Free Molecular Limits Research Articles

Modeling drag coefficients of spheroidal particles in rarefied flow conditions

Transport of particles in flows is often modeled in a combined Eulerian–Lagrangian framework. The flow is evaluated on an Eulerian grid, while particles are modeled as Lagrangian points whose positions and velocities are evolved in time, resulting in particle trajectories embedded in the time-dependent flow field. The method essentially resolves the flow field in complex geometries in detail but uses a closure model for the particle dynamics aimed at including the essential particle–fluid interactions at the cost of detailed small-scale physics. Rarefaction effects are usually included through the phenomenological Cunningham correction on the drag force experienced by the particles. In this Lagrangian point-particle approach, any explicit reference to the finite size and the shape of the particles, and their local orientation in the flow field, is typically ignored. In this work we aim to address this gap by deriving, from fully-resolved Direct Simulation Monte Carlo (DSMC) studies, heuristic or closure models for the drag force acting on prolate and oblate spheroidal particles with different aspect ratios, and a fixed orientation, in uniform ambient rarefied flows covering the transition regime between the continuum and free-molecular limits. These closure models predict the drag in the transition regime for all considered parameter settings (validated with DSMC data). The continuum limit is enforced a priori and we retrieve the free-molecular limit with reasonable accuracy (based on comparisons with literature data). We also include in the models the capability to predict effects related to basic gas-surface interactions via the tangential momentum accommodation coefficient. We furthermore assess the validity of the proposed closure model for particle dynamics in proximity to solid walls. This investigation extends our previous work, which focused on small aspect ratio spheroids with exclusively diffusive gas-surface interactions [see Livi et al. (2022)]. The derived models are obtained for isothermal, subsonic flows relevant for particle contamination control in semiconductor manufacturing.

Simplified hydrodynamic-wave particle method for the multiscale rarefied flow

Coupling the Computational Fluid Dynamics (CFD) with the stochastic particle method to address multi-scale rarefied flows remains of great interest to many researchers. Rather than involving the field decomposition and the information exchange, the simplified hydrodynamic-wave particle method (SHWPM) has been proposed in this paper to automatically switch from the stochastic particle solver to the conventional CFD method in the multi-scale simulation. The weights to couple the respective numerical fluxes are derived based on the integral solution of the Boltzmann equation. By introducing the simplified hydrodynamic-wave flux that can be computed by the linear combination of inviscid and viscous fluxes from the conventional CFD solver, the collision effects of particles are calculated in a macroscopic approach and the number of sampling particles could be significantly reduced in the near-continuum regime. Error analysis demonstrates that the simplified treatment can preserve the second-order accuracy and asymptotic preserving property of the present method in both the continuum and free molecular limits. Several numerical cases over a wide range of Knudsen and Mach numbers exhibit the capability of SHWPM for modeling the multiscale flow accurately and efficiently. The example of hypersonic flow passing a cylinder shows that the SHWPM could achieve tens of times speedup ratio compared to the discrete velocity method (DVM) in the continuum regime.

Inversion of the transverse force on a spinning sphere moving in a rarefied gas

The flow around a spinning sphere moving in a rarefied gas is considered in the following situation: (i) the translational velocity of the sphere is small (i.e. the Mach number is small); (ii) the Knudsen number, the ratio of the molecular mean free path to the sphere radius, is of the order of unity (the case with small Knudsen numbers is also discussed); and (iii) the ratio between the equatorial surface velocity and the translational velocity of the sphere is of the order of unity. The behaviour of the gas, particularly the transverse force acting on the sphere, is investigated through an asymptotic analysis of the Boltzmann equation for small Mach numbers. It is shown that the transverse force is expressed as $\boldsymbol{F}_L = {\rm \pi}\rho a^3 (\boldsymbol{\varOmega} \times \boldsymbol{v}) \bar{h}_L$ , where $\rho$ is the density of the surrounding gas, a is the radius of the sphere, $\boldsymbol {\varOmega }$ is its angular velocity, $\boldsymbol {v}$ is its velocity and $\bar {h}_L$ is a numerical factor that depends on the Knudsen number. Then, $\bar {h}_L$ is obtained numerically based on the Bhatnagar–Gross–Krook model of the Boltzmann equation for a wide range of Knudsen number. It is shown that $\bar {h}_L$ varies with the Knudsen number monotonically from 1 (the continuum limit) to $-\tfrac {2}{3}$ (the free molecular limit), vanishing at an intermediate Knudsen number. The present analysis is intended to clarify the transition of the transverse force, which is previously known to have different signs in the continuum and the free molecular limits.

