Nonequilibrium Gas Research Articles

General synthetic iterative scheme for non-equilibrium dense gas flows

The recently-developed general synthetic iterative scheme (GSIS), which is tailored for non-equilibrium dilute gas, has been extended to find the steady-state solutions of the non-equilibrium dense gas flows based on the Shakhov-Enskog model, resolving the problems of slow convergence and requirement of ultra-fine grids in near-continuum flows that exist in the conventional iterative scheme. The key ingredient of GSIS is the tight coupling of the mesoscopic kinetic equation and the macroscopic synthetic equations that are exactly derived from the kinetic equation. On the one hand, high-order terms computed from the velocity distribution function provide the higher-order constitutive relations describing the non-equilibrium effects for the macroscopic synthetic equations. On the other hand, the macroscopic quantities obtained from the macroscopic synthetic equations are used to guide the evolution of the velocity distribution function in the kinetic equation. The efficiency and accuracy of GSIS are demonstrated in several test cases, including the shock wave passing through a cylinder and the pressure-driven dense gas flows passing through parallel plates and porous media. The effects of denseness are analyzed in a wide range of gas rarefaction.

Kinetic model of multi-component polyatomic gas mixture considering internal energy excitation

In this paper, a kinetic Boltzmann model equation with internal degrees of freedom is established for thermodynamic non-equilibrium multi-component polyatomic gas mixture by using continuous energy levels to deal with rotational energy and vibrational energy. The normalization and conservativeness of the kinetic model are analyzed and proved. Then, numerical algorithm and wall boundary conditions for solving the model equation are given. Then, normal shock-wave structure flow problem and pressure/temperature gradient driven micro-channel flow problem for the two-component gas mixture are used to verify the reliability of the model. The simulation results are in good agreement with those obtained by other methods, which verifies the reliability of the model and algorithm in this paper. Finally, taking 25 N attitude control engine two-dimensional profile as the object, the study for two-dimensional profile nozzle internal and external mixed flow problem is carried out, and the influence of different inlet rarefaction levels on the molecular transports of mixed flow field and the molecular transport phenomena of mixed flow field with different number of components are discussed.

Capturing non-equilibrium in hypersonic flows: Insights from a two-temperature model in polyatomic rarefied gases

The study utilizes a two-temperature model to analyze non-equilibrium in normal shocks within hypersonic flows in polyatomic rarefied gases. Derived from the extended second law of thermodynamics, this model separates translational and internal temperatures in polyatomic gases, providing a more accurate depiction of non-equilibrium gas flow compared to classical theories like the Navier–Stokes and Fourier (NSF) system. Notably, the analysis reveals that the two-temperature model incorporates an additional contribution to the heat flux due to the gradient of the dynamic temperature, resulting in improved accuracy, especially for high Mach numbers. Results show that the model gives satisfactory shock density and temperature profiles up to Mach 10, with very good agreement observed up to Mach 6.1 compared to the classical NSF model. We conduct an order of magnitude analysis on the dynamic temperature and heat flux gradients appearing in the new constitutive equation using the Mott-Smith method. This analysis highlights the impact of these terms on accurately modeling polyatomic gas behavior in high-speed flows. The effects of bulk viscosity and incoming temperature on shock profiles are also investigated, contributing to a better understanding of shock wave structures in polyatomic gases and their implications for hypersonic flow dynamics.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

An Open-Source Code for High-Speed Non-Equilibrium Gas–Solid Flows in OpenFOAM

This paper details the development and verification tests of an open-source code named hy2LPTFoam intended for solving high-speed non-equilibrium gas-particle flows in OpenFOAM. The solver, based on hy2Foam for high-speed non-equilibrium gas flow, integrates multiple particle force models, heat transfer models, the diffusion-based smoothing method, and the MPPIC method. The verification tests incorporate interactions between shock waves and particle curtains with varying particle volume fractions, a JPL nozzle generating a two-phase gas–particle flow, and a Mars entry body with two particle inflow mass fractions. The tests yield good physical agreement with numerical and experimental data from the literature.

