Rotational-translational Energy 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.

Peak heat flux prediction of hypersonic flow over compression ramp under vibrationally excited free-stream condition

In hypersonic shock tunnel experiments, the high-temperature reservoir gas expands and accelerates so rapidly that there is not enough time for vibrational energy relaxation. As a result, thermal nonequilibrium gas flow is frequently encountered in the test section, and this significantly affects the measured heat flux. In this paper, hypersonic compression-ramp flows are studied numerically to investigate the effect of incomplete vibrational energy accommodation on the separation flow structure and peak heat flux in the reattachment region under low-to-medium Reynolds number and high Mach number conditions. Numerical results and theoretical analysis suggest that the vibrational energy accommodation has no noticeable impact on the length scale of the separation zone, but strongly influences the peak heat flux of the separated ramp flows. Decomposing the peak heat flux into translational–rotational energy and vibrational energy components, qtr and qv, respectively, we find that qv/qtr characterizes the nonequilibrium degree of the vibrational energy accommodation. A formula for predicting the peak heat flux is then proposed, taking the effect of incomplete vibrational energy accommodation into consideration. Finally, surface heat flux measurements in a hypersonic shock tunnel indicate that a deviation of up to 13% in total peak heat flux could arise if vibrational energy accommodation is not considered under the vibrationally excited free-stream condition.

An atomic-level study of the N2–N2 collision process at temperatures up to 2000 K

This article studies the mechanics of the N2–N2 collision process at temperatures up to 2000 K through an extensive set of classical trajectory calculations of binary collisions. It is found that key postcollision characteristics, namely, the deflection angle and the rotational–translational energy exchange rate, are significantly affected by precollision values of the rotational energies of the molecules, which is not addressed in commonly used collision models. On the macroscopic scale, such a behavior will lead to viscosity collision cross section and relaxation rate becoming dependent on both translational and rotational temperatures, as well as on the form of the nonequilibrium rotational energy distribution.

Decaying compressible turbulence with thermal non-equilibrium

The interaction of decaying turbulence with thermal non-equilibrium (TNE) is studied using direct numerical simulations. The focus is on energy exchanges and decay rates in decaying flows with initial vibrational excitation. A key finding is the identification of different regimes in the interaction and the nondimensional parameter (β) that controls it. The latter accounts for the degree of initial TNE as well as the ratio of timescales of turbulence and vibrational relaxation. For β < 1, TNE is essentially frozen and turbulence is largely unaffected by the decay of vibrational energy. For β > 1, TNE relaxation is relatively fast and produces an increase in translational–rotational energy, which, through changes in transport coefficients, leads to a temporary increase in dissipation leading to faster turbulence decay rates. Theoretical arguments are put forth to determine the asymptotic limits of this effect. TNE relaxation is also affected by turbulent fluctuations in unexpected ways. For example, although initial conditions are always vibrationally hot, the flow may undergo vibrationally cold transients, which are explained through simple models. The results presented here help explain disagreement between previous experimental and numerical data.

Interplay between polydispersity, inelasticity, and roughness in the freely cooling regime of hard-disk granular gases.

A polydisperse granular gas made of inelastic and rough hard disks is considered. Focus is laid on the kinetic-theory derivation of the partial energy production rates and the total cooling rate as functions of the partial densities and temperatures (both translational and rotational) and of the parameters of the mixture (masses, diameters, moments of inertia, and mutual coefficients of normal and tangential restitution). The results are applied to the homogeneous cooling state of the system and the associated nonequipartition of energy among the different components and degrees of freedom. It is found that disks typically present a stronger rotational-translational nonequipartition but a weaker component-component nonequipartition than spheres. A noteworthy "mimicry" effect is unveiled, according to which a polydisperse gas of disks having common values of the coefficient of restitution and of the reduced moment of inertia can be made indistinguishable from a monodisperse gas in what concerns the degree of rotational-translational energy nonequipartition. This effect requires the mass of a disk of component i to be approximately proportional to 2σ_{i}+〈σ〉, where σ_{i} is the diameter of the disk and 〈σ〉 is the mean diameter.

