Showing papers on "Knudsen number published in 2013"

Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case

Abstract: Based on the Boltzmann-BGK (Bhatnagar-Gross-Krook) equation, in this paper a discrete unified gas kinetic scheme (DUGKS) is developed for low-speed isothermal flows. The DUGKS is a finite-volume scheme with the discretization of particle velocity space. After the introduction of two auxiliary distribution functions with the inclusion of collision effect, the DUGKS becomes a fully explicit scheme for the update of distribution function. Furthermore, the scheme is an asymptotic preserving method, where the time step is only determined by the Courant-Friedricks-Lewy condition in the continuum limit. Numerical results demonstrate that accurate solutions in both continuum and rarefied flow regimes can be obtained from the current DUGKS. The comparison between the DUGKS and the well-defined lattice Boltzmann equation method (D2Q9) is presented as well.

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

Abstract: Due 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.

Morphological Analyses of Polymer Electrolyte Fuel Cell Electrodes with Nano-Scale Computed Tomography Imaging

Abstract: We report a three-dimensional (3D), pore-scale analysis of morphological and transport properties for a polymer electrolyte fuel cell (PEFC) catalyst layer. The 3D structure of the platinum/carbon/Nafion electrode was obtained using nano-scale resolution X-ray computed tomography (nano-CT). The 3D nano-CT data was analyzed according to several morphological characteristics, with particular focus on various effective pore diameters used in modeling gas diffusion in the Knudsen transition regime, which is prevalent in PEFC catalyst layers. The pore diameter metrics include those based on chord length distributions, inscribed spheres, and surface area. Those pore diameter statistics are evaluated against computational pore-scale diffusion simulations with local gas diffusion coefficients determined from the local pore size according to the Bosanquet formulation. According to our comparison, simulations that use local pore diameters defined by inscribed spheres provide effective diffusion coefficients that are consistent with chord-length based estimations for an effective Knudsen length scale. By evaluating transport rates in regions of varying porosity within the nano-CT data, we identified a Bruggeman correction scaling factor for the effective diffusivity.

Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows

Abstract: A thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small Mach number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct Monte Carlo and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately 0:5. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations that have been chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased.

Electro-thermal vibration of visco-elastically coupled BNNT systems conveying fluid embedded on elastic foundation via strain gradient theory

Abstract: Electro-thermal vibration of the double of Boron Nitride nanotubes (BNNTs) which are coupled by visco-Pasternak medium is carried out based on strain gradient theory. Two BNNTs are placed in uniform temperature and electric fields, the latter being applied through attached electrodes at both ends. Moreover, one of the BNNT is under conveying fluid. Considering Euler-Bernoulli beam (EBB) model and Knudsen number, the higher-order equations of motion are derived base on the Hamilton's principle where differential quadrature method (DQM) is applied to obtain the frequency of coupled BNNTs system. The detailed parametric study is conducted, focusing on the combined effects of the Knudsen number, aspect ratio, thermal and electric fields, velocity of conveying fluid and visco-Pasternak coefficients on the stability of coupled system. Also, it is found that trend of figures have good agreement with the other studies.

Performance Analysis of the Lattice Boltzmann Model Beyond Navier-Stokes

Abstract: The lattice Boltzmann method is increasingly important in facilitating large-scale fluid dynamics simulations. To date, these simulations have been built on discretized velocity models of up to 27 neighbors. Recent work has shown that higher order approximations of the continuum Boltzmann equation enable not only recovery of the Navier-Stokes hydrodynamics, but also simulations for a wider range of Knudsen numbers, which is especially important in micro- and nanoscale flows. These higher-order models have significant impact on both the communication and computational complexity of the application. We present a performance study of the higher-order models as compared to the traditional ones, on both the IBM Blue Gene/P and Blue Gene/Q architectures. We study the tradeoffs of many optimizations methods such as the use of deep halo level ghost cells that, alongside hybrid programming models, reduce the impact of extended models and enable efficient modeling of extreme regimes of computational fluid dynamics.

A Parallel DSMC Investigation of Monatomic/Diatomic Gas Flows in a Micro/Nano Cavity

Abstract: In the current study, we performed a DSMC investigation to study the nonequilibrium effects on monatomic and diatomic rarefied flows in micro/nano lid-driven cavity. Our DSMC solver is parallel and benefits from a variable time step algorithm. The results of our simulation showed that growing rarefaction effects increase the maximum temperature of the cavity flow. As the inter-molecular rate of collision diminishes in the nonequilibrium regime, molecules manage to conserve their obtained energy from diffused surfaces; consequently, the flow temperature increases. We also investigate the nonequilibrium effects on the shear stress profile in the micro/nano cavity. Although increasing the Knudsen number decreases flow shear stress, the nondimensional shear stress, which shows the molecular potential for performing collision in the rarefied flow, increases.

A lattice Boltzmann study of gas flows in a long micro-channel

Abstract: In this paper, the pressure-driven flow in a long micro-channel is studied via a lattice Boltzmann equation (LBE) method. With the inclusion of the gas-wall collision effects, the LBE is able to capture the flow behaviors in the transition regime. The numerical results are compared with available data of other methods. Furthermore, the effects of rarefaction and compressibility on the deviation of the pressure distribution from the linear one are also investigated.

