Search

Due to the presence of the steep slope and concentration of the gas distribution function in the velocity space, local velocity grid adaptation becomes necessary for high speed aerospace applications and/or other high Knudsen flow. However, the adaptation of velocity grid complicates the quadrature of discrete distribution function, degrades the accuracy even with a large number of velocity grid points. This study presents a simple technique, termed as equilibrium offset, to modify quadrature in the velocity space, which separates the quadrature into an analytical integral of equilibrium distribution and a numerical quadrature of the derivation from the equilibrium. Modified quadrature provides accurate moments of the reference equilibrium state. As the numbers of velocity points increases, the modified quadrature converges to the real value of the integral regardless of the reference equilibrium state. We test several kinds of quadrature on nonuniform velocity grid. The results show that the proposed modification is superior as the distribution function is close to the equilibrium state. For the highly non-equilibrium distribution function the modified quadrature is equally good as the original quadrature. The equilibrium offset is implemented on the adaptive velocity grid. Several numerical simulations demonstrate the efficiency of the modified quadrature on adaptive velocity grid.

Hypersonic separated flows over the so-called “tick” geometry have been studied using the time-accurate direct simulation Monte Carlo (DSMC) method and global linear theory. The free stream condition for two experimental cases studied in the free-piston shock tunnel (named T-ADFA) was modeled. These two cases span a Knudsen number from transitional to continuum, a Mach number of about 10, a free stream enthalpy from 10 to 3 MJ/kg, a Reynolds number varying by a factor of four, and a leading edge geometry varied from sharp to one with a bevel of 0.2 mm. For the first time, the time dependence of flow macroparameters on the leading edge nose radius and the Reynolds number are studied using global linear theory. High-fidelity DSMC simulations showed that the temporal behavior of the separation region, which has significant effects on the surface parameters, depends closely on the leading edge bluntness and wall temperature. The formation of a secondary vortex was seen in about 2 ms for the sharp leading edge, whereas in the rounded leading edge geometry, it formed at earlier 0.7 ms. At a steady state, the size and structure of the separation zone, vortex structures, and surface parameters predicted by DSMC were found to be in good agreement with computational fluid dynamics for the higher density case. Finally, linear stability theory showed that for some leading edge shapes and flow densities, the time to reach the steady state was longer than the facility measurement time.

A fundamental and yet computationally feasible parameter based on the characteristic function of the velocity distribution function (VDF) is proposed for determining the deviation from near-equilibrium conditions in rarefied flow simulations using the direct simulation Monte Carlo (DSMC) method. The proposed parameter utilizes the one-to-one correspondence between the VDF and its characteristic function (or Fourier transform), thereby correlating the deviation of the VDF (from a Chapman-Enskog VDF) with the deviation of the characteristic function (also from that of a Chapman-Enskog VDF). The results are first presented for an unsteady Bobylev solution for approach to equilibrium in 0-D, free-molecular Fourier-Couette flow problem and the Mott-Smith solution for the shock wave all of which have analytical solutions for the VDF, thereby confirming that the proposed parameter indeed captures the deviation from near-equilibrium conditions accurately. The utility of the proposed parameter is then demonstrated using two benchmark problems—Couette flow (over a range of Knudsen numbers) and structure of a normal shock (for upstream Mach numbers of 1.5, 3, and 5)—solved using the DSMC method. While the current work only presents results for benchmark one-dimensional DSMC simulations, the approach can be extended easily to rarefied flows in higher dimensions. Therefore, the proposed parameter has the potential to be used for understanding the nature of VDF and its deviation from near-equilibrium conditions at all locations in a flow field without the need for explicitly sampling the VDF.A fundamental and yet computationally feasible parameter based on the characteristic function of the velocity distribution function (VDF) is proposed for determining the deviation from near-equilibrium conditions in rarefied flow simulations using the direct simulation Monte Carlo (DSMC) method. The proposed parameter utilizes the one-to-one correspondence between the VDF and its characteristic function (or Fourier transform), thereby correlating the deviation of the VDF (from a Chapman-Enskog VDF) with the deviation of the characteristic function (also from that of a Chapman-Enskog VDF). The results are first presented for an unsteady Bobylev solution for approach to equilibrium in 0-D, free-molecular Fourier-Couette flow problem and the Mott-Smith solution for the shock wave all of which have analytical solutions for the VDF, thereby confirming that the proposed parameter indeed captures the deviation from near-equilibrium conditions accurately. The utility of the proposed parameter is then demonstrated...

This work investigates the species separation in the rarefied flow of the argon-helium mixture through convergent-divergent micronozzles. Imposing a molecular mass ratio in the order of 10, the flow of this mixture can lead to the formation of serious nonhomogeneous phenomena such as the species separation. This study is performed in the ranges of 2.0–4.0 for the geometrical expansion ratio, 200–400 K for the wall temperature, and 0.003–1.454 for the inlet Knudsen number. The effects of these parameters are examined on the separative performances of micronozzle. The direct simulation Monte Carlo method is selected as the solution method because it can provide reliable solutions in the current rarefied flow regime study. The current study reveals two important separative effects in the mixture flow through micronozzles. The first effect is the lateral species separation, which results in the enrichment of heavier species near the centerline. The second effect is the streamwise separation, which leads to the enrichment of one species, mostly the lighter one, as the mixture passes through the micronozzle. The current results show that increasing the expansion ratio will enhance the lateral separation monotonically. However, there are specific wall temperature and Knudsen values, which can result in optimum lateral separative effects. In addition, it is observed that the expansion ratio has little effect on the streamwise separation. However, increasing either the wall temperature or the Knudsen number will enhance the streamwise separation, albeit with a limiting value at very high Knudsen numbers.This work investigates the species separation in the rarefied flow of the argon-helium mixture through convergent-divergent micronozzles. Imposing a molecular mass ratio in the order of 10, the flow of this mixture can lead to the formation of serious nonhomogeneous phenomena such as the species separation. This study is performed in the ranges of 2.0–4.0 for the geometrical expansion ratio, 200–400 K for the wall temperature, and 0.003–1.454 for the inlet Knudsen number. The effects of these parameters are examined on the separative performances of micronozzle. The direct simulation Monte Carlo method is selected as the solution method because it can provide reliable solutions in the current rarefied flow regime study. The current study reveals two important separative effects in the mixture flow through micronozzles. The first effect is the lateral species separation, which results in the enrichment of heavier species near the centerline. The second effect is the streamwise separation, which leads to th...

