The stochastic weighted particle method (SWPM) is a generalization of the Direct Simulation Monte Carlo (DSMC) method where particle weights are variable and dynamic. SWPM is backed by a strong theoretical foundation but has not been critically evaluated for problems of practical interest. A thorough assessment of SWPM for boundary-driven flows reveals significant numerical artifacts near the boundary, notably a diverging heat flux. To correct the boundary heat flux, two modifications to SWPM are proposed: separated grouping and a spatially-dependent weight transfer function. To gauge the relative efficiency of SWPM in comparison to DSMC, a high-Mach-number wheel flow which forms a strong density gradient is also simulated.
We investigate hydrodynamic fluctuations in the flow past a circular cylinder near the critical Reynolds number Rec for the onset of vortex shedding. Starting from the fluctuating Navier-Stokes equations, we perform a perturbation expansion around Rec to derive analytical expressions for the statistics of the fluctuating lift force. Molecular-level simulations using the direct simulation Monte Carlo method support the theoretical predictions of the lift power spectrum and amplitude distribution. Notably, we have been able to collect sufficient statistics at distances Re/Rec-1=O(10-3) from the instability that confirm the appearance of non-Gaussian fluctuations, and we observe that they are associated with intermittent vortex shedding. These results emphasize how unavoidable thermal-noise-induced fluctuations become dramatically amplified in the vicinity of oscillatory flow instabilities and that their onset is fundamentally stochastic.
This report summarizes the work towards developing stochastic weighted particle methods (SWPM) for future application in hypersonic flows. Extensive changes to Sandia’s direct simulation Monte Carlo (DSMC) solver, SPARTA (Stochastic Particle Real Time Analyzer), were made to enable the necessary particle splitting and reduction capabilities for SWPM. The results from one-dimensional Couette and Fourier flows suggest that SWPM can reproduce the correct transport for a large range of Knudsen numbers with adequate accuracy. The associated velocity and temperature profiles are in good agreement with DSMC. An issue with particle placement during particle number reduction, is identified, to which, a simple but effective solution based on minimizing the center of mass error is proposed. High Mach wheel flows are simulated using the SWPM and DSMC methods. SWPM is capable of providing nearly an order of magnitude increase in efficiency over DSMC while retaining high accuracy.
We report flow statistics and visualizations from molecular-gas-dynamics simulations using the direct simulation Monte Carlo (DSMC) method for turbulent Couette flow in a minimal domain where the lower wall is replaced by an idealized permeable fibrous substrate representative of thermal-protection-system materials for which the Knudsen number is O(10-1). Comparisons are made with smooth-wall DSMC simulations and smooth-wall direct numerical simulations (DNS) of the Navier-Stokes equations for the same conditions. Roughness, permeability, and noncontinuum effects are assessed. In the range of Reynolds numbers considered herein, the scalings of the skin friction on the permeable substrate and of the mean flow within the substrate suggest that they are dominated by viscous effects. While the regenerative cycle characteristic of smooth-wall turbulence remains intact for all cases considered, we observe that the near-wall velocity fluctuations are modulated by the permeable substrate with a wavelength equal to the pore spacing. Additionally, the flow within the substrate shows significant rarefaction effects, resulting in an apparent permeability that is 13% larger than the intrinsic permeability. In contrast, the smooth-wall DSMC and DNS simulations exhibit remarkably good agreement for the statistics examined, despite the Knudsen number based on the viscous length scale being as large as O(10-1). This latter result is at variance with classical estimates for the breakdown of the continuum assumption and calls for further investigations into the interaction of noncontinuum effects and turbulence.
Kolmogorov's theory of turbulence assumes that the small-scale turbulent structures in the energy cascade are universal and are determined by the energy dissipation rate and the kinematic viscosity alone. However, thermal fluctuations, absent from the continuum description, terminate the energy cascade near the Kolmogorov length scale. Here, we propose a simple superposition model to account for the effects of thermal fluctuations on small-scale turbulence statistics. For compressible Taylor-Green vortex flow, we demonstrate that the superposition model in conjunction with data from direct numerical simulation of the Navier-Stokes equations yields spectra and structure functions that agree with the corresponding quantities computed from the direct simulation Monte Carlo method of molecular gas dynamics, verifying the importance of thermal fluctuations in the dissipation range.
For decades, it has been observed that the commonly used Borgnakke-Larsen method for energy redistribution in Direct Simulation Monte Carlo codes fails to satisfy the principle of detailed balance when coupled to a wide variety of temperature dependent relaxation models, while seemingly satisfying detailed balance when coupled to others. Many attempts have been made to remedy the issue, yet much ambiguity remains, and no consensus appears in the literature regarding the root cause of the intermittent compatibility of the Borgnakke-Larsen method with temperature dependent relaxation models. This paper alleviates that ambiguity by presenting a rigorous theoretical derivation of the Borgnakke-Larsen method's requirement for satisfying detailed balance. Specifically, it is shown that the Borgnakke-Larsen method maintains detailed balance if and only if the probability of internal-energy exchange during a collision depends only on collision invariants (e.g., total energy). The consequences of this result are explored in the context of several published definitions of relaxation temperature, including translational, total, and cell-averaged temperatures. Of particular note, it is shown that cell-averaged temperatures, which have been widely discussed in the literature as a way to ensure equilibrium is reached, also fail in a similar, although less dramatic, fashion when the aforementioned relationship is not enforced. The developed theory can be used when implementing existing or new relaxation models and will ensure that detailed balance is satisfied.