Propagation of two-dimensional vibroacoustic disturbances in a rarefied gas

The effect of gas rarefaction on the propagation of two-dimensional vibroacoustic disturbances generated by a nonuniformly oscillating plane is studied. Closed-form descriptions are obtained in the free-molecular (left panel of the figure) and continuum (right panel) limits and complemented by direct simulation Monte Carlo results. The impacts of gas rarefaction on signal decay rate and directivity pattern are highlighted and rationalized.

Rarefied Gas Flow between Two Coaxial Cylinders Driven by Temperature Gradient in the Case of Specular-Diffuse Reflection

The slow longitudinal flow of a rarefied gas between two infinitely long coaxial cylinders driven by a prescribed temperature gradient is studied. The problem is formulated for a linearized kinetic model with specular-diffuse boundary conditions. The reduced heat and mass fluxes through the channel are obtained as functions of the tangential momentum accommodation coefficient and the Knudsen number for various ratios of the inner to outer radius of the cylinders. The values of these fluxes are found using Chebyshev polynomials. The resulting expressions are analyzed in the free-molecular and hydrodynamic limits.

Numerical investigation of the heat and mass transfer analogy in rarefied gas flows

Transport of heat and mass to surfaces in fluid flows lies at the heart of a vast array of engineering problems. The Colburn analogy that relates the transport of heat and mass to the surface is often applied to approximate mass transport using correlations developed for heat transport. Here we use a numerical method that solves the Shakhov model Boltzmann equation of fluid mechanics coupled to a simple monolayer adsorption model to show that this analogy does not apply for selected rarefied flows. We consider steady flow of gas through a micro-channel and the flow in a lid-driven cavity. We define a breakdown parameter indicating the departure of the relative rates of heat and mass transport from that predicted using the analogy, and give simple analytic expressions for this parameter in the free-molecular and continuum limits. Results show that the analogy breaks down even for Knudsen numbers corresponding to slip and transition region flows, and that the magnitude of this departure is such that mass transfer is many times slower than predicted using the analogy. These results have important application to the design of micro-scale devices involving mass transport to surfaces.

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows III: Microflow Simulations

AbstractDue to the rapid advances in micro-electro-mechanical systems (MEMS), the study of microflows becomes increasingly important. Currently, the molecular-based simulation techniques are the most reliable methods for rarefied flow computation, even though these methods face statistical scattering problem in the low speed limit. With discretized particle velocity space, a unified gas-kinetic scheme (UGKS) for entire Knudsen number flow has been constructed recently for flow computation. Contrary to the particle-based direct simulation Monte Carlo (DSMC) method, the unified scheme is a partial differential equation-based modeling method, where the statistical noise is totally removed. But, the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space. Due to the multiscale modeling in the unified method, i.e., the update of both macroscopic flow variables and microscopic gas distribution function, the conventional constraint of time step being less than the particle collision time in many direct Boltzmann solvers is released here. The numerical tests show that the unified scheme is more efficient than the particle-based methods in the low speed rarefied flow computation. The main purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena. In the continuum and free molecular limits, the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions. In the transition regime, the DSMC solution will be used for the validation of UGKS results. This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows. It not only recovers the DSMC results whenever available, but also provides high resolution results in cases where the DSMC can hardly afford the computational cost. In thermal creep flow simulation, surprising solution, such as the gas flowing from hot to cold regions along the wall surface, is observed for the first time by the unified scheme, which is confirmed later through intensive DSMC computation.