The hydrodynamic mix of fusion fuel and inert pusher can simultaneously generate smaller fuel pockets and finer pusher layers that separate them. Smaller fuel pockets have greater local Knudsen numbers, which tend to exacerbate the Knudsen layer reactivity reduction. A thinner pusher layer separating the neighboring fuel pockets, on the other hand, can enable the diffusive tunneling of Gamow fuel ions through the pusher layer and hence alleviate the Knudsen layer reactivity degradation. Here, the diffusive tunneling phenomenon describes a random walk process by which the Gamow fuel ions from one fuel pocket can traverse the inert pusher layer to join a neighboring fuel pocket without losing much of their energy. This is made possible by the much slower collisional slowing down rate compared with the pitch angle scattering rate of light fuel ions with heavier pusher ions. In an isobaric target mixture where fuel and pusher segments can have distinct temperatures, due to their different compressibilities, the temperature effect on the critical pusher layer areal density below which diffusive tunneling can occur, which is a property of the hydrodynamic mix, is understood by computing the ion charge state distribution using a collisional radiative model. This information is fed into the collisionality evaluation, enabling a parametric scan of the diffusive tunneling physics in terms of the target pressure, fuel, and pusher temperatures. It is found that when the gold pusher layer has a temperature above 1 keV, the variation of the pusher temperature has little effect on the critical areal mass density below which diffusive tunneling can occur. If the pusher layer is 1 keV or below, the critical areal mass density rises sharply, indicating that for a stronger fuel-pusher temperature disparity, the onset of diffusive tunneling will be at an earlier stage of the hydrodynamic mix when the fuel-pusher mixing structures are of less reduced size.The hydrodynamic mix of fusion fuel and inert pusher can simultaneously generate smaller fuel pockets and finer pusher layers that separate them. Smaller fuel pockets have greater local Knudsen numbers, which tend to exacerbate the Knudsen layer reactivity reduction. A thinner pusher layer separating the neighboring fuel pockets, on the other hand, can enable the diffusive tunneling of Gamow fuel ions through the pusher layer and hence alleviate the Knudsen layer reactivity degradation. Here, the diffusive tunneling phenomenon describes a random walk process by which the Gamow fuel ions from one fuel pocket can traverse the inert pusher layer to join a neighboring fuel pocket without losing much of their energy. This is made possible by the much slower collisional slowing down rate compared with the pitch angle scattering rate of light fuel ions with heavier pusher ions. In an isobaric target mixture where fuel and pusher segments can have distinct temperatures, due to their different compressibilities, t...

The general theory of slip flow established in the late 1960s is revisited. For a long time, the complete set of data of the slip and jump coefficients up to the second order of the small Knudsen number have been available only for the Bhatnagar-Gross-Krook model. The present paper provides the complete set of data of those coefficients for a hard-sphere gas on the diffuse reflection boundary. The data are obtained by using the general identities that have been deduced from recently developed symmetry arguments. A few simple application examples are also presented., "28th International Symposium on Rarefied Gas Dynamics 2012"; Conference date: 9–13 July 2012; Location: Zaragoza, Spain

A gas separator that can induce a large variation of mole fraction by a small temperature difference in a single membrane is proposed. The separator makes use of two effects: The first effect is the thermal transpiration through microchannels in the membrane; the second is the accumulation of the effect of micro-channels in the counter flow of gas mixture that flows around the membrane. The numerical results show that small gas separation effects of numerous micro-channels are accumulated to induce a large variation of the mole fraction along the membrane. The numerical calculation is carried out by the direct simulation Monte Carlo (DSMC) method and a fluid model based on the mass conservation which is shown to be able to simulate the DSMC result. The performance of the device is investigated for several temperature differences between the two sides of the membrane. The relation to the membrane gas separation by Knudsen diffusion is also discussed., "28th International Symposium on Rarefied Gas Dynamics 2012"; Conference date: 9–13 July 2012; Location: Zaragoza, Spain

This paper investigates the thickness distribution of substrate coating of electron-beam physical vapor deposition (EBPVD) using a novel spatial-distribution-function, which is different from the empirical modified law. The Knudsen cosine law and empirical modified law were usually used for predicting coating thickness in EBPVD. However, these two laws obtained from experiments and geometric concepts cannot provide the elucidation for physical point of view and all-inclusive spatial-distribution-functions. Therefore, instead of the spatial-distribution-functions based on the Knudsen cosine law for low evaporation rates and empirical modified law for the higher evaporation rates, the spatial distribution function based on the mass diffusion theory is proposed to analyze the coating thickness of EB-induced physical vapor deposition. The coating thicknesses predicted by these spatial-distribution-functions are compared with the data measured by the published papers. The results reveal that the coating thicknesses predicted by the spatial distribution function of the diffusion theory are more consistent with the measured data than these obtained from the spatial distribution function of the empirical modified law. The spatial distribution function proposed by this study is validated by correctly predicting the measured coating thickness for EBPVD of titanium and aluminum.

