Free Molecular Regime Research Articles

DSMC 및 IP 방법을 이용한 2차원 아음속 희박기체 유동 해석

In the present study, to investigate the characteristics of subsonic rarefied gas flows efficiently, the information preservation(IP) method was applied to the direct simulation Monte-Carlo(DSMC) method to reduce statistical scatters from the DSMC method for low speed rarefied gas flows. To find the effects of the IP method, the results of the IP method were compared to the results of the DSMC method for the same numbers of sampling. Before the comparison, it was necessary to verify the DSMC solver. The verification of the solver was conducted by simulations of the micro-channel flows by comparing to the other researchers’ DSMC results. The velocity profiles at the 2/3 section of the channel agreed well with the other researchers’ results. In addition, the normalized slip velocity distributions on the wall and the pressure distributions along the centerline also agreed well with the other researchers’ results. After the DSMC solvers were verified, two types of Couette flows were considered. One is the flows that have temperature differences between two plates, and the other is the flows that have moving plates. For the first Couette flows, the temperature of 373K was used for the upper plate, and of 173K for lower plate. For the second Couette flows, the upper plate is moving at 300m/s, and the temperature of 273K was used for both two plates. All simulations of Couette flows were conducted from near-continuum to free-molecular regimes. From both Couette flow simulations, it was found that the temperature jump and the velocity slip occurred on the plate surface. In addition, it was also shown that the temperature and the velocity differences between the plate and the gas became larger as the flow fields became more rarefied. Lastly, flow simulations around a NACA0012 airfoil were conducted to identify the effects of the IP method. The freestream Mach number of 0.8 was used, and the Knudsen number of 0.014 was considered. It was observed that more clear contours were obtained from the IP method than the DSMC method, since the statistical scatter error was reduced by the IP method. In addition, it was found that the velocity slip on the airfoil surface occurred due to the effects of the rarefied atmospheric environment.

Rarefied isothermal gas flow in a long circular tube due to oscillating pressure gradient

The time-dependent isothermal fully developed rarefied gas flow in a circular tube driven by harmonically oscillating pressure gradient is investigated, based on the linearized unsteady BGK kinetic model equation. The flow is characterized by the gas rarefaction parameter, which is proportional to the inverse Knudsen number and the oscillation parameter, defined as the ratio of the collision frequency over the pressure gradient oscillation frequency. Computational results of the amplitude and the phase angle of the flow rates and the velocity distributions, as well as of the periodic time evolution of these macroscopic quantities, are provided, covering the whole range of the two parameters. The kinetic results properly recover the limiting solutions in the slip and free molecular regimes for low- and high-speed oscillations. At low frequencies, the time-dependent flow becomes quasi-steady and gradually tends to the corresponding steady-steady one, which is reached faster when the flow is more rarefied. As the frequency is increased, the amplitude of the macroscopic quantities is decreased and their phase angle lag with respect to the pressure gradient is increased approaching asymptotically the limiting value of $$\pi /2$$ . In terms of the gas rarefaction, there is a non-monotonic behavior and the maximum flow rate amplitude may be observed at some intermediate value of the gas rarefaction parameter depending upon the oscillation parameter. At high frequencies, the flow consists of an inviscid piston flow in the core and the frictional Stokes wall layer with a velocity overshoot. These effects, well known in the viscous regime, are also present here in the transition regime and depend on both the gas rarefaction and oscillation parameters. As the gas rarefaction is increased, higher oscillation frequencies are needed to trigger these phenomena. Oscillatory rarefied flows are of main interest in sensors, controllers and resonators, which may be present in various microfluidic applications (e.g., microcooling, microseparators and micropropulsion).

A comparative study of discrete velocity methods for low-speed rarefied gas flows

In the study of rarefied gas dynamics, the discrete velocity method (DVM) has been widely employed to solve the gas kinetic equations. Although various versions of DVM have been developed, their performance, in terms of modeling accuracy and computational efficiency, is yet to be comprehensively studied in all the flow regimes. Here, the traditional third-order time-implicit Godunov DVM (GDVM) and the recently developed discrete unified gas-kinetic scheme (DUGKS) are analysed in finding steady-state solutions of the low-speed force-driven Poiseuille and lid-driven cavity flows. With the molecular collision and free streaming being treated simultaneously, the DUGKS preserves the second-order accuracy in the spatial and temporal discretizations in all flow regimes. Towards the hydrodynamic flow regime, not only is the DUGKS faster than the GDVM when using the same spatial mesh, but also requires less spatial resolution than that of the GDVM to achieve the same numerical accuracy. From the slip to free molecular flow regimes, however, the DUGKS is slower than the GDVM, due to the complicated flux evaluation and the restrictive time step which is smaller than the maximum effective time step of the GDVM. Therefore, the DUGKS is preferable for problems involving different flow regimes, particularly when the hydrodynamic flow regime is dominant. For highly rarefied gas flows, if the steady-state solution is mainly concerned, the implicit GDVM, which can boost the convergence significantly, is a better choice.

