Translational Nonequilibrium Research Articles

An improved continuum model for hypersonic thermal nonequilibrium flow in the near-continuum regime

In this work, the rarefied Couette flow of diatomic gases with thermal nonequilibrium effects is investigated by the direct simulation Monte Carlo (DSMC) method, and a macroscopic computational model is developed to consider the local rarefaction effects for diatomic gases in the near-continuum regime. The nonlinear transport properties of the diatomic gases are studied, indicating that effective viscosity and effective translational thermal conductivity in the shear nonequilibrium state are affected by translational nonequilibrium effects, which obey the same laws for both monatomic and diatomic gases. The transport coefficients of internal energy modes are affected by both translational nonequilibrium and internal energy relaxation, therefore, the effective rotational and vibrational thermal conductivities are related to the effective viscosity through a modified Eucken relation that accounts for internal energy relaxation. Conclusively, effective constitutive relations are newly established as a function of the shear nonequilibrium parameter and the modified Eucken factors for thermal nonequilibrium flows, and these are integrated into the macroscopic two-temperature model. Subsequently, it is assessed in the simulation of hypersonic flows over flat plates and cylinders at various Knudsen numbers. The results show that the surface shear stress and heat flux obtained by the proposed model agree well with the DSMC results, indicating significantly improved performance compared to the conventional Navier–Stokes two-temperature model for hypersonic flows in the near-continuum regime.

Collision enhancement in shocks and its implication on gas-phase detonations: A molecular dynamics and gas-kinetic theory study

Conventional assumption in gas-phase detonations, where shock compression is decoupled from chemical kinetics, predicates on the shock and triple point structures being treated as perfect discontinuities. However, the shock is a region of high translational nonequilibrium three to five mean free paths in thickness. In this study, we use molecular dynamics simulations to probe Ar and N2 shocks focusing on the collision statistics in the shock front. Translationally superheated molecules were identified, as suggested by Zeldovich (Dokl. Akad. Nauk 248 (1979) 349–351), which raise the collision temperature and potentially enhance chemical reaction rates within and ahead of the shock front. We evaluated this reaction rate enhancement effect on stoichiometric H2/O2 ZND detonation and found the effect to be negligible. The triple point region is observed to have a similar distribution of translationally superheated molecules. The temperature in the triple point region in Ar is substantially higher than that in N2; the difference could impact detonation and deflagration-to-detonation characteristics due to diluent differences.

A kinetic approach to studying low-frequency molecular fluctuations in a one-dimensional shock

Low-frequency molecular fluctuations in the translational nonequilibrium zone of one-dimensional strong shock waves are characterized for the first time in a kinetic collisional framework in the Mach number range 2≤M≤10. Our analysis draws upon the well-known bimodal nature of the probability density function (PDF) of gas particles in the shock, as opposed to their Maxwellian distribution in the freestream, the latter exhibiting two orders of magnitude higher dominant frequencies than the former. Inside the (finite-thickness) shock region, the strong correlation between perturbations in the bimodal PDF and fluctuations in the normal stress suggests introducing a novel two-bin model to describe the reduced-order dynamics of a large number of collision interactions of gas particles. Our model correctly predicts two orders of magnitude differences in fluctuation frequencies in the shock vs those in the freestream and is consistent with the small-amplitude fluctuations obtained from the highly resolved direct simulation Monte Carlo computations of the same configuration. The variation of low-frequency fluctuations with changes in the conditions upstream of the shock revealed that these fluctuations can be described by a Strouhal number, based on the bulk velocity upstream of the shock and the shock-thickness based on the maximum density gradient inside the shock, that remains practically independent of Mach number in the range examined.

Unified gas-kinetic wave-particle methods V: Diatomic molecular flow

In this paper, the unified gas-kinetic wave-particle (UGKWP) method is further developed for diatomic gas with the energy exchange between translational and rotational modes for flow study in all regimes. The multiscale transport mechanism in UGKWP is coming from the direct modelling in a discretized space, where the cell's Knudsen number, defined by the ratio of particle mean free path over the numerical cell size, determines the flow physics simulated by the wave particle method. The non-equilibrium distribution function in UGKWP is tracked by the discrete particle and analytical wave. The weights of distributed particle and wave in different regimes depend on cell's Knudsen number, where distinguishable macroscopic flow variables of particle and wave are updated inside each control volume. The UGKWP becomes a particle method in the highly rarefied flow regime and converges to the gas-kinetic scheme (GKS) for the Navier-Stokes solution in the continuum flow regime without particles. In comparison with discrete velocity method (DVM)-based unified gas-kinetic scheme (UGKS), the computational cost and memory requirement in UGKWP have been reduced by several orders of magnitude for the high speed and high temperature flow simulation, where the translational and rotational non-equilibrium can be captured accurately in the transition and rarefied regime. As a result, for the hypersonic flow around a flying vehicle, the computation can be conducted using a personal computer for the studies in all regimes. The UGKWP method for diatomic gas will be validated in various cases from one dimensional shock structure to three dimensional flow over a sphere, and the numerical solutions will be compared with the reference DSMC results and experimental measurements.