Normal stress effects are investigated on tube flow of a single-component non-Newtonian fluid under a constant pressure gradient in a constant temperature field. The generalized hydrodynamic equations are employed, which are consistent with the laws of thermodynamics. In the cylindrical tube flow configuration, the solutions of generalized hydrodynamic equations are exactly solvable and the flow velocity is obtained in a simple one-dimensional integral quadrature. Unlike the case of flow in the absence of normal stresses, the flow develops an anomaly in that the flow in the boundary layer becomes stagnant and the thickness of such a stagnant velocity boundary layer depends on the pressure gradient, the aspect ratio of the radius to the length of the tube, and the pressure (or density and temperature) at the entrance of the tube. The volume flow rate formula through the tube is derived for the flow. It generalizes the Knudsen flow rate formula to the case of a non-Newtonian stress tensor in the presence of normal stress differences. It also reduces to the Navier–Stokes theory formula in the low shear rate limit near equilibrium.

In this study, the novel method to predict the performance of a multi-stage Knudsen pump is proposed. The performance prediction method is carried out in two steps numerically with the assistance of a simple experimental result. In the first step, the performance of a single-stage Knudsen pump was measured experimentally under various pressure conditions, and the relationship of the mass flow rate was obtained with respect to the average pressure between the inlet and outlet of the pump and the pressure difference between them. In the second step, the performance of a multi-stage pump was analyzed by a one-dimensional model derived from the mass conservation law. The performances predicted by the 1D-model of 1-stage, 2-stage, 3-stage, and 4-stage pumps were validated by the experimental results for the corresponding number of stages. It was concluded that the proposed prediction method works properly.

Three-dimensional unsteady flows between two infinite walls are simulated by using the direct simulation Monte Carlo (DSMC) method. An artificial forcing that mimics the centrifugal force in the Taylor problem has been applied to the flow. The sampled behaviors of the resulting flow, including the long time average and the disturbance components, are studied. The computations have been preformed using parallel computer clusters. The results presented are for two different channel heights with various values for the forcing coefficient. The change in the channel height, which also results in changes in the flow Knudsen number and Reynolds number, affects the development of both the mean flows and the disturbances. Spatially coherent mean flow patterns, which are dominated by a hierarchy of harmonic modes, can be identified in the DSMC solutions. Temporally, the evolution of the Fourier amplitudes of the harmonic modes shows that these modes grow in a sequential manner. Disturbances with energy spectra that are significantly higher than the statistical noises are resolved. Their pathline patterns indicate that the disturbance flow fields are three dimensional and spatially coherent. These results suggest that the discrete DSMC approach is capable of capturing unsteady, three-dimensional flow disturbances that evolve around a stationary mean flow.

We introduce and discuss Grad’s moment equations for dilute granular systems of hard spheres with dissipative collisions and variable coefficient of restitution, under the assumption of weak inelasticity. An important by-product is that in this way we obtain the hydrodynamic description of a system of nearly elastic particles by a direct procedure from the Boltzmann equation, without resorting to any homogeneous cooling state assumption. Several crucial results of the pertinent literature are recovered in the present physical context in which deviation from elastic scattering is of the same order as the Knudsen number. In particular, the correlation function plays a fundamental role in the decay of the temperature, and the latter is described asymptotically, in space homogeneous conditions, by a corrected Haff’s law.

An approach is presented to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen number. The method focuses on the order of magnitude of the moments of the phase density, and the order of accuracy of the transport equations, both measured in powers of the Knudsen number. The method is developed up to the third order, and it is shown that it yields the Euler equations at zeroth order, the Navier–Stokes–Fourier equations at first order, Grad’s 13 moment equations (with omission of a nonlinear term) at second order, and a regularization of these at third order. The method is discussed in detail, and compared with the classical methods of kinetic theory, i.e., Chapman–Enskog expansion and Grad moment method. The advantages of this method above the classical approaches are discussed conclusively. An important feature of the method presented is that the equations of any order are stable, other than in the Chapman–Enskog method, where the second and third ...

A steady flow of a vapor–vapor mixture between two parallel plane condensed phases for small Knudsen numbers is investigated on the basis of the kinetic theory of gases. By a systematic asymptotic analysis of the Boltzmann equation, it is shown that there are two distinct types of behavior of the mixture: the Euler-type behavior and the convection-diffusion-type behavior. Both types of behavior are confirmed numerically for the Boltzmann equation by the direct simulation Monte Carlo method and for the model Boltzmann equation proposed by Garzó, Santos, and Brey by the standard finite-difference method. Finally, the continuum limit (Kn=0₊) is considered, and it is shown that the ghost effect that some of the gas rarefaction effects still have an influence in the continuum limit manifests itself in the case of the convection-diffusion-type. This result shows that the infinitesimal jump of pressure at the surface of the condensed phase must be taken into account correctly for the description of the behavior of the vapors in the continuum limit.

The shear flow of a binary mixture of rarefied gases over a plane wall is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. This fundamental problem in rarefied gas dynamics is analyzed numerically by a finite-difference method, in which the complicated collision integrals are computed by the extension to the case of a gas mixture of the method proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)]. As a result, the behavior of the mixture is clarified not only at the level of the macroscopic variables but also at the level of the velocity distribution function. In addition, an accurate formula of the shear-slip (viscous-slip) coefficient for arbitrary values of the concentration of a component gas is constructed by the use of the Chebyshev polynomial approximation.