Effects of pressure on suspended micromechanical membrane arrays

The effects of pressure on micromechanical air-filled cavities made by a pair of suspended, parallel silicon nitride membranes are investigated in the free molecular and quasi-molecular regimes. Variations of the fundamental drummode mechanical resonant frequencies and damping with air pressure are determined by means of optical interferometry. A kinetic damping linear friction force and a positive resonant frequency shift due to the compression of the fluid between the membranes are observed to be proportional to pressure in the range of 0.01–10 mbars. For resonators with near-degenerate modes, hybridization of the modes due to this squeeze film effect is also observed and well accounted for by a simple spring-coupled oscillator model.

Bivariate Taylor-series expansion method of moment for particle population balance equation in Brownian coagulation

In this study, we extend the Taylor-series expansion method of moment to two-component aggregation problem undergoing Brownian coagulation with kernels that are independent of composition. A set of closed particle population balance equation for lower-order moments is then derived. Numerical results and its asymptotic solutions are validated by comparing with Monte Carlo simulation method both in free molecular regime and continuum regime. It is shown that three dimensionless particle moments MC1,MC2,MC3 almost approach to a same value over large evolution time. The normalized variance of excess component A decreases as 1/v¯ and it tends to zero over large evolution time.

Comparative study of the discrete velocity and lattice Boltzmann methods for rarefied gas flows through irregular channels.

Rooted from the gas kinetics, the lattice Boltzmann method (LBM) is a powerful tool in modeling hydrodynamics. In the past decade, it has been extended to simulate rarefied gas flows beyond the Navier-Stokes level, either by using the high-order Gauss-Hermite quadrature, or by introducing the relaxation time that is a function of the gas-wall distance. While the former method, with a limited number of discrete velocities (e.g., D2Q36), is accurate up to the early transition flow regime, the latter method (especially the multiple relaxation time (MRT) LBM), with the same discrete velocities as those used in simulating hydrodynamics (i.e., D2Q9), is accurate up to the free-molecular flow regime in the planar Poiseuille flow. This is quite astonishing in the sense that less discrete velocities are more accurate. In this paper, by solving the Bhatnagar-Gross-Krook kinetic equation accurately via the discrete velocity method, we find that the high-order Gauss-Hermite quadrature cannot describe the large variation in the velocity distribution function when the rarefaction effect is strong, but the MRT-LBM can capture the flow velocity well because it is equivalent to solving the Navier-Stokes equations with an effective shear viscosity. Since the MRT-LBM has only been validated in simple channel flows, and for complex geometries it is difficult to find the effective viscosity, it is necessary to assess its performance for the simulation of rarefied gas flows. Our numerical simulations based on the accurate discrete velocity method suggest that the accuracy of the MRT-LBM is reduced significantly in the simulation of rarefied gas flows through the rough surface and porous media. Our simulation results could serve as benchmarking cases for future development of the LBM for modeling and simulation of rarefied gas flows in complex geometries.

Experimental investigation of the effect of inlet particle properties on the capture efficiency in an exhaust particulate filter

The impact of inlet particle properties on the filtration performance of clean and particulate matter (PM) laden cordierite filter samples was evaluated using PM generated by a spark-ignition direct-injection (SIDI) engine fuelled with tier II EEE certification gasoline. Prior to the filtration experiments, an advanced aerosol characterization system that comprised of a scanning mobility particle spectrometer (SMPS), centrifugal particle mass analyzer (CPMA), a differential mobility analyzer (DMA), and a single particle mass spectrometer (SPLAT II) was used to obtain a wide range of information on the SIDI PM emissions including particle size distribution (PSD), composition, mass, and dynamic shape factors (DSFs) in the transition (χt) and free-molecular (χv) flow regimes. During the filtration experiments, real-time measurements of PSDs upstream and downstream of the filter sample were used to estimate the filtration performance and the total trapped mass within the filter using an integrated particle size distribution method. The filter loading process was paused multiple times to evaluate the filtration performance in the partially loaded state. The change in vacuum aerodynamic diameter (dva) distribution of mass-selected particles was examined for flow through the filter to identify whether preferential capture of particles of certain shapes occurred in the filter. The filter was also probed using different inlet PSDs. Pausing the filter loading process and subsequently performing the filter probing experiments did not impact the overall evolution of filtration performance. Within the present distribution of particle sizes, filter efficiency was independent of particle shape potentially due to the diffusion-dominant filtration process. Particle mobility diameter and trapped mass within the filter appeared to be the dominant parameters that impacted filter performance.