Plasma induced vibrational excitation of CH4—a window to its mode selective processing

Vibrational excitation of methane is believed to promote chemistry and improve product selectivity, compared to thermal conversion methods. We report on unique direct measurements of vibrational–rotational non-equilibrium in methane plasma. The non-equilibrium is sustained for 50 μs, after which the gas temperature equilibrates with the vibrational temperature at around 900 K. The plasma is generated by applying 200 μs, 30 Hz pulses of microwave radiation to methane at 25 mBar. We demonstrate that in microwave discharges, power transfer to vibrational modes of CH4 is the dominant power transfer mechanism, which leads to creation of a vibrational–translational (VT) non-equilibrium. VT relaxation is determined to be the dominant translational heating mechanism in the discharge. However, the high electron temperature at breakdown also leads to strong electronic excitation which may be responsible for some of the heating. Furthermore, we find that the CH4 vibrational levels are in equilibrium with each other due to fast intra-polyad relaxation (VV), and therefore bending vibrational modes population density is greatly in excess of stretching vibrational modes. The window of opportunity to exploit this non-equilibrium is limited by the VT relaxation timescale, which is approximately 50 μs in our experiment.

Hypersonic approximation for distribution function of pairs of molecules in shock wave

New results of an analytical study of translational nonequilibrium in a shock wave are presented that are associated with the use of the exact expression for the longitudinal kinetic temperature Tx in a simple gas. It is shown that the relative magnitude of the overshoot effect for the distribution function of the pairs of molecules inside the shock front satisfies the principle of independence from the Mach number. Asymptotically exact expressions are obtained for the effect of high-speed overshoot of the pair distribution function: “light” -“heavy” component.

Boltzmann–Curtiss Description for Flows Under Translational Nonequilibrium

Abstract Continuum-based theories, such as Navier–Stokes (NS) equations, have been considered inappropriate for flows under nonequilibrium conditions. In part, it is due to the lack of rotational degrees-of-freedom in the Maxwell–Boltzmann distribution. The Boltzmann–Curtiss formulation describes gases allowing both rotational and translational degrees-of-freedom and forms morphing continuum theory (MCT). The first-order solution to Boltzmann–Curtiss equation yields a stress tensor that contains a coupling coefficient that is dependent on the particles number density, the temperature, and the total relaxation time. A new bulk viscosity model derived from the Boltzmann–Curtiss distribution is employed for shock structure and temperature profile under translational and rotational nonequilibrium. Numerical simulations of argon and nitrogen shock profiles are performed in the Mach number range of 1.2–9. The current study, when comparing with experimental measurements and direct simulation Monte Carlo (DSMC) method, shows a significant improvement in the density profile, normal stresses, and shock thickness at nonequilibrium conditions than NS equations. The results indicate that equations derived from the Boltzmann–Curtiss distribution are valid for a wide range of nonequilibrium conditions than those from the Maxwell–Boltzmann distribution.

Lp time asymptotic decay for general hyperbolic–parabolic balance laws with applications

We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term for relaxation, damping or chemical reaction. The viscosity matrix and the Jacobian matrix of the lower-order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper, we obtain optimal [Formula: see text] decay rates for [Formula: see text]. Our result is general and applies to models such as Keller–Segel equations with logarithmic chemotactic sensitivity and logistic growth, and gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic–parabolic conservation laws and hyperbolic balance laws, respectively.

Influence of Small Xe Additives on the Detonation Threshold for O2−H2−He Mixtures

The effect of a small Xe additive on the conditions of detonation initiation in incident shock waves of various intensities is studied. The experiments are carried out on a shock tube facility with 10% H2 + 5% O2 + 85% He, 10% H2 + 5% O2 + 84.75% He + 0.25% Xe, and 10% H2 + 5% O2 + 84.5% He + 0.5% Xe mixtures. The addition of Xe led to a shift in the detonation threshold toward weaker shock waves. This effect is probably due to a significant increase in the frequency of high-energy collisions between O2 and Xe molecules in the shock wave front in comparison with that characteristic of the equilibrium behind the wave, a factor that significantly accelerates the chemical reaction between O2 and H2 behind the front. The effect is a consequence of the formation of a specific translational nonequilibrium in the wave front. A previously performed numerical study of the distributions of pairs of O2 and Xe molecules in the shock wave front shows that this effect can be enhanced by decreasing the Xe concentration from 0.5 to 0.25%. The experiment performed indirectly confirms this conclusion. It turns out that, for the mixture with 0.25% Xe, the detonation threshold shifts more strongly to the region of weaker shock waves than for the mixture with 0.5% Xe. This result gives additional arguments in favor of the assumption that this effect is due to the specifics of the translational nonequilibrium in the wave front.