The present Brief Communication analyzed and demonstrated the failures of the molecular block (MB) model for the direct simulation Monte Carlo (DSMC) method. In the MB model, the gas viscosity and Knudsen number remained the same but the Mach number increased with the MB factor α. Therefore, the flow predicted by the MB model failed to agree with the original flow. Micro Couette flows and microchannel flows were simulated to demonstrate this point. Significant differences were found between results from the MB model and the original DSMC method. When α was large enough so that the Mach number in the MB model exceeded unity, a shock wave appeared in the channel, when there were no shocks in the original subsonic flow.

We present here a tractable theory of transport of simple fluids in cylindrical nanopores, which is applicable over a wide range of densities and pore sizes. In the Henry law low-density region the theory considers the trajectories of molecules oscillating between diffuse wall collisions, while at higher densities beyond this region the contribution from viscous flow becomes significant and is included through our recent approach utilizing a local average density model. The model is validated by means of equilibrium as well nonequilibrium molecular dynamics simulations of supercritical methane transport in cylindrical silica pores over a wide range of temperature, density, and pore size. The model for the Henry law region is exact and found to yield an excellent match with simulations at all conditions, including the single-file region of very small pore size where it is shown to provide the density-independent collective transport coefficient. It is also shown that in the absence of dispersive interactions the model reduces to the classical Knudsen result, but in the presence of such interactions the latter model drastically overpredicts the transport coefficient. For larger micropores beyond the single-file region the transport coefficient is reduced at high density because of intermolecular interactions and hindrance to particle crossings leading to a large decrease in surface slip that is not well represented by the model. However, for mesopores the transport coefficient increases monotonically with density, over the range studied, and is very well predicted by the theory, though at very high density the contribution from surface slip is slightly overpredicted. It is also seen that the concept of activated diffusion, commonly associated with diffusion in small pores, is fundamentally invalid for smooth pores, and the apparent activation energy is not simply related to the minimum pore potential or the adsorption energy as generally assumed.

A complete mathematical description of oscillatory Couette flows within the framework of kinetic theory is not available in the literature. Motivated by this and their vast engineering applications, we present a parametric study of time-periodic oscillatory Couette flows using the unsteady direct simulation Monte Carlo (DSMC) method. Computations are performed as a function of the Knudsen (Kn) and Stokes (β) numbers, in the entire Knudsen regime (Kn⩽100) and a wide range of Stokes numbers (β⩽7.5). The DSMC results are validated using a recently developed semianalytical/empirical model that is applicable for quasisteady flows (β⩽0.25) in the entire Knudsen regime, and for any Stokes number flow in the slip flow regime (Kn⩽0.1). In addition, we derived an analytical solution of the linearized collisionless Boltzmann equation for oscillatory Couette flows, and utilized this to validate the DSMC results in the free-molecular flow regime. Dynamic response of the flow, including the velocity profiles, phase ang...

On the basis of Langmuir’s theory of adsorption of gases on solids, a robust gaseous slip model is presented. The concept of accommodation coefficient and the difference of gas particles are explained within the new framework. It turned out that the Langmuir model recovers the Maxwell model in the first-order approximation in the case of the microchannel gas flow. In order to validate the new approach, the model is applied to problems of technical interests: pressure-driven microchannel gas flow and low Reynolds number gas flow past a sphere. With the help of previous theoretical and experimental results it is shown that with an adjustable parameter the model in low-speed creeping regime with moderate Knudsen numbers yields a prediction in qualitative agreement with the data.

Microchannel based systems have emerged as a critical design trend in development of precise control and maneuvering of small devices. In microelectronics, space propulsion and biomedical areas, these systems are especially useful. Nanoscale pores are recently becoming of great interest due to their beneficial drag and heat transfer properties. However it is difficult to predict the flow performance of these microsystems and nanosystems numerically since the standard assumptions of using Navier–Stokes equations break down at micrometer scales, while the computational times of applicable molecular-dynamics codes become exorbitant. A two-dimensional finite-element based microscale flow model is developed to efficiently predict the overall flow characteristics up to the transition regime for reasonably high Knudsen number flow inside microchannels and nanopores. Presented two-dimensional numerical results for Poiseuille flow of a simple fluid through the microchannel are comparable to the numerical and exper...

A numerical experiment is carried out to prove the difference between the value of the kinetic temperature and that of the thermodynamic temperature in a gas in the presence of molecular transport. As a reference situation the velocity distribution function is evaluated between two concentric cylinders at different wall temperatures. A direct simulation Monte Carlo method (DSMC) is adopted for various Knudsen numbers Kn and comparisons are made with existing data. The results prove quantitatively that the difference between the two differently defined temperatures increases with Kn and with the temperature gradient, as predicted by the theory for systems which are almost in thermodynamical equilibrium. The present article does not aim to validate DSMC, but rather to illustrate how the method can be used to address fundamental issues in gas dynamics.

The gas-dynamic Euler equations require two boundary conditions to be specified at the surface of evaporated condensed phase and one condition at the surface of condensation. In the commonly considered three-parameter space of the temperature and pressure ratios and the Mach number this corresponds to a three-dimensional curve in the case of evaporation and to a surface in the case of condensation. To obtain the conditions of evaporation and condensation the steady-state Knudsen layer is numerically studied by the discrete velocity method applied to a Boltzmann equation with a relaxation collision term. Simple models of Mott-Smith type based on the conservation laws and analytical approximations of the velocity distribution function in the Knudsen layer may give satisfactory description of the gas-dynamic evaporation and condensation conditions while in general they inadequately represent the detailed structure of the distribution function. One of the reasons why the models deviate from the calculations is that they do not allow different parallel and perpendicular temperatures of the velocity distribution. Under evaporation, the Knudsen layer thickness increases with the Mach number M. Under condensation, it is inversely proportional to M when M is low. Numerical results are obtained and an analytical model is proposed for the vapor temperature considerably less than the condensed phase one (up to 10 times) what is typical for back condensation under pulsed laser ablation.

