diffusion, mixture in the force field, Kelvin-Helmholtz instability, flame at constant pressure, opposing chemical reaction, and steady detonation.

Plasma kinetics: Discrete Boltzmann modeling and Richtmyer–Meshkov instability

In this paper, a discrete Boltzmann method (DBM) for plasma kinetics is proposed and further used to investigate the non-equilibrium characteristics in Orszag–Tang (OT) vortex and Richtmyer–Meshkov instability (RMI) problems. The construction of DBM mainly considers two aspects. The first is to build a physical model with sufficient capability to capture underlying physics. The second is to devise schemes for extracting more valuable information from massive data. For the first aspect, the generated model is equivalent to a magnetohydrodynamic model, and a coarse-grained model for extracting the most relevant thermodynamic non-equilibrium (TNE) behaviors including the entropy production rate. For the second aspect, the DBM uses non-conserved kinetic moments of (f−feq) to describe the non-equilibrium states and behaviors of complex systems. It is found that (i) for OT vortex, the entropy production rate and compression difficulty first increase and then decrease with time. (ii) For RMI with interface inversion and re-shock process, the influence of magnetic field on TNE effects shows stages: before the interface inversion, the TNE strength is enhanced by delaying the interface inversion; while after the interface inversion, the TNE strength is significantly reduced. Both the global average TNE strength and entropy production rate contributed by non-organized energy flux can be used as physical criteria to identify whether or not the magnetic field is sufficient to prevent the interface inversion. In general, this paper proposes a generalized physical modeling and analysis scheme that has the potential for investigating the kinetic physics in plasma.

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.

The influences of acceleration on compressible Rayleigh–Taylor instability with non-equilibrium effects

Rayleigh–Taylor (RT) instability is a classical interface instability of great importance in nature and engineering fields. In this paper, the influences of acceleration on the compressible RT instability is investigated by using the discrete Boltzmann method based on non-equilibrium statistical physics. The differences in RT systems with various accelerations have been analyzed through the physical gradients and non-equilibrium measures. It is interesting to find that the global temperature gradient, the maximum Mach number, and the non-equilibrium strength increase initially, reduce afterwards, and have peaks in the dynamic process. Specifically, in the early stage, the global temperature gradient of the fluid system is higher for a case with larger acceleration, and there is an exponential relationship between them. Moreover, the maximum Mach number is located at the heavy fluid that drops fast in the system’s midsection, rises more sharply, and reaches the peak earlier for a larger acceleration. In addition, for a larger acceleration, the non-equilibrium strength rises (reduces) faster in the early (later) stage, and presents an exponential relationship between them in the early stage as well. From the kinetic perspective, these results further enrich our understanding of the physical mechanism of the compressible RT instability with both hydrodynamic and thermodynamic non-equilibrium effects.

A discrete Boltzmann model with symmetric velocity discretization for compressible flow

A discrete Boltzmann model (DBM) with symmetric velocity discretization is constructed for compressible systems with an adjustable specific heat ratio in the external force field. The proposed two-dimensional (2D) nine-velocity scheme has better spatial symmetry and numerical accuracy than the discretized velocity model in literature [Acta Aerodyn. Sin. 40 98108 (2022)] and owns higher computational efficiency than the one in literature [Phys. Rev. E 99 012142 (2019)]. In addition, the matrix inversion method is adopted to calculate the discrete equilibrium distribution function and force term, both of which satisfy nine independent kinetic moment relations. Moreover, the DBM could be used to study a few thermodynamic nonequilibrium effects beyond the Euler equations that are recovered from the kinetic model in the hydrodynamic limit via the Chapman–Enskog expansion. Finally, the present method is verified through typical numerical simulations, including the free-falling process, Sod’s shock tube, sound wave, compressible Rayleigh–Taylor instability, and translational motion of a 2D fluid system.

High-order modeling of multiphase flows: Based on discrete Boltzmann method

The high-order kinetic model for compressible multiphase flow is presented within the framework of the discrete Boltzmann method (DBM). Based on the Carnahan–Starling state equation, this model can describe the phase transition by introducing a source term of molecular interaction on the right-hand side of the Boltzmann equation. Meanwhile, the force term is incorporated to describe the external force. Through Hermite polynomial expansion, the equilibrium distribution function is expressed. Compared to the Navier–Stokes equations, the DBM provides more detailed and accurate information on both hydrodynamic and thermodynamic non-equilibrium effects. Finally, the model is verified through several typical benchmarks, including the liquid–vapor coexistence curve, free-falling process, shock wave, sound wave, thermal phase separation, and two-bubble oblique collision.