<i>L</i><sup><i>P</i></sup> decay for general hyperbolic-parabolic systems of balance laws

We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.

Numerical study of translational nonequilibrium in an He–Xe mixture by direct simulation Monte Carlo

The distributions of pairs of particles over relative velocities at the shock wave front in He with a small Xe additive have been studied. It has turned out that the values of the distributions over relative velocities for an Xe–Xe atomic pair far (up to 109 times) exceed their equilibrium values behind a shock wave within a narrow part of its front at high velocities of the wave and small Mach numbers (M = 2). This feature is lacking in the distributions of He–Xe atomic pairs over relative velocities.

Global existence theory for general hyperbolic-parabolic balance laws with application

We study a general system of hyperbolic-parabolic balance laws in [Formula: see text] space dimensions ([Formula: see text]). The system has rank deficient viscosity matrices and a lower order term whose Jacobian matrix is rank deficient as well. We consider the Cauchy problem when initial data are small perturbations of a constant equilibrium state. Under a set of reasonable assumptions including Kawashima–Shizuta condition, we establish the existence of solution global in time via energy method. The proposed assumptions are sufficiently general for applications to physical models such as electro-magneto flows and physical gas flows. In particular, we study the gas flow with an internal non-equilibrium mode besides the translational non-equilibrium. The general result in this paper recovers the existing results in literature on hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively, as two special cases.

On the role of 'hot' atoms in plasma-assisted ignition.

This paper discusses the processes leading to the formation of 'hot' atoms and radicals possessing excessive translational energy in high-voltage NS pulse discharges. It is shown that the formation of such 'hot' atoms occurs efficiently both in the dissociation of molecules by direct electron impact, and in the collisional quenching of electronically excited states. Depending on the magnitude of the reduced electric field in the discharge, reactions of these 'hot' atoms increase the initial concentration of radicals in the discharge afterglow two to three times when compared with the values calculated without effects of translational non-equilibrium. The role of thermally non-equilibrium excitation has been demonstrated in the formation of the initial distribution of the chemically active components in the mixture and its influence on the kinetics of ignition initiation at low and high temperatures. It was found that in undiluted mixtures the presence of 'hot' atoms can significantly decrease an ignition threshold and accelerate a low-temperature oxidation.

Numerical Prediction of Hypersonic Flowfields Including Effects of Electron Translational Nonequilibrium

The degree of electron thermal nonequilibrium occurring in continuum, hypersonic, slender body, and blunt body flows is investigated. The effect of thermal nonequilibrium between the electron translational and vibrational-electronic modes on the predicted electron density and electron temperature is quantified at flight conditions characteristic of a slender hypersonic vehicle, as well as at a higher energy, superorbital flight condition of a blunt reentry vehicle. The most significant effect of electron thermal nonequilibrium on the flowfield is through the influence of the electron temperature on the magnitude of the chemical reaction rates in the high density shock layer. A twofold reduction in peak plasma density is predicted in the flow around the slender body when the electron nonequilibrium model is used, and this results in better agreement between the simulation results and the experimental electron density measurements. A change in the shape of the electron density profile with the use of the electron nonequilibrium model is predicted along the stagnation streamline of the blunt body flow. In both cases, the changes in predicted electron density are much more pronounced at the higher density flight conditions examined. In all cases, the rise of the electron temperature precedes the rise of the vibrational-electronic temperature along the stagnation streamline. These results have potentially important implications for the prediction of the onset of radio frequency blackout during the flights of next generation hypersonic vehicles, a phenomenon that is governed by the properties of the electron gas.

Generalized Boltzmann solution for non-equilibrium flows and the computation of flowfields of binary gas mixture

Hypersonic flows about space vehicles produce flowfields in thermodynamic non-equilibrium with the local Knudsen numbers Kn which may lie in all the three regimes: continuum, transition and rarefied. Continuum flows can be modeled accurately by solving the Navier–Stokes (NS) equations; however, the flows in transition and rarefied regimes require a kinetic approach such as the direct simulation Monte Carlo (DSMC) method or the solution of the Boltzmann equation. The Boltzmann equation and the general solution approach, using the splitting method, will be introduced in this paper. Details of the method used for solving both the classical Boltzmann equation (CBE) and the generalized Boltzmann equation (GBE) are also provided. The gas mixture discussed in this paper may consist of both monoatomic and diatomic gases. In particular, the method is applied to simulate two of the three primary constituents of air (N2, O2, and Ar) in a binary mixture at 1:1 density ratio at Mach 2 and 5, with gases in translational, rotational and vibrational non-equilibrium.