Adaptive orthogonal collocation for aerosol dynamics under coagulation

The fast algorithm for the calculation of the orthogonal collocation-based coagulation tensor (J. Aerosol Sci. 37 (2006) 1356) is used to implement an adaptive scheme for the numerical solution of the general dynamic equation of a univariate population of fractal aggregates under Brownian coagulation. Two cases corresponding to the continuum and free molecular limits are solved using, both, the Whittaker and the Laguerre bases, for several values of the aggregate fractal dimension. In this adaptive scheme the general dynamic equation is solved using orthogonal collocation in terms of a scaled variable, defined as the particle volume over the (time dependent) standard deviation, which is previously computed using QMOM. This way, in terms of the scaled variable the population is always located on an order-one-width interval, thus allowing for an efficient numerical solution using orthogonal collocation, even for very long times. This scheme allows for a very accurate and efficient calculation of the self-similar distribution function reached in the long-time limit. However, higher numbers of spectral components were needed in the cases corresponding to the free molecular regime, as compared to the results corresponding to the continuum regime. This result is explained as being a consequence of the smoothness of the number density distribution function in the small particle limit.It is also shown that the fast algorithm can be easily generalized to aerosols depending on state variables following arbitrary constitutive laws. This enables the use of this algorithm in a general case, and it also enables the use of more complex time-dependent mappings. Finally, it is shown that the fast algorithm leads to the correct evolution equations of the moments, which explains the good numerical behavior observed in the calculations.

Rarefied gas flow in a triangular duct based on a boundary fitted lattice

The rarefied fully developed flow of a gas through a duct of a triangular cross section is solved in the whole range of the Knudsen number. The flow is modelled by the BGK kinetic equation, subject to Maxwell diffuse boundary conditions. The numerical solution is based on the discrete velocity method, which is applied for first time on a triangular lattice in the physical space. The boundaries of the flow and computational domains are identical deducing accurate results with modest computational effort. Results on the velocity profiles and on the flow rates for ducts of various triangular cross sections are reported and they are valid in the whole range of gas rarefaction. Their accuracy is validated in several ways, including the recovery of the analytical solutions at the free molecular and hydrodynamic limits. The successful implementation of the triangular grid elements is promising for generalizing kinetic type solutions to rarefied flows in domains with complex boundaries using adaptive and unstructured grids.

Rarefied gas flow in concentric annular tube: Estimation of the Poiseuille number and the exact hydraulic diameter

The fully developed flow of rarefied gases through circular ducts of concentric annular cross sections is solved via kinetic theory. The flow is due to an externally imposed pressure gradient in the longitudinal direction and it is simulated by the BGK kinetic equation, subject to Maxwell diffuse-specular boundary conditions. The approximate principal of the hydraulic diameter is investigated for first time in the field of rarefied gas dynamics. For the specific flow pattern, in addition to the flow rates, results are reported for the Poiseuille number and the exact hydraulic diameter. The corresponding parameters include the whole range of the Knudsen number and various values of the accommodation coefficient and the ratio of the inner over the outer radius. The accuracy of the results is validated in several ways, including the recovery of the analytical solutions at the hydrodynamic and free molecular limits.

Effect of acceleration by internal and external force fields on particle motion in intermediate regimes between the hydrodynamic and free-molecular limits

The kinetic theory of gases is applied to analyze slow translational motion of low-concentration particles driven by an external force in a homogeneous gas. The analysis takes into account the diffusion due to the difference in acceleration between particles and molecules in internal and external force fields. A general expression is derived for the particle drag force in hydrodynamic, free-molecular, and intermediate regimes. This expression reduces to a simple relation between the drag force and its values in the hydrodynamic and free-molecular limits and the force of intermolecular interaction between particles and gas molecules. In the case of spherically symmetric potential of interaction between the particle and molecules, the drag force is the harmonic mean of its limit values.

Generalization of an asymptotic model for constant-rate aerosol reactors

An asymptotic model was previously developed to predictbehaviors of constant-rate aerosol reactors operating with particles in the free-molecular or continuum limits. This model considered the limiting cases of having either condensation or nucleation dominate during a nucleation burst that occurs from steady addition of condensable monomer. In the present article, this model is generalized to allow condensation and nucleation to both be important during a nucleation burst. Criteria are derived to predict the relative magnitudes of nucleation and condensation, and scaling relations are presented for particle number densities, particle sizes, and onset times and durations of nucleation bursts. Comparison of the asymptotic results with numerical integration of the governing equations is favorable both qualitatively and quantitatively.

