Showing papers on "Knudsen number published in 2019"

Uniformly accurate machine learning-based hydrodynamic models for kinetic equations.

Abstract: A framework is introduced for constructing interpretable and truly reliable reduced models for multiscale problems in situations without scale separation. Hydrodynamic approximation to the kinetic equation is used as an example to illustrate the main steps and issues involved. To this end, a set of generalized moments are constructed first to optimally represent the underlying velocity distribution. The well-known closure problem is then solved with the aim of best capturing the associated dynamics of the kinetic equation. The issue of physical constraints such as Galilean invariance is addressed and an active-learning procedure is introduced to help ensure that the dataset used is representative enough. The reduced system takes the form of a conventional moment system and works regardless of the numerical discretization used. Numerical results are presented for the BGK (Bhatnagar–Gross–Krook) model and binary collision of Maxwell molecules. We demonstrate that the reduced model achieves a uniform accuracy in a wide range of Knudsen numbers spanning from the hydrodynamic limit to free molecular flow.

Diffusion in Porous Media: Phenomena and Mechanisms

Abstract: Two distinct but interconnected approaches can be used to model diffusion in fluids; the first focuses on dynamics of an individual particle, while the second deals with collective (effective) motion of (infinitely many) particles. We review both modeling strategies, starting with Langevin’s approach to a mechanistic description of the Brownian motion in free fluid of a point-size inert particle and establishing its relation to Fick’s diffusion equation. Next, we discuss its generalizations which account for a finite number of finite-size particles, particle’s electric charge, and chemical interactions between diffusing particles. That is followed by introduction of models of molecular diffusion in the presence of geometric constraints (e.g., the Knudsen and Fick–Jacobs diffusion); when these constraints are imposed by the solid matrix of a porous medium, the resulting equations provide a pore-scale representation of diffusion. Next, we discuss phenomenological Darcy-scale descriptors of pore-scale diffusion and provide a few examples of other processes whose Darcy-scale models take the form of linear or nonlinear diffusion equations. Our review is concluded with a discussion of field-scale models of non-Fickian diffusion.

Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow

Abstract: We consider causal higher order theories of relativistic viscous hydrodynamics in the limit of one-dimensional boost-invariant expansion and study the associated dynamical attractor. We obtain evolution equations for the inverse Reynolds number as a function of Knudsen number. The solutions of these equations exhibit attractor behavior which we analyze in terms of Lyapunov exponents using several different techniques. We compare the attractors of the second-order M\"uller-Israel-Stewart (MIS), transient Denicol-Niemi-Molnar-Rischke (DNMR), and third-order theories with the exact solution of the Boltzmann equation in the relaxation-time approximation. It is shown that for Bjorken flow the third-order theory provides a better approximation to the exact kinetic theory attractor than both MIS and DNMR theories. For three different choices of the time dependence of the shear relaxation rate we find analytical solutions for the energy density and shear stress and use these to study the attractors analytically. By studying these analytical solutions at both small and large Knudsen numbers we characterize and uniquely determine the attractors and Lyapunov exponents. While for small Knudsen numbers the approach to the attractor is exponential, strong power-law decay of deviations from the attractor and rapid loss of initial state memory are found even for large Knudsen numbers. Implications for the applicability of hydrodynamics in far-off-equilibrium situations are discussed.

Mass and heat transfer between evaporation and condensation surfaces: Atomistic simulation and solution of Boltzmann kinetic equation.

Abstract: Boundary conditions required for numerical solution of the Boltzmann kinetic equation (BKE) for mass/heat transfer between evaporation and condensation surfaces are analyzed by comparison of BKE results with molecular dynamics (MD) simulations. Lennard-Jones potential with parameters corresponding to solid argon is used to simulate evaporation from the hot side, nonequilibrium vapor flow with a Knudsen number of about 0.02, and condensation on the cold side of the condensed phase. The equilibrium density of vapor obtained in MD simulation of phase coexistence is used in BKE calculations for consistency of BKE results with MD data. The collision cross-section is also adjusted to provide a thermal flux in vapor identical to that in MD. Our MD simulations of evaporation toward a nonreflective absorbing boundary show that the velocity distribution function (VDF) of evaporated atoms has the nearly semi-Maxwellian shape because the binding energy of atoms evaporated from the interphase layer between bulk phase and vapor is much smaller than the cohesive energy in the condensed phase. Indeed, the calculated temperature and density profiles within the interphase layer indicate that the averaged kinetic energy of atoms remains near-constant with decreasing density almost until the interphase edge. Using consistent BKE and MD methods, the profiles of gas density, mass velocity, and temperatures together with VDFs in a gap of many mean free paths between the evaporation and condensation surfaces are obtained and compared. We demonstrate that the best fit of BKE results with MD simulations can be achieved with the evaporation and condensation coefficients both close to unity.