Macroscopic modeling of gas permeability in hierarchical micro/nanoporous media: A unified characterization of rarefaction using Klinkenberg theory and equivalent diameter

Estimating gas transport through a hierarchical micro/nanoporous system is challenging due to non-equilibrium gas dynamics. The primary difficulty lies in determining the rarefaction level, because identifying a representative flow dimension in a complex porous system with multiple pore scales is not straightforward. Our study performed a pore-level analysis for gas permeability in dual-scale porous media with varying porosity, throat size, and secondary pore size under different rarefaction conditions. We found that secondary porosity negatively affects permeability due to increased friction forces, with this influence growing as the secondary pore size and porosity increase until the secondary pore becomes comparable to the throat. However, rarefaction reduces the effects of secondary pores due to boundary slip. Traditional Knudsen number (Kn) calculations based on Darcy-defined height failed to accurately describe the rarefaction effects on gas permeability. Instead, we introduced an equivalent diameter to calculate the Kn, which provided an accurate normalization of apparent gas permeability independent of pore geometry. The extended Kozeny–Carman–Klinkenberg model developed in our previous study successfully yielded a macroscopic model for apparent gas permeability in hierarchical micro/nanoporous systems as a function of the traditional Darcy height and porosity.

Numerical investigation of thermochemical non-equilibrium effects in Mach 10 scramjet nozzle

Abstract High-temperature non-equilibrium effects are prominent in scramjet nozzle flows at high Mach numbers. Hence, the thermochemical non-equilibrium gas model incorporating the vibrational relaxation process of molecules in the hydrocarbon-air reaction is developed to numerically simulate the flow of a hydrocarbon fuel scramjet nozzle at Mach 10. Besides, the results computed by the models of the thermally perfect gas, chemically non-equilibrium gas, and thermally non-equilibrium chemically frozen gas are applied for comparative studies. Results indicate that chemical non-equilibrium effects are more significant for the flow-field structure and parameters compared to thermal non-equilibrium effects. Meanwhile, vibrational relaxation and chemical reactions interact in the flow-field. The heat released from the chemical reactions in the flow-field of the thermochemical non-equilibrium gas model makes the thermal non-equilibrium effects weaker compared to the thermally non-equilibrium chemically frozen gas model; the chemical reactions in the thermochemical non-equilibrium gas model are more intense than in the chemically non-equilibrium gas model. Due to the slow relaxation of vibrational energy, the thermal non-equilibrium models predicted nozzle thrust lower than the thermal equilibrium models by approximately 1.11% to 1.33%; when considering the chemical reactions, the chemical non-equilibrium models predicted nozzle thrust higher than the chemical frozen models by approximately 7.30% to 7.54%. Hence, the structural design and performance study of the high Mach numbers scramjet nozzle must consider thermochemical non-equilibrium effects.

Numerical investigations on flow and heat transfer characteristics of a high-enthalpy double-cone in thermal and chemical non-equilibrium for hypersonic propulsion

There are strong interests in hypersonic propulsion communities in predicting the flow and heat transfer characteristics over a hypersonic vehicle. However, this is quite challenging due to the emergence of thermal and chemical out-of-equilibrium effects as prominent in a high-temperature flow. In this work, we conduct 2D numerical investigations on the hypersonic double cone flow in a surrounding of the stagnation enthalpy of 15.23 MJ/kg by solving the laminar Navier–Stokes equation systems with the vibrational non-equilibrium processes and seven species air reactions. We then conduct the sensitivities analyses on the flow and heat transfer features, as different transport models and different thermochemical models are applied. It is shown that the aerodynamic heat is more sensitive to the transport models than the flow, and the transport coefficients computed by the Gupta-Yos model are more suitable for predicting the double cone flow at the high enthalpy. On the other hand, the flow field of the high enthalpy double cone is more sensitive to the thermochemical models than the heat transfer on the wall excluding the peak position. Behind the detached shock, chemical reactions are found to enhance thermal or vibrational out-of-equilibrium effects. However, vibrational non-equilibrium effects are revealed to weaken the degree of molecule dissociation reactions. Two flow separations occur in the flow field computed by using the thermal non-equilibrium gas models, but the flows of thermal equilibrium gas models separate only once. The maximum values of wall parameters without chemical reactions are found to deviate greatly from the experimental measurements. The thermochemical out-of-equilibrium gas model can predict the separation region size closer to the experiment than the two reported numerical results. The wall static pressure and heat flux are also shown to well match the experimental results. The present work should shed lights on better understanding thermochemical non-equilibrium effects of complex flow phenomenon like the shock wave/boundary layer interaction in hypersonic propulsion with high enthalpy surroundings.