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.

Computational and Experimental Assessment of a MW-Scale Supercritical CO2 Compressor Operating in Multiple Near-Critical Conditions

Abstract This work illustrates the results of a wide experimental campaign in the frame of the EU-funded project sCO2-Flex, which focused on the investigation of a MW-scale sCO2 compressor operating in plant-representative conditions. The experimental tests were carried out for four temperature levels between 304.15 K and 309.15 K at a fixed pressure of 79.79 bar, hence covering an extended thermodynamic region close to the critical point. The experimental results are thoroughly discussed with the support of steady computational fluid-dynamics simulations, assuming homogeneous flows and thermodynamic equilibrium for the two-phase flow description. Changing the upstream total state, two peculiar variabilities in the compressor pressure ratio and choking flow rate are experimentally and computationally observed. While the former is mainly related to the single-phase flow thermodynamics, the latter originates from the onset of two-phase flows. As the simulations predict the experimental choking with a maximum error of 3%, the corresponding two-phase speed of sound is analyzed to infer the underlying equilibria between phases. It is found that, for the tested conditions, two-phase flows quickly achieve thermodynamic equilibrium, and non-equilibrium or metastable effects arguably play a marginal role in the process.

Discrete Boltzmann modeling of high-speed compressible flows with various depths of non-equilibrium

The non-equilibrium high-speed compressible flows present wealthy applications in engineering and science. With the deepening of Thermodynamic Non-Equilibrium (TNE), higher-order non-conserved kinetic moments of the distribution function are needed to capture the main feature of the flow state and the evolution process. Based on the ellipsoidal statistical Bhatnagar–Gross–Krook model, Discrete Boltzmann Models (DBMs) that consider various orders of TNE effects are developed to study flows in various depths of TNE. In numerical tests, DBMs including the first up to the sixth order TNE effects are demonstrated. Specifically, at first, the model's capability to capture large flow structures with zeroth-order TNE effects in two types of one-dimensional Riemann problems is demonstrated. The ability to capture large flow structures with first-order TNE effects is also shown in the Couette flow. Then, a shock wave structure given by Direct simulation Monte Carlo is used to verify the model's capability to capture fine structures at the level of the mean free path of gas molecules. Furthermore, we focus on the TNE degree of two colliding fluids mainly decided by two parameters: the relaxation time τ and relative speeds Δu of two fluids. Consequently, three numerical tests for flows with various depths of TNE are constructed. Due to any definition of the TNE strength is dependent on the perspective of investigation, we propose to use a N-component vector STNE to describe the TNE system from N perspectives. As specific applications, we use a three-component vector STNE=(τ,Δu,Δ2*) to roughly characterize three cases for numerical tests in this work. Then, we check the system TNE behavior from the perspective of the xx component of the TNE quantity, viscous stress Δ2*. It is found that, for the first two cases, at least up to the second-order TNE effects, i.e., the second-order terms in Knudsen number in the CE expansion, should be included in the model construction, while for the third case, at least up to the third-order TNE effects should be included. Similar to Δ2*, three numerical tests for flows in various depths of Δ3,1* are constructed. It is found that from the perspective of Δ3,1,x*, for case 1 and case 3, at least up to the second-order TNE effects should be required, while for case 2, the first-order TNE effects are enough. These findings demonstrate that the inadequacy of focusing only on the few kinetic moments appearing in Navier–Stokes increases with the degree of discreteness and deviation from thermodynamic equilibrium. Finally, a two-dimensional free jet is simulated to indicate that, to obtain satisfying hydrodynamic quantities, the DBM should include at least up to the third-order TNE effects. This study is meaningful for the understanding of the TNE behavior of complex fluid systems and the choice of an appropriate fluid model to handle desired TNE effects.

