Chonglin Zhang

Determination of the Scalar Friction Factor for Nonspherical Particles and Aggregates Across the Entire Knudsen Number Range by Direct Simulation Monte Carlo (DSMC)

The friction factor of an aerosol particle depends upon the Knudsen number (Kn), as gas molecule–particle momentum transfer occurs in the transition regime. For spheres, the friction factor can be calculated using the Stokes–Millikan equation (with the slip correction factor). However, a suitable friction factor relationship remains sought-after for nonspherical particles. We use direct simulation Monte Carlo (DSMC) to evaluate an algebraic expression for the transition regime friction factor that is intended for application to arbitrarily shaped particles. The tested friction factor expression is derived from dimensional analysis and is analogous to Dahnekes adjusted sphere expression. In applying this expression to nonspherical objects, we argue for the use of two previously developed drag approximations in the continuum (Kn → 0) and free molecular (Kn → ∞) regimes: the Hubbard–Douglas approximation and the projected area (PA) approximation, respectively. These approximations lead to two calculable geometric parameters for any particle: the Smoluchowski radius, R S, and the projected area, PA. Dimensional analysis reveals that Kn should be calculated with PA/πR S as the normalizing length scale, and with Kn defined in this manner, traditional relationships for the slip correction factor should apply for arbitrarily shaped particles. Furthermore, with this expression, Kn-dependent parameters, such as the dynamic shape factor, are readily calculable for nonspherical objects. DSMC calculations of the orientationally averaged drag on spheres and test aggregates (dimers, and open and dense 20-mers) in the range Kn = 0.05–10 provide strong support for the proposed method for friction factor calculation in the transition regime. Experimental measurements of the drag on aggregates composed of 2–5 primary particles further agree well with DSMC results, with differences of less than 10% typically between theoretical predictions, numerical calculations, and experimental measurements. Copyright 2012 American Association for Aerosol Research

Rovibrational coupling in molecular nitrogen at high temperature: An atomic-level study

This article contains an atomic-level numerical investigation of rovibrational relaxation in molecular nitrogen at high temperature (>4000 K), neglecting dissociation. We conduct our study with the use of pure Molecular Dynamics (MD) and Classical Trajectory Calculations (CTC) Direct Simulation Monte Carlo (DSMC), verified to produce statistically identical results at the conditions of interest here. MD and CTC DSMC solely rely on the specification of a potential energy surface: in this work, the site-site Ling-Rigby potential. Additionally, dissociation is prevented by modeling the N–N bond either as a harmonic or an anharmonic spring. The selected molecular model was shown to (i) recover the shear viscosity (obtained from equilibrium pure MD Green-Kubo calculations) of molecular nitrogen over a wide range of temperatures, up to dissociation; (ii) predict well the near-equilibrium rotational relaxation behavior of N2; (iii) reproduce vibrational relaxation times in excellent accordance with the Millikan-...

Molecular dynamics simulation of rotational relaxation in nitrogen: Implications for rotational collision number models

We study the rotational relaxation process in nitrogen using all-atom molecular dynamics (MD) simulations and direct simulation Monte Carlo (DSMC). The intermolecular model used in the MD simulations is shown to (i) reproduce very well the shear viscosity of nitrogen over a wide range of temperatures, (ii) predict the near-equilibrium rotational collision number in good agreement with published trajectory calculations done on ab initio potential energy surfaces, and (iii) produce shock wave profiles in excellent accordance with the experimental measurements. We find that the rotational relaxation process is dependent not only on the near-equilibrium temperature (i.e., when systems relax to equilibrium after a small perturbation), but more importantly on both the magnitude and direction of the initial deviation from the equilibrium state. The comparison between MD and DSMC, based on the Borgnakke-Larsen model, for shock waves (both at low and high temperatures) and one-dimensional expansions shows that a j...

A Three-Level Cartesian Geometry Based Implementation of the DSMC Method

The data structures and overall algorithms of a newly developed 3-D direct simulation Monte Carlo (DSMC) program are outlined. The code employs an embedded 3-level Cartesian mesh, accompanied by a cut-cell algorithm to incorporate triangulated surface geometry into the adaptively refined Cartesian mesh. Such an approach enables decoupling of the surface mesh from the flow fie ld mesh, which is desirable for near-continuum flows, flows with large density variation, and also for adapti ve mesh refinement (AMR). Two separate data structures are proposed in order to separate geometry data from cell and particle information. The geometry data structure requires little memory so that each partition in a parallel simulation can store the entire mesh, potentially leading to better scalability and efficient AMR for parallel simulations. A simple and efficient AMR algorithm that maintains local cell size and time step consistent with the local mean-free-path and local mean collision time is detailed. The 3-level embedded Cartesian mesh combined with AMR allows increased flexibility for precise control of local mesh size and time-step, both vital for accurate and efficient DSMC simulation. Simulations highlighting the benefits of AMR and variable lo cal time steps will be presented along with DSMC results for 3-D flows with large density variations.