Influence of thermochemical non-equilibrium effects on non-uniform entrance scramjet nozzle

The high Mach number scramjet nozzle flow has prominent non-uniform characteristics and high-temperature non-equilibrium effects. Thus, the thermochemical non-equilibrium gas model was developed for simulating flows in the non-uniform and uniform entrance of a Mach 10 hydrocarbon fuel scramjet nozzle. The results show that the non-uniform entrance has a higher non-uniform degree in the flow-field compared to the uniform entrance. Additionally, the expansion degree inside the nozzle is greater and the thermal non-equilibrium effects downstream of the nozzle exit are stronger. The non-uniform entrance condition has a more intense chemical reaction near the nozzle exit but has less impact on the flow parameters. For the nozzle performance, under the influence of the non-uniform entrance, the greater gas expansion degree inside the nozzle results in a lower pressure acting on the upper wall, and then the nozzle thrust decreases by 1.68%. Hence, the structural design and performance optimization of the high Mach number scramjet nozzle must consider entrance non-uniform characteristics and thermochemical non-equilibrium effects.

Spin Rotations in a Bose-Einstein Condensate Driven by Counterflow and Spin-Independent Interactions.

We observe spin rotations caused by atomic collisions in a nonequilibrium Bose-condensed gas of ^{87}Rb. Reflection from a pseudomagnetic barrier creates counterflow in which forward- and backward-propagating matter waves have partly transverse spin directions. Even though inter-atomic interaction strengths are state independent, the indistinguishability of parallel spins leads to spin dynamics. A local magnetodynamic model, which captures the salient features of the observed spin textures, highlights an essential connection between four-wave mixing and collisional spin rotation. The observed phenomenon is commonly thought not to occur in Bose condensates; our observations and model clarify the nature of these effective-magnetic spin rotations.

Constructing structure-gradient silicone rubber/CNTs foam with desirable resilience and strength via green supercritical CO2 foaming of non-equilibrium gas concentration profiles

In comparison to uniform foams, structure-gradient foams can offer practical solutions that optimize their resilience and strength. However, controlling the gradient structure is a challenge, especially in silicone rubber (PMVS) based foams. Herein, a facile and eco-friendly one-step supercritical CO2 (scCO2) foaming technique is presented for fabricating structure-gradient PMVS/CNTs composite foams utilizing a non-equilibrium gas concentration profile. Compared with conventional scCO2 foaming, this method features a more energy-efficient process (lower saturation temperature and time) and results in lighter foam products (density as low as 0.11 g/cm3). Moreover, the formation mechanism of gradient cellular structures in PMVS/CNTs foams and the means of adjusting the cell gradient degree are further discussed. Finally, cyclic compression results demonstrate that PMVS/CNTs foams can be optimized to achieve desirable mechanical strength and resilience by adjusting cell gradient degree.

