Larsen Borgnakke Research Articles

Post-reaction internal energy distributions of quantum-kinetics model for simulating chemical reactions of polyatomic molecules

Chemical reactions significantly influence aerodynamic performance during spacecraft entry into the Martian atmosphere. Several chemical reaction models have been proposed in the direct simulation Monte Carlo simulation. The quantum-kinetics (Q-K) model has been applied in the case of diatomic molecules. Given that the Martian atmosphere consists primarily of CO2, it is crucial to find ways of implementing the Q-K model for polyatomic molecules. Although the chemical reaction rates involving CO2 have been investigated using the Q-K model, the problem of achieving detailed balance remains. Multiple vibrational modes exist for polyatomic molecules. Under the Q-K distribution, the average vibrational level of each mode is higher than that under the equilibrium distribution, and the total energy may be insufficient. Hence, its applicability to polyatomic molecules needs to be revealed. In this study, a comparison is made of the respective results obtained using the Larsen–Borgnakke (L–B) and Q-K distribution methods for the energy distribution of the reaction CO2+O⇋CO+O2, and the detailed balance is achieved with the Q-K method but not the L–B method. Under the conditions assumed in this study, the vibrational energy distribution of CO consumed by the reverse reaction is not in good agreement with that generated by the forward reaction, leading to the failure of the L–B method. Finally, the results indicate that the detailed balance is reached only when the collision temperature, based on the translational and vibrational energy, is employed to adjust the activation energy rather than the translational temperature generally adopted in the literature.

A universal method of redistributing relaxation energies in inelastic molecular collisions

The Larsen–Borgnakke (L–B) model is most widely used to redistribute relaxation energies in direct simulation Monte Carlo (DSMC) method; the relaxation energies in an inelastic collision are determined beforehand and redistributed by sampling from the equilibrium distributions (not locally equilibrium distributions) of these energies that are appropriate to this collision. The present concern focuses on how to redistribute the selected energies when they have been determined to require relaxation. When certain energies are determined to require relaxation and redistribution in an inelastic collision, the hierarchical method and the serial or sequential method are used to complete the L–B procedure. Although the hierarchical method is theoretically accurate, it needs to complete the integration of the joint energy distribution function, so it is not applicable when the energy distribution function has a complex form or contains more than one discontinuous energy. Hence, researchers turn to the serial or sequential method, but the rationality proof of this method implies an assumption of locally thermal equilibrium, so it is obviously not appropriate in the DSMC method usually applied to hypersonic nonequilibrium flows. Then, a new method of redistributing relaxation energies to complete the L–B procedure is proposed, which does not include any integration process; the original form of the energy distribution function is used directly, so it can be expected to have excellent adaptability, especially for complex cases such as energy redistribution of reaction products. The effectiveness, efficiency, and extensive applicability of the method are proved by multiple examples.

Machine learning product state distributions from initial reactant states for a reactive atom-diatom collision system.

A machine-learned model for predicting product state distributions from specific initial states (state-to-distribution or STD) for reactive atom-diatom collisions is presented and quantitatively tested for the N(4S) + O2(X3Σg -) → NO(X2Π) + O(3P) reaction. The reference dataset for training the neural network consists of final state distributions determined from quasi-classical trajectory (QCT) simulations for ∼2000 initial conditions. Overall, the prediction accuracy as quantified by the root-mean-squared difference (∼0.003) and the R2 (∼0.99) between the reference QCT and predictions of the STD model is high for the test set, for off-grid state-specific initial conditions, and for initial conditions drawn from reactant state distributions characterized by translational, rotational, and vibrational temperatures. Compared with a more coarse grained distribution-to-distribution (DTD) model evaluated on the same initial state distributions, the STD model shows comparable performance with the additional benefit of the state resolution in the reactant preparation. Starting from specific initial states also leads to a more diverse range of final state distributions, which requires a more expressive neural network compared with DTD. A direct comparison between QCT simulations, the STD model, and the widely used Larsen-Borgnakke (LB) model shows that the STD model is quantitative, whereas the LB model is qualitative at best for rotational distributions P(j') and fails for vibrational distributions P(v'). As such, the STD model can be well-suited for simulating nonequilibrium high-speed flows, e.g., using the direct simulation Monte Carlo method.