In turbulent flows, kinetic energy is transferred from the largest scales to progressively smaller scales, until it is ultimately converted into heat. The Navier-Stokes equations are almost universally used to study this process. Here, by comparing with molecular-gas-dynamics simulations, we show that the Navier-Stokes equations do not describe turbulent gas flows in the dissipation range because they neglect thermal fluctuations. We investigate decaying turbulence produced by the Taylor-Green vortex and find that in the dissipation range the molecular-gas-dynamics spectra grow quadratically with wave number due to thermal fluctuations, in agreement with previous predictions, while the Navier-Stokes spectra decay exponentially. Furthermore, the transition to quadratic growth occurs at a length scale much larger than the gas molecular mean free path, namely in a regime that the Navier-Stokes equations are widely believed to describe. In fact, our results suggest that the Navier-Stokes equations are not guaranteed to describe the smallest scales of gas turbulence for any positive Knudsen number.
Most studies of vortex shedding from a circular cylinder in a gas flow have explicitly or implicitly assumed that the no-slip condition applies on the cylinder surface. To investigate the effect of slip, vortex shedding is simulated using molecular gas dynamics (the direct simulation Monte Carlo method) and computational fluid dynamics (the incompressible Navier-Stokes equations with a slip boundary condition). A Reynolds number of 100, a Mach number of 0.3, and a corresponding Knudsen number of 0.0048 are examined. For these conditions, compressibility effects are small, and periodic laminar vortex shedding is obtained. Slip on the cylinder is varied using combinations of diffuse and specular molecular reflections with accommodation coefficients from zero (maximum slip) to unity (minimum slip). Although unrealistic, bounce-back molecular reflections are also examined because they approximate the no-slip boundary condition (zero slip). The results from both methods are in reasonable agreement. The shedding frequency increases slightly as the accommodation coefficient is decreased, and shedding ceases at low accommodation coefficients (large slip). The streamwise and transverse forces decrease as the accommodation coefficient is decreased. Based on the good agreement between the two methods, computational fluid dynamics is used to determine the critical accommodation coefficient below which vortex shedding ceases for Reynolds numbers of 60-100 at a Mach number of 0.3. Conditions to observe the effect of slip on vortex shedding appear to be experimentally realizable, although challenging.
Few of those who read the 1963 research note by Graeme A. Bird in Physics of Fluids could have imagined that 50 years later the proposed new numerical technique would have become the dominant numerical technique in molecular gas dynamics. The introduction of the Direct Simulation Monte Carlo (DSMC) method not only has altered the field of molecular gas dynamics but also has influenced fields such as physical chemistry, mathematics, computer science, and aerothermodynamics. Further, the DSMC method has been used to probe into previously uninvestigated theoretical aspects of the Boltzmann equation and has also served as a platform for the development of nonequilibrium chemistry models. The DSMC method’s most noteworthy achievement is that molecular gas dynamics became a practical tool in the hands of aerospace engineers in many situations of spacecraft design and mission analysis.
The gold-standard definition of the Direct Simulation Monte Carlo (DSMC) method is given in the 1994 book by Bird [Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon Press, Oxford, UK, 1994)], which refined his pioneering earlier papers in which he first formulated the method. In the intervening 25 years, DSMC has become the method of choice for modeling rarefied gas dynamics in a variety of scenarios. The chief barrier to applying DSMC to more dense or even continuum flows is its computational expense compared to continuum computational fluid dynamics methods. The dramatic (nearly billion-fold) increase in speed of the largest supercomputers over the last 30 years has thus been a key enabling factor in using DSMC to model a richer variety of flows, due to the method's inherent parallelism. We have developed the open-source SPARTA DSMC code with the goal of running DSMC efficiently on the largest machines, both current and future. It is largely an implementation of Bird's 1994 formulation. Here, we describe algorithms used in SPARTA to enable DSMC to operate in parallel at the scale of many billions of particles or grid cells, or with billions of surface elements. We give a few examples of the kinds of fundamental physics questions and engineering applications that DSMC can address at these scales.
Here, we provide a demonstration that gas-kinetic methods incorporating molecular chaos can simulate the sustained turbulence that occurs in wall-bounded turbulent shear flows. The direct simulation Monte Carlo method, a gas-kinetic molecular method that enforces molecular chaos for gas-molecule collisions, is used to simulate the minimal Couette flow at Re = 500 . The resulting law of the wall, the average wall shear stress, the average kinetic energy, and the continually regenerating coherent structures all agree closely with corresponding results from direct numerical simulation of the Navier-Stokes equations. Finally, these results indicate that molecular chaos for collisions in gas-kinetic methods does not prevent development of molecular-scale long-range correlations required to form hydrodynamic-scale turbulent coherent structures.
A heterogeneous run on the full Trinity supercomputer at LANL was performed using SPARTA during March 9-12, 2018. Over 19,000 nodes (9200+ Haswell and 9900+ KNL) and 1.2 million MPI processes were used. The run was successful, with SPARTA running for several hours with good performance (better than the same simulation running on full Sequoia). However, several challenges were encountered, and some unresolved issues remain.
Modeling of chemical and ionization reactions at the extreme conditions of upper-atmosphere hypersonic flow has been critical for spacecraft design from the Apollo era to the present because chemical activity in the flow reduces heat transfer. Nitrogen, which behaves as an inert gas in ambient flows, becomes chemically active under conditions of hypersonic reentry (-10,000 K). Atmospheric chemical reactions during hypersonic reentry are dominated by dissociation of diatomic nitrogen and oxygen molecules and exchange reactions involving diatomic molecules and single atoms. At higher temperatures, ionization also occurs.
