Knudsen Number Research Articles

Thin-airfoil aerodynamics in a rarefied gas wind tunnel: A theoretical study

We study the steady aerodynamic field and loadings about a thin flat plate placed in a wind tunnel under non-continuum conditions. Considering a two-dimensional straight tunnel configuration, the flow is driven by either density or temperature differences between the inlet and outlet tunnel reservoirs, producing a pressure gradient across the channel. Focusing on highly rarefied conditions, we derive a semi-analytic description for the gas flow field in the free-molecular limit for diffuse- and specular-wall configurations. The solution is valid at arbitrary differences between the inlet and outlet reservoirs as well as plate angles of attack α. The results are compared with direct simulation Monte Carlo calculations, indicating that the free-molecular description is valid through O(1) plate-size-based Knudsen numbers. The aerodynamic lift and drag forces are evaluated and their variations with α, reservoir conditions, and tunnel size are analyzed. At a fixed pressure ratio between the outlet and inlet reservoirs, the density-driven flow generates higher aerodynamic loads compared with its counterpart temperature-driven configuration, in line with the associated larger mass flow rate in the former. The results are discussed in light of the existing rarefied-gas description of the free-stream (non-confined) problem.

An improved continuum model for hypersonic thermal nonequilibrium flow in the near-continuum regime

In this work, the rarefied Couette flow of diatomic gases with thermal nonequilibrium effects is investigated by the direct simulation Monte Carlo (DSMC) method, and a macroscopic computational model is developed to consider the local rarefaction effects for diatomic gases in the near-continuum regime. The nonlinear transport properties of the diatomic gases are studied, indicating that effective viscosity and effective translational thermal conductivity in the shear nonequilibrium state are affected by translational nonequilibrium effects, which obey the same laws for both monatomic and diatomic gases. The transport coefficients of internal energy modes are affected by both translational nonequilibrium and internal energy relaxation, therefore, the effective rotational and vibrational thermal conductivities are related to the effective viscosity through a modified Eucken relation that accounts for internal energy relaxation. Conclusively, effective constitutive relations are newly established as a function of the shear nonequilibrium parameter and the modified Eucken factors for thermal nonequilibrium flows, and these are integrated into the macroscopic two-temperature model. Subsequently, it is assessed in the simulation of hypersonic flows over flat plates and cylinders at various Knudsen numbers. The results show that the surface shear stress and heat flux obtained by the proposed model agree well with the DSMC results, indicating significantly improved performance compared to the conventional Navier–Stokes two-temperature model for hypersonic flows in the near-continuum regime.

Numerical analysis of a gas flow in a square cavity driven by spanwise lid motion on the basis of kinetic theory: Behavior of the gas near a sharp corner

A gas flow in a square cavity driven by a lid sliding in the direction of its line of contact with the cavity wall is considered. The steady behavior of the gas is numerically investigated based on the linearized Bhatnagar–Gross–Krook kinetic equation and the diffuse reflection boundary condition. When one applies the Stokes equation and the no-slip boundary condition to the system considered here, the flow velocity becomes multivalued at the corner between the lid and the cavity wall, and the shear stress diverges at the corner inversely proportionally to the distance from there, which is known as the so-called corner singularity. In the present work, the behavior of the gas near the corner is examined based on numerical results obtained from the kinetic theory. Although the range of the flow velocity value in the kinetic solution is limited due to the significant velocity slip near the corner, the flow velocity is, nevertheless, multivalued at the corner. The shear stress varies inversely proportionally to the distance from the corner up to the position that is a few tens of mean free paths away from there. The increase in the stress is suppressed at positions closer to the corner and its magnitude remains bounded. Thus, the total forces acting on the lid and the side cavity walls are bounded as well. Due to the distinctive behavior of the stress near the corner, the resulting nondimensional total forces behave with an unconventional rate Kn ln Kn for small Knudsen numbers Kn.

Probing Turbulent Scattering Effects on Suprathermal Electrons in the Solar Wind: Modeling, Observations, and Implications

This study explores the impact of a turbulent scattering mechanism, akin to those influencing solar and galactic cosmic rays propagating in the interplanetary medium, on the population of suprathermal electrons in the solar wind. We employ a Fokker–Planck equation to model the radial evolution of electron pitch angle distributions under the action of magnetic focusing, which moves the electrons away from isotropy, and of a diffusion process that tends to bring them back to it. We compare the steady-state solutions of this Fokker–Planck equation with data obtained from the Solar Orbiter and Parker Solar Probe missions and find a remarkable agreement, varying the turbulent mean free path as the sole free parameter in our model. The obtained mean free paths are of the order of the astronomical unit, and display weak dependence on electron energy within the 100 eV–1 keV range. This value is notably lower than Coulomb collision estimates but aligns well with observed mean free paths of low-rigidity solar energetic particle events. The strong agreement between our model and observations leads us to conclude that the hypothesis of turbulent scattering at work on electrons at all heliospheric distances is justified. We discuss several implications, notably the existence of a low Knudsen number region at large distances from the Sun, which offers a natural explanation for the presence of an isotropic “halo” component at all distances from the Sun—electrons being isotropized in this distant region before traveling back into the inner part of the interplanetary medium.