Discrete Boltzmann modeling of detonation: Based on the Shakhov model

A Discrete Boltzmann Model(DBM) based on the Shakhov model for detonation is proposed. Compared with the DBM based on the Bhatnagar–Gross–Krook (BGK) model, the current model has a flexible Prandtl numbers and consequently can be applied to a much wider range of detonation phenomena. Besides the Hydrodynamic Non-Equilibrium (HNE) behaviors usually investigated by the Navier–Stokes model, the most relevant Thermodynamic Non-Equilibrium (TNE) effects can be probed by the current model. The model is validated by some well-known benchmarks, and some steady and unsteady detonation processes are investigated. As for the von Neumann peak relative to the wave front, it is found that (i) (within the range of numerical experiments) the peak heights of pressure, density, and flow velocity increase exponentially with the Prandtl number, the maximum stress increases parabolically with the Prandtl number, and the maximum heat flux decreases exponentially with the Prandtl number and (ii) the peak heights of pressure, density, temperature and flow velocity, and the maximum stress within the peak are parabolically increase with the Mach number, the maximum heat flux decreases exponentially with the Mach number.

Discrete Boltzmann modeling of plasma shock wave

Plasma shock waves widely exist and play an important role in high-energy-density environment, especially in the inertial confinement fusion. Due to the large gradient of macroscopic physical quantities and the coupled thermal, electrical, magnetic, and optical phenomena, there exist not only hydrodynamic non-equilibrium (HNE) effects but also strong thermodynamic non-equilibrium (TNE) effects around the wavefront. In this work, a two-dimensional single-fluid discrete Boltzmann model is proposed to investigate the physical structure of ion shock. The electron is assumed inertialess and always in thermodynamic equilibrium. The Rankine–Hugoniot relations for single-fluid theory of plasma shock wave is derived. It is found that the physical structure of shock wave in plasma is significantly different from that in normal fluid and somewhat similar to that of detonation wave from the sense that a peak appears in the front. The non-equilibrium effects around the shock front become stronger with increasing Mach number. The charge of electricity deviates oppositely from neutrality in upstream and downstream of the shock wave. The large inertia of the ions causes them to lag behind, so the wavefront charge is negative and the waverear charge is positive. The variations of HNE and TNE with Mach number are numerically investigated. The characteristics of TNE can be used to distinguish plasma shock wave from detonation wave.

Nonequilibrium effects of reactive flow based on gas kinetic theory* *This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51806116 and 11875001.

How to accurately probe chemically reactive flows with essential thermodynamic nonequilibrium effects is an open issue. Via the Chapman–Enskog analysis, the local nonequilibrium particle velocity distribution function is derived from the gas kinetic theory. It is demonstrated theoretically and numerically that the distribution function depends on the physical quantities and derivatives, and is independent of the chemical reactions directly as the chemical time scale is longer than the molecular relaxation time. Based on the simulation results of the discrete Boltzmann model, the departure between equilibrium and nonequilibrium distribution functions is obtained and analyzed around the detonation wave. In addition, it has been verified for the first time that the kinetic moments calculated by summations of the discrete distribution functions are close to those calculated by integrals of their original forms.

A weakly nonlinear wave equation for damped acoustic waves with thermodynamic non-equilibrium effects

The problem of propagating nonlinear acoustic waves is considered; the solution to which, both with and without damping, having been obtained to-date starting from the Navier–Stokes–Duhem equations together with the continuity and thermal conduction equation. The novel approach reported here adopts instead, a discontinuous Lagrangian approach, i.e. from Hamilton’s principle together with a discontinuous Lagrangian for the case of a general viscous flow. It is shown that ensemble averaging of the equation of motion resulting from the Euler–Lagrange equations, under the assumption of irrotational flow, leads to a weakly nonlinear wave equation for the velocity potential: in effect a generalisation of Kuznetsov’s well known equation with an additional term due to thermodynamic non-equilibrium effects.

Specific heat ratio effects of compressible Rayleigh—Taylor instability studied by discrete Boltzmann method

Rayleigh-Taylor (RT) instability widely exists in nature and engineering fields. How to better understand the physical mechanism of RT instability is of great theoretical significance and practical value. At present, abundant results of RT instability have been obtained by traditional macroscopic methods. However, research on the thermodynamic non-equilibrium (TNE) effects in the process of system evolution is relatively scarce. In this paper, the discrete Boltzmann method based on non-equilibrium statistical physics is utilized to study the effects of the specific heat ratio on compressible RT instability. The evolution process of the compressible RT system with different specific heat ratios can be analyzed by the temperature gradient and the proportion of the non-equilibrium region. Firstly, as a result of the competition between the macroscopic magnitude gradient and the non-equilibrium region, the average TNE intensity first increases and then reduces, and it increases with the specific heat ratio decreasing; the specific heat ratio has the same effect on the global strength of the viscous stress tensor. Secondly, the moment when the total temperature gradient in y direction deviates from the fixed value can be regarded as a physical criterion for judging the formation of the vortex structure. Thirdly, under the competition between the temperature gradients and the contact area of the two fluids, the average intensity of the non-equilibrium quantity related to the heat flux shows diversity, and the influence of the specific heat ratio is also quite remarkable.