We provide the first demonstration that molecular-level methods based on gas kinetic theory and molecular chaos can simulate turbulence and its decay. The direct simulation Monte Carlo (DSMC) method, a molecular-level technique for simulating gas flows that resolves phenomena from molecular to hydrodynamic (continuum) length scales, is applied to simulate the Taylor-Green vortex flow. The DSMC simulations reproduce the Kolmogorov -5/3 law and agree well with the turbulent kinetic energy and energy dissipation rate obtained from direct numerical simulation of the Navier-Stokes equations using a spectral method. This agreement provides strong evidence that molecular-level methods for gases can be used to investigate turbulent flows quantitatively.
The Rayleigh-Taylor instability (RTI) is investigated using the direct simulation Monte Carlo (DSMC) method of molecular gas dynamics. Here, fully resolved two-dimensional DSMC RTI simulations are performed to quantify the growth of flat and single-mode perturbed interfaces between two atmospheric-pressure monatomic gases as a function of the Atwood number and the gravitational acceleration. The DSMC simulations reproduce many qualitative features of the growth of the mixing layer and are in reasonable quantitative agreement with theoretical and empirical models in the linear, nonlinear, and self-similar regimes. In some of the simulations at late times, the instability enters the self-similar regime, in agreement with experimental observations. For the conditions simulated, diffusion can influence the initial instability growth significantly.
Quasi-classical trajectory (QCT) calculations are used to study state-specific ro-vibrational energy exchange and dissociation in the O2 + O system. Atom-diatom collisions with energy between 0.1 and 20 eV are calculated with a double many body expansion potential energy surface by Varandas and Pais [Mol. Phys. 65, 843 (1988)]. Inelastic collisions favor mono-quantum vibrational transitions at translational energies above 1.3 eV although multi-quantum transitions are also important. Post-collision vibrational favoring decreases first exponentially and then linearly as Δv increases. Vibrationally elastic collisions (Δv = 0) favor small ΔJ transitions while vibrationally inelastic collisions have equilibrium post-collision rotational distributions. Dissociation exhibits both vibrational and rotational favoring. New vibrational-translational (VT), vibrational-rotational-translational (VRT) energy exchange, and dissociation models are developed based on QCT observations and maximum entropy considerations. Full set of parameters for state-to-state modeling of oxygen is presented. The VT energy exchange model describes 22 000 state-to-state vibrational cross sections using 11 parameters and reproduces vibrational relaxation rates within 30% in the 2500–20 000 K temperature range. The VRT model captures 80 × 106 state-to-state ro-vibrational cross sections using 19 parameters and reproduces vibrational relaxation rates within 60% in the 5000–15 000 K temperature range. The developed dissociation model reproduces state-specific and equilibrium dissociation rates within 25% using just 48 parameters. The maximum entropy framework makes it feasible to upscale ab initio simulation to full nonequilibrium flow calculations.
The Richtmyer-Meshkov instability (RMI) is investigated using the Direct Simulation Monte Carlo (DSMC) method of molecular gas dynamics. Due to the inherent statistical noise and the significant computational requirements, DSMC is hardly ever applied to hydrodynamic flows. Here, DSMC RMI simulations are performed to quantify the shock-driven growth of a single-mode perturbation on the interface between two atmospheric-pressure monatomic gases prior to re-shocking as a function of the Atwood and Mach numbers. The DSMC results qualitatively reproduce all features of the RMI and are in reasonable quantitative agreement with existing theoretical and empirical models. The DSMC simulations indicate that there is a universal behavior, consistent with previous work in this field that RMI growth follows.
This report presents the test cases used to verify, validate and demonstrate the features and capabilities of the first release of the 3D Direct Simulation Monte Carlo (DSMC) code SPARTA (Stochastic Real Time Particle Analyzer). The test cases included in this report exercise the most critical capabilities of the code like the accurate representation of physical phenomena (molecular advection and collisions, energy conservation, etc.) and implementation of numerical methods (grid adaptation, load balancing, etc.). Several test cases of simple flow examples are shown to demonstrate that the code can reproduce phenomena predicted by analytical solutions and theory. A number of additional test cases are presented to illustrate the ability of SPARTA to model flow around complicated shapes. In these cases, the results are compared to other well-established codes or theoretical predictions. This compilation of test cases is not exhaustive, and it is anticipated that more cases will be added in the future.
The development of advanced thermal barrier coatings (TBCs) of yttria stabilized zirconia (YSZ) that exhibit lower thermal conductivity through better control of electron beam - physical vapor deposition (EB-PVD) processing is of prime interest to both the aerospace and power industries. This report summarizes the work performed under a two-year Lab-Directed Research and Development (LDRD) project (38664) to produce lower thermal conductivity, graded-layer thermal barrier coatings for turbine blades in an effort to increase the efficiency of high temperature gas turbines. This project was sponsored by the Nuclear Fuel Cycle Investment Area. Therefore, particular importance was given to the processing of the large blades required for industrial gas turbines proposed for use in the Brayton cycle of nuclear plants powered by high temperature gas-cooled reactors (HTGRs). During this modest (~1 full-time equivalent (FTE)) project, the processing technology was developed to create graded TBCs by coupling ion beam-assisted deposition (IBAD) with substrate pivoting in the alumina-YSZ system. The Electron Beam - 1200 kW (EB-1200) PVD system was used to deposit a variety of TBC coatings with micron layered microstructures and reduced thermal conductivity below 1.5 W/m.K. The use of IBAD produced fully stoichiometric coatings at a reduced substrate temperature of 600°C and a reduced oxygen background pressure of 0.1 Pa. IBAD was also used to successfully demonstrate the transitioning of amorphous PVD-deposited alumina to the -phase alumina required as an oxygen diffusion barrier and for good adhesion to the substrate Ni2Al3 bondcoat. This process replaces the time consuming thermally grown oxide formation required before the YSZ deposition. In addition to the process technology, Direct Simulation Monte Carlo plume modeling and spectroscopic characterization of the PVD plumes were performed. The project consisted of five tasks. These included the production of layered periodic microstructures in the coating, the Direct Simulation Monte Carlo (DSMC) modeling of particle transport in the PVD plume, functional graded layer development, the deposition of all layers to form a complete coating, and materials characterization including thermal testing. Ion beam-assisted deposition, beam sharing through advanced digital rastering, substrate pivoting, hearth calorimetry, infrared imaging, fiber optic-enabled optical emission spectroscopy and careful thermal management were used to achieve all the milestones outlined in the FY02 LDRD proposal.