Controlled growth of transition metal dichalcogenide monolayers using Knudsen-type effusion cells for the precursors

Abstract: Controlling the flow rate of precursors is essential for the growth of high quality monolayer single crystals of transition metal dichalcogenides (TMDs) by chemical vapor deposition (CVD). Thus, introduction of an excess amount of the precursors affects reproducibility of the growth process and results in the formation of TMD multilayers and other unwanted deposits. Here we present a simple method for controlling the precursor flow rates using the Knudsentype effusion cells [1]. This method results in a highly reproducible growth of large area and high density TMD monolayers. The size of the grown crystals can be adjusted between 10 and 200 μm. We characterized the grown monolayers by optical, atomic force and transmission electron microscopies as well as by X-ray photoelectron, Raman and photoluminescence spectroscopies, and by electrical transport measurements showing their high optical and electronic quality based on the single crystalline nature. The quality of these CVD grown TMDs can be compared to the best quality exfoliated materials [2].

Unified Gas-kinetic Wave-Particle Methods II: Multiscale Simulation on Unstructured Mesh

Abstract: In this paper, we present a unified gas-kinetic wave-particle (UGKWP) method on unstructured mesh for multiscale simulation of continuum and rarefied flow. Inheriting from the multicale transport in the unified gas-kinetic scheme (UGKS), the integral solution of kinetic model equation is employed in the construction of UGKWP method to model the flow physics in the cell size and time step scales. A novel wave-particle adaptive formulation is introduced in the UGKWP method to describe the flow dynamics in each control volume. The local gas evolution is constructed through the dynamical interaction of the deterministic hydrodynamic wave and the stochastic kinetic particle. Within the resolution of cell size and time step, the decomposition, interaction, and evolution of the hydrodynamic wave and the kinetic particle depend on the ratio of the time step to the local particle collision time. In the rarefied flow regime, the flow physics is mainly recovered by the discrete particles and the UGKWP method performs as a stochastic particle method. In the continuum flow regime, the flow behavior is solely followed by macroscopic variable evolution and the UGKWP method becomes a gas-kinetic hydrodynamic flow solver for the viscous and heat-conducting Navier--Stokes solutions. In different flow regimes, many numerical test cases are computed to validate the UGKWP method on unstructured mesh. The UGKWP method can get the same UGKS solutions in all Knudsen regimes without the requirement of the time step and mesh size being less than than the particle collision time and mean free path. With an automatic wave-particle decomposition, the UGKWP method becomes very efficient. For example, at Mach number 30 and Knudsen number 0.1, in comparison with UGKS several-order-of-magnitude reductions in computational cost and memory requirement have been achieved by UGKWP.

Radiometric flow in periodically patterned channels: fluid physics and improved configurations

Abstract: With the aid of direct simulation Monte Carlo (DSMC), we conduct a detailed investigation pertaining to the fluid and thermal characteristics of rarefied gas flow with regard to various arrangements for radiometric pumps featuring vane and ratchet structures. For the same, we consider three categories of radiometric pumps consisting of channels with their bottom or top surfaces periodically patterned with different structures. The structures in the design of the first category are assumed to be on the bottom wall and consist of either a simple vane, a right-angled triangular fin or an isosceles triangular fin. The arrangements on the second category of radiometric pumps consist of an alternating diffuse–specular right-angled fin and an alternating diffuse–specular isosceles fin on the bottom wall. The third category contains either a channel with double isosceles triangular fins on its lowermost surface or a zigzag channel with double isosceles triangular fins on both walls. In the first and the third categories, the surfaces of the channel and its structures are considered as diffuse reflectors. The temperature is kept steady on the horizontal walls of the channel; thus, radiometric flow is created by subjecting the adjacent sides of the vane/ratchet to constant but unequal temperatures. On the other hand, for the second category, radiometric flow is introduced by specifying different top/bottom channel wall temperatures. The DSMC simulations are performed at a Knudsen number based on the vane/ratchet height of approximately one. The dominant mechanism in the radiometric force production is clarified for the examined configurations. Our results demonstrate that, at the investigated Knudsen number, the zigzag channel experiences maximum induced velocity with a parabolic velocity profile, whereas its net radiometric force vanishes. In the case of all other configurations, the flow pattern resembles a Couette flow in the open section of the channel situated above the vane/ratchet. To further enhance the simulations, the predictions of the finite volume discretization of the Boltzmann Bhatnagar–Gross–Krook (BGK)–Shakhov equation for the mass flux dependence on temperature difference and Knudsen number are also reported for typical test cases.