Chemical reaction mechanism related vibrational nonequilibrium effect on the Zel'dovich−von Neumann−Döring (ZND) detonation model

Discrepancies in detonation cell size are always found between experiments and numerical simulations. The recent numerical simulation by Shi et al. (2016) included the vibrational relaxation effect in molecules and revealed that the vibrational nonequilibrium effect may be one of the reasons for the above discrepancy. In this study, a steady one-dimensional Zel'dovich–von Neumann–Döring (ZND) model is modified to account for the vibrational nonequilibrium effect in gaseous detonation. By introducing a new parameter α, which is defined as the ratio of translational-rotational mode of the specific heat at constant volume to the overall specific heat at constant volume, the translational-rotational energy and the vibrational energy are treated separately in the energy conservation equation. Both the single step and the two step Arrhenius models are used to describe the chemical reaction progress, and the Landau–Teller model and Park's two-temperature model are applied to describe the vibrational energy relaxation and the coupling between the molecular vibration and the chemical reaction, respectively. The parametric study on the modified ZND model is conducted and compared with the conventional ZND one, including the activation energy in chemical model, the ratio of the chemical time scale to the vibrational relaxation time scale and the characteristic vibrational temperature. The results show that the half reaction length is relatively not sensitive to the variation of characteristic vibrational temperature applied in the model (compared with the activation energy and the time ratio), but is increased to a significant extent with the increase of the activation energy and the decrease of the time ratio, when considered the vibrational nonequilibrium. The simulated half reaction lengths in the modified model are in good agreement with those in the conventional model, if the chemical time scale is larger than the vibrational relaxation time scale above a critical ratio when the vibrational nonequilibrium effect is insignificant.

On the Relaxation of Molecules’ Rotational Energy in the Direct Simulation Monte Carlo Method

Rotational-translational energy exchange simulation in the direct simulation Monte Carlo method is considered for the problem of the entry of hypersonic space vehicles in the atmosphere of the Earth, Mars, Venus, Titan, and gas giants. The diatomic and polyatomic molecules’ discrete quantum energy levels are systematized. The energy exchange Larsen-Borgnakke algorithm is described for the molecules with discrete rotational energy levels. The parameters of the model of hard spheres of variable diameters (VHS) are derived for molecules present in the atmospheres from the experimental data on the viscosity of gases and calculation of cross sections of clashing molecules elastic. Analogously, the parameters in Parker’s formula, describing the rotational-translational relaxation, are chosen from the experimental data on the times of relaxation of the rotational energy of the molecules. Close agreement between The calculated and experimental data are in close agreement for moving away and the width of the shock wave before the sphere in nitrogen gas. This means that the used models and their parameters are adequate.

Dynamics and Hydrodynamics of Molecular Superrotors.

In recent years, several femtosecond laser techniques have been developed that can make gas molecules rotate extremely fast, whereas the gas stays translationally cold. Herein we use molecular-dynamics simulations to investigate the collisional dynamics of gases of such molecules ("superrotors"). We found that the common route of superrotors to equilibrium is rather generic. It starts with a long-lasting "gyroscopic stage", during which the molecules keep their fast rotation and the orientation of their angular momentum despite the many collisions they undergo. The inhibited rotational relaxation is characterized by a persistent anisotropy in the molecular angular distribution, manifested in long-lasting optical birefringence and in anisotropic diffusion of the gas. Later, the gyroscopic stage is abruptly terminated by a self-accelerating explosive rotational-translational energy exchange that generates sound and macroscopic vortices with a hot rotating core.

Collisional dynamics in a gas of molecular super-rotors.

Recently, femtosecond laser techniques have been developed that are capable of bringing gas molecules to extremely fast rotation in a very short time, while keeping their translational motion relatively slow. Here we study collisional equilibration dynamics of this new state of molecular gases. We show that the route to equilibrium starts with a metastable ‘gyroscopic stage' in the course of which the molecules maintain their fast rotation and orientation of the angular momentum through many collisions. The inhibited rotational–translational relaxation is characterized by a persistent anisotropy in the molecular angular distribution, and is manifested in the optical birefringence and anisotropic diffusion in the gas. After a certain induction time, the ‘gyroscopic stage' is abruptly terminated by an explosive rotational–translational energy exchange, leading the gas towards the final equilibrium. We illustrate our conclusions by direct molecular dynamics simulation of several gases of linear molecules.

Sound emission from the gas of molecular superrotors.

We use an optical centrifuge to deposit a controllable amount of rotational energy into dense molecular ensembles. Subsequent rotation-translation energy transfer, mediated by thermal collisions, results in the localized heating of the gas and generates strong sound wave, clearly audible to the unaided ear. For the first time, the amplitude of the sound signal is analyzed as a function of the experimentally measured rotational energy and linear proportionality between the two observables is established.