The Quantum-Kinetic (Q-K) chemical reaction model is implemented in a Navier-Stokes solver, US3D, and tested on the Bow Shock UltraViolet flight experiments. The chemical reaction rates predicted by the Q-K model are compared to a commonly used Park model for flows in thermal non-equilibrium. The results show that in thermal equilibrium the reaction rates between these two models are comparable. The Q-K model predicts greater rates for some chemical reactions and lesser rates for other reactions in an five species air chemistry model. In thermal non-equilibrium, the Q-K model maintains comparable rates near thermal equilibrium, while avoiding issues of strong thermal non-equilibrium seen in the Park model. The application of the Q-K model to the Bow Shock UltraViolet flight experiments show that the model remains consistent with previous Navier-Stokes and DSMC computations over altitudes ranging from 53:5 km up to 87:5 km despite the enforcement of translational-rotational equilibrium. The commonly used Park model was unable to match this performance.
The Quantum-Kinetic (Q-K) chemical reaction model is implemented in a Navier-Stokes solver, US3D, and tested on the Bow Shock UltraViolet flight experiments. The chemical reaction rates predicted by the Q-K model are compared to a commonly used Park model for flows in thermal non-equilibrium. The results show that in thermal equilibrium the reaction rates between these two models are comparable. The Q-K model predicts greater rates for some chemical reactions and lesser rates for other reactions in an five species air chemistry model. In thermal non-equilibrium, the Q-K model maintains comparable rates near thermal equilibrium, while avoiding issues of strong thermal non-equilibrium seen in the Park model. The application of the Q-K model to the Bow Shock UltraViolet flight experiments show that the model remains consistent with previous Navier-Stokes and DSMC computations over altitudes ranging from 53:5 km up to 87:5 km despite the enforcement of translational-rotational equilibrium. The commonly used Park model was unable to match this performance.
The effect of collision-partner selection schemes on the accuracy and the efficiency of the Direct Simulation Monte Carlo (DSMC) method of Bird is investigated. Several schemes to reduce the total discretization error as a function of the mean collision separation and the mean collision time are examined. These include the historically first sub-cell scheme, the more recent nearest-neighbor scheme, and various near-neighbor schemes, which are evaluated for their effect on the thermal conductivity for Fourier flow. Their convergence characteristics as a function of spatial and temporal discretization and the number of simulators per cell are compared to the convergence characteristics of the sophisticated and standard DSMC algorithms. Improved performance is obtained if the population from which possible collision partners are selected is an appropriate fraction of the population of the cell.
We will present experimental and computational investigations of the thermal performance of microelectromechanical systems (MEMS) as a function of the surrounding gas pressure. Lowering the pressure in MEMS packages reduces gas damping, providing increased sensitivity for certain MEMS sensors; however, such packaging also dramatically affects their thermal performance since energy transfer to the environment is substantially reduced. High-spatial-resolution Raman thermometry was used to measure the temperature profiles on electrically heated, polycrystalline silicon bridges that are nominally 10 microns wide, 2.25 microns thick, 12 microns above the substrate, and either 200 or 400 microns long in nitrogen atmospheres with pressures ranging from 0.05 to 625 Torr. Finite element modeling of the thermal behavior of the MEMS bridges is performed and compared to the experimental results. Noncontinuum gas effects are incorporated into the continuum finite element model by imposing temperature discontinuities at gas-solid interfaces that are determined from noncontinuum simulations. The experimental and simulation results indicate that at pressures below 0.5 Torr the gas-phase heat transfer is negligible compared to heat conduction through the thermal actuator legs. As the pressure increases above 0.5 Torr, the gas-phase heat transfer becomes more significant. At ambient pressures, gas-phase heat transfer drastically impacts the thermal performance. The measured and simulated temperature profiles are in qualitative agreement in the present study. Quantitative agreement between experimental and simulated temperature profiles requires accurate knowledge of temperature-dependent thermophysical properties, the device geometry, and the thermal accommodation coefficient.
