Rotational Nonequilibrium Research Articles

Effect of multi-temperature models on heat transfer and electron behavior in hypersonic flows

In hypersonic computational fluid dynamics, the two-temperature (2-T) model is widely used to simulate thermochemical nonequilibrium. The 2-T model incorporates translational-rotational and electron-electronic-vibrational energies, assuming that the integrated energies have the equivalent temperature. In this study, multi-T models are constructed to accurately predict the effects on heat flux and free electrons due to the separation of energy modes under hypersonic environments. The three-temperature (3-T) model separates the electron-electronic energy from the electron-electronic-vibrational energy of the 2-T model. The 3-T model can accurately predict the distribution and temperature of free electrons by separating the energy of free electrons, which has different characteristics from heavy particles. The four-temperature model treats rotational energy as a nonequilibrium energy mode, distinct from translational-rotational energy. While the rotational temperature reaches equilibrium rapidly at low temperatures, at high-temperature regime rotational temperature shows a relaxation time similar to that of vibrational temperature, which cannot be ignored. To develop multi-T models, electron-vibrational relaxation and translational-rotational relaxation, which are omitted in the 2-T model, are considered. Various flight test and ground facility conditions are analyzed to verify the effects of electron and heat flux under circumstances that include shock, expansion, and shock wave boundary layer interaction. The results of the multi-T models show significant differences in electron temperature and distribution caused by electron-electronic nonequilibrium. Additionally, rotational nonequilibrium increases the shock standoff distance and alters the electron distribution at high altitudes. The heat flux difference across multi-T models is found to be negligible, except in the high degree of ionization condition.

Adaptive unified gas-kinetic scheme for diatomic gases with rotational and vibrational nonequilibrium

Multiscale nonequilibrium physics at large variations of local Knudsen number are encountered in applications of aerospace engineering and micro-electro-mechanical systems, such as high-speed flying vehicles and low pressure of the encapsulation. An accurate description of flow physics in all flow regimes within a single computation requires a genuinely multiscale method. The adaptive unified gas-kinetic scheme (AUGKS) is developed for such multiscale flow simulation. The AUGKS applies discretized velocity space to accurately capture the non-equilibrium physics in the multiscale UGKS, and adaptively employs continuous distribution functions following Chapman–Enskog expansion to efficiently recover near-equilibrium flow region in GKS. The UGKS and GKS are dynamically connected at the cell interface through the fluxes from the discretized and continuous gas distribution functions, which avoids any buffer zone between them. In this study, the AUGKS with rotation and vibration non-equilibrium is developed based on a multiple temperature relaxation model. The real gas effect in different flow regimes has been properly captured. To capture aerodynamic heating accurately, the heat flux modifications from the rotation and vibration modes are also included in the current scheme. Unstructured discrete particle velocity space is adopted to further improve the computational performance of the AUGKS. Numerical tests, including Sod tube, normal shock structure, high-speed flow around the two-dimensional cylinder and three-dimensional sphere and space vehicles, and an unsteady nozzle plume flow from the continuum flow to the background vacuum, have been conducted to validate the current scheme. In comparison with the original UGKS, the current scheme speeds up the computation, reduces the memory requirement, and maintains the equivalent accuracy for multiscale flow simulation, which provides an effective tool for nonequilibrium flow simulations, especially for the flows at low and medium speed.

Unified gas-kinetic wave-particle methods VII: Diatomic gas with rotational and vibrational nonequilibrium

Hypersonic flow around a vehicle in near space flight is associated with multiscale non-equilibrium physics at a large variation of local Knudsen number from the leading edge highly compressible flow to the trailing edge particle free transport. To accurately capture the solution in all flow regimes from the continuum Navier–Stokes solution to the rarefied gas dynamics in a single computation requires genuinely multiscale method. The unified gas-kinetic wave-particle (UGKWP) method targets on the simulation of such a multiscale transport. Due to the wave-particle decomposition, the dynamics in the Navier–Stokes wave and kinetic particle transport has been unified systematically and efficiently under the unified gas-kinetic scheme (UGKS) framework. In this study, the UGKWP method with the non-equilibrium among translation, rotation and vibration modes, is developed based on a multiple temperature relaxation model. The real gas effect for high speed flow in different flow regimes has been properly captured. Numerical tests, including Sod tube, normal shock structure, hypersonic flow around two-dimensional cylinder and three-dimensional flow around a sphere and space vehicle, have been conducted to validate the UGKWP method. In comparison with the discrete velocity method (DVM)-based Boltzmann solver and particle-based direct simulation Monte Carlo (DSMC) method, the UGKWP method shows remarkable advantages in terms of computational efficiency, memory reduction, and automatic recovering of multiscale solution in the high speed flow simulations.