Hybrid method of moments with interpolation closure–Taylor-series expansion method of moments scheme for solving the Smoluchowski coagulation equation

This paper presents a hybrid method of moments with interpolation closure–Taylor-series expansion method of moments (MoMIC–TEMoM) scheme for solving the Smoluchowski coagulation equation. In the proposed scheme, the exponential function, which arises in the conversion from a particle size distribution space to a space of moments, is expressed in an additive form using the third-order Taylor-series expansion; the implicit moments are approximated using two Lagrange interpolation functions, namely the newly defined normalized moment function and the normalized moment function defined by Frenklach and Harris (1987). The new hybrid scheme allows implementation of the method of moments with an arbitrary type of moment sequence, and it overcomes the shortcomings of the Taylor-series expansion moment method proposed by Frenklach and Harris. The proposed scheme is verified with three aerosol dynamics, namely Brownian coagulation in the free molecular regime, Brownian coagulation in the continuum-slip regime, and turbulence coagulation. The results reveal that the hybrid MoMIC–TEMoM scheme has similar accuracy to currently recognized methods including the quadrature method of moments, MoMIC, and TEMoM, and its accuracy can be further enhanced as the fractional moment sequence type is used for Brownian coagulation in the free molecular regime. Thus, the proposed scheme is a reliable for solving the Smoluchowski coagulation equation.

Mathematical simulation of transfer processes in an elliptical channel in a free molecular regime

In the framework of a free molecular regime, we consider the problem of heat and mass transfer in a long channel of constant elliptical cross-section. The diffuse reflection model is used as the boundary condition. The distributions over the channel cross-section are obtained for the mass velocity of a gas and the heat flux vector. Some analytical method is proposed for finding the heat and mass fluxes through the channel cross-section.

Dry and Direct Deposition of Aerosol-Synthesized Single-Walled Carbon Nanotubes by Thermophoresis.

Single-walled carbon nanotubes (SWCNTs) show great potential as an active material in electronic and photonic devices, but their applicability is currently limited by shortcomings in existing deposition methods. SWCNTs can be dispersed from liquid solutions; however, their poor solubility requires the use of surfactants and ultrasonication, causing defects and degradation in device performance. Likewise, the high temperatures required by their chemical vapor deposition growth limit substrates on which SWCNTs can be directly grown. Here, we present a systematic study of the direct deposition of pristine, aerosol-synthesized SWCNTs by thermophoresis. The density of the deposited nanotube film can be continuously adjusted from individual, separated nanotubes to multilayer thin films by changing the deposition time. Depending on the lateral flow inside the thermophoretic precipitator, the angular distribution of the deposited SWCNT film can be changed from uniform to nonuniform. Because the substrate is kept at nearly ambient temperature, deposition can be thus carried out on practically any flat substrate with high efficiencies close to unity. The thermophoretic terminal velocity of SWCNTs, determined by aerosol loss measurements, is found to be approximately one-third of the usual prediction in the free molecular regime and shows a weak dependence on the nanotube diameter. As a demonstration of the applicability of our technique, we have used thermophoretic deposition in the fabrication of carbon nanotube thin-film transistors with uniform electrical properties and a high, over 99.5%, yield.

Aerothermodynamic correlations for high-speed flow

Heat flux and drag correlations are developed for high-speed flow over spherical geometries that are accurate for any Knudsen number ranging from continuum to free-molecular conditions. A stagnation point heat flux correlation is derived as a correction to the continuum (Fourier model) heat flux and also reproduces the correct heat flux in the free-molecular limit by use of a bridging function. In this manner, the correlation can be combined with existing continuum correlations based on computational fluid dynamics simulations, yet it can now be used accurately in the transitional and free-molecular regimes. The functional form of the stagnation point heat flux correlation is physics based, and was derived via the Burnett and super-Burnett equations in a recent article, Singh & Schwartzentruber (J. Fluid Mech., vol. 792, 2016, pp. 981–996). In addition, correlation parameters from the literature are used to construct simple expressions for the local heat flux around the sphere as well as the integrated drag coefficient. A large number of direct simulation Monte Carlo calculations are performed over a wide range of conditions. The computed heat flux and drag data are used to validate the correlations and also to fit the correlation parameters. Compared to existing continuum-based correlations, the new correlations will enable engineering analysis of flight conditions at higher altitudes and/or smaller geometry radii, useful for a variety of applications including blunt body planetary entry, sharp leading edges, low orbiting satellites, meteorites and space debris.