Flow Mechanism and Simulation Approaches for Shale Gas Reservoirs: A Review

Abstract: The past two decades have borne remarkable progress in our understanding of flow mechanisms and numerical simulation approaches of shale gas reservoir, with much larger number of publications in recent 5 years compared to that before year 2012. In this paper, a review is constructed with three parts: flow mechanism, reservoir models and numerical approaches. In mechanism, it is found that gas adsorption process can be concluded into different isotherm models for various reservoir basins. Multi-component adsorption mechanisms are taken into account in recent years. Flow mechanism and equations vary with different Knudsen numbers, which could be figured out in two ways: molecular dynamics (MD) and lattice Boltzmann method (LBM). MD has been successfully applied in the study of adsorption, diffusion, displacement and other mechanisms. LBM has been introduced in the study of slippage, Knudsen diffusion and apparent permeability correction. The apparent permeability corrections are introduced to improve classic Darcy’s model in matrix with low velocities and fractures with high velocities. At reservoir-scale simulation, gas flow models are presented with multiple porosity classified into organic matrix with nanopores, organic matrix with micropores, inorganic matrix and natural fractures. A popular trend is to incorporate geomechanism with flow model in order to better understand the shale gas production. Finally, to solve the new models based on enhanced flow mechanisms, improved macroscopic numerical approaches, including the finite difference method and finite element method, are commonly used in this area. Other approaches like finite volume method and fast matching method are also developed in recent years.

From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight

Abstract: We study the Boltzmann equation on the d-dimensional torus in a perturbative setting around a global equilibrium under the Navier-Stokes lineari- sation. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a C0-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently on the Knudsen number. Finally we show a Cauchy theory and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal and furthermore, this result only requires derivatives in the space variable and allows to connect to solutions to the incompressible Navier-Stokes equations in these spaces.

Conserved discrete unified gas-kinetic scheme with unstructured discrete velocity space

Abstract: Discrete unified gas-kinetic scheme (DUGKS) is a multiscale numerical method for flows from continuum limit to free molecular limit, and is especially suitable for the simulation of multiscale flows, benefiting from its multiscale property. To reduce integration error of the DUGKS and ensure the conservation property of the collision term in isothermal flow simulations, a conserved-DUGKS (C-DUGKS) is proposed. On the other hand, both DUGKS and C-DUGKS adopt Cartesian-type discrete velocity space, in which Gaussian and Newton-Cotes numerical quadrature are used for calculating the macroscopic physical variables in low-speed and high-speed flows, respectively. However, the Cartesian-type discrete velocity space leads to huge computational cost and memory demand. In this paper, the isothermal C-DUGKS is extended to the nonisothermal case by adopting coupled mass and inertial energy distribution functions. Moreover, since the unstructured mesh, such as the triangular mesh in the two-dimensional case, is more flexible than the structured Cartesian mesh, it is introduced to the discrete velocity space of C-DUGKS, such that more discrete velocity points can be arranged in the velocity regions that enclose a large number of molecules, and only a few discrete velocity points need to be arranged in the velocity regions with a small amount of molecules in it. By using the unstructured discrete velocity space, the computational efficiency of C-DUGKS is significantly increased. A series of numerical tests in a wide range of Knudsen numbers, such as the Couette flow, lid-driven cavity flow, two-dimensional rarefied Riemann problem, and the supersonic cylinder flows, are carried out to examine the validity and efficiency of the present method.

Microstructure Effects on Effective Gas Diffusion Coefficient of Nanoporous Materials

Abstract: In this work, we develop a numerical framework for gas diffusion in nanoporous materials including a random generation-growth algorithm for microstructure reconstruction and a multiple-relaxation-time lattice Boltzmann method for solution of diffusion equation with Knudsen effects carefully considered. The Knudsen diffusion is accurately captured by a local diffusion coefficient computed based on a corrected Bosanquet-type formula with the local pore size determined by the largest sphere method. A robust validation of the new framework is demonstrated by predicting the effective gas diffusion coefficient of microporous layer and catalyst layer in fuel cell, which shows good agreement with several recent experimental measurements. Then, a detailed investigation is made of the influence on effective gas Knudsen diffusivity by many important microstructure factors including morphology category, size effect, structure anisotropy, and layering structure effect. A widely applicable Bosanquet-type empirical relation at the Darcy scale is found between the normalized effective gas diffusion coefficient and the average Knudsen number. The present work will promote the understanding and modeling of gas diffusion in nanoporous materials and also provide an efficient platform for the optimization design of nanoporous systems.