Molecular beam scattering of a diatomic molecule from a solid surface in case of strong rotational non-equilibrium

In this work, a numerical study on molecular beam scattering of a nitrogen molecule from a graphite surface has been performed. The study was carried out using the molecular dynamics method. The focus of the study is mainly placed on investigating the scattering dynamics in the case of strong rotational non-equilibrium, defined here as a state in which rotational temperature of a molecule strongly deviates from the room temperature. To that end, the incident beam velocity and initial rotational energy of nitrogen molecules have been varied greatly in order to capture a broad range of possible initial states. The obtained results provide valuable insight into the nature of energy transfer occurring during collisions and help to quantify the intensity of rotational–translational coupling between inner kinetic modes in gas–surface collisions. Consequently, the collected data can potentially be used for more accurate characterization of the respective phenomena and improve the quality of boundary models used in rarefied gas flow simulations.

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.

Prediction of shock standoff distance with modified rotational relaxation time of air mixture

The rotational relaxation time of an air mixture is modified as an approach to improve accuracy when predicting hypersonic shock standoff distance. A novel atomistic quasi-classical trajectory (QCT) method with a modified approach is devised to drastically reduce computational cost, and rigorously model the rotational relaxation time of N2 in N2–N and N2–N2 collisions. The calculated full sets of rotational state-to-state transition rates obtained by the QCT method are fed into the rotational state-resolved master equations to determine the rotational relaxation time of N2. Clear discrepancies are observed when the present rotational relaxation time is compared with existing empirical data for N2. The existing empirical model is utilized to determine the rotational relaxation time of other atmospheric gas species. Then the present set of rotational relaxation times for the air mixture is employed to predict the hypersonic shock standoff distance over a blunt body of the ground and flight experiments. Compared with the results from the two-temperature model, the rotational nonequilibrium enlarges the hypersonic shock standoff distance. This increase in shock standoff distance by the rotational nonequilibrium is attributed to the delay in chemical reactions inside the shock layers. The accuracy of the predicted measured shock standoff distance is improved by considering the present rotational relaxation time of the air mixture.

An ES-BGK model for polyatomic gases in rotational and vibrational nonequilibrium

We propose an extension of the Ellipsoidal–Statistical BGK model to account for discrete levels of vibrational energy in a rarefied polyatomic gas. This model satisfies an H-theorem and contains parameters that allow to fit almost arbitrary values for the Prandtl number and the relaxation times of rotational and vibrational energies. With the reduced distribution technique, this model can be reduced to a three distribution system that could be used to simulate polyatomic gases with rotational and vibrational energy for a computational cost close to that of a simple monoatomic gas.

Preliminary numerical study of three-temperature model investigation of hypersonic oxygen flow under rotational nonequilibrium

The effect of rotational nonequilibrium on the macroscopic parameters of the flow behind a normal shock wave in oxygen gas flow has been examined. The electron thermal equilibrium was taken into account where the electron temperature was equal to the vibrational temperature according to Park’s assumption. Therefore, only the effect of rotational nonequilibrium on the translational and vibrational temperature was analyzed. Rotational and vibrational relaxation time for the O2-O2 and O2-O collisions proposed recently by Andrienko and Boyd are used. Also, the O2 dissociation rates proposed by Kim and Park are used. The results obtained with the three-temperature model well reproduce the data obtained in shock tube for the shock velocity of 4.44 km/s.

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.

Effect of rotational nonequilibrium behind a normal shock wave in oxygen and nitrogen flow

In this study, the effect of rotational nonequilibrium on the macroscopic parameters of the flow behind a normal shock wave in oxygen and nitrogen gas flow has been examined. Comparison between three chemical kinetic models including Park89, Park93, and Dunn-Kang was used to describe the chemical nonequilibrium. The obtained results show the significant effect of the rotational nonequilibrium on the relaxation zone and also on the translational and vibrational temperatures and were found to be in a good agreement with the experimental data particularly when the Park89 model was used.