Motility-induced coexistence of a hot liquid and a cold gas

If two phases exist at the same time, such as a gas and a liquid, they have the same temperature. This fundamental law of equilibrium physics is known to apply even to many non-equilibrium systems. However, recently, there has been much attention in the finding that inertial self-propelled particles like Janus colloids in a plasma or microflyers could self-organize into a hot gas-like phase that coexists with a colder liquid-like phase. Here, we show that a kinetic temperature difference across coexisting phases can occur even in equilibrium systems when adding generic (overdamped) self-propelled particles. In particular, we consider mixtures of overdamped active and inertial passive Brownian particles and show that when they phase separate into a dense and a dilute phase, both phases have different kinetic temperatures. Surprisingly, we find that the dense phase (liquid) cannot only be colder but also hotter than the dilute phase (gas). This effect hinges on correlated motions where active particles collectively push and heat up passive ones primarily within the dense phase. Our results answer the fundamental question if a non-equilibrium gas can be colder than a coexisting liquid and create a route to equip matter with self-organized domains of different kinetic temperatures.

Effect of Hyperthermal Hybrid Gas Composition on the Interfacial Oxidation and Nitridation Mechanisms of Graphene Sheet.

Graphene is one of the most promising thermal protection materials for high-speed aircraft due to its lightweight and excellent thermophysical properties. At high Mach numbers, the extremely high postshock temperature would dissociate the surrounding air into a mixture of atomic and molecular components in a highly thermochemical nonequilibrium state, which greatly affects the subsequent thermal chemical reactions of the graphene interface. Through establishing a reactive gas-solid interface model, the reactive molecular dynamics method is employed in this study to reveal the influences of the thermochemical nonequilibrium gas mixture on the thermal oxidation and nitridation mechanisms of graphene sheet. The results show that three distinctive stages can be recognized during bombardment of various nonequilibrium gas components toward the graphene sheet: (i) collision and surface adsorption stage, (ii) gas-solid heterogeneous reaction stage, and (iii) gas phase homogeneous reaction stage. The surface catalysis effect is found to be dominant during the first two stages, which can influence the following ablation behavior of graphene significantly at high-temperature conditions. Moreover, surface catalysis, oxidation, nitridation, and ablation mechanisms between nonequilibrium gas and graphene interface are revealed, which is of high relevance for future interfacial design and application of graphene as a thermal protection material.

Fully Parabolized Hypersonic Sonic Boom Prediction with Real Gas and Viscous Effects

We present a methodology to predict the aerodynamic near-field and sonic boom signature from slender bodies and waveriders using a fully parabolized approach. We solve the parabolized Navier–Stokes equations, which are integrated via spatial marching in the streamwise direction. We find that unique physics must be accounted for in the hypersonic regime relative to the supersonic, which includes viscous, nonequilibrium, and real gas effects. The near-field aerodynamic pressure is propagated through the atmosphere to the ground via the waveform parameter method. To illustrate the approach, three bodies are analyzed: the Sears–Haack geometry, the HIFiRE-5, and a power-law waverider. Ambient Mach numbers range from 4 through 15. The viscous stress tensor is essential for accurate hypersonic prediction. For example, viscous effects increase near-field and sonic boom overpressure by 15.7 and 8.49%, respectively, for the Sears–Haack geometry. The difference between viscous and inviscid predictions of the near-field is due to the hypersonic boundary layer. The computational cost for predicting the near-field is approximately 6.6% relative to fully nonlinear computational fluid dynamics.

A unified stochastic particle method with spatiotemporal adaptation for simulating multiscale gas flows

Recently, the Unified Stochastic Particle (USP) method has emerged as a promising approach for multiscale particle simulations, tailored specifically for non-equilibrium gas flow scenarios. However, in typical simulations of multiscale flow fields, the time step and grid size of the original USP method remain constrained by the smallest spatiotemporal scale of local flow field, limiting its true multiscale simulation potential. To overcome this critical issue, this work introduces a spatiotemporal adaptive USP method. Initially, we analyze the sources of errors in the traditional single-scale stochastic particle methods under large time steps, which leads to the derivation of the governing equations of USP. Building upon this foundation, we propose principles that dictate the rational time step and grid size for an arbitrary USP simulation. Subsequently, we introduce a spatiotemporal adaptive approach primarily based on the local gradient length, leading to the development of a USP solver endowed with spatiotemporal adaptability. The universality, reliability, and efficiency of the algorithm have been verified through an examination of three cases: two-dimensional flow over a cylinder, three-dimensional flow over a blunted body, and flow over a scaled X38-like model. This work forms an integral component of the development initiative for the general USP solver—SPARTACUS—with relevant source code being open-source in an online repository under the GNU General Public License (GPL).