Dimensional homogeneity constrained gene expression programming for discovering governing equations

Data-driven discovery of governing equations is of great significance for helping us understand intrinsic mechanisms and build physical models. Recently, numerous highly innovative algorithms have emerged, aimed at inversely discovering the underlying governing equations from data, such as sparse regression-based methods and symbolic regression-based methods. Along this direction, a novel dimensional homogeneity constrained gene expression programming (DHC-GEP) method is proposed in this work. The DHC-GEP simultaneously discovers the forms and coefficients of functions using basic mathematical operators and physical variables, without requiring preassumed candidate functions. The constraint of dimensional homogeneity is capable of filtering out the overfitting equations effectively. The key advantages of DHC-GEP compared with the original gene expression programming, including being more robust to hyperparameters, the noise level and the size of datasets, are demonstrated on two benchmark studies. Furthermore, DHC-GEP is employed to discover the unknown constitutive relations of two representative non-equilibrium flows. Galilean invariance and the second law of thermodynamics are imposed as constraints to enhance the reliability of the discovered constitutive relations. Comparisons, both quantitative and qualitative, indicate that the derived constitutive relations are more accurate than the conventional Burnett equations in a wide range of Knudsen numbers and Mach numbers, and are also applicable to the cases beyond the parameter space of the training data.

Modeling of dual-skin asymmetric hollow fiber microporous membranes for predicting the skins parameters separately using gas permeation test

This study presented a comprehensive gas permeation model to predict the skin parameters of a complex system, specifically a hollow fiber membrane with an asymmetric structure consisting of two external and internal skin layers. According to the work of Haefer for n segments of pores connected in series as in hollow fiber membranes, applying the Wakao et al. model, a comprehensive model for the transition regime was derived. The model was examined against the various fabricated PES and PEI hollow fiber membranes. Unlike conventional models, the proposed method predicted the mean pore size and mean pore length independent of surface porosity for each skin layer. The study revealed valuable insights into i) the middle pressure, ii) the difference between the skin parameters of the two membrane skins, and iii) the impact of the feeding direction on gas permeation behavior. It was shown that the dependency of permeation behavior on the feeding direction is linked to the influence of middle pressure on viscous flow, which is closely related to surface porosities of each side. It is evident that even the slightest variations in the fabrication parameters can result in different structures. Furthermore, it is evident that the assumption of neglecting slip flow was reasonable within the experienced Knudsen number range. The model also highlighted that the skin parameters of the two skin layers may not be identical, a distinction not captured by the current models. The traditional approach of setting the middle pressure as the average of the upstream and downstream pressures has been improved by expressing it as a linear function of both pressures. In the feed side, higher pressures led to dominant viscous flow, whilst the permeate side was more influenced by the free molecular regime due to lower pressures. There existed a competition between the Knudsen flow and the viscous flow at a specific Knudsen number for the feeding side pores, whether it was the inner or outer surface. The finding emphasized the significance of porosity values on each side in determining the proximity of middle pressure to feed and permeate side pressures, underscoring the importance of accurate knowledge of surface porosities, mean pore sizes, and pore lengths for both skin layers in understanding processes involving porous dual-skin hollow fiber membranes.

Rarefied gas flow in functionalized microchannels

The interaction of rarefied gases with functionalized surfaces is of great importance in technical applications such as gas separation membranes and catalysis. To investigate the influence of functionalization and rarefaction on gas flow rate in a defined geometry, pressure-driven gas flow experiments with helium and carbon dioxide through plain and alkyl-functionalized microchannels are performed. The experiments cover Knudsen numbers from 0.01 to 200 and therefore the slip flow regime up to free molecular flow. To minimize the experimental uncertainty which is prevalent in micro flow experiments, a methodology is developed to make optimal use of the measurement data. The results are compared to an analysis-based hydraulic closure model (ACM) predicting rarefied gas flow in straight channels and to numerical solutions of the linearized S-model and BGK kinetic equations. The experimental data shows that if there is a difference between plain and functionalized channels, it is likely obscured by experimental uncertainty. This stands in contrast to previous measurements in smaller geometries and demonstrates that the surface-to-volume ratio of 0.4 μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upmu$$\\end{document}m-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{-1}$$\\end{document} seems to be too small for the functionalization to have a strong influence and highlights the importance of geometric scale for surface effects. These results also shed light on the molecular reflection characteristics described by the TMAC.