Extension of a Modular Particle-Continuum Method to Vibrationally Excited, Hypersonic Flows

A modular particle-continuum method is extended to include vibrationally excited energy modes to simulate hypersonic steady-state flows that exhibit small regions of translational nonequilibrium in a mainly continuum flowfield. This method loosely couples an existing direct simulation Monte Carlo code to a Navier–Stokes solver (computational fluid dynamics) while allowing both time step and cell size to be completely decoupled between each method. A new information-transfer scheme that controls the inherently large statistical scatter of vibrational energies in low-temperature regions is described and tested. Two vibrational-relaxation models are implemented to test the sensitivity in agreement between direct simulation Monte Carlo and the modular particle-continuum method.By limiting the size of the direct simulationMonteCarlo region to only areas in translational nonequilibrium andmaintaining consistent physicalmodels in both computationalfluid dynamics anddirect simulationMonteCarlo modules, the modular particle-continuummethod is able to reproduce full direct simulationMonte Carlo results for flowwith globalKnudsen number of 0.01while decreasing the computational time required by a factor of about four.

Multiple temperature model for the information preservation method and its application to nonequilibrium gas flows

The information preservation (IP) method has been successfully applied to various nonequilibrium gas flows. Comparing with the direct simulation Monte Carlo (DSMC) method, the IP method dramatically reduces the statistical scatter by preserving collective information of simulation molecules. In this paper, a multiple temperature model is proposed to extend the IP method to strongly translational nonequilibrium gas flows. The governing equations for the IP quantities have been derived from the Boltzmann equation based on an assumption that each simulation molecule represents a Gaussian distribution function with a second-order temperature tensor. According to the governing equations, the implementation of IP method is divided into three steps: molecular movement, molecular collision, and update step. With a reasonable multiple temperature collision model and the flux splitting method in the update step, the transport of IP quantities can be accurately modeled. We apply the IP method with the multiple temperature model to shear-driven Couette flow, external force-driven Poiseuille flow and thermal creep flow, respectively. In the former two cases, the separation of different temperature components is clearly observed in the transition regime, and the velocity, temperature and pressure distributions are also well captured. The thermal creep flow, resulting from the presence of temperature gradients along boundary walls, is properly simulated. All of the IP results compare well with the corresponding DSMC results, whereas the IP method uses much smaller sampling sizes than the DSMC method. This paper shows that the IP method with the multiple temperature model is an accurate and efficient tool to simulate strongly translational nonequilibrium gas flows.

Theoretical investigation of nonequilibrium processes in shock wave in bubbly liquid

The effects related to a translational nonequilibrium at the shock wave front in a bubbly liquid flow with volume gas contents within 0.3 ≤ φ ≤ 0.98 have been theoretically studied. Analytical expressions for the longitudinal and transverse pressures, longitudinal and transverse temperatures, viscous stresses, and heat flux are obtained and the corresponding values are calculated for various Mach numbers and gas contents in the bubbly flow. The influence of the translational nonequilibrium at the shock wave front on the bubble fragmentation process is considered.

A hybrid particle approach for continuum and rarefied flow simulation

A hybrid particle scheme is presented for the simulation of compressible gas flows involving both continuum regions and rarefied regions with strong translational nonequilibrium. The direct simulation Monte Carlo (DSMC) method is applied in rarefied regions, while remaining portions of the flowfield are simulated using a DSMC-based low diffusion particle method for inviscid flow simulation. The hybrid scheme is suitable for either steady state or unsteady flow problems, and can simulate gas mixtures comprising an arbitrary number of species. Numerical procedures are described for strongly coupled two-way information transfer between continuum and rarefied regions, and additional procedures are outlined for the determination of continuum breakdown. The hybrid scheme is evaluated through a comparison with DSMC simulation results for a Mach 6 flow of N 2 over a cylinder, and good overall agreement is observed. Large potential efficiency gains (over three orders of magnitude) are estimated for the hybrid algorithm relative to DSMC in a simple example involving a rarefied expansion flow through a small nozzle into a vacuum chamber.

Effect of an Xe additive on the degree of translational nonequilibrium of an He-O2-H2 mixture in a shock wave front

Computer simulations of the experimentally observed detonation threshold shift upon adding small amounts of Xe with the use of the direct simulation Monte Carlo method once again confirmed that this effect is associated with a substantial increase in the degree of translational nonequilibrium in the shock wave front. It was demonstrated that, when the presence of hydrogen in the mixture is taken into account, the calculated degree of translational nonequilibrium becomes higher. In addition, the effect of Xe was observed to grow stronger as the Xe concentration was decreased from 0.5 to 0.25%.