Unified gas-kinetic wave-particle methods. II. Multiscale simulation on unstructured mesh

Abstract: In this paper, we present a unified gas-kinetic wave-particle (UGKWP) method on unstructured mesh for multiscale simulation of continuum and rarefied flow. Inheriting from the multicale transport in the unified gas-kinetic scheme (UGKS), the integral solution of kinetic model equation is employed in the construction of UGKWP method to model the flow physics in the cell size and time step scales. A novel wave-particle adaptive formulation is introduced in the UGKWP method to describe the flow dynamics in each control volume. The local gas evolution is constructed through the dynamical interaction of the deterministic hydrodynamic wave and the stochastic kinetic particle. Within the resolution of cell size and time step, the decomposition, interaction, and evolution of the hydrodynamic wave and the kinetic particle depend on the ratio of the time step to the local particle collision time. In the rarefied flow regime, the flow physics is mainly recovered by the discrete particles and the UGKWP method performs as a stochastic particle method. In the continuum flow regime, the flow behavior is solely followed by macroscopic variable evolution and the UGKWP method becomes a gas-kinetic hydrodynamic flow solver for the viscous and heat-conducting Navier--Stokes solutions. In different flow regimes, many numerical test cases are computed to validate the UGKWP method on unstructured mesh. The UGKWP method can get the same UGKS solutions in all Knudsen regimes without the requirement of the time step and mesh size being less than than the particle collision time and mean free path. With an automatic wave-particle decomposition, the UGKWP method becomes very efficient. For example, at Mach number 30 and Knudsen number 0.1, in comparison with UGKS several-order-of-magnitude reductions in computational cost and memory requirement have been achieved by UGKWP.

Effect of distributed axial loading on dynamic stability and buckling analysis of a viscoelastic DWCNT conveying viscous fluid flow

Abstract: In this article, dynamic stability and buckling analysis of a double-walled carbon nanotube (DWCNT) under distributed axial force are investigated. The visco-Pasternak model is used to simulate the elastic medium between nanotubes considering the effects of spring, shear and damping of the elastic medium. This system is conveying viscous fluid, and the relevant force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. The nanostructure is modeled as two orthotropic moderately thick cylindrical shells, and the effects of small-scale and structural damping are accounted based on modified couple stress and Kelvin–Voigt theories. The governing equations and boundary conditions are developed using Hamilton’s principle and solved with the aid of Navier and generalized differential quadrature methods. In this research, the dynamic instability occurs in the viscoelastic DWCNT conveying viscous fluid flow as the natural frequency becomes equal to zero. The results show that the velocity of viscous fluid flow, axial load, mode number, length-to-radius ratio, radius-to-thickness ratio, visco-Pasternak foundation and the boundary conditions play important roles on the critical pressure and natural frequency of the viscoelastic DWCNT conveying viscous fluid flow under axial force.

A slip-flow model for multi-component shale gas transport in organic nanopores

Abstract: With the rapid development of shale gas resources, the accurate simulation of shale gas development process is becoming more and more important. Shale gas transport through nanopores of shale formation is the basic of shale gas development simulation. At present, the effect of impurities on methane transport through nanopores is neglected. In this paper, a novel model is presented for simulating multi-component shale gas transport through nanopores of shale formation. The effects of multi-component, slippage flow, and Knudsen diffusion are considered in the model. Results show that when the shale gas in nanopores is very thin, the Knudsen diffusion plays the dominant role over wide range of nanopore radius. While the effect of multi-component on Knudsen number and contribution degree can be neglected, both of the slippage flow rate and the Knudsen diffusion rate increase with increasing of CO2 content. Under medium pressure condition, there exists two turning points where the slippage flow and Knudsen diffusion take turns in charge of the shale gas transmission. Under high pressure condition, the slippage flow is the dominant factor over wide range of nanopore sizes. While the conductivities increase with decreasing methane content, the effect of multi-component on contribution degrees of slippage and Knudsen diffusion can be neglected.

Numerical study of species separation in rarefied gas mixture flow through micronozzles using DSMC

Abstract: 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...

Elastic waves in fluid-conveying carbon nanotubes under magneto-hygro-mechanical loads via a two-phase local/nonlocal mixture model

Abstract: The nonlocal elasticity has been widely utilized for carbon nanotubes in its differential form. However, recently, it has been indicated that using the integral form of this size-dependent theory yields more accurate and reliable results. In this paper, an integral form of nonlocal elasticity with two distinct phases is used to analyze wave propagations in carbon nanotubes conveying nanofluid. The nanotube is subjected to a magneto-hygro-mechanical loading. Furthermore, the nanotube is surrounded by a two-parameter polymer matrix. To incorporate slip effects between the fluid and tube at nanoscales, a correction factor is employed on the basis of Beskok-Karniadakis model. The wave propagation in carbon nanotubes conveying nanofluid is examined incorporating the influences of various parameters such as phase fractions, magnetic strength, initial stress and Knudsen number. Comparing the present results with those of molecular dynamics simulations, it is shown that the limitation of the differential nonlocal elasticity for high wave numbers is overcome by using the two-phase integral form.

On the flow regime model for fast estimation of tight sandstone gas apparent permeability in high-pressure reservoirs

Abstract: A flow regime-based gas apparent permeability model in high-pressure tight sandstone reservoirs is established by bridging molecular kinetics, gas transport mechanisms, and apparent permeability. T...