Michael A. Gallis

The usefulness of higher-order constitutive relations for describing the Knudsen layer

The Knudsen layer is an important rarefaction phenomenon in gas flows in and around microdevices. Its accurate and efficient modeling is of critical importance in the design of such systems and in predicting their performance. In this paper we investigate the potential that higher-order continuum equations may have to model the Knudsen layer, and compare their predictions to high-accuracy DSMC (direct simulation Monte Carlo) data, as well as a standard result from kinetic theory. We find that, for a benchmark case, the most common higher-order continuum equation sets (Grads 13 moment, Burnett, and super-Burnett equations) cannot capture the Knudsen layer. Variants of these equation families have, however, been proposed and some of them can qualitatively describe the Knudsen layer structure. To make quantitative comparisons, we obtain additional boundary conditions (needed for unique solutions to the higher-order equations) from kinetic theory. However, we find the quantitative agreement with kinetic theory and DSMC data is only slight.

New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows

Fluid flows that do not have local equilibrium are characteristic of some of the new frontiers in engineering and technology, for example, high–speed high–altitude aerodynamics and the development of micrometre–sized fluid pumps, turbines and other devices. However, this area of fluid dynamics is poorly understood from both the experimental and simulation perspectives, which hampers the progress of these technologies. This paper reviews some of the recent developments in experimental techniques and modelling methods for non–equilibrium gas flows, examining their advantages and drawbacks. We also present new results from our computational investigations into both hypersonic and microsystem flows using two distinct numerical methodologies: the direct simulation Monte Carlo method and extended hydrodynamics. While the direct simulation approach produces excellent results and is used widely, extended hydrodynamics is not as well developed but is a promising candidate for future more complex simulations. Finally, we discuss some of the other situations where these simulation methods could be usefully applied, and look to the future of numerical tools for non–equilibrium flows.

Capturing the Knudsen Layer in Continuum-Fluid Models of Nonequilibrium Gas Flows

In hypersonic aerodynamics and microflow device design, the momentum and energy fluxes to solid surfaces are often of critical importance. However, these depend on the characteristics of the Knudsen layer - the region of local non-equilibrium existing up to one or two molecular mean free paths from the wall in any gas flow near a surface. While the Knudsen layer has been investigated extensively using kinetic theory, the ability to capture it within a continuum-fluid formulation (in conjunction with slip boundary conditions) suitable for current computational fluid dynamics toolboxes would offer distinct and practical computational advantages.

A kinetic-theory approach for computing chemical-reaction rates in upper-atmosphere hypersonic flows

Recently proposed molecular-level chemistry models that predict equilibrium and nonequilibrium reaction rates using only kinetic theory and fundamental molecular properties (i.e., no macroscopic reaction-rate information) are investigated for chemical reactions occurring in upper-atmosphere hypersonic flows. The new models are in good agreement with the measured Arrhenius rates for near-equilibrium conditions and with both measured rates and other theoretical models for far-from-equilibrium conditions. Additionally, the new models are applied to representative combustion and ionization reactions and are in good agreement with available measurements and theoretical models. Thus, molecular-level chemistry modeling provides an accurate method for predicting equilibrium and nonequilibrium chemical-reaction rates in gases.

An Experimental Assembly for Precise Measurement of Thermal Accommodation Coefficients.

An experimental apparatus has been developed to determine thermal accommodation coefficients for a variety of gas-surface combinations. Results are obtained primarily through measurement of the pressure dependence of the conductive heat flux between parallel plates separated by a gas-filled gap. Measured heat-flux data are used in a formula based on Direct Simulation Monte Carlo (DSMC) simulations to determine the coefficients. The assembly also features a complementary capability for measuring the variation in gas density between the plates using electron-beam fluorescence. Surface materials examined include 304 stainless steel, gold, aluminum, platinum, silicon, silicon nitride, and polysilicon. Effects of gas composition, surface roughness, and surface contamination have been investigated with this system; the behavior of gas mixtures has also been explored. Without special cleaning procedures, thermal accommodation coefficients for most materials and surface finishes were determined to be near 0.95, 0.85, and 0.45 for argon, nitrogen, and helium, respectively. Surface cleaning by in situ argon-plasma treatment reduced coefficient values by up to 0.10 for helium and by ∼0.05 for nitrogen and argon. Results for both single-species and gas-mixture experiments compare favorably to DSMC simulations.

Ab initio-informed maximum entropy modeling of rovibrational relaxation and state-specific dissociation with application to the O2 + O system

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 × 10(6) 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.