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases

AbstractA kinetic model of the Boltzmann equation for non-vibrating polyatomic gases is proposed, based on the Rykov model for diatomic gases. We adopt two velocity distribution functions (VDFs) to describe the system state; inelastic collisions are the same as in the Rykov model, but elastic collisions are modelled by the Boltzmann collision operator (BCO) for monatomic gases, so that the overall kinetic model equation reduces to the Boltzmann equation for monatomic gases in the limit of no translational–rotational energy exchange. The free parameters in the model are determined by comparing the transport coefficients, obtained by a Chapman–Enskog expansion, to values from experiment and kinetic theory. The kinetic model equations are solved numerically using the fast spectral method for elastic collision operators and the discrete velocity method for inelastic ones. The numerical results for normal shock waves and planar Fourier/Couette flows are in good agreement with both conventional direct simulation Monte Carlo (DSMC) results and experimental data. Poiseuille and thermal creep flows of polyatomic gases between two parallel plates are also investigated. Finally, we find that the spectra of both spontaneous and coherent Rayleigh–Brillouin scattering (RBS) compare well with DSMC results, and the computational speed of our model is approximately 300 times faster. Compared to the Rykov model, our model greatly improves prediction accuracy, and reveals the significant influence of molecular models. For coherent RBS, we find that the Rykov model could overpredict the bulk viscosity by a factor of two.

Kernels of the linear Boltzmann equation for spherical particles and rough hard sphere particles

Kernels for the collision integral of the linear Boltzmann equation are presented for several cases. First, a rigorous and complete derivation of the velocity kernel for spherical particles is given, along with reductions to the smooth, rigid sphere case. This combines and extends various derivations for this kernel which have appeared previously in the literature. In addition, the analogous kernel is derived for the rough hard sphere model, for which a dependence upon both velocity and angular velocity is required. This model can account for exchange between translational and rotational degrees of freedom. Finally, an approximation to the exact rough hard sphere kernel is presented which averages over the rotational degrees of freedom in the system. This results in a kernel depending only upon velocities which retains a memory of the exchange with rotational states. This kernel tends towards the smooth hard sphere kernel in the limit when translational-rotational energy exchange is attenuated. Comparisons are made between the smooth and approximate rough hard sphere kernels, including their dependence upon velocity and their eigenvalues.

Theoretical Investigation of Two H2 Molecules Inside the Cages of the Structure H Clathrate Hydrate

We report 10-dimensional quantum dynamic calculations of the translation-rotation energy levels of two hydrogen molecules inside the cages of the structure H clathrate hydrate. For the small and medium cages, positive ground state energies are reported supporting their single occupancy, while for the large cage negative energy levels are calculated and some low-lying states are characterized for two para-H2, ortho-H2, para-D2, ortho-D2, and HD molecules inside the cage. The calculations are performed using the block improved relaxation method implemented in the multiconfiguration time-dependent Hartree (MCTDH) code.

Modeling the liquid–gas interface using self-assembled monolayers

Stochastic classical trajectory simulations were used to study the efficiency of the energy exchange at the gas–liquid interface. Self-assembled monolayers (SAM) of long-chain functionalized molecules were used to mimic the liquid surface. Since the molecules in the monolayers are anchored by only one end, they retain some of the mobility that they have in the liquid but lose all their fluidity. The corrugation of the surface and the stiffness of the interface were tuned by varying the length of the molecules in the monolayers. The use of longer molecules leads to increased corrugation of the surface and provides additional dissipation channels that promote more efficient momentum and energy accommodation, increase the translational–rotational energy interconversion and enhance trapping. However, this “length effect” appears to saturate, as no further significant changes are observed in those properties when the monolayer's molecules's length is elongated from six to nine carbons. This saturation effect suggests that, even though monolayers can provide some of the mobility observed in liquid surfaces, they lack the energy dissipation channel provided by the fluidity of the liquid.

The effect of rotational and translational energy exchange on tracer diffusion in rough hard sphere fluids

A study is presented of tracer diffusion in a rough hard sphere fluid. Unlike smooth hard spheres, collisions between rough hard spheres can exchange rotational and translational energy and momentum. It is expected that as tracer particles become larger, their diffusion constants will tend toward the Stokes-Einstein hydrodynamic result. It has already been shown that in this limit, smooth hard spheres adopt "slip" boundary conditions. The current results show that rough hard spheres adopt boundary conditions proportional to the degree of translational-rotational energy exchange. Spheres for which this exchange is the largest adopt "stick" boundary conditions while those with more intermediate exchange adopt values between the "slip" and "stick" limits. This dependence is found to be almost linear. As well, changes in the diffusion constants as a function of this exchange are examined and it is found that the dependence is stronger than that suggested by the low-density, Boltzmann result. Compared with smooth hard spheres, real molecules undergo inelastic collisions and have attractive wells. Rough hard spheres model the effect of inelasticity and show that even without the presence of attractive forces, the boundary conditions for large particles can deviate from "slip" and approach "stick."