Direct simulation Monte Carlo modeling of H2–O2 deflagration waves

Combustion at extreme conditions such as high-speed and microscale involve nonequilibrium transport and chemical reactions that require atomistic treatment of molecular processes. We present a framework for applying the direct simulation Monte Carlo method (DSMC) to model combustion at the molecular scale. We show that the standard DSMC approach employing Total Collision Energy (TCE) chemistry and Larsen–Borgnakke (LB) energy exchange models is not applicable for combustion simulations which are dominated by exchange and recombination reactions. A methodology for modifying the TCE-LB approach is developed to ensure detailed balance and relaxation towards thermal equilibrium regardless of the internal energy relaxation rates. A simplified 6-species and 7-reversible reaction mechanism with rates modified to account for discrete vibrational levels in DSMC is used for the benchmark flame study. The laminar flame structure of H2–O2 premixed systems and the corresponding deflagration wave speeds by DSMC are compared with PREMIX results. The DSMC simulations based on the extended TCE-LB framework correctly reproduces the 1-D flame structure and its propagation speed is consistent with continuum modeling predictions for the same reaction mechanism and similar flow conditions. The DSMC approach presents opportunities to study combustion phenomena at the molecular scale including state-to-state processes in conditions far from thermal equilibrium for improved combustion diagnostics and control.

A low noise discrete velocity method for the Boltzmann equation with quantized rotational and vibrational energy

A discrete velocity method is developed for gas mixtures of diatomic molecules with both rotational and vibrational energy states. A full quantized model is described, and rotation–translation and vibration–translation energy exchanges are simulated using a Larsen–Borgnakke exchange model. Elastic and inelastic molecular interactions are modeled during every simulated collision to help produce smooth internal energy distributions. The method is verified by comparing simulations of homogeneous relaxation by our discrete velocity method to numerical solutions of the Jeans and Landau–Teller equations, and to direct simulation Monte Carlo. We compute the structure of a 1D shock using this method, and determine how the rotational energy distribution varies with spatial location in the shock and with position in velocity space.

State-to-state models of vibrational relaxation in Direct Simulation Monte Carlo (DSMC)

In the present work, the application of state-to-state models of vibrational energy exchanges to the Direct Simulation Monte Carlo (DSMC) is considered. A state-to-state model for VT transitions of vibrational energy in nitrogen and oxygen, based on the application of the inverse Laplace transform to results of quasiclassical trajectory calculations (QCT) of vibrational energy transitions, along with the Forced Harmonic Oscillator (FHO) state-to-state model is implemented in DSMC code and applied to flows around blunt bodies. Comparisons are made with the widely used Larsen–Borgnakke model and the in uence of multi-quantum VT transitions is assessed.

Rayleigh–Brillouin scattering in molecular Oxygen by CT-DSMC simulations

Rayleigh–Brillouin scattering spectra (RBS) in molecular Oxygen have been simulated by DSMC. Different scattering models have been implemented based either on the Larsen–Borgnakke relaxation model and on the Classical Trajectories technique. Results are compared with recent experimentally measured spectra showing good agreement. It is suggested that DSMC-based models be used in the interpretation of light scattering experiments in place of the simplified kinetic models, widely used for the interpretation of RBS experiments. Actually, the former have a firmer physical ground and are readily extended to treat gas mixtures of arbitrary complexity.

Consistent post-reaction vibrational energy redistribution in DSMC simulations using TCE model

The direct simulation Monte Carlo (DSMC) method has been widely applied to study shockwaves, hypersonic reentry flows, and other nonequilibrium flow phenomena. Although there is currently active research on high-fidelity models based on ab initio data, the total collision energy (TCE) and Larsen-Borgnakke (LB) models remain the most often used chemistry and relaxation models in DSMC simulations, respectively. The conventional implementation of the discrete LB model, however, may not satisfy detailed balance when recombination and exchange reactions play an important role in the flow energy balance. This issue can become even more critical in reacting mixtures involving polyatomic molecules, such as in combustion. In this work, this important shortcoming is addressed and an empirical approach to consistently specify the post-reaction vibrational states close to thermochemical equilibrium conditions is proposed within the TCE framework. Following Bird’s quantum-kinetic (QK) methodology for populating post-reaction states, the new TCE-based approach involves two main steps. The state-specific TCE reaction probabilities for a forward reaction are first pre-computed from equilibrium 0-D simulations. These probabilities are then employed to populate the post-reaction vibrational states of the corresponding reverse reaction. The new approach is illustrated by application to exchange and recombination reactions relevant to H2–O2 combustion processes.