Precipitation in reactor pressure vessel steels under ion and neutron irradiation: On the role of segregated network dislocations

The ultrahigh density of Mn-Ni-Si-Cu precipitates that form in irradiated reactor pressure vessels (RPV) result in hardening and embrittlement, which may limit the service life of aging nuclear reactors. Here, we quantitatively analyze solute segregation to, and precipitation on, network dislocations in irradiated low alloy RPV steels. The analysis is based on a broad thermodynamic framework, which allows proper identification of possible non-equilibrium perturbative effects of irradiation, like radiation induced segregation (RIS). For the first time, we quantify the segregation of Cu, Mn, Ni and Si to ≈ 5–10 nm network dislocation segments, typically separated by a string-of-pearl type array of MnNiSi precipitates (MNSPs); the MNSPs are appended to Cu rich co-precipitates (CRPs) in steels bearing this element. Detailed atom probe tomography (APT) data are reported for 16 RPV steels, with a wide matrix of tailored compositions, irradiated by both neutrons and 6.4 MeV Fe3+ ions at 320 and 290 °C, respectively. A key aspect of this study is that the very high displacement per atom (dpa) doses and radiation enhanced diffusion (RED) result in nearly full phase separation, allowing isolation of thermodynamic (equilibrium or non-equilibrium) effects. We characterize segregation and precipitates with a comprehensive set of descriptors under both ion and neutron irradiation, including: a) solute enrichment factors (EFs) at dislocations, as a function of alloy composition; and, b) matrix versus dislocation precipitates and their respective characteristics. Formation of these dislocation features can be due to RIS, or thermodynamic in origin, or both. Coupled with simple models, we show that the nucleation MNSP features in low Cu steels is enhanced by segregation. However, in typical supersaturated RPV steels, rapid growth of the precipitates is thermodynamically driven to a quantitatively predictable, nearly full phase separation mole fraction.

Three-dimensional multiple-relaxation-time discrete Boltzmann model of compressible reactive flows with nonequilibrium effects

Based on the kinetic theory, a three-dimensional multiple-relaxation-time discrete Boltzmann model (DBM) is proposed for nonequilibrium compressible reactive flows where both the Prandtl number and specific heat ratio are freely adjustable. There are 30 kinetic moments of the discrete distribution functions, and an efficient three-dimensional thirty-velocity model is utilized. Through the Chapman–Enskog analysis, the reactive Navier–Stokes equations can be recovered from the DBM. Unlike existing lattice Boltzmann models for reactive flows, the hydrodynamic and thermodynamic fields are fully coupled in the DBM to simulate combustion in subsonic, supersonic, and potentially hypersonic flows. In addition, both hydrodynamic and thermodynamic nonequilibrium effects can be obtained and quantified handily in the evolution of the discrete Boltzmann equation. Several well-known benchmarks are adopted to validate the model, including chemical reactions in the free falling process, thermal Couette flow, one-dimensional steady or unsteady detonation, and a three-dimensional spherical explosion in an enclosed cube. It is shown that the proposed DBM has the capability to simulate both subsonic and supersonic fluid flows with or without chemical reactions.

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.

Kinetic Simulation of Unsteady Detonation with Thermodynamic Nonequilibrium Effects

Thanks to its mesoscopic kinetic nature, the discrete Boltzmann method has a capability of investigating unsteady detonation with essential hydrodynamic and thermodynamic nonequilibrium effects. In this work, an efficient and precise reactive discrete Boltzmann method is employed to investigate the impact of the amplitude and wave length of the initial perturbation, as well as of the chemical heat on the evolution of unsteady detonation with nonequilibrium effects. It is shown that the initial perturbation amplitude only affects the unsteady detonation in the early period, and the detonation becomes self-similar with minor phase differences later on. For small wave lengths, the pressure increases faster with a higher oscillation frequency in the early period, but decreases soon afterwards. With increasing chemical heat release, the pressure and its oscillations increase, and the nonequilibrium effects become more pronounced, but the oscillatory period decreases. When the wave length or chemical heat release is small enough, there is no transverse wave or cellular pattern, and the two-dimensional unsteady detonation reduces to the one-dimensional case.

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.