Revised Knudsen-layer reduction of fusion reactivity

Abstract: Recent work by Molvig et al. [Phys. Rev. Lett. 109, 095001 (2012)] examined how fusion reactivity may be reduced from losses of fast ions in finite assemblies of fuel. In this paper, this problem is revisited with the addition of an asymptotic boundary-layer treatment of ion kinetic losses. This boundary solution, reminiscent of the classical Milne problem from linear transport theory, obtains a free-streaming limit of fast ion losses near the boundary, where the diffusion approximation is invalid. Thermonuclear reaction rates have been obtained for the ion distribution functions predicted by this improved model. It is found that while Molvig's “Knudsen distribution function” bounds from above the magnitude of the reactivity reduction, this more accurate treatment leads to less dramatic losses of tail ions and associated reduction of thermonuclear reaction rates for finite fuel volumes.

A Fokker–Planck based kinetic model for diatomic rarefied gas flows

Abstract: A Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

A fast hybrid fourier-boltzmann transport equation solver for nongray phonon transport

Abstract: Nongray phonon transport solvers based on the Boltzmann transport equation (BTE) are being increasingly employed to simulate submicron thermal transport in semiconductors and dielectrics. Typical sequential solution schemes encounter numerical difficulties because of the large spread in scattering rates. For frequency bands with very low Knudsen numbers, strong coupling between other BTE bands result in slow convergence of sequential solution procedures. This is due to the explicit treatment of the scattering kernel. In this paper, we present a hybrid BTE-Fourier model which addresses this issue. By establishing a phonon group cutoff Knc, phonon bands with low Knudsen numbers are solved using a modified Fourier equation which includes a scattering term as well as corrections to account for boundary temperature slip. Phonon bands with high Knudsen numbers are solved using the BTE. A low-memory iterative solution procedure employing a block-coupled solution of the modified Fourier equations and a sequential solution of BTEs is developed. The hybrid solver is shown to produce solutions well within 1% of an all-BTE solver (using Knc = 0.1), but with far less computational effort. Speedup factors between 2 and 200 are obtained for a range of steady-state heat transfer problems. The hybrid solver enables efficient and accurate simulation of thermal transport in semiconductors and dielectrics across the range of length scales from submicron to the macroscale.

Characteristics of plasma properties in an ablative pulsed plasma thruster

Abstract: Pulsed plasma thrusters are electric space propulsion devices which create a highly transient plasma bulk in a short-time arc discharge that is expelled to create thrust. The transitional character and the dependency on the discharge properties are yet to be elucidated. In this study, optical emission spectroscopy and Mach-Zehnder interferometry are applied to investigate the plasma properties in variation of time, space, and discharge energy. Electron temperature, electron density, and Knudsen numbers are derived for the plasma bulk and discussed. Temperatures were found to be in the order of 1.7 to 3.1 eV, whereas electron densities showed maximum values of more than 1017 cm−3. Both values showed strong dependency on the discharge voltage and were typically higher closer to the electrodes. Capacitance and time showed less influence. Knudsen numbers were derived to be in the order of 10−3−10−2, thus, indicating a continuum flow behavior in the main plasma bulk.

Invariant Manifolds for Steady Boltzmann Flows and Applications

Abstract: We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green’s function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of −1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne’s problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves.

Ubiquitous non-thermals in astrophysical plasmas: restating the difficulty of maintaining maxwellians

Abstract: This paper outlines the rather narrow conditions on a radiatively decoupled plasma where a Maxwell-Boltzmann (MB) distribution can be assumed with confidence. The complementary non-thermal distribution with non-perturbative kurtosis is argued to have a much broader purview than has previously been accepted. These conditions are expressed in terms of the electron Knudsen number, Ke , the ratio of the electron mean free path to the scale length of electron pressure. Rather generally, f(v 0.01 is common in all main-sequence stellar atmospheres above approximately 0.05 stellar radii from the surface. The entire solar corona and wind are included in this regime where non-thermal distributions with kurtosis are shown to be ubiquitous, heat flux is not well modeled by Spitzer-Braginskii closure, and fluid modeling is qualitative at best.

DSMC Simulation of Low Knudsen Micro/Nanoflows Using Small Number of Particles per Cells

Abstract: Direct simulation Monte Carlo (DSMC) method in low Knudsen rarefied flows at micro/ nanoscales remains a big challenge for researchers due to large computational requirements. In this article, the application of the simplified Bernoulli-trials (SBT)/dual grid collision scheme is extended for solving low Knudsen/low speed and low Knudsen/high gradient rarefied micro/nanoflows. The main advantage of the SBT algorithm is to provide accurate calculations using much smaller number of particles per cell, i.e., hN i� 2, which is quite beneficial for near continuum DSMC simulations where the requirement of fine meshes faces the simulation with serious memory restrictions. Comparing the results of the SBT/dual grid scheme with the no time counter (NTC) scheme and majorant frequency scheme (MFS), it is shown that the SBT/dual grid scheme could successfully predict the thermal pattern and hydrodynamics field as well as surface parameters such as velocity slip, temperature jump and wall heat fluxes. Therefore, we present SBT/dual grid algorithm as a suitable alternative of the standard collision schemes in the DSMC method for typical micro/nanoflows solution. Nonlinear flux-corrected transport (FCT) algorithm is also employed as a filter to extract the smooth solution from the noisy DSMC calculation for low speed/low Knudsen number DSMC calculations. [DOI: 10.1115/1.4024505]

Excess-entropy scaling for gas diffusivity in nanoporous materials.