The thermophoretic force on a near-wall, spherical particle in a rarefied, monatomic gas flow is calculated numerically. The rarefied gas flow is calculated with the Direct Simulation Monte Carlo (DSMC) method, which provides the molecular velocity distribution. The force is calculated from the molecular velocity distribution using a force Green’s function. Calculations are performed over a Knudsen-number range from 0.0475 to 4.75 using Maxwell and hard-sphere collision models. Results are presented for the thermophoresis parameter, ξ, a dimensionless quantity proportional to the thermophoretic force. The spatial profiles of ξ show a clear progression from free-molecular conditions (ξ is constant throughout the domain) to near-continuum conditions (ξ is constant in the interior but increases in the Knudsen layers). For near-continuum conditions, the DSMC calculations and Chapman–Enskog theory are in excellent agreement in the interior, suggesting that their velocity distributions are similar in this region. For all conditions examined, ξ lies between the continuum and free-molecular limits, which differ by only 10%. Moreover, the near-wall ξ values differ from the interior values by less than 5% for a fully diffuse wall, in sharp contrast with most previous studies. An approximate theory for the wall effect is presented that agrees reasonably well with the calculations.

The ability to measure extremely low thrust levels with unusual precision is becoming more critical as attempts are made to characterize the performance of emerging micropropulsion systems. Many new attitude control concepts for nanospacecraft involve the production of thrust below 1 μN. A simple, but uniquely successful thrust stand has been developed and used to measure thrust levels as low as 86.2 nano-Newtons (nN) with an estimated accuracy of ±11%. Thrust levels in the range of 712 (nN) to 1 μN have been measured with an estimated accuracy of ±2%. Thrust is measured from an underexpanded orifice operating in the free molecule flow regime with helium, argon, and nitrogen propellants. The thrust stand is calibrated using results from direct simulation Monte Carlo numerical models and analytical solutions for free molecule orifice flow. The accuracy of the gas dynamic calibration technique, using free molecule orifice flow, has also been investigated. It is shown that thrust stand calibration using high...

The ratio of the mean time between collisions of a typical molecule to the scale time of the density change of an element moving with the flow has long been used as a criterion for the breakdown of translational equilibrium in gaseous expansions. The pressure tensor becomes anisotropic when this parameter ceases to be very small in comparison with unity. It is proposed in this paper that a breakdown parameter for an internal mode is obtained simply by multiplying the translational parameter by the relaxation collision number for that mode. Emphasis is placed on the breakdown of vibrational equilibrium because this is of practical significance in many applications. For steady flows, the vibrational breakdown parameter is the product of a constant of order unity, the relaxation collision number, the speed ratio or Mach number, and the Knudsen number based on the scale length of the density variation. The Mach number term becomes nonlinear if the scale length of the area variation is employed in place of the density scale length. The direct simulation Monte Carlo (DSMC) method is used to make a series of calculations that support the use of this parameter to predict the breakdown of vibrational equilibrium in expansions.

The direct simulation Monte Carlo method is used to evaluate the improved slip boundary condition and slip-corrected Reynolds equation for gas lubrication in magnetic recording storage system. The study shows that in transition regime, the slip velocity is affected not only by shear stress, but also by density itself. In this ultra-thin film lubrication environment, the slip velocity has a linear relationship with the ratio of shear stress to density near the wall. This observation is particularly true when Knudsen number is around 1 or larger. Through this study, the authors have developed a new slip velocity model, named stress-density ratio model, particularly suitable to transition regime. Based on this novel model, a slip-corrected Reynolds equation is also derived and solved numerically. The stress-density ratio model is confirmed to offer a better applicability than the previous slip-flow models for modern computer magnetic storage device, where the Knudsen number of slider air bearing is in the range of unity or larger.

Fast ion transport and the resulting fusion yield reduction are computed at a gas-metal interface. The extent of fusion yield reduction is observed to depend sensitively on the charge state of the surrounding pusher material and the width of the atomically mixed region. These sensitivities suggest that idealized boundary conditions often implemented at the gas-pusher interface for the purpose of estimating fast ion loss will likely overestimate fusion reactivity reduction in several important limits. In addition, the impact of a spatially complex material interface is investigated by considering a collection of droplets of the pusher material immersed in a DT plasma. It is found that for small Knudsen numbers, the extent of fusion yield reduction scales with the surface area of the material interface. As the Knudsen number is increased, however, the simple surface area scaling is broken, suggesting that hydrodynamic mix has a nontrivial impact on the extent of fast ion losses.

In the continuum flow regime, the Navier-Stokes equations are usually used for the description of gas dynamics. On the other hand, the Boltzmann equation is applied for the rarefied gas dynamics. Both equations are constructed from modeling flow physics in different scales. Fortunately, due to the distinct separation of scales, i.e., the hydrodynamic and kinetic ones, both Navier-Stokes equations and the Boltzmann equation are valid in their respectable domains. However, in real physical application, there may not have such a distinctive scale separation. For example, around a hypersonic flying vehicle, the flow physics at different regions may correspond to different regimes, where the local Knudsen number can be changed in several order of magnitudes. With a variation of modeling scale, theoretically a continuous governing equation from kinetic Boltzmann equation to the hydrodynamic Navier-Stokes equations should exist. However, due to the difficulties of a direct modeling of flow physics in the scale between the kinetic and hydrodynamic ones, there is basically no reliable theory or valid governing equation to cover the whole transition regime. In fact, it is an unresolved problem about the exact scale for the validity of the NS equations as Reynolds number decreases. The traditional computational fluid dynamics (CFD) is based on the numerical solution of partial differential equations (PDE), and it targets on the recovering of the exact solution of the PDEs as mesh size and time step converging to zero. This methodology can be hardly applied here because there is no such a complete PDE for flow physics in all scales. It may be the time to combine the modeling and computation together without going through the process of constructing PDEs. In other words, the CFD research is not only to obtain the numerical solution of governing equation, but also to construct a valid discrete governing equation to identify the flow physics in the mesh size and time step scales. In this paper, we are going to present the idea for the modeling and computation. This methodology leads to the unified gas-kinetic scheme (UGKS) for flow simulation in all flow regimes. Based on UGKS, the boundary for the validation of the Navier-Stokes equations can be quantitatively evaluated. The combination of modeling and computation provides a paradigm for the description of multiscale transport process., 17 pages, 12 figures