DSMC and Navier-Stokes Predictions for Hypersonic Laminar Interacting Flows

Direct Simulation Monte Carlo (DSMC) and NavierStokes 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 re

Navier-Stokes and direct simulation Monte Carlo predictions for laminar hypersonic separation

Axisymmetric direct simulation Monte Carlo (DSMC) and Navier‐Stokes simulations are performed as part of a code validation effort for hypersonice ows. The e owe eld examined herein is the Mach 11 laminar e ow over a 25 ‐ 55-deg blunted biconic. Experimental data are available for surface pressure and heat e ux at a Knudsen number Kn=0.019 based on the nose radius. Simulations at a reduced freestream density (Kn=0.057) are performed to explore the region of viability of the numerical methods for hypersonic separated e ows. A detailed and careful effort is made to address the numerical accuracy of these simulations, including iterative and grid convergence studiesforNavier ‐Stokesandtemporal,grid,andparticleconvergencestudiesforDSMC.Goodagreementisfound between the DSMC and Navier ‐Stokes simulation approaches for surface properties as well as velocity proe les within the recirculation zone for the reduced density case. The results obtained indicate that the failure of earlier DSMC simulations at Kn=0.019 is due to insufe cient grid ree nement within the recirculation zone. Furthermore, it is shown that accurate simulations of the biconic at the experimental conditions with the DSMC method are not yet possible due to the extreme computational cost. Nomenclature d = molecular diameter, m f = general solution variable Kn = Knudsen number based on nose radius, ¸=RN L = characteristic length scale, m n = number density, particles/m 3 p = pressure, N/m 2 , order of accuracy q = heat e ux, W/m 2

Assessment of Collisional-Energy-Based Models for Atmospheric Species Reactions in Hypersonic Flows

A recently proposed set of direct simulation Monte Carlo chemical reaction models, based solely on the collisional energy and the vibrational energy levels of the species involved, is applied to calculate equilibrium and nonequilibrium chemical reaction rates for atmospheric reactions in hypersonic flows. The direct simulation Monte Carlo model predictions are in good agreement with Parks model, several theoretical models, and experimental measurements. Physically plausible modifications to some of the direct simulation Monte Carlo models are presented that improve agreement. The observed agreement provides strong evidence that modeling of chemical reactions based on collisional energy and vibrational energy levels provides an accurate method for predicting equilibrium and nonequilibrium chemical reaction rates.

Navier-Stokes and DSMC simulations for hypersonic laminar shock-shock interaction flows

DSMC and Navier-Stokes simulations are performed as part of a code validation effort for hypersonic flows. The flowfield examined herein is the Mach 11, laminar flow over a 25 deg 55 deg blunted biconic for which experimental data are available for surface pressure and heat flux. Considerable effort is made to address the numerical accuracy of all simulations including iterative and grid convergence studies for Navier-Stokes and temporal, grid, and particle convergence studies for DSMC. Simulations of the biconic at a reduced freestream density are performed to explore the region of viability of the numerical methods for hypersonic separated flows. Excellent agreement is found between the DSMC and Navier-Stokes simulation approaches for surface properties as well as velocity profiles within the recirculation zone. The results of the rarefied biconic study indicate that the failure of prior DSMC simulations at the experimental densities is due to insufficient grid refinement within the recirculation zone. Additional DSMC and Navier-Stokes simulations are performed for the blunted 25 deg forecone using fine computational meshes to address discrepancies between the simulations and the experiment for the forecone heating. The results of this highly refined forecone study provide strong evidence for the presence of a bias error in the freestream conditions.  1 † Senior Member of Technical Staff, MS 0825, E-mail: cjroy@sandia.gov, Member AIAA ‡ Member of Technical Staff, MS 0827, E-mail: magalli@sandia.gov, Member AIAA § Principal Member of Technical Staff, MS 0820, E-mail: tjbarte@sandia.gov, Member AIAA # Principal Member of Technical Staff, MS 0825, E-mail: jlpayne@sandia.gov, Member AIAA * Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. This paper is declared a work of the U. S. Government and is not subject to copyright protection in the United States. Nomenclature Cp specific heat at constant pressure, J/kgK f general solution variable h specific enthalpy, J/kg p pressure, N/m2, order of accuracy q heat flux, W/m2 R specific gas constant, J/kgK (= 296.8 for N2) RN nose radius, m (= 0.00635) r grid refinement factor s specific entropy, J/kgK T translational temperature, K t time, ms u axial velocity component, m/s V velocity magnitude, m/s x axial coordinate, m y radial coordinate, m γ ratio of specific heats μ absolute viscosity, Ns/m2