Calculations of the near-wall thermophoretic force in rarefied gas flow

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.

Statistical Simulation of Low-Speed Rarefied Gas Flows

Molecular-based numerical schemes, such as the direct simulation Monte Carlo (DSMC) method, are more physically appropriate for rarefied gas flows in microelectromechanical systems (MEMS). It is difficult for them to be statistically convergent, however, because the statistical fluctuation becomes insurmountably large at the low Mach numbers that are characteristic of MEMS. An information preservation (IP) technique is proposed to address this issue. This technique assigns each simulated molecule in the DSMC method two velocities. One is the molecular velocity used to compute the molecular motion following the same steps as the DSMC method. The other is called information velocity. It corresponds to the collective velocity of an enormous number of real molecules that the simulated molecule represents. Using the information velocity to compute macroscopic velocity and shear stress may remove the statistical fluctuation source inherent in the DSMC method that results from the randomness of the thermal velocity. The IP technique has been applied to benchmark problems, namely Couette, Poiseuille, and Rayleigh flows, in the entire Knudsen regime. The characteristic velocities in these flows range from 0.01 to 1 m/s, much smaller than the thermal velocity of about 340 m/s at room temperature. The meaningful results are obtained at a sample size of 103–104, in comparison with a sample size of 108 or more required for the DSMC method at such a range of flow velocity. This results in a tremendous gain in CPU time. The velocity distributions, surface shear stress, and mass flux given by the IP calculations compare quite well with exact solutions at the continuum and free molecular limits, and with the numerical solutions of the linearized Boltzmann equation and experimental data in the transition regime.

Monte Carlo computation of rarefied supersonic flow into a pitot probe

Pitot probes are commonly used to make measurements in high-speed gas flows in wind tunnels and vacuum chambers. In supersonic flow, a shock wave forms in front of the probe. In the continuum flow regime, the Rayleigh pitot probe equation is used to describe nonisentropic pressure losses across this shock. Under the rarefied flow conditions typical of supersonic plumes produced by spacecraft propulsion systems, the continuum theory is inaccurate. Hence, pressure corrections are needed to make use of pitot probe measurements in these flows. Currently, there is no general analytical theory describing this behavior. In the present study, the direct simulation Monte Carlo method is used to simulate rarefied, supersonic flow of nitrogen into a representative pitot probe geometry in order to predict these pressure corrections. Numerical results are compared with experimental data and are found to show good agreement qualitatively and quantitatively. Pressure corrections computed at very low probe Reynolds numbers are found to be consistent with free molecular limits calculated using collisionless gas theory.

Analysis of Constant-Rate Aerosol Reactors

Aerosol nucleation and growth in constant-rate aerosol reactors are considered for the free-molecular and continuum limits. Asymptotic techniques are used to predict supersaturation histories, nucleation rates, and particle number densities in nucleation bursts that result from increasing the supersaturation by continuously adding condensable monomer. A variable present in expressions for homogeneous nucleation is treated as a large parameter, and asymptotic analyses are developed in terms of this parameter. Scaling relations for average particle sizes, particle number densities, and onset times and durations of nucleation bursts are presented. Numerical and asymptotic solutions compare well, demonstrating the usefulness of asymptotic techniques for analyzing problems of this type.

Modeling of a small hydrazine thruster plume in the transition flow regime

Three different techniques commonly employed in the calculation of rocket and thruster expansion plumes are assessed. These techniques vary both in computational expense and in the accuracy and detail of the solutions that they provide. The assessment is made with reference to the plume expanding from a small monopropellant hydrazine thruster and includes comparison with experimental data. Two of the modeling techniques, the Simons model and the Method of Characteristi cs, rely on the continuum equations. The third, the Direct Simulation Monte Carlo method (DSMC), adopts a discrete particle approach. The validity in employing continuum methods in the flowfield between the continuum and free molecular limits (i.e., the transition flow regime) is investigated. It is noted that the more computationally intensive DSMC solution method is the proper technique in this region of the expansion plume. Additional results provided solely by the DSMC calculations, such as thermal nonequilibrium effects, are presented. The consequences arising from the apparent differences in the results obtained with the continuum and discrete particle methods at the free molecular limit are assessed in terms of impingement effects.