A contribution to the amaranthine quarrel between true and average electrical mobility in the free molecular regime

ABSTRACTLandau and Lipschitz's approach—termed here H&B due to the use of Happel and Brenner's slow rotation approximation—for calculating the average electrical mobility over all orientations of an ion in the free molecular regime is shown in this manuscript to be an invalid assumption for non-globular ions when a fixed electrical field is present. The reason behind the invalidity seems to be the confusion between average “settling” velocity (the calculation intended by H&B) and the average mobility (drag) in the direction of the field. When a missing orientation is taken into account by rotating the drag tensor, the average mobility obtained through Landau's approach coincides with well-known orientationally averaged Kinetic Theory Methods such as those of Mason and McDaniel (M&M). H&B's averaging approach, however, can be related to the true mobility displacement of the ion or, in other words, the displacement occurring in the direction of the velocity. This true mobility displacement only agrees with ...

New scheme for implementing the method of moments with interpolative closure

ABSTRACTThis article presents a new scheme for implementing the method of moments with interpolative closure (MOMIC), which was proposed by Frenklach and Harris in 1987. The new scheme can be implemented with arbitrary moment sequences. A new interpolation scheme, which is implemented among the square root of logarithms of newly defined normalized moments, was verified to be suitable for interpolation with only three known data points. The new scheme was applied to studies on Brownian coagulation in both free-molecular and continuum-slip regimes. The new MOMIC with integer moment sequence was verified to exhibit nearly identical accuracy to its original version as well as other state-of-the-art numerical methods, including QMOM, TEMOM, and log MM, and higher accuracy than these referenced numerical methods as the fractional moment sequence was employed. The study enhances the capability of the MOMIC for solving population balance equations.© 2017 American Association for Aerosol Research

Numerical analysis of surface roughness effects on the Poiseuille flow caused by a small pressure drop

The rarefied gas flow caused by a pressure drop in a planar microchannel of finite length is analysed numerically to investigate the effect of surface roughness on thermal and hydrodynamic flow characteristics. The kinetic approach based on the nonlinear S-model of the Boltzmann kinetic equation coupled with the Maxwell’s diffuse gas-surface interaction model imposed at the wall is used. An implicit scheme for the solution of the S-model kinetic equation is applied and the algorithm has been optimised for the use of massive parallelization in the velocity space. A surface roughness is configured with series of triangular, semicircular, trapezoidal and rectangular obstructions. A competition between rarefaction and effects of roughness geometry is estimated by varying a rarefaction regime of flow from continuum up to free–molecular regime. The results indicate that the global flow performances in terms of the Poiseuille number, mass and thermal energy flow rates are affected by the wall roughness geometry when compared to the flow performances for a smooth channel. The triangular roughness produces the smallest impact on a gas flow, while the rectangular roughness the largest one. Rarefaction plays a significant role enhancing the effect of the surface roughness. In the continuum flow regime the roughness elements induce strong recirculation and flow separation regions. Moreover, the weakly rarefied flow strongly depends on the shape and the spacing of roughness elements. On the other hand, for the highly rarefied flow the global parameters are almost independent of the shape of elements. The increase in the spacing of roughness elements results in the reduction of the Poiseuille number and an enhancement of mass and thermal energy flow rates for all shapes of roughness elements.

Gas flow mechanisms under the effects of pore structures and permeability characteristics in source rocks of coal measures in Qinshui Basin, China

The gas flow mechanisms in source rocks of coal measures under the effects of the pore structures and permeability characteristics were investigated by field-emission scanning electron microscopy, low-pressure nitrogen gas adsorption, high-pressure mercury intrusion, and pressure pulse decay permeability method. Various flow regimes were distinguished in the pores and fractures of differing scales, and the mass fluxes through the same were calculated using the data obtained by the numerical and experimental investigations. Results indicated that mesopores predominated in shale, while coal contained well-developed mesopores and macropores. In addition, the permeabilities of coal and shale were observed to be significantly anisotropic and highly stress dependent. The cross-sectional area proportions of the pores per unit cross-sectional area of the matrix in the free molecular, transition, and slip flow regimes in shale and coal were determined to be, respectively, 0.2:0.7:0.1 and 0.15:0.6:0.25. In the free molecular and transition flow regimes, the mass flux decreased with increasing reservoir depth, while the reverse was the case in the slip flow regime. Further, in the continuum flow regime, the mass flux was unimodally distributed with respect to the reservoir depth. The total mass flux in coal was greater in the direction perpendicular to the bedding compared to the direction parallel to the bedding, while the reverse was the case in shale. In addition, the continuum flow regime predominated in coal in both the directions perpendicular and parallel to the bedding, but only in the direction parallel to the bedding in shale. This work presents a comprehensive model for the analysis of all the flow regimes in pores and fractures of differing scales, as well as the anisotropy. Findings of the study are meaningful for establishing the coupling accumulation mechanism of the Three Coal Gases and developing a unified exploration and exploitation program.