Thermal accommodation coefficients have been derived for a variety of gas-surface combinations using an experimental apparatus developed to measure the pressure dependence of the conductive heat flux between parallel plates at unequal temperature separated by a gas-filled gap. The heat flux is inferred from temperature-difference measurements across the plates in a configuration where the plate temperatures are set with two carefully controlled thermal baths. Temperature-controlled shrouds provide for environmental isolation of the opposing test plates. Since the measured temperature differences in these experiments are very small (typically 0.3 C or less over the entire pressure range), high-precision thermistors are used to acquire the requisite temperature data. High-precision components have also been utilized on the other control and measurement subsystems in this apparatus, including system pressure, gas flow rate, plate alignment, and plate positions. The apparatus also includes the capability for in situ plasma cleaning of the installed test plates. Measured heat-flux results are used in a formula based on Direct Simulation Monte Carlo (DSMC) code calculations to determine the thermal accommodation coefficients. Thermal accommodation coefficients have been determined for three different gases (argon, nitrogen, helium) in contact with various surfaces. Materials include metals and alloys such as aluminum, gold, platinum, and 304 stainless steel. A number of materials important to fabrication of Micro Electro Mechanical Systems (MEMS) devices have also been examined. For most surfaces, coefficient values are near 0.95, 0.85, and 0.45 for argon, nitrogen, and helium, respectively. Only slight differences in accommodation as a function of surface roughness have been seen. Surface contamination appears to have a more significant effect: argon plasma treatment has been observed to reduce thermal accommodation by as much as 0.10 for helium. Mixtures of argon and helium have also been examined, and the results have been compared to DSMC simulations incorporating thermal-accommodation values from single-species experiments.
The effect of collision-partner selection schemes on the accuracy and the efficiency of the Direct Simulation Monte Carlo (DSMC) method of Bird is investigated. Several schemes to reduce the total discretization error as a function of the mean collision separation and the mean collision time are examined. These include the historically first sub-cell scheme, the more recent nearest-neighbor scheme, and various near-neighbor schemes, which are evaluated for their effect on the thermal conductivity for Fourier flow. Their convergence characteristics as a function of spatial and temporal discretization and the number of simulators per cell are compared to the convergence characteristics of the sophisticated and standard DSMC algorithms. Improved performance is obtained if the population from which possible collision partners are selected is an appropriate fraction of the population of the cell.
A recently proposed approach for the Direct Simulation Monte Carlo (DSMC) method to calculate chemical-reaction rates is assessed for high-temperature atmospheric species. The new DSMC model reproduces measured equilibrium reaction rates without using any macroscopic reaction-rate information. Since it uses only molecular properties, the new model is inherently able to predict reaction rates for arbitrary non-equilibrium conditions. DSMC non-equilibrium reaction rates are compared to Park's phenomenological nonequilibrium reaction-rate model, the predominant model for hypersonic-flow-field calculations. For near-equilibrium conditions, Park's model is in good agreement with the DSMC-calculated reaction rates. For far-from-equilibrium conditions, corresponding to a typical shock layer, significant differences can be found. The DSMC predictions are also found to be in very good agreement with measured and calculated non-equilibrium reaction rates, offering strong evidence that this is a viable and reliable technique to predict chemical reaction rates.
The effect of collision-partner selection schemes on the accuracy and the efficiency of the Direct Simulation Monte Carlo (DSMC) method of Bird is investigated. Several schemes to reduce the total discretization error as a function of the mean collision separation and the mean collision time are examined. These include the historically first sub-cell scheme, the more recent nearest-neighbor scheme, and various near-neighbor schemes, which are evaluated for their effect on the thermal conductivity for Fourier flow. Their convergence characteristics as a function of spatial and temporal discretization and the number of simulators per cell are compared to the convergence characteristics of the sophisticated and standard DSMC algorithms. Improved performance is obtained if the population from which possible collision partners are selected is an appropriate fraction of the population of the cell.
A set of Direct Simulation Monte Carlo (DSMC) chemical-reaction models recently proposed by Bird and based solely on the collision energy and the vibrational energy levels of the species involved is applied to calculate nonequilibrium chemical-reaction rates for atmospheric reactions in hypersonic flows. The DSMC non-equilibrium model predictions are in good agreement with theoretical models and experimental measurements. The observed agreement provides strong evidence that modeling chemical reactions using only the collision energy and the vibrational energy levels provides an accurate method for predicting non-equilibrium chemical-reaction rates.
This report summarizes the work completed under the Laboratory Directed Research and Development (LDRD) project 09-1351, 'Computational Investigation of Thermal Gas Separation for CO{sub 2} Capture'. Thermal gas separation for a binary mixture of carbon dioxide and nitrogen is investigated using the Direct Simulation Monte Carlo (DSMC) method of molecular gas dynamics. Molecular models for nitrogen and carbon dioxide are developed, implemented, compared to theoretical results, and compared to several experimental thermophysical properties. The molecular models include three translational modes, two fully excited rotational modes, and vibrational modes, whose degree of excitation depends on the temperature. Nitrogen has one vibrational mode, and carbon dioxide has four vibrational modes (two of which are degenerate). These models are used to perform a parameter study for mixtures of carbon dioxide and nitrogen confined between parallel walls over realistic ranges of gas temperatures and nominal concentrations of carbon dioxide. The degree of thermal separation predicted by DSMC is slightly higher than experimental values and is sensitive to the details of the molecular models.
This report summarizes the work completed during FY2009 for the LDRD project 09-1332 'Molecule-Based Approach for Computing Chemical-Reaction Rates in Upper-Atmosphere Hypersonic Flows'. The goal of this project was to apply a recently proposed approach for the Direct Simulation Monte Carlo (DSMC) method to calculate chemical-reaction rates for high-temperature atmospheric species. The new DSMC model reproduces measured equilibrium reaction rates without using any macroscopic reaction-rate information. Since it uses only molecular properties, the new model is inherently able to predict reaction rates for arbitrary nonequilibrium conditions. DSMC non-equilibrium reaction rates are compared to Park's phenomenological non-equilibrium reaction-rate model, the predominant model for hypersonic-flow-field calculations. For near-equilibrium conditions, Park's model is in good agreement with the DSMC-calculated reaction rates. For far-from-equilibrium conditions, corresponding to a typical shock layer, the difference between the two models can exceed 10 orders of magnitude. The DSMC predictions are also found to be in very good agreement with measured and calculated non-equilibrium reaction rates. Extensions of the model to reactions typically found in combustion flows and ionizing reactions are also found to be in very good agreement with available measurements, offering strong evidence that this is a viable and reliable technique to predict chemical reaction rates.