A post-collision internal energy model for O(3P) + SO2([formula omitted][formula omitted]) in DSMC based on Molecular Dynamics computations

A model is developed for determining molecular internal energies after O(3P)+SO2(X,1A1) collisions in the Direct Simulation Monte Carlo (DSMC) method in order to improve modeling of the hyperthermal interactions occurring in the upper atmosphere of Io. Molecular Dynamics/Quasi-Classical Trajectory (MD/QCT) studies are conducted to generate post-collision SO2 and post-dissociation SO internal energy distributions as a function of initial SO2 internal energy and relative collision velocity, which are found to be an improvement over the baseline Larsen–Borgnakke (LB) method that often predicts unphysical internal energies above the dissociation energy for non-reacting collisions and under-predicts post-dissociation SO internal energy. An approach for sampling from the MD/QCT-based internal energy distributions in DSMC is developed and DSMC simulations are then conducted for a time-dependent thermal nonequilibrium heat bath using both the MD/QCT-based distributions and the LB model. When only SO2–O collisions are considered, noticeable differences are observed for post-collisional SO2 and SO internal temperatures.

Modeling of Unsteady Shock Tube Flows Using Direct Simulation Monte Carlo

The simulation of the UV spectra of moderate Mach number flows is the strongest test possible of the complex thermochemical models in hypersonic flows. To take advantage of low-density ground-based measurements, however, one needs to be able to model unsteady flows using kinetic particle approaches. The direct simulation Monte Carlo approach is used in this paper to model the thermal relaxation in nonreacting nitrogen shocks corresponding to the data and the higher Mach number air shocks and radiation of previous research. In the first case, it is found that better agreement with experimental data can be achieved when assuming a constant rotational collision number instead of a temperature-dependent one. For the second data set, the nonequilibrium NO UV radiation from unsteady reflected air shocks are modeled and compared with experimental data. The Larsen–Borgnakke and total collision energy models are used to simulate the internal energy relaxation and chemistry of five chemical species (, , N, O, and NO), respectively. Unlike steady shock layer flows, the long period of equilibration behind the shock wave occurs and potential factors that prevent complete thermochemical equilibrium are explored. The main goal of this paper, however, is to compare the spectrally averaged peak shock radiation with data. Because there are insufficient collisions to assume a Boltzmann distribution of the NO electronic states, a set of excitation/deexcitation mechanisms are proposed and the quasi-steady-state assumption is used in calculating the neutral-impact excitation of NO and the associated radiation. The calculated spectrally averaged peak radiations in the shock are in good agreement with the experimental data of Gorelov for shock speeds below , but are lower at higher shock speeds.

Modeling of Nitrogen Monoxide Formation and Radiation in Nonequilibrium Hypersonic Flows

The interaction between nitrogen vibrational-translational relaxation with different chemistry models in a hypersonic shock is studied. The vibrational-translational transition models considered include the Larsen–Borgnakke, Schwartz-Slawsky-Herzfeld, and forced harmonic oscillator models; and the nitrogen monoxide formation reaction was studied using the total collision energy chemistry model and reaction cross sections obtained from quasi-classical trajectory simulations. Using direct simulation Monte Carlo solutions that incorporate these models, improved nitrogen monoxide spectral radiation modeling was performed by employing quenching rates from recent experiments. Comparisons of electronic state populations and the corresponding radiative spectra for the different vibrational-translational and chemistry models are presented, and it is shown that the vibronic spectral structure of the nitrogen monoxide radiation could provide a way to test the fidelity of the internal energy transfer and chemistry models.

On potential energy landscape and combustion chemistry modeling

Combustion at extreme conditions such as high-speed and microscale involve nonequilibrium transport and chemical reactions that require atomistic treatment of molecular processes. We present a framework for applying the direct simulation Monte Carlo method (DSMC) to model combustion at the molecular scale. We show that the standard DSMC approach employing Total Collision Energy (TCE) chemistry and Larsen–Borgnakke (LB) energy exchange models is not applicable for combustion simulations which are dominated by exchange and recombination reactions. A methodology for modifying the TCE-LB approach is developed to ensure detailed balance and relaxation towards thermal equilibrium regardless of the internal energy relaxation rates. A simplified 6-species and 7-reversible reaction mechanism with rates modified to account for discrete vibrational levels in DSMC is used for the benchmark flame study. The laminar flame structure of H2–O2 premixed systems and the corresponding deflagration wave speeds by DSMC are compared with PREMIX results. The DSMC simulations based on the extended TCE-LB framework correctly reproduces the 1-D flame structure and its propagation speed is consistent with continuum modeling predictions for the same reaction mechanism and similar flow conditions. The DSMC approach presents opportunities to study combustion phenomena at the molecular scale including state-to-state processes in conditions far from thermal equilibrium for improved combustion diagnostics and control.