Influence of Gas-Surface and Gas-Gas interactions on the energy accommodation coefficient in non-equilibrium hypersonic gas flows

Accurate accommodation coefficients are crucial in predicting aerodynamic force and heating in hypersonic rarefied flows. This study employs molecular dynamics simulations to investigate the characteristics of non-equilibrium hypersonic flow and its impact on the energy accommodation coefficient (EAC). Our findings reveal that the non-equilibrium gas flow in the gas-surface interaction region comprises an equilibrium subsonic flow, an incident hypersonic flow, and its reflected counterpart. Gas-surface and gas–gas interactions occur simultaneously as gas atoms incident toward the solid surface, with the latter enhancing the EAC by altering the ratios of collision types. Overall, an increase in the ratios of trapping-desorption and permanent-adsorption collisions enhances the EAC at the interface. Results indicate that the number density of the subsonic flow determines the transition from immediate-reflection collisions to trapping-desorption collisions, while the ratio between the potential well depth and the internal energy of the subsonic flow determines the occurrence of permanent-adsorption collisions. These findings disclose the influence of the gas-surface and gas–gas interactions on the atomic collision types and EAC in a hypersonic flying interface, providing valuable insights for predicting the aerodynamic heating in rarefied gas flow.

An efficient solution of the multi-term multi-harmonic electron Boltzmann equation for use in global models

Solving the electron Boltzmann equation is an essential but costly step in simulating low-temperature plasma kinetics. This work addresses the problem by introducing a solution of the general multi-term multi-harmonic Boltzmann equation (MTMH-BE) optimized for electrons in time-dependent non-equilibrium gases and electric fields. This is accomplished by configuring the numerical Jacobian as the product of a time-independent sparse matrix and an array of time-dependent coefficients. As this approach requires a fixed energy grid, the MTMH-BE is discretized using a finite volume scheme with support for non-uniform cell size, enabling robust solutions across a wide range of reduced field strengths. The new MTMH-BE solver is tested against a range of benchmarks, demonstrating excellent agreement with existing methods. Performance with time-dependent electron energy is evaluated using LoKI-B's pulsed electric field model, with the new solver demonstrating improved wall time scaling and an improvement in grid convergence of over three orders of magnitude. Alternatively, performance with quasi-stationary electrons in an evolving non-equilibrium gas is tested using a nitrogen cross-section set with up to 58 vibrational states and 3,570 processes. Here, the MTMH-BE solver achieves a wall time reduction of up to 99.6% over MultiBolt for error-controlled simulations in the two-term limit, reducing the average solution time from 1-2 seconds to less than 10 ms. These results indicate that the new MTMH-BE solver can improve accuracy and performance in state-to-state global models without the use of grid refinement, cross-section pruning, and other time-saving procedures.

Fully time-dependent cloud formation from a non-equilibrium gas-phase in exoplanetary atmospheres