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.

State-to-state chemical kinetic mechanism for HF chemical lasers

State-to-state rotational relaxation processes play a critical role in predicting rotational nonequilibrium, which can significantly affect total laser power and spectral distribution. In this paper, the state-to-state rotational relaxation rate coefficients for the pumping reaction of F + H2, the rotational energy transfer processes of HF(υ, j = 0–10) + HF(υ′, j′ = 0–10) and HF(υ, j = 0–10) + H2 are calculated. Subsequently, a detailed state-to-state chemical kinetic mechanism is constructed including 61 species and 124,482 elementary reactions for HF chemical lasers. All of these rotational relaxation rate coefficients have been expressed with Arrhenius-like formulas. A comparison of the Boltzmann distribution model with the detailed state-to-state chemical mechanism has been analysed and discussed. These results show that rotational nonequilibrium processes can significantly affect the distribution of ro-vibrational excited HF molecules. The rotational distributions of HF(υ = 0–2, j) are far from equilibrium distribution. Conversely, the distributions of HF(υ = 3, 4, j) seem to be a Boltzmann distribution. These results indicate that numerical models must include the rotational nonequilibrium processes and numerical models using a Boltzmann distribution for the rotational populations cannot predict performance of real HF chemical lasers.

Thermal relaxation of molecular oxygen in collisions with nitrogen atoms

Investigation of O2-N collisions is performed by means of the quasi-classical trajectory method on the two lowest ab initio potential energy surfaces at temperatures relevant to hypersonic flows. A complete set of bound-bound and bound-free transition rates is obtained for each precollisional rovibrational state. Special attention is paid to the vibrational and rotational relaxations of oxygen as a result of chemically non-reactive interaction with nitrogen atoms. The vibrational relaxation of oxygen partially occurs via the formation of an intermediate NO2 complex. The efficient energy randomization results in rapid vibrational relaxation at low temperatures, compared to other molecular systems with a purely repulsive potential. The vibrational relaxation time, computed by means of master equation studies, is nearly an order of magnitude lower than the relaxation time in N2-O collisions. The rotational nonequilibrium starts to play a significant effect at translational temperatures above 8000 K. The present work provides convenient relations for the vibrational and rotational relaxation times as well as for the quasi-steady dissociation rate coefficient and thus fills a gap in data due to a lack of experimental measurements for this system.

Master Equation Analysis of Post Normal Shock Waves of Nitrogen

One-dimensional post normal shock flow calculations are carried out using state-of-the-art thermochemical nonequilibrium models. Two-temperature, four-temperature, and electronic master equation coupling models are adopted in the present work. In the four-temperature model, the rotational nonequilibrium is described by Parker and modified Park models. In the electronic master equation coupling model, recently evaluated electron and heavy-particle impacts and radiative transition cross-sections are employed in constructing the system of electronic master equations. In analyzing the shock-tube experiments, the results calculated by the state-of-the-art thermochemical nonequilibrium models are compared with existing shock-tube experimental data. The four-temperature and electronic master equation coupling models with rotational nonequilibrium described by the modified Park model approximately reproduce the measured rotational, vibrational, and electronic temperatures.

Computation of Rarefied Hypersonic Flows Using Modified Form of Conventional Burnett Equations

This paper describes the computations of hypersonic flows in a diatomic gas in rotational nonequilibrium using a newly developed simplified set of Burnett equations designated as simplified conventional Burnett equations. Since the original formulation by Burnett, a number of variations to the original Burnett equations have been proposed, and the differences among these variants and their merits/shortcomings have been described in the literature. A new variant is created based on the conventional Burnett equations for hypersonic flows by neglecting higher-order terms that are inversely proportional to the Mach number. This set of simplified conventional Burnett equations is linearly stable for small disturbances in contrast to the conventional Burnett equations that suffer from Bobylev instability. To simulate the rotational nonequilibrium effect in a diatomic gas, both the Navier–Stokes and the simplified conventional Burnett equations are modified by including a rotational nonequilibrium relaxation model. The flow variables (density and translational and rotational temperature) for one-dimensional shock structure, flow past a two-dimensional cylinder, and an axisymmetric bicone in nitrogen are computed using both the simplified conventional Burnett and Navier–Stokes equations in the continuum-transition regime and are compared with the direct simulation Monte Carlo results. Simplified conventional Burnett calculations are in close agreement with the direct simulation Monte Carlo results at high Mach numbers.