The convergence rate of a new direct simulation Monte Carlo (DSMC) method, termed 'sophisticated DSMC', is investigated for one-dimensional Fourier flow. An argon-like hard-sphere gas at 273.15K and 266.644Pa is confined between two parallel, fully accommodating walls 1mm apart that have unequal temperatures. The simulations are performed using a one-dimensional implementation of the sophisticated DSMC algorithm. In harmony with previous work, the primary convergence metric studied is the ratio of the DSMC-calculated thermal conductivity to its corresponding infinite-approximation Chapman-Enskog theoretical value. As discretization errors are reduced, the sophisticated DSMC algorithm is shown to approach the theoretical values to high precision. The convergence behavior of sophisticated DSMC is compared to that of original DSMC. The convergence of the new algorithm in a three-dimensional implementation is also characterized. Implementations using transient adaptive sub-cells and virtual sub-cells are compared. The new algorithm is shown to significantly reduce the computational resources required for a DSMC simulation to achieve a particular level of accuracy, thus improving the efficiency of the method by a factor of 2.
This report documents technical work performed to complete the ASC Level 2 Milestone 2841: validation of thermal models for a prototypical MEMS thermal actuator. This effort requires completion of the following task: the comparison between calculated and measured temperature profiles of a heated stationary microbeam in air. Such heated microbeams are prototypical structures in virtually all electrically driven microscale thermal actuators. This task is divided into four major subtasks. (1) Perform validation experiments on prototypical heated stationary microbeams in which material properties such as thermal conductivity and electrical resistivity are measured if not known and temperature profiles along the beams are measured as a function of electrical power and gas pressure. (2) Develop a noncontinuum gas-phase heat-transfer model for typical MEMS situations including effects such as temperature discontinuities at gas-solid interfaces across which heat is flowing, and incorporate this model into the ASC FEM heat-conduction code Calore to enable it to simulate these effects with good accuracy. (3) Develop a noncontinuum solid-phase heat transfer model for typical MEMS situations including an effective thermal conductivity that depends on device geometry and grain size, and incorporate this model into the FEM heat-conduction code Calore to enable it to simulate these effects with good accuracy. (4) Perform combined gas-solid heat-transfer simulations using Calore with these models for the experimentally investigated devices, and compare simulation and experimental temperature profiles to assess model accuracy. These subtasks have been completed successfully, thereby completing the milestone task. Model and experimental temperature profiles are found to be in reasonable agreement for all cases examined. Modest systematic differences appear to be related to uncertainties in the geometric dimensions of the test structures and in the thermal conductivity of the polycrystalline silicon test structures, as well as uncontrolled nonuniform changes in this quantity over time and during operation.
An experimental apparatus is described that measures gas-surface thermal accommodation coefficients from the pressure dependence of the conductive heat flux between parallel plates separated by a gas-filled gap. Heat flux between the plates is inferred from measurements of temperature drop between the plate surface and an adjacent temperature-controlled water bath. Thermal accommodation coefficients are determined from the pressure dependence of the heat flux at a fixed plate separation. The apparatus is designed to conduct tests with a variety of gases in contact with interchangeable, well-characterized surfaces of various materials (e.g., metals, ceramics, semiconductors) with various surface finishes (e.g., smooth, rough). Experiments are reported for three gases (argon, nitrogen, and helium) in contact with pairs of 304 stainless steel plates prepared with one of two finishes: lathe-machined or mirror-polished. For argon and nitrogen, the measured accommodation coefficients for machined and polished plates are near unity and independent of finish to within experimental uncertainty. For helium, the accommodation coefficients are much lower and show a slight variation with surface roughness. Two different methods are used to determine the accommodation coefficient from experimental data: the Sherman-Lees formula and the GTR formula. These approaches yield values of 0.87 and 0.94 for argon, 0.80 and 0.86 for nitrogen, 0.36 and 0.38 for helium with the machined finish, and 0.40 and 0.42 for helium with the polished finish, respectively, with an uncertainty of ±0.02. The GTR values for argon and nitrogen are generally in better agreement with the results of other investigators than the Sherman-Lees values are, and both helium results are in reasonable agreement with values in the literature.
The convergence behavior of the Direct Simulation Monte Carlo (DSMC) method is investigated for transient flows. Two types of flows are considered: a Couette-like flow, in which an initial velocity profile decays in time, and a Fourier-like flow, in which an initial temperature profile decays in time. DSMC results are presented for hard-sphere argon with Knudsen numbers in the range 0.01-0.4. Low-Knudsen-number DSMC results are compared with Navier-Stokes results. The DSMC discretization errors from finite time step and finite cell size (in the limit of infinite number of computational molecules per cell) are compared with the predictions of Green-Kubo theory for conditions in this regime.
