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.