Translation-rotation energy levels of one H2molecule inside the small, medium and large cages of the structure H clathrate hydrate

We report quantum dynamics calculations of the translation-rotation energy levels of one hydrogen molecule inside the small, medium and large cages of the structure H clathrate hydrate. The calculations are performed using the multiconfiguration time-dependent Hartree (MCTDH) method. Some low-lying states are computed for para-H(2), ortho-H(2), para-D(2), ortho-D(2) and HD, by block improved relaxation.

Homogeneous Bose gas of dipolar molecules in the mean field approximation

We present a mean field analysis of the effects of molecular rotation on the excitation spectrum and stability of ultracold dipolar gases. For an unpolarized homogeneous gas interacting with a pure dipole-dipole interaction, we find that for the rotational state L = 1 the dipole-dipole interaction causes a splitting of the translation-rotation energy levels into a single M = 0 and a doubly degenerate M = ±1 excitation. For all other rotational states, the dipole-dipole interaction does not lead to coupling of translations and rotations and therefore has no effect on the rotational degeneracy of the excitations. The addition of arbitrarily small electric fields is found to introduce instabilities similar to those known to arise in the fully polarized dipolar gas. As in the case of a fully polarized gas, addition of a large enough short range repulsive potential is seen to stabilize the system, with the critical value of the repulsive interaction required for stabilization being larger when rotations are included.

Electrostrictive laser-induced gratings for time-resolved observation of translational-rotational energy transfer in H2

Electrostrictive laser-induced gratings (LIG) have been formed in H2 of various pressures (0.5…5 bar) and with frequencies of the Brillouin modes (sound waves) of about 14 MHz. Under these conditions, the rotational degrees of freedom can only partially follow the temperature variation from the acoustic oscillation. A model is given, which completes the linearized fluid dynamical equations with equations for the rotational state relaxation, separately for the average relaxation of the states for Ortho-H2 and Para-H2. Using such a model, the dispersion of sound frequency is reproduced with an accuracy of a few percent. However, the total dissipation of sound and the observed strong alternation in peak heights in the temporal evolution of the LIG intensity is only approximately predicted by the model.

Theoretical Studies of CH4 Inside an Open-Cage Fullerene: Translation−Rotation Coupling and Thermodynamic Effects

Molecules trapped inside fullerenes exhibit interesting quantum behavior, including quantization of their translational degrees of freedom. In this study, a theoretical framework for predicting quantum properties of nonlinear small molecules in nonsymmetric open-cage fullerenes (OCFs) has been described along the lines of similar theories which treat small molecules inside C(60) and clathrate cages. As an example, the coupled translational-rotational energy structure has been calculated for the case of CH(4) inside a known OCF. The calculated energy levels have been used to calculate the equilibrium fraction of incorporated CH(4) as well as the translational heat capacity for the encapsulated molecule. The heat capacity shows an anomalous maximum at 239 K for CH(4) and 215 K for CD(4) which are not present in free methane.

Effects of multitemperature nonequilibrium on compressible homogeneous turbulence

We study the effects of the rotational-translational energy exchange on the compressible decaying homogeneous isotropic turbulence (DHIT) in three dimensions through direct numerical simulations. We use the gas-kinetic scheme coupled with multitemperature nonequilibrium based on the Jeans-Landau-Teller model. We investigate the effects of the relaxation time of rotational temperature, ZR, and the initial ratio of the rotational and translational temperatures, TR0/TL0, on the dynamics of various turbulence statistics including the kinetic energy K(t), the dissipation rate epsilon(t), the energy spectrum E(k,t), the root mean square of the velocity divergence theta'(t), the skewness Su(t) and the flatness Fu(t) of the velocity derivatives, and the probability distribution functions of the local Mach number Ma and the shocklet strength chi. The larger the ZR is, the faster the compressibility decays after an initial time. Similarly, with a fixed TL0, the higher the initial energy ratio TR0/TL0, the weaker is the compressibility in the flow. It is also observed that the effect of TR0/TL0 is strong in all times in the decay, while the effect of ZR is severe only in the later times passing through the stage with strong nonlinearity. We also observe that the multitemperature model does not affect the self-similarities obeyed by the probability distribution functions of Ma and chi , which appear to be a robust feature of the compressible DHIT.