The loss of fuel ions in the Gamow peak and other kinetic effects related to the α particles during ignition, run-away burn, and disassembly stages of an inertial confinement fusion D-T capsule are investigated with a quasi-1D hybrid volume ignition model that includes kinetic ions, fluid electrons, Planckian radiation photons, and a metallic pusher. The fuel ion loss due to the Knudsen effect at the fuel-pusher interface is accounted for by a local-loss model by Molvig et al. [Phys. Rev. Lett. 109, 095001 (2012)] with an albedo model for ions returning from the pusher wall. The tail refilling and relaxation of the fuel ion distribution are captured with a nonlinear Fokker-Planck solver. Alpha heating of the fuel ions is modeled kinetically while simple models for finite alpha range and electron heating are used. This dynamical model is benchmarked with a 3 T hydrodynamic burn model employing similar assumptions. For an energetic pusher (∼40 kJ) that compresses the fuel to an areal density of ∼1.07g/cm2 at ignition, the simulation shows that the Knudsen effect can substantially limit ion temperature rise in runaway burn. While the final yield decreases modestly from kinetic effects of the α particles, large reduction of the fuel reactivity during ignition and runaway burn may require a higher Knudsen loss rate compared to the rise time of the temperatures above ∼25 keV when the broad D-T Gamow peak merges into the bulk Maxwellian distribution.

In hypersonic flows about space vehicles in low earth orbits or flows in microchannels of microelectromechanical devices, the local Knudsen number lies in the continuum–transition regime. Navier–Stokes equations are not adequate to model these flows since they are based on small deviation from local thermodynamic equilibrium. To model these flows, a number of extended hydrodynamics or generalized hydrodynamics models have been proposed over the past fifty years, along with the direct simulation Monte Carlo (DSMC) approach. One of these models is the Burnett equations which are obtained from the Chapman–Enskog expansion of the Boltzmann equation [with Knudsen number (Kn) as a small parameter] to O(Kn2). With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with the Burnett model. As a result, several improvements to the conventio...

The pressure-driven gas flow through micro- and nano-porous structures is particularly interesting for innovative technologies such as microelectromechanical and nano-mechanical-electrical systems. The classical continuum assumption breaks down for rarefied flow through channels with a characteristic dimension comparable to the mean free path of the gas. Theories based on molecular interactions have been formulated to predict the flow at high Knudsen numbers. Measuring rarefied gas flow experimentally is a challenge since only a few studies have been able to determine flowrates in the molecular flow regime. Here we present the design of an experimental apparatus, which can be used to measure the flow of gases through nano- and microscale channels in the flow regimes where molecular effects are critical. The equations used to design the apparatus are given, focusing on the slip and transition flow regimes (together sometimes called "Intermediate flow regime"). A channel with a diameter of 325 μm ± 5μm and a length of 2 mm was tested experimentally with the apparatus for a wide range of Knudsen numbers (10-2 < Kn < 1 × 105) demonstrating its suitability through the slip and transition regime (2.23 × 10-2 < Kn < 2.26).

The mass, heat, and momentum transfer through a rarefied gas confined between two coaxial rotating cylinders is studied. The evaporation and condensation of the gas on the cylinder surfaces is assumed. The state of equilibrium is perturbed by small deviations of pressure, temperature, and angular velocity on the inner cylinder. The S model of the Boltzmann equation has been solved numerically by the discrete velocity method. The fields of the pressure, temperature, velocity, heat flux vector, and shear stress are calculated in the wide range of the Knudsen number. Further, the Onsager reciprocity relations between the cross effects are verified.

From consideration of the length scales characteristic of molecular and turbulent phenomena, it is proposed that flow instabilities and structural motions should be generated under certain rarefied, hypersonic flow conditions. This proposal is investigated using the direct simulation Monte Carlo (DSMC) method for hypersonic, rarefied flow over a cylinder. The overall mean flow field contains a number of regions including an undisturbed free stream, a bow shock, a recompression in the wake, and a recirculation zone behind the cylinder. Analysis of the fluctuations in velocity and number density predicted by the DSMC technique for this flow is conducted. An important aspect of this analysis is the clarification of the nature of the observed fluctuations. It is found that different types of fluctuations are found in the bow shock and in the wake. These are attributed to different types of physical phenomena. In the bow shock, large fluctuations are caused by the macroscopic field gradients in the shock front. In the wake, one can also observe an area with enhanced scatter of the mean flow characteristics caused by a macroscopic structural motion. The calculated instantaneous flow fields show small oscillations of the vortices immediately downstream of the cylinder for a Mach number of 26 and a Knudsen number of 0.0025. When the Knudsen number is reduced to 0.001, an irregular secondary flow motion containing vortex structures is exhibited in the far wake.