In many direct simulation Monte Carlo (DSMC) simulations, the majority of computation time is consumed after the flowfield reaches a steady state. This situation occurs when the desired output quantities are small compared to the background fluctuations. For example, gas flows in many microelectromechanical systems (MEMS) have mean speeds more than two orders of magnitude smaller than the thermal speeds of the molecules themselves. The current solution to this problem is to collect sufficient samples to achieve the desired resolution. This can be an arduous process because the error is inversely proportional to the square root of the number of samples so we must, for example, quadruple the samples to cut the error in half. This work is intended to improve this situation by employing more advanced techniques, from fields other than solely statistics, for determining the output quantities. Our strategy centers on exploiting information neglected by current techniques, which collect moments in each cell without regard to one another, values in neighboring cells, nor their evolution in time. Unlike many previous acceleration techniques that modify the method itself, the techniques examined in this work strictly post-process so they may be applied to any DSMC code without affecting its fidelity or generality. Many potential methods are drawn from successful applications in a diverse range of areas, from ultrasound imaging to financial market analysis. The most promising methods exploit relationships between variables in space, which always exist in DSMC due to the absence of shocks. Disparate techniques were shown to produce similar error reductions, suggesting that the results shown in this report may be typical of what is possible using these methods. Sample count reduction factors of approximately three to five were found to be typical, although factors exceeding ten were shown on some variables under some techniques.
Modeling microscale heat transfer with the computational-heat-transfer code Calore is discussed. Microscale heat transfer problems differ from their macroscopic counterparts in that conductive heat transfer in both solid and gaseous materials may have important noncontinuum effects. In a solid material, three noncontinuum effects are considered: ballistic transport of phonons across a thin film, scattering of phonons from surface roughness at a gas-solid interface, and scattering of phonons from grain boundaries within the solid material. These processes are modeled for polycrystalline silicon, and the thermal-conductivity values predicted by these models are compared to experimental data. In a gaseous material, two noncontinuum effects are considered: ballistic transport of gas molecules across a thin gap and accommodation of gas molecules to solid conditions when reflecting from a solid surface. These processes are modeled for arbitrary gases by allowing the gas and solid temperatures across a gas-solid interface to differ: a finite heat transfer coefficient (contact conductance) is imposed at the gas-solid interface so that the temperature difference is proportional to the normal heat flux. In this approach, the behavior of gas in the bulk is not changed from behavior observed under macroscopic conditions. These models are implemented in Calore as user subroutines. The user subroutines reside within Sandia's Source Forge server, where they undergo version control and regression testing and are available to analysts needing these capabilities. A Calore simulation is presented that exercises these models for a heated microbeam separated from an ambient-temperature substrate by a thin gas-filled gap. Failure to use the noncontinuum heat transfer models for the solid and the gas causes the maximum temperature of the microbeam to be significantly underpredicted.
A combined experimental/modeling study was conducted to better understand the critical role of gas-surface interactions in rarefied gas flows. An experimental chamber and supporting diagnostics were designed and assembled to allow simultaneous measurements of gas heat flux and inter-plate gas density profiles in an axisymmetric, parallel-plate geometry. Measurements of gas density profiles and heat flux are made under identical conditions, eliminating an important limitation of earlier studies. The use of in situ, electron-beam fluorescence is demonstrated as a means to measure gas density profiles although additional work is required to improve the accuracy of this technique. Heat flux is inferred from temperature-drop measurements using precision thermistors. The system can be operated with a variety of gases (monatomic, diatomic, polyatomic, mixtures) and carefully controlled, well-characterized surfaces of different types (metals, ceramics) and conditions (smooth, rough). The measurements reported here are for 304 stainless steel plates with a standard machined surface coupled with argon, helium, and nitrogen. The resulting heat-flux and gas-density-profile data are analyzed using analytic and computational models to show that a simple Maxwell gas-surface interaction model is adequate to represent all of the observations. Based on this analysis, thermal accommodation coefficients for 304 stainless steel coupled with argon, nitrogen, and helium are determined to be 0.88, 0.80, and 0.38, respectively, with an estimated uncertainty of {+-}0.02.
A general, approximate expression is described that can be used to predict the thermophoretic force on a free-molecular, motionless, spherical particle suspended in a quiescent gas with a temperature gradient. The thermophoretic force is equal to the product of an order-unity coefficient, the gas-phase translational heat flux, the particle cross-sectional area, and the inverse of the mean molecular speed. Numerical simulations are used to test the accuracy of this expression for monatomic gases, polyatomic gases, and mixtures thereof. Both continuum and noncontinuum conditions are examined; in particular, the effects of low pressure, wall proximity, and high heat flux are investigated. The direct simulation Monte Carlo (DSMC) method is used to calculate the local molecular velocity distribution, and the force-Green's-function method is used to calculate the thermophoretic force. The approximate expression is found to predict the calculated thermophoretic force to within 10% for all cases examined.
A Micro Electro Mechanical System (MEMS) typically consists of micron-scale parts that move through a gas at atmospheric or reduced pressure. In this situation, the gas-molecule mean free path is comparable to the geometric features of the microsystem, so the gas flow is noncontinuum. When mean-free-path effects cannot be neglected, the Boltzmann equation must be used to describe the gas flow. Solution of the Boltzmann equation is difficult even for the simplest case because of its sevenfold dimensionality (one temporal dimension, three spatial dimensions, and three velocity dimensions) and because of the integral nature of the collision term. The Direct Simulation Monte Carlo (DSMC) method is the method of choice to simulate high-speed noncontinuum flows. However, since DSMC uses computational molecules to represent the gas, the inherent statistical noise must be minimized by sampling large numbers of molecules. Since typical microsystem velocities are low (< 1 m/s) compared to molecular velocities ({approx}400 m/s), the number of molecular samples required to achieve 1% precision can exceed 1010 per cell. The Discrete Velocity Gas (DVG) method, an approach motivated by radiation transport, provides another way to simulate noncontinuum gas flows. Unlike DSMC, the DVG method restricts molecular velocities to have only certain discrete values. The transport of the number density of a velocity state is governed by a discrete Boltzmann equation that has one temporal dimension and three spatial dimensions and a polynomial collision term. Specification and implementation of DVG models are discussed, and DVG models are applied to Couette flow and to Fourier flow. While the DVG results for these benchmark problems are qualitatively correct, the errors in the shear stress and the heat flux can be order-unity even for DVG models with 88 velocity states. It is concluded that the DVG method, as described herein, is not sufficiently accurate to simulate the low-speed gas flows that occur in microsystems.