Thermochemical nonequilibrium analysis of O2+Ar based on state-resolved kinetics

The thermochemical nonequilibrium of the three lowest lying electronic states of molecular oxygen, O2(X3Σg-,a1Δg,b1Σg+), through interactions with argon is studied in the present work. The multi-body potential energy surfaces of O2+Ar are evaluated from the semi-classical RKR potential of O2 in each electronic state. The rovibrational states and energies of each electronic state are calculated by the quantum mechanical method based on the present inter-nuclear potential of O2. Then, the complete sets of the rovibrational state-to-state transition rate coefficients of O2+Ar are calculated by the quasi-classical trajectory method including the quasi-bound states. The system of master equations constructed by the present state-to-state transition rate coefficients are solved to analyze the thermochemical nonequilibrium of O2+Ar in various heat bath conditions. From these studies, it is concluded that the vibrational relaxation and coupled chemical reactions of each electronic state needs to be treated as a separate nonequilibrium process, and rotational nonequilibrium needs to be considered at translational temperatures above 10,000K.

Computation of 1-D shock structure in a gas in rotational non-equilibrium using a new set of simplified Burnett equations

This paper describes the computations of hypersonic shock wave structure in a gas in rotational non-equilibrium using a newly developed simplified set of Burnett equations designated as Simplified Conventional Burnett (SCB) equations. Since the original formulation by Burnett, a number of variations to the original Burnett equations have been proposed and the differences among these variants and their merits/shortcomings have been described in the literature. A new variant is created based on the conventional Burnett equations for hypersonic flows by neglecting terms that are inversely proportional to the Mach number. This simplified set of conventional Burnett equations is linearly stable for small disturbances in contrast to the conventional Burnett equations which suffer from Bobylev instability. To simulate the rotational non-equilibrium effect in a diatomic gas, both the Navier–Stokes (NS) and the SCB equations are modified by including a rotational non-equilibrium relaxation model. The flow variables (density, translational and rotational temperature) in a typical 1-D shock at different Mach numbers (1.2, 5, and 10) in Nitrogen are computed using SCB and NS equations and are compared with the DSMC results. SCB calculations are in close agreement with the DSMC results at high Mach numbers.

State-resolved master equation analysis of thermochemical nonequilibrium of nitrogen

Using a NASA database of state-to-state transition rates for N+N2, master equation studies are performed for various nonequilibrium heat bath conditions. In these master equation studies, relaxation of the rotational and vibrational modes, time variation of chemical composition, reaction rate coefficients, and average rotational and vibrational energy losses due to dissociation are each considered in strong and weak nonequilibrium conditions. A system of master equations is coupled with one-dimensional flow equations to analyze the relaxation of N2 in post-normal shock flows. From the results of master equations and the post-normal shock calculations, it is recommended that the rotational nonequilibrium of N2 should be treated as a nonequilibrium mode in hypersonic re-entry calculations.

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.

Accounting for rotational non-equilibrium effects in subsonic DSMC boundary conditions

From study of the kinetic theory of dilute gases, it becomes clear that the subsonic, implicit boundary conditions which are commonly used in direct simulation Monte Carlo simulations of micro-scale channel flows do not take any form of rotational non-equilibrium into account. The consequences of not considering non-equilibration of the rotational mode are discussed and a new form of the boundary conditions which attempt to take this effect into account are proposed. It is shown that for a highly compressible test case involving these implicit boundaries in the literature that a degree of rotational non-equilibrium is present. The changes in the recovered macroscopic properties when the new boundary is applied in the same case include an improved rotational temperature distribution and a slightly lower velocity at the channel exit. A further test case exhibiting no rotational non-equilibrium was performed and it is shown that the macroscopic properties which are recovered are essentially the same for both boundaries. The results indicate that the newly proposed outlet boundary condition increases the range of applicability of these types of boundary conditions, without adding any significant computational expense.