Drag coefficient for micron-sized particle in high-speed flows

The drag force on the small particle in high-speed flows is influenced by the combined effects of fluid viscosity, compressibility, and rarefaction. The existing drag coefficient models are still insufficient in accuracy and efficiency for gas-particle flow simulation. This study comprehensively considers these effects and conducts high-fidelity numerical simulations. A new drag coefficient is generated using a symbolic regression method reasonably based on the particle Mach number, Reynolds number, and Knudsen number, which are related to particle diameter, gas-particle relative velocity, and other parameters. The new drag coefficient possesses clear physical significance, high predictive accuracy, low computational cost, and consistency with theory in limiting conditions. The application of the new drag coefficient to three typical gas-solid two-phase flow cases demonstrated its excellent performance.

Unified gas-kinetic wave–particle method for polydisperse gas–solid particle multiphase flow

The gas-particle flow with multiple dispersed solid phases is associated with a complicated multiphase flow dynamics. In this paper, a unified algorithm is proposed for the gas-particle multiphase flow. The gas-kinetic scheme (GKS) is used to simulate the gas phase and the multiscale unified gas-kinetic wave–particle (UGKWP) method is developed for the multiple dispersed solid particle phase. For each disperse solid particle phase, the decomposition of deterministic wave and statistic particle in UGKWP is based on the local cell's Knudsen number. The method for solid particle phase can become the Eulerian fluid approach at the small cell's Knudsen number and the Lagrangian particle approach at the large cell's Knudsen number. This becomes an optimized algorithm for simulating dispersed particle phases with a large variation of Knudsen numbers due to different physical properties of the individual particle phase, such as the particle diameter, material density, etc. The GKS-UGKWP method for gas-particle flow unifies the Eulerian–Eulerian and Eulerian–Lagrangian methods. The particle and wave decompositions for the solid particle phase and their coupled evolution in UGKWP come from the consideration to balance the physical accuracy and numerical efficiency. Two cases of a gas–solid fluidization system, i.e. one circulating fluidized bed and one turbulent fluidized bed, are simulated. The typical flow structures of the fluidized particles are captured, and the time-averaged variables of the flow field agree well with the experimental measurements. In addition, the shock particle–bed interaction is studied by the proposed method, which validates the algorithm for the polydisperse gas-particle system in the highly compressible case, where the dynamic evolution process of the particle cloud is investigated.

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

Hilbert expansion for Coulomb collisional kinetic models

The relativistic Vlasov-Maxwell-Landau (r-VML) system and the relativistic Landau (r-LAN) equation are fundamental models that describe the dynamics of an electron gas. In this paper, we introduce a novel weighted energy method and establish the validity of the Hilbert expansion for the one-species r-VML system and r-LAN equation. As the Knudsen number shrinks to zero, we rigorously demonstrate the relativistic Euler-Maxwell limit and relativistic Euler limit, respectively. This successfully resolves the long-standing open problem regarding the hydrodynamic limits of Landau-type equations.

Nonequilibrium Flow Simulations Using Unified Gas-Kinetic Wave-Particle Method

Nonequilibrium flows are commonly encountered in aerospace engineering applications such as spacecraft reentry, rocket launch, and satellite attitude control. Numerical simulations help greatly in the understanding of nonequilibrium flow, multiscale effects, and the dynamics in these applications. Coupling particle transport and collision within the gas evolution process, the unified gas-kinetic wave-particle (UGKWP) method has been developed for multiscale flow simulation, offering a strategy that strikes a balance between accuracy and efficiency. In the present study, the UGKWP method simulates challenging flow problems with large Knudsen number variations, including supersonic flow around a sphere, hypersonic flow around a space vehicle, nozzle plume into vacuum, and side-jet impingement on the hypersonic flow. The UGKWP accurately captures complex flow structures and has been verified through experimental data or direct simulation Monte Carlo results. Regarding computational cost, the UGKWP method requires only 60 GiB of memory to simulate the three-dimensional space vehicle with 560,593 cells under various flow conditions, which is manageable even on personal workstations. Given its excellent efficiency and accuracy, the UGKWP method shows its substantial benefits in simulating multiscale flow for aerospace engineering applications.