An improved model for the gas damping of out-of-plane motion of a microbeam is developed based on the Reynolds equation (RE). A boundary condition for the RE is developed that relates the pressure at the beam perimeter to the beam motion. The two coefficients in this boundary condition are determined from Navier-Stokes (NS) simulations with the slip boundary condition for small slip lengths (relative to the gap height) and from Direct Simulation Monte Carlo (DSMC) molecular gas dynamics simulations for larger slip lengths. This boundary condition significantly improves the accuracy of the RE for cases where the beam width is only slightly greater than the gap height.
An approach is presented to compute the force on a spherical particle in a rarefied flow of a monatomic gas. This approach relies on the development of a Green's function that describes the force on a spherical particle in a delta-function molecular velocity distribution function. The gas-surface interaction model in this development allows incomplete accommodation of energy and tangential momentum. The force from an arbitrary molecular velocity distribution is calculated by computing the moment of the force Green's function in the same way that other macroscopic variables are determined. Since the molecular velocity distribution function is directly determined in the DSMC method, the force Green's function approach can be implemented straightforwardly in DSMC codes. A similar approach yields the heat transfer to a spherical particle in a rarefied gas flow. The force Green's function is demonstrated by application to two problems. First, the drag force on a spherical particle at arbitrary temperature and moving at arbitrary velocity through an equilibrium motionless gas is found analytically and numerically. Second, the thermophoretic force on a motionless particle in a motionless gas with a heat flux is found analytically and numerically. Good agreement is observed in both situations.
Direct Simulation Monte Carlo (DSMC) and Navier-Stokes calculations are performed for a Mach 11 25 deg.-55 deg. spherically blunted biconic. The conditions are such that flow is laminar, with separation occurring at the cone-cone juncture. The simulations account for thermochemical nonequilibrium based on standard Arrhenius chemical rates for nitrogen dissociation and Millikan and White vibrational relaxation. The simulation error for the Navier-Stokes (NS) code is estimated to be 2% for the surface pressure and 10% for the surface heat flux. The grid spacing for the DSMC simulations was adjusted to be less than the local mean-freepath (mfp) and the time step less than the cell transient time of a computational particle. There was overall good agreement between the two simulations; however, the recirculation zone was computed to be larger for the NS simulation. A sensitivity study is performed to examine the effects of experimental uncertainty in the freestream properties on the surface pressure and heat flux distributions. The surface quantities are found to be extremely sensitive to the vibrational excitation state of the gas at the test section, with differences of 25% found in the surface pressure and 25%-35% for the surface heat flux. These calculations are part of a blind validation comparison and thus the experimental data has not yet been released.
Icarus is a 2D Direct Simulation Monte Carlo (DSMC) code which has been optimized for the parallel computing environment. The code is based on the DSMC method of Bird[11.1] and models from free-molecular to continuum flowfields in either cartesian (x, y) or axisymmetric (z, r) coordinates. Computational particles, representing a given number of molecules or atoms, are tracked as they have collisions with other particles or surfaces. Multiple species, internal energy modes (rotation and vibration), chemistry, and ion transport are modeled. A new trace species methodology for collisions and chemistry is used to obtain statistics for small species concentrations. Gas phase chemistry is modeled using steric factors derived from Arrhenius reaction rates or in a manner similar to continuum modeling. Surface chemistry is modeled with surface reaction probabilities; an optional site density, energy dependent, coverage model is included. Electrons are modeled by either a local charge neutrality assumption or as discrete simulational particles. Ion chemistry is modeled with electron impact chemistry rates and charge exchange reactions. Coulomb collision cross-sections are used instead of Variable Hard Sphere values for ion-ion interactions. The electro-static fields can either be: externally input, a Langmuir-Tonks model or from a Green's Function (Boundary Element) based Poison Solver. Icarus has been used for subsonic to hypersonic, chemically reacting, and plasma flows. The Icarus software package includes the grid generation, parallel processor decomposition, post-processing, and restart software. The commercial graphics package, Tecplot, is used for graphics display. All of the software packages are written in standard Fortran.
A new scheme to simulate elastic collisions in particle simulation codes is presented. The new scheme aims at simulating the collisions in the highly collisional regime, in which particle simulation techniques typically become computationally expensive. The new scheme is based on the concept of a grid-based collision field. According to this scheme, the particles perform a single collision with the background grid during a time step. The properties of the background field are calculated from the moments of the distribution function accumulated on the grid. The collision operator is based on the Langevin equation. Based on comparisons with other methods, it is found that the Langevin method overestimates the collision frequency for dilute gases.
This paper presents an investigation of a technique for using two-dimensional bodies composed of simple polygons with a body-decoupled uniform Cartesian grid in the Direct Simulation Monte Carlo method (DSMC). The method employs an automated grid preprocessing scheme beginning from a CAD geometry definition file, and is based on polygon triangulation using a trapezoid algorithm. A particle-body intersection time comparison is presented between the Icarus DSMC code using a body-fitted structured grid, and using a structured body-decoupled Cartesian grid with both linear and logarithmic search techniques. A comparison of neutral flow over a cylinder is presented using the structured body fitted grid, and the Cartesian body de-coupled grid.