Context. Recent observations suggest the presence of clouds in exoplanet atmospheres, but they have also shown that certain chemical species in the upper atmosphere might not be in chemical equilibrium. Present and future interpretation of data from, for example, CHEOPS, JWST, PLATO, and Ariel require a combined understanding of the gas-phase and the cloud chemistry. Aims. The goal of this work is to calculate the two main cloud formation processes, nucleation, and bulk growth consistently from a non-equilibrium gas phase. The aim is also to explore the interaction between a kinetic gas-phase and cloud microphysics. Methods. The cloud formation is modelled using the moment method and kinetic nucleation, which are coupled to a gas-phase kinetic rate network. Specifically, the formation of cloud condensation nuclei is derived from cluster rates that include the thermochemical data of (TiO2)N from N = 1 to 15. The surface growth of nine bulk Al, Fe, Mg, O, Si, S, and Ti binding materials considers the respective gas-phase species through condensation and surface reactions as derived from kinetic disequilibrium. The effect of the completeness of rate networks and the time evolution of the cloud particle formation is studied for an example exoplanet, HD 209458 b. Results. A consistent, fully time-dependent cloud formation model in chemical disequilibrium with respect to nucleation, bulk growth, and the gas-phase is presented and first test cases are studied. This model shows that cloud formation in exoplanet atmospheres is a fast process. This confirms previous findings that the formation of cloud particles is a local process. Tests on selected locations within the atmosphere of the gas-giant HD 209458 b show that the cloud particle number density and volume reach constant values within 1 s. The complex kinetic polymer nucleation of TiO2 confirms results from classical nucleation models. The surface reactions of SiO[s] and SiO2[s] can create a catalytic cycle that dissociates H2 to 2 H, resulting in a reduction of the CH4 number densities.

Regression models for calculating state-to-state coefficients of the rate of vibrational energy exchanges

The paper proposes an effective algorithm for solving problems of nonequilibrium gas dynamics taking into account detailed state-to-state vibrational kinetics. One of the problems of traditional methods is their high computational complexity, which requires a lot of time and memory. The work explored the possibilities of using relaxation rate prediction to improve the performance of numerical simulations of nonequilibrium oxygen flows instead of direct calculations. For this purpose, an approach based on nonlinear regression analysis was used, which made it possible to obtain computationally efficient approximation formulas for the energy exchange rate coefficients in the model of a forced harmonic oscillator, taking into account free rotations (FHO-FR), to significantly increase the calculation speed while maintaining accuracy, and to construct an optimized model FHO-FR-reg. Using the obtained regression formulas, numerical modeling was carried out, which made it possible to validate the model for the problem of oxygen flow behind an incident and reflected shock wave. A comparison between the forced harmonic oscillator (FHO) and FHO-FR models is not possible due to the high computational complexity of the second model. With the advent of a common approximation model, it became possible to compare simulation results for these models. Numerical calculations have shown that the FHO-FR-reg model gives values of gas-dynamic parameters close to the FHO model. The developed regression models make it possible to speed up the solution of the problem of modeling oxygen relaxation several times compared to other models of similar accuracy.

B3C2P3 monolayers based highly sensitive and selective room-temperature gas sensors for reusable NO and NO2 detection

Motivated by the development of high-performance gas sensors using two-dimensional (2D) nanomaterials, herein using the combination methods of density functional theory (DFT) and nonequilibrium Green's function (NEGF), the structural, electronic, transport, and gas sensing properties of the B3C2P3 monolayer adsorbed with various gases (NO2, NO, NH3, SO2, H2S, CO2, CO, H2O, O2 and CH2O) have been systematically studied. The NO2 (or NO) molecules were chemically adsorbed (or strong physisorbed) on the B3C2P3 monolayer, while other molecules were weak-physically adsorbed on the substrate. Furthermore, only the NO2 and NO adsorption results in the remarkable change in electronic properties of B3C2P3. The transport properties such as current-voltage (I–V) characteristics and transmission functions indicate the B3C2P3 monolayer is highly sensitive and selective towards NO2 and NO gases. The recovery times of NO2 and NO were calculated to be 2.27 and 0.0785 ms at 300 K, indicating the reusability of the B3C2P3-based NO2 (NO) sensors. The existence of moisture hardly influences the adsorption of NO and NO2 on the B3C2P3 monolayer. Our results indicated that the B3C2P3 monolayer, as a room-temperature reusable gas sensor, is highly sensitive and selective for NO2 and NO detection regardless of the existence of moisture, O2 and CO2.