Abstract: We present an efficient computational procedure for the rapid prediction of the self-diffusivity of gas molecules in nanoporous materials by a combination of the Knudsen model, Rosenfeld’s excess-entropy scaling method, and a classical density functional theory (DFT). The self-diffusivity conforms to the Knudsen model at low density, and the effects of intermolecular interactions at higher densities are accounted for by Rosenfeld’s excess-entropy scaling method. The classical DFT provides a convenient way to calculate the excess entropy used in the scaling analysis. The hybrid computational procedure has been calibrated with MD simulation for the adsorption of H2, He, Ne, and Ar gases in several nanoporous materials over a broad range of pressure. It predicts adsorption isotherms and different types of diffusion behavior in excellent agreement with the simulation results. Although the simulation of gas diffusion in nanoporous materials is extremely time-consuming, the new procedure is computationally very...

Analytical solution to predicting gaseous mass flow rates of microchannels in a wide range of Knudsen numbers.

Abstract: To predict the gaseous mass flow rate of microchannels, conventional analytical solutions based on the Navier-Stokes equation or volume diffusion hydrodynamics (bivelocity hydrodynamics) associated with first-order or second-order slip boundary condition are not very successful, especially in high-Knudsen-number flow. An analytical solution which agrees with experimental data to a Knudsen number of 50 is presented in this paper. To achieve this goal, a concept of effective volume diffusion is defined. Then, with a general slip boundary condition, the gaseous mass flow rate of microchannel is derived by solving the momentum equation of this effective volume diffusion hydrodynamics. Compared with six other analytical solutions and one group of numerical solutions of the linearized Boltzmann equation, this solution is validated by three groups of experimental data. The results not only illustrate an improvement of this solution compared with other analytical solutions but also show the importance of the effective volume diffusion hydrodynamics for compressible microfluids.

Thermal Escape in the Hydrodynamic Regime: Reconsideration of Parker's Isentropic Theory Based on Results of Kinetic Simulations

Abstract: The one-dimensional steady-state problem of thermal escape from a single-component atmosphere of mon- and diatomic gases is studied in the hydrodynamic (blow-off) regime using the direct simulation Monte Carlo method for an evaporative-type condition at the lower boundary. The simulations are performed for various depths into an atmosphere, indicated by a Knudsen number, Kn 0, equal to the ratio of the mean free path of molecules to the radial position of the source surface, ranging from 10 to 10–5, and for the range of the source Jeans parameter, λ0, equal to the ratio of gravitational and thermal energies, specific to blow-off. The results of kinetic simulations are compared with the isentropic model (IM) and the Navier-Stokes model. It is shown that the IM can be simplified if formulated in terms of the local Mach number and Jeans parameter. The simulations predict that at Kn 0 < ~ 10–3 the flow includes a near-surface non-equilibrium Knudsen layer, a zone where the flow can be well approximated by the IM, and a rarefied far field. The corresponding IM solutions, however, only approach Parker's critical solution as λ0 approaches the upper limit for blow-off. The IM alone is not capable for predicting the flow and requires boundary conditions at the top of the Knudsen layer. For small Kn 0, the scaled escape rate and energy loss rate are found to be independent of λ0. The simulation results can be scaled to any single-component atmosphere exhibiting blow-off if the external heating above the lower boundary is negligible, in particular, to sublimation-driven atmospheres of Kuiper belt objects.

Rarefied gas flow through a cylindrical tube due to a small pressure difference

Abstract: Flow of a rarefied gas through a cylindrical tube connecting two reservoirs maintained at a small pressure difference is considered using the axisymmetric version of the linearised BGK kinetic model equation subject to Maxwell diffuse–specular boundary conditions. This is a problem of five dimensions in phase space, solved in a fully deterministic manner using a parallelised discrete velocity algorithm. Results include flow rates as well as distributions of density and velocity perturbations, from the free molecular up to the slip regime and for length-over-radius ( L / R ) ratios ranging from zero (orifice flow) up to 20. The dependency of the results on gas rarefaction, wall accommodation and tube length is analysed and discussed. It is found that the Knudsen minimum appears only at L / R = 20 . Furthermore, in the case of L / R = 0 it is confirmed that the results are practically independent of the accommodation coefficient. Comparing the present linear results with corresponding non-linear ones, it is seen that linearised analysis can capture the correct behaviour of the flow field not only for infinitesimally small but also for small but finite pressure differences and that its range of applicability is wider than expected. Also, the error introduced by the assumption of fully developed flow for channels of moderate length is estimated through a comparison with the present corresponding results.