Improved sampling techniques for the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is a widely used approach for flow simulations having rarefied or nonequilibrium effects. It involves heavily to sample instantaneous values from prescribed distributions using random numbers. In this note, we briefly review the sampling techniques typically employed in the DSMC method and present two techniques to speedup related sampling processes. One technique is very efficient for sampling geometric locations of new particles and the other is useful for the Larsen–Borgnakke energy distribution.

Direct simulation of pumping characteristics in a fully 3D model of a single-stage turbomolecular pump

The pumping characteristics of a single-stage turbomolecular pump from free molecule region to transition region are calculated by the DSMC method. The formulation is fully three-dimensional and derived on the rotating frame. Variable soft sphere (VSS) model and No Time Counter (NTC) collision scheme are used. The intermolecular collisions are regarded as inelastic by applying the Larsen–Borgnakke model. Numerical results show good quantitative agreement with existing experiments in the free molecular region, and have greater discrepancy as the Knudsen number decreases.

Pumping performance analyses of a turbo booster pump by direct simulation Monte Carlo method

The pumping performance of a new type of turbo booster pump in the free molecular and transitional flow regimes is investigated using the direct simulation Monte Carlo (DSMC) method. The problem is modeled as a 3D channel with variable pitch and ditch depth screw grooves. A real 3D analysis in a rotating frame is proposed to simulate the flow field. Additional terms arising from the Coriolis and centripetal accelerations are formulated, and the variable soft sphere (VSS) model and no time counter (NTC) collision scheme are used. The intermolecular collisions are regarded as inelastic by applying the Larsen–Borgnakke model. Numerical results show good quantitative agreement compared with the experimental data.

Interaction between Shock Waves and Boundary Layer in Non-Equilibrium Hypersonic Rarefield Flow. 1st Report. Comparison with Continuum Model and Larsen-Borgnakke Model.

A DSMC (direct simulation Monte Carlo) simulation using Dynamic Molecular Collision (DMC) Model based on Molecular Dynamics (MD) calculation is applied for solving a two-dimensional non-equilibrium hypersonic rarefied flow over a flat plate with and without angle of leading edge. Numerical results show that non-equilibrium characteristics between translational and rotational temperature are obtained behind leading edge over the plate. The results of the DSMC agree well with those of the experiments, Howere, the results of the Navier-Stokes are quite different from those of the experiments. The comparison between Larsen-Borgnakke (LB) model and DMC model reveals that the DMC model obtains excellent results without a parameter such as an inelastic parameter.

Construction of Collision Model of Diatomic Molecules. Improvement of Energy Transfer Model and Its Verification.

In our 3rd report, we introduced a Lennard-Jones (LJ) potential parameter based on the potential obtained by ab initio calculation and collision cross section from the Wang-Chang, Uhlenbeck and Taxman's theory and molecular dynamics (MD) simulation. In the present study, we have improved the dynamic molecular collision (DMC) model to callculate the property of MD simulation better than the previous model. To confirm its validity we calculated the equilibrium state, the transport coefficient (viscosity coefficient and heat conductivity) at various temperatures and the normal shock wave by the direct simulation Monte Carlo (DSMC) method using the DMC model and compared the results with other theoretical and experimental results. Consequently, we found that the diatomic rarefied gas flows could be simulated very well using our model. These results were compared with those obtained by the Larsen Borgnakke model. It was found that this model was more efficient than the previous model.

Conservative and Entropy Schemes for Boltzmann Collision Operator of Polyatomic Gases

We propose two discrete velocity models derived from the Boltzmann equation of Larsen–Borgnakke type for polyatomic gases. These two models are natural extensions of previously discussed discrete velocity models used for monoatomic gases. These two models have the same properties as the continuous one, which are conservation of mass, momentum and energy, discrete Maxwellians as equilibrium states and H-theorems.

Probability of inelastic collisions for the Larsen-Borgnakke model to the Monte Carlo simulation method

Introduction T HE Larsen-Borgnakke (LB) model has become a standard for calculating inelastic collisions in the Monte Carlo direct simulation method. The essential assumption of the LB model is that the postcollision state of a collision pair is subject to a temporal equilibrium. The model is not a collision-dynamical one but a statistical one. The problem in using the model is how to assign the probabilities of inelastic events in collision. Owing to the very nature of the model, the probabilities of inelastic collisions have no direct connection with measured probabilities or relaxation times, which are of course the outcome of collision dynamics of real molecules. In this Note, the probabilities of inelastic collisions for the LB model are determined in such a way that the difference between the Monte Carlo solution for a very small Knudsen number and the Navier-Stokes solution takes a minimum.