Research on the thermal properties and interface resistance of multilayer multiscale semiconductor structures

The investigation of the thermal characteristics of high-power chips is mainly based on Fourier's law, and it does not adequately explore the microscale effects of the heat transfer process. In pursuit of a more profound comprehension of the size-dependent traits manifesting in the thermal conduction across multilayer, multiscale materials, this manuscript incorporates the phonon transport's scattering mismatch attributes, introducing a MBDE methodology aimed at delineating the thermal characteristics inherent to multilayer, multiscale structures. This approach was applied to elucidate the energy transfer dynamics among bi-layer (Si/Si, Diamond/Si, and Cu/Si) configurations of differing scales. To enhance thermal transfer efficiency, this work prioritizes the minimization of interface thermal resistance and system temperature gradients as optimization objectives, employing a genetic algorithm to identify scales favorable for energy transfer. The results indicate: 1) An increase in the Knudsen number (Kn) correlates with a more pronounced energy discontinuity at the interface, thereby elevating the interface thermal resistance; 2) The disparate material combinations exhibit varied energy transfer efficiencies, where the phononic compatibility between Cu/Si is inferior to that of Diamond/Si, culminating in a notably larger energy discontinuity and, consequently, augmented energy transfer losses for Cu/Si interfaces; 3) Through the application of a genetic algorithm, it was determined that transitioning from Kn1 to Kn2 yields an optimal thermal conduction scenario, characterized by an interface thermal resistance of 1.83 and a comprehensive temperature gradient of 6.69. This work holds significant implications for guiding the thermal design of high-power integrated chips.

Examination of Couette Flow with a Pressure Gradient and Heat Conduction Using Molecular Dynamics Simulation

As computer capabilities improve, Molecular Dynamics simulations are becoming more important for solving various flow problems. In this study, Couette and Poiseuille flows at different wall temperatures were investigated using a hard-sphere Molecular Dynamics simulation approach. Although a low spacing ratio was used in the simulations, the results are valid for rarefied gas flows when proper scaling based on the Knudsen number was used because only binary collisions with a hard-sphere model were considered. The main focus of this study was the examination of the effects of various wall speeds, pressure gradients, and wall temperatures. A pressure gradient was generated by developing a modified selective periodicity condition in the flow direction. With the combined effect of the pressure gradient and the wall velocities, subsonic, transonic, and supersonic speeds in nanochannels were examined. With the combination of different parameters, 1260 simulation cases were conducted. The results showed that there are temperature and velocity slips that are dependent on not only the temperature and velocity values but also on the magnitudes of a pressure gradient. The pressure gradient also caused nonlinearities in temperature and velocity profiles.

Viscous dissipation effects on slip flow heat transfer in rhombic microchannels

This study aimed to numerically investigate the viscous dissipation effect on forced convection in rhombic microchannels for gases in a slip flow regime. The numerical analysis was carried out by assuming a 2D steady-state flow. The solution of governing equations was obtained by adopting the finite element method and assuming that the fluid is Newtonian, with constant thermophysical properties, and in a fully developed laminar flow regime. The solution of the momentum equation is obtained by considering a first-order boundary condition, while the solution of the energy balance equation is obtained by assuming a constant wall heat flux (H2 boundary condition) and taking into account the wall temperature jump. The validation of the numerical model was carried out using the data available in the scientific literature. The numerical outcomes obtained for several values of the acute angle of the rhombus, the Knudsen number (i.e., rarefaction effects), and the Brinkman number (i.e., viscous dissipation effects) reveal that viscous forces play an opposite role with respect to rarefaction and significantly affect the convective heat transfer coefficient, especially for low Knudsen numbers and for high values of the acute angle. In particular, it was observed that Nu is significantly affected by the Brinkman number for acute angles higher than 50°.

Influence of boundary conditions on non-equilibrium heat transport under ultrafast laser action based on the lattice Boltzmann method

Based on the lattice Boltzmann method, this paper simulated the non-equilibrium heat transfer process in nano-silicon thin film under ultrafast laser irradiation. The influence of boundary conditions on heat transport was investigated. Results show that under rebound and diffuse boundary conditions, the energy distribution within the film is non-uniform due to interface effects. However, the specular boundary condition is equivalent to eliminating the interface, resulting in a smooth energy distribution. Under convective boundary conditions, the thermal wave phenomenon disappears owing to the open interface. When the energy tends to be stable, the energy density under convective boundary conditions is reduced to 50% of that under adiabatic boundary conditions. As the film size decreases, the differences between boundary conditions become more significant. Particularly when the film size is smaller than the phonon mean free path, the influence of boundary conditions cannot be neglected. Therefore, boundary conditions and size effects are important for the design of nanodevices. Furthermore, compared with the results obtained by the Cattaneo-Vernotte model, it is found that at smaller Knudsen numbers, the Cattaneo-Vernotte model has a better match with the lattice Boltzmann method.