Starting from the Boltzmann kinetic equation, a system is constructed which allows one to calculate flow fields with strong deviation from equilibrium and for Knudsen numbers 0

The condensation coefficient of water vapor on liquid water and the thermal accommodation coefficient of air on liquid water are poorly known despite their importance in many applications, such as cloud physics. We have developed a new technique for determining the condensation and thermal accommodation coefficients experimentally. The technique consists of simultaneously measuring the homogeneous nucleation rate of ice and the evaporation rate of liquid water droplets as a function of pressure (droplet Knudsen number). As the Knudsen number increases, surface kinetic processes limit mass and energy fluxes and, as a result, the equilibrium temperature of an evaporating droplet is a function of the condensation and thermal accommodation coefficients. The homogeneous freezing nucleation rate is used as a sensitive measure of the droplet temperature. The nucleation and evaporation rates are determined by observing the scattered light from evaporating water droplets suspended in an electrodynamic levitation s...

In high-Knudsen-number flows nonequilibrium effects become dominant and the use of Navier–Stokes–Fourier equations becomes questionable since they are based on small deviation from local thermodynamic equilibrium. In this paper new hydrodynamic computational models are proposed for modeling gases in the transition regime. They are based on Eu’s generalized hydrodynamic equations and it turns out that they apply in all Mach numbers and satisfy the second law of thermodynamics to every order of approximation. In order to learn more about the new equations a model equation similar to the Burgers’ equation is studied. From this analysis new insight into constitutive relations of various hydrodynamic equations has been gained. In addition, a convergent iterative method for solving the highly nonlinear constitutive equations is developed. Finally, the shock structure and slip flow problems are computed by using high resolution numerical schemes and issues of extending the one-dimensional solver to multidimensional problems are discussed.

The Burnett equations have been shown to provide improved descriptions, relative to the Navier–Stokes equations, of flow structure in high-velocity (i.e., hypersonic) gases. We examine here the accuracy of the Burnett constitutive equation for fluid stress as applied to stationary gases. Specifically, we investigate the effects of “thermal stress” (fluid stress induced by a temperature gradient), as predicted by the Burnett equation, on the pressure distributions and normal stress in a stationary, buoyancy-free, hard-sphere gas for the case of one-dimensional heat transfer. We show, using first-law principles and the Burnett equation, that thermal stress results in a reduction in normal stress in the nonisothermal gas relative to that in the equilibrium state. The normal stress, in turn, can be obtained as an eigenvalue to a second-order ordinary differential equation, representing the Burnett equation, for the pressure distribution in the gas. Simple asymptotic solutions to the Burnett equation are developed, and are used in combination with order-Kn pressure slip relations to formulate pressure boundary conditions at the heated and cooled surfaces. The approximate solutions, as well as exact numerical calculations, are compared with pressure distributions generated from the direct-simulation Monte Carlo (DSMC) method. The Burnett and DSMC predictions of pressure are in good agreement for effective Knudsen numbers (based on the temperature gradient in the gas) less than 0.1. In particular, the Burnett equations can provide a reasonable description of the Knudsen (or rarefaction) layers adjacent to the heated and cooled surfaces that bound the gas, and can also describe the variation in pressure in the bulk gas. In addition, theoretical predictions of the reduction in normal stress correspond well to DSMC-derived values.

The rarefied gas flow between two coaxial cylinders caused by the evaporation and condensation on their surfaces is investigated on the basis of the kinetic and hydrodynamical equations. It is assumed that the cylinders may rotate with equal angular velocity. The equilibrium is perturbed by the small density difference of evaporated particles on the inner cylinder. The hydrodynamical equation has been solved analytically and the kinetic equation has been solved by the discrete velocity method modified for the discontinuous distribution function. The fields of the density, the velocity and the temperature are presented for a wide range of the Knudsen number. An analysis of the influence of the rotation of the cylinders on the flow field is given.

The present review, containing 178 references, is dedicated to one of the largest and most important branches of the rarefied gas dynamics, namely internal flows. A critical analysis of the corresponding numerical data and analytical results available in the literature was made. The most reliable data were selected and tabulated. The review will be useful as a reference for mathematicians, physicists and aerodynamicists interested in rarefied gas flows. In this paper the complete ranges of the main parameters, determining rarefied gas flows through a capillary, are covered. The capillary length varies from zero, when the capillary degenerates into a thin orifice, to infinity when the end effects can be neglected. The Knudsen number, characterizing the gas rarefaction, varies from zero when the gas is considered as a continuous medium to infinity when the intermolecular collisions can be discounted. The pressure and temperature drops on the capillary ends vary from the small values when the linear theory i...

A comparative analysis of two approaches (molecular and continuum) for three one-dimensional unsteady-state rarefied gas flows with small Knudsen number is presented. Numerical results have been obtained by using Direct Simulation Monte Carlo (DSMC) method for the molecular and a finite difference method for the continuum approaches, respectively.

The asymptotic theory proposed by Sone [in Rarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, 1971), p. 737] is applied to the investigation of the accuracy of the lattice Boltzmann method (LBM) for small Knudsen number with finite Reynolds number. The S-expansion procedure of the asymptotic theory is applied to LBM with the nine-velocity model and fluid-dynamic type equations are obtained. From the fluid-dynamic type equations it is found that by using the LBM we can obtain the macroscopic flow velocities and the pressure gradient for incompressible fluid with relative errors of O(e′2) where e′ is a modified Knudsen number which is of the same order as the lattice spacing and is related to a dimensionless relaxation time. In two problems, the Couette flow with flow injection and suction through porous walls and a three-dimensional flow through a square duct, the accuracy of LBM is examined for relaxation times between 0.8 and 1.7 and the validity of the asymptotic theory for LB...