Inelastic collision selection procedures for direct simulation Monte Carlo calculations of gas mixtures

A modification to existing phenomenological inelastic collision selection procedures suitableformodeling theinternalenergy exchange processesofgasmixturesindirect simulation Monte Carlo calculations is presented. The selection procedure does not depend on the relative order of rotational and vibrational relaxation processes and does not require the solution of a quadratic equation for every collision to determine the inelastic collision probability. The simulated relaxation process resulting from the selection procedure is analytically proven to be equivalent to the procedures of Haas et al. [“Rates of thermal relaxation in direct simulation Monte Carlo methods,” Phys. Fluids 6, 2191‐2201 (1994)] and the modified procedure of Gimelshein et al. [“Vibrational relaxation rates in the direct simulation Monte Carlo method,” Phys. Fluids 14, 4452‐4455 (2002)]. The implementation and computational efficiency of each of the procedures are discussed. The proposed selection procedure is verified to accurately simulate rotational and vibrational processes for gas mixtures through isothermal relaxation simulations compared with analytical solutions using the Jeans equation. C 2013 AIP Publishing LLC .[ http://dx.doi.org/10.1063/1.4825340]

Consistent implementation of state-to-state collision models for direct simulation Monte Carlo

A general framework for implementing state-to-state collision models in the DSMC method is presented. In developing a state-to-state DSMC collision model, one key aspect is to propose efficient algorithms to correctly simulate the state-to-state collision cross-sections, hence the state-to-state collision (or transition) rates. This includes the calculation of the total collision rate, the selection of potential collision pairs, and the procedure to perform actual state-to-state collisions in the DSMC method. To achieve these tasks in a computationally efficient manner, we proposed the detailed implementation of a general rovibrational state-to-state collision model for the DSMC method. The proposed model implementation successfully achieved microscopic reversibility, detailed balance, and equipartition of energy under equilibrium conditions. This was first demonstrated using qualitatively-constructed state-to-state cross-sections. With the algorithms verified, we further developed a vibrational state-to-state DSMC collision model using the transition probabilities of the forced harmonic oscillator (FHO) model, where the transition probabilities are modified to satisfy microscopic reversibility, and a power law temperature dependent viscosity is imposed. Furthermore, DSMC simulation results of isothermal vibration relaxation using the modified FHO cross-sections are compared with master equation simulation results, where the transition rates in the master equation are obtained from integrating the state-to-state cross-sections used in the DSMC simulation. Overall, excellent agreement is observed for both the vibrational temperature relaxation history and the time dependent vibrational energy distribution functions, between DSMC and master equation simulations.

Numerical Assessment of Vibration and Dissociation Models in DSMC for Hypersonic Stagnation Line Flows

Existing DSMC phenomenological models for vibrational relaxation and dissociation reactions are implemented in a three dimensional direct simulation Monte Carlo code. A modication to an existing inelastic collision selection procedure is proposed. The proposed selection procedure simplies the calculation of the collision probability. The selection procedure no longer depends on the relative order of rotational and vibrational relaxation and no longer requires the solution of a quadratic equation for every collision to determine the inelastic collision probability. The proposed selection technique is validated to accurately simulate rotational and vibrational processes for gas mixtures through isothermal relaxation simulations compared with analytical solutions. Using the proposed selection procedure as a consistent framework, four existing DSMC models for vibrational relaxation are compared using both isothermal and adiabatic relaxation calculations as well as 1D stagnation line simulations for hypersonic ow. Furthermore, using a consistent vibrational relaxation model, the Quantum-Kinetic, Total Collision Energy, and Vibrationally Favored dissociation models are compared for 1D stagnation line hypersonic ow.