A Monte Carlo method for the simulation of coagulation and nucleation based on weighted particles and the concepts of stochastic resolution and merging

Monte Carlo simulations based on weighted simulation particles can solve a variety of population balance problems and allow thus to formulate a solution-framework for many chemical engineering processes. This study presents a novel concept for the calculation of coagulation rates of weighted Monte Carlo particles by introducing a family of transformations to non-weighted Monte Carlo particles. The tuning of the accuracy (named ‘stochastic resolution’ in this paper) of those transformations allows the construction of a constant-number coagulation scheme. Furthermore, a parallel algorithm for the inclusion of newly formed Monte Carlo particles due to nucleation is presented in the scope of a constant-number scheme: the low-weight merging. This technique is found to create significantly less statistical simulation noise than the conventional technique (named ‘random removal’ in this paper). Both concepts are combined into a single GPU-based simulation method which is validated by comparison with the discrete-sectional simulation technique. Two test models describing a constant-rate nucleation coupled to a simultaneous coagulation in 1) the free-molecular regime or 2) the continuum regime are simulated for this purpose.

Mathematical simulation of heat transfer in an elliptic channel under the action of a pressure gradient

A method for calculating the heat flux through the cross section of an elliptic channel in the presence of a pressure gradient that is normal to the symmetry axis of the channel has been suggested. As a basic equation that describes the process kinetics, the Williams equation is taken, and the model of diffuse reflection sets boundary conditions at the channel wall. The deviation of the state of the gas from equilibrium is assumed to be small. The dependence of the heat flux through the cross section of the channel on the Knudsen number has been derived. The heat fluxes in the limiting cases of free-molecular and hydrodynamic regimes have been compared.

Soot particle sizing based on analytical formula derived from laser-induced incandescence decay signals

The laser-induced incandescence (LII) signal during a heat-conduction-dominated cooling process was used to derive an analytical formula to describe the relationship between the soot particle size and the LII signal decay time by exponential fitting. The formula was used to determine particle sizes based on the experimental LII signals at different detection wavelengths for an atmospheric C2H4/air diffusion flame. The results agree with those obtained from temporal temperature measurements. The measurements and numerical calculations demonstrate that particle sizing depends weakly on the maximum temperature in the formula within a typical heat-up temperature range. The results show that based on this formula, a compact single-color LII detection system can be used for particle sizing with low uncertainty under most practical combustion conditions, at least in cases where heat conduction is dominant and occurs in a free molecular regime during particle cooling.

Diffusive-to-ballistic transition of the modulated heat transport in a rarefied air chamber

Modulated heat transfer in air subject to pressures from 760 Torr to 10-4 Torr is experimentally studied by means of a thermal-wave resonant cavity placed in a vacuum chamber. This is done through the analysis of the amplitude and phase delay of the photothermal signal as a function of the cavity length and pressure through of the Knudsen’s number. The viscous, transitional, and free molecular regimes of heat transport are observed for pressures P>1.5 Torr, 25 mTorr<P<1.5 Torr, and P<25 mTorr; respectively. It is shown that the fingerprint of each regime is determined by the concavity of the amplitude decay in a length scan, which is concave upward for the viscous regime and concave downward in the free molecular one. Furthermore, the increase of the radiative contribution on both the amplitude and phase is also observed as the pressure reduces. The obtained results show that the proposed methodology can be used to study the molecular dynamics in gases supporting diffusive and ballistic heat transport.

Conditional maximums of heat fluxes in cylindrically and spherically symmetric problems of heat transfer in rarefied gas for the free molecular and transitional regimes

We consider the heat transfer in gas between surfaces of cylindrical and spherical symmetry for the free molecular and transitional regimes. For the free molecular heat transfer we derive the analytical expressions for the surface temperatures ratios at which the heat fluxes to the surfaces reach the conditional maximums provided that the temperatures of the opposite surfaces remain constant. For the transitional regime we study conditional maximums of the heat fluxes at different Knudsen numbers by the Direct Simulation Monte Carlo method.