A new low-pressure flow reactor operated as a Knudsen cell and intended for chemical kinetic studies is described. The reactor is specifically designed to study the kinetics of heterogeneous reactions. Gas-phase species are detected either by mass-spectrometric sampling or by in situ optical techniques, e.g., laser-induced fluorescence, resonantly enhanced multiphoton ionization. A feature of the reactor is its modular design, allowing full interchangeability of several sample holders at minimal effort, allowing the measurement of uptake coefficients ranging from 10−7 to 1.0. Sample supports operating at low and high temperatures have been developed which cover the stated temperature range. Several experimental examples of the utility of the reactor are detailed. The reliability and error bars of the kinetic results due to the errors and uncertainties associated with the experimental procedures are discussed, in particular for fast heterogeneous processes. It is found that even in the molecular flow regim...

A uniform supersonic flow of a rarefied gas past a flat plate at zero angle of attack is considered, and the steady behavior of the gas around the plate is investigated numerically on the basis of the Boltzmann–Krook–Welander equation (or the so-called BGK model) and the diffuse reflection boundary condition. An accurate finite-difference analysis, which gives the correct description of the discontinuity of the velocity distribution function of the gas molecules occurring in the gas, is carried out, and the features of the flow field (the velocity distribution function and the macroscopic variables such as the density, temperature, and flow velocity of the gas), in particular, those around the leading and trailing edges, are clarified for a wide range of the Knudsen number. The drag acting on the plate and the energy transferred to it are also obtained accurately. In addition, on the basis of the results for small Knudsen numbers, the behavior of the gas around the leading edge of a semi-infinite plate is...

Steady evaporation from a spherical and heterogeneous condensed body into vacuum is considered. Inert solid particles, initially embedded in the condensed phase, are liberated and collisionally accelerated by the evaporating gas molecules. A simple mathematical model for the flow is proposed and executed by the Direct Simulation Monte Carlo (DSMC) method. For small Knudsen numbers, the results are compared with a continuum solution including Knudsen layer corrections obtained by a moment method, demonstrating that these corrections are leading order effects in the treatment.

A rarefied gas in a channel of two parallel plane walls with a spatially periodic steady temperature distribution, on which a series of ditches is dug periodically, is considered, and the behavior of a flow induced by the temperature variation on the walls is investigated numerically for various Knudsen numbers and ditch sizes by the standard direct simulation Monte Carlo method (Bird’s method). It is found that a steady one‐way flow is induced through the channel without any pressure gradient being applied externally. The pumping effect owing to the one‐way flow is discussed.

The methods of qualitative theory of dynamical systems are used to provide new information of the Navier–Stokes solutions for gas flows driven by evaporation and condensation at interphase surfaces. It is shown that these solutions correspond to separatrixes of the saddle point in the (u,T) plane. The classification of solutions is given, and some special cases are studied in detail. The qualitative methods are applied to the problem of evaporation/condensation between two plates. It is shown that the topology of the saddle point implies the following situation: one of the functions u(x) (velocity) or T(x) (temperature) is always non‐monotone for sufficiently small Knudsen number. This explains some previously published numerical results. The case of finite Reynolds number is considered separately, and relationship with the kinetic‐theory results is discussed throughout.

The effect of boundary geometry on the properties of a rarefied vapor flow emerging from a narrow and infinitely long source was studied. The boundary consisted of walls, along both sides of the source which were parallel to the bulk velocity direction and of a collimator which plane was perpendicular to this direction. Two‐dimensional numerical solutions for the flow field were obtained based on the Bhatnagar–Gross–Krook model. It was found that the rarefied flow is affected mainly by the size of the collimator aperture and by the walls separation. The streamwise length of the walls and the location of the collimator have a minor influence on the far field density.

In this study, the grand (translational) friction tensor for free‐molecule (large Knudsen number) flows over two, unequal‐size cylinders is obtained analytically considering both ‘‘shielding’’ effects and multiple collisions off cylinder surfaces, as well as including both diffuse and specular molecule‐surface boundary conditions. A perturbation solution is used where the leading‐order term accounts for molecular collisions from the free‐streaming gas including all shielding effects. The first‐order correction term is also obtained here and accounts for collisions by molecules reflected off neighboring surfaces from the free‐stream gas. It is shown that the grand friction tensor is symmetric under the conditions of small ratios of the free‐stream gas velocity to the thermal gas velocity, the same type of molecular reflection boundary condition on both cylinders, and constant temperature. It is also shown that the two‐term expansion formula is accurate under isothermal conditions in the so‐called ‘‘far‐fie...

The initial expansion of a copper plasma generated at reduced pressure (20 mTorr air) by XeCl excimer laser irradiation of 308 nm has been studied by fast intensified charge‐coupled device photography of the overall visible plume emission with a time resolution of 3 ns. The spatial distribution and its temporal development along the target normal as well as in a plane parallel to the sample surface were measured in a single laser shot (28 ns). We report the first in situ measurement and observation of a one‐dimensional plasma expansion with very small divergence during the initial stage (δt

It is shown analytically and numerically on the basis of the kinetic theory that the heat‐conduction equation is not suitable for describing the temperature field of a gas in the continuum limit around bodies at rest in a closed domain or in an infinite domain without flow at infinity, where the flow vanishes in this limit. The behavior of the temperature field is first discussed by asymptotic analysis of the time‐independent boundary‐value problem of the Boltzmann equation for small Knudsen numbers. Then, simple examples are studied numerically: as the Knudsen number of the system approaches zero, the temperature field obtained by the kinetic equation approaches that obtained by the asymptotic theory and not that of the heat‐conduction equation, although the velocity of the gas vanishes.