Study of Two-Fluid Effects in Weakly Ionized Plasma Under Electromagnetic Fields

The multifluid effects caused by the relative motion and interaction between plasma components have a significant effect on electromagnetohydrodynamic wave propagation and the related mass and energy transport of flows. In this work, the charged particle (ion and electron)–neutral two-fluid model is analyzed and employed to study the two-fluid effects (e.g., velocity drift and thermodynamic nonequilibrium between plasma components) of the weakly ionized plasma flows under the electromagnetic fields. In a hypersonic E×B crossed-field configuration acceleration problem, the increased electromagnetic field strength significantly enhances the two-fluid effects. When the load factor equals 3, which measures the strength of the electrode voltage and induced electromotive force, the relative velocity difference between the charged particles and neutrals can reach 28%. The influences of two-fluid effects appreciably increase with the Mach number and Knudsen number but decrease with the degree of ionization. This suggests that, as the vehicle speed and altitude increase, the two-fluid effects become more significant.

Evaluation of a New Droplet Growth Model for Small Droplets in Condensing Steam Flows

As the condensation phenomenon occurs in the low-pressure stages of steam turbines, an accurate modelling of the condensing flows is very crucial and has a significant impact on the development of highly efficient steam turbines. In order to accurately simulate condensing steam flows, it is essential to choose the right condensation model. Further research to enhance condensation models is of special importance because the outcomes of numerical studies of condensation models in recent years have not been entirely compatible with the experiments and there are still uncertainties in this area. Therefore, the main aim of this paper is to evaluate a proposed droplet growth model for modelling condensation phenomenon in condensing steam flows. The new model is derived to profit from the advantages of models based on the continuum approach for large droplets and those based on the kinetic theorem for small droplets, which results in the model being robust for a wide range of Knudsen numbers. The model is implemented into a commercial CFD tool, ANSYS Fluent 2022 R1, using UDFs. The results of the CFD simulations are validated against experimental data for linear cascades within the rotor and stator blade geometries of low-pressure steam turbine stages. The findings clearly demonstrate the superiority of the new model in capturing droplet growth, particularly for very small droplets immediately following nucleation. In contrast, widely used alternative droplet growth models tend to either underpredict or overpredict the droplet growth rate. This research significantly contributes to the ongoing efforts to enhance condensation modeling, providing a more accurate tool for optimizing the design and operation of low-pressure steam turbines, ultimately leading to a higher energy efficiency and a reduced environmental impact.

Measurements of Turbulence in Compressible Low-Density Flows at the Inlet of a Transonic Linear Cascade With and Without Unsteady Wakes

Abstract In the present work, hot-wire anemometry was employed for the characterization of the turbulent field at the inlet of a high-speed low-pressure turbine cascade, in terms of turbulence intensity and integral length scales. This work addresses two major topics relevant to the turbomachinery field: the application of hot-wire anemometry in transonic and rarefied flow regimes and the decoupling of the deterministic and the stochastic fluctuations when measuring unsteady phenomena. In compressible and rarefied flows, a hot-wire is strongly sensitive to both density and velocity fluctuations, and the commonly used Nusselt–Reynolds correlations are not valid. In this article, a nondimensional calibration methodology, based on Nusselt, Reynolds, and Knudsen numbers, was coupled with a sensitivity analysis and employed to postprocess the experimental dataset, allowing to decouple the fluctuations of density and velocity and to compute the turbulence parameters. In the presence of unsteady wakes generated upstream of the cascade, two different phase-locked averaging techniques were employed to distinguish the wake deterministic fluctuations from the background turbulence intensity.

Impact of hydraulic tortuosity on microporous and nanoporous media flow.

Using two-dimensional porous structures made up of homogeneously arranged solid obstacles, we examine the effects of rarefaction on the hydraulic tortuosity in the slip and early transition flow regimes via extended lattice Boltzmann method. We observed that modification in either the obstacle's arrangement or the porosity led to a power-law relation between the porosity-tortuosity. Along with this, we also found that in the slip-flow regime, the exponent of this relation contains the effect of finite Knudsen number (Kn). In addition, we observed that on properly scaling Kn with porosity and hydraulic tortuosity, a generalized correlation can be obtained for apparent permeability.