Assessment of Impingement Effects in the Isentropic Core of a Small Satellite Control Thruster Plume

Two computational techniques commonly employed in the calculation of rocket and thruster expansion plumes are assessed. These are the method of characteristics (MOC), which is derived from the continuum Euler equations, and the direct simulation Monte Carlo (DSMC) method, which adopts a discrete particle approach. These techniques vary both in the computational expense and in the accuracy and detail of the solutions that they provide, depending upon the regime of application. The assessment is made with reference to the plume expanding from a small monopropellant hydrazine thruster and concentrates on the isentropic core of the jet for the flow regime lying between the continuum and free molecular limits. It is found that the more numerically intensive DSMC method offers the better correspondence to the available experimental data. In addition, large differences in typical impingement effects such as drag force and heat transfer are found at the free molecular limit of the plume expansion for the two predictive techniques. It is concluded that accurate estimation of impingement potential may only be achieved through application of the discrete particle method.

The Brownian coagulation of aerosols over the entire range of Knudsen numbers: Connection between the sticking probability and the interaction forces

The existing models for Brownian coagulation of aerosols account for the effect of interparticle forces through a phenomenological sticking probability, i.e., the probability of coagulation upon collision. This probability is usually assumed to be equal to unity. A model for Brownian coagulation of equal-sized electrically neutral aerosol particles is proposed, which takes into account explicitly the van der Waals attraction and Born repulsion, instead of the phenomenological sticking probability. In this model, the relative motion between two particles in the vicinity of the sphere of influence is considered to be free molecular, the thickness of this region is taken to be equal to the average correlation length for the relative Brownian motion. The relative motion of the particles outside this region is described by the Fokker-Plank equation. The sticking probability predicted by the model, in terms of the interaction potential between two particles, becomes vanishingly small for very small particles. The collisions with the medium that a particle experiences during its escape from the potential well and the corresponding energy dissipation, neglected in this model, will increase the coagulation coefficient, in particular for the large particles because they stay longer in the region in which the interaction potential is acting. For this reason the model is likely to be valid for sufficiently small particles and to provide a lower bound for the coagulation coefficient of the larger particles. However, for large particles the interaction potential is deep and the decay length of the potential is small compared to the particle sizes. As a result, the region of the interaction potential can be replaced by a sink. This is equivalent to considering the sticking probability as equal to unity. In contrast to the earlier models, the expression derived for the coagulation coefficient in the latter case displays the proper continuum and free molecular limits and agrees well with the Fuchs empirical formula. It yields an upper bound for sufficiently small particles but can be exceeded for particles of intermediate sizes. Comparison with the experimental data seem to indicate the validity of this upper bound for particles as small as 0.01 μm.

Shock-wave structure in a gas of ideally elastic and rigid spherical molecules: FG Cheremisin, Chislen Metody Teorii Razrezh Gazov, AN SSSR, Moscow 1969, 65–78 ( Russian)

Transport of heat and mass to surfaces in fluid flows lies at the heart of a vast array of engineering problems. The Colburn analogy that relates the transport of heat and mass to the surface is often applied to approximate mass transport using correlations developed for heat transport. Here we use a numerical method that solves the Shakhov model Boltzmann equation of fluid mechanics coupled to a simple monolayer adsorption model to show that this analogy does not apply for selected rarefied flows. We consider steady flow of gas through a micro-channel and the flow in a lid-driven cavity. We define a breakdown parameter indicating the departure of the relative rates of heat and mass transport from that predicted using the analogy, and give simple analytic expressions for this parameter in the free-molecular and continuum limits. Results show that the analogy breaks down even for Knudsen numbers corresponding to slip and transition region flows, and that the magnitude of this departure is such that mass transfer is many times slower than predicted using the analogy. These results have important application to the design of micro-scale devices involving mass transport to surfaces.