Analysis of rovibrational relaxation in nitrogen via direct atomic simulation

Pure Molecular Dynamics (MD) and Classical Trajectory Calculations (CTC) Direct Simulation Monte Carlo (DSMC) are used to analyze the rovibrational behavior of molecular nitrogen for temperatures greater than 4,000 K. Both techniques are shown to produce statistically identical results at the conditions of interest here. Furthermore, they solely rely on the specification of a potential energy surface (PES). In this work, we used the site-site Ling-Rigby potential, and modeled the N-N bond either as a harmonic spring or an anharmonic spring (for bound states ) or with the Morse potential (to model bond breaking). Selected preliminary results, obtained with a global fit of a quantum-chemistry PES, are also included. We show that the Ling-Rigby molecular model (i) recovers the shear viscosity (obtained from equilibrium pure MD Green-Kubo calculations) of molecular nitrogen over a wide range of temperatures, up to dissociation; (ii) predicts well the near-equilibrium rotational relaxation behavior of N2; (iii) reproduces vibrational relaxation times in excellent accordance with the Millikan-White correlation and previous semiclassical trajectory calculations in the low temperature range, i.e., between 4,000 K and 10,000 K. By simulating isothermal relaxations in a periodic box, we found that the traditional two-temperature model assumptions become invalid at high temperatures (> 10, 000 K), due to a significant coupling between rotational and vibrational modes for bound states. This led us to add a modification to both the Jeans and the Landau-Teller equations to include a coupling term, essentially described by an additional relaxation time for internal energy equilibration. The model thus obtained was parametrized by fitting temperature histories obtained with molecular-level calculations. The degree of anharmonicity of the N-N bond determines the strength of the rovibrational coupling, with possible implication on rovibration/chemistry interaction at the onset of N2 dissociation. Initial results for vibrational relaxation times are also obtained with the quantum-chemistry based PES of Paukku and co-workers in the low temperature range (< 10, 000 K) and were found to agree well with the experimental data. Although the bound assumption is largely unrealistic under equilibrium conditions at temperatures above about 10,000 K, high-temperature extrapolations are nonetheless very important for flows characterized by extreme nonequilibrium. This is demonstrated through the direct MD simulation of a reflected shock wave in dissociating N2.

Investigation of rotational relaxation in nitrogen via Molecular Dynamics simulation

The objective of this study is to use Molecular Dynamics simulation to evaluate the rotational relaxation process in nitrogen from a first-principles perspective. The intermolecular model used in the MD simulations is shown to (i) reproduce very well the shear viscosity of nitrogen over a wide range of temperatures, (ii) predict the near-equilibrium rotational collision number in good agreement with published trajectory calculations done on ab initio Potential Energy Surfaces, and (iii) produce shock wave profiles in excellent accordance with the experimental measurements. The rotational relaxation process is found to be dependent not only on the near-equilibrium temperature (i.e., when systems relax to equilibrium after a small perturbation), but more importantly on both the magnitude and direction of the initial deviation from the equilibrium state. Although this dependence has been previously recognized, it is here investigated systematically. The comparison between MD and DSMC, based on the Larsen-Borgnakke model, for shock waves (both at low and high temperatures) and one-dimensional expansions shows that a judicious choice of a constant Zrot can produce DSMC results which are in relatively good agreement with MD. However, the selection of the rotational collision number is case-specific, depending not only on the temperature range, but more importantly on the type of flow (compression or expansion), with significant limitations for more complex simulations characterized both by expansion and compression zones. Parker’s model, parametrized for nitrogen, overpredicts the magnitude of Zrot for temperatures above about 300 K. It is also unable to describe the dependence of the relaxation process on the direction to equilibrium. Finally, based on the MD data, a preliminary formulation for a novel rotational relaxation model, which includes a dependence on both the rotational and the translational state of the gas, is presented.

Molecular Dynamics Simulations of Normal Shock Waves in Dilute Gas Mixtures

The structure of a normal shock waves in dilute gas mixtures of various compositions (Xenon-Helium and Argon-Helium) are studied using Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC). The MD solutions reproduce well the experimental data, with the exception of the parallel temperature profile in the 24.7% Ar-He mixture, despite the satisfactory agreement between the parallel velocity profiles. The Generalized Hard Sphere (GHS) DSMC solutions match almost perfectly the MD data, for all the cases considered. Using textbook parameters for like particles and their arithmetic averages for unlike particles, the Variable Hard Sphere (VHS) model fails to describe the shock wave structure in the Xe-He mixture. For the Ar-He cases, the VHS diffusion cross-section better approximates that obtained with the Lennard-Jones potential. Therefore, the VHS DSMC solutions exhibit less significant differences with those obtained with MD. If the VHS model is properly parametrized, the agreement between DSMC and MD solutions increases significantly. These cases exemplify situations where MD simulations based on a reliable interatomic potential can be used to inform simplified collision schemes used in DSMC.