Knudsen Layer Research Articles

Kinetic multiscale scheme based on the discrete-velocity and lattice-Boltzmann methods

A novel hybrid computational method based on the discrete-velocity (DV) approximation, including the lattice-Boltzmann (LB) technique, is proposed. Numerical schemes for the kinetic equations are used in regions of rarefied flows, and LB schemes are employed in continuum flow zones. The schemes are written under the finite-volume (FV) formulation to achieve the flexibility of local mesh refinement. The truncated Hermite polynomial expansion is used for matching of DV and LB solutions. Special attention is paid to preserving conservation properties in the coupling algorithm. The test results obtained for the Couette flow of a rarefied gas are in excellent agreement with the benchmark solutions, mostly thanks to mesh refinement (both in the physical and velocity spaces) in the Knudsen layer.

Calculation the efficiency of energy transfer from nanosecond CO2-laser pulses to the iron target during plasma appearance

The steel target evaporation and the gas-dynamic processes were calculated, thenfollowed by the plasma creation on in the metal vapor under the influence of nanosecond pulses were calculated. Mathematical model based on the Euler equations for the inviscid gas flow in conjunction statement with the radiation transfer equation in the diffusion multi-group approximation and the heat equation in solid metal was formulated. Nonlinear evaporation, depending on the energy entering to the target surface, was calculated using analytical expressions for the Knudsen layer. The structures of gas-dynamic flows with the formation of shock waves in an ambient gas (air) in the active vaporization mode and during plasma formation are were studied, also the thermal state of the target is was determined in the one-dimensional formulation. The comparison of the efficiency of energy transfer to the target is was made in both cases, : from laser radiation directly and when plasma appearsappeared, by means of thermal radiation. The significant increase in the adsorption coefficient of energy in the case of the appearance of plasma is shown.

Transport phenomena in the Knudsen layer near an evaporating surface.

Using the combination of the kinetic theory of gases (KTG), Boltzmann transport equation (BTE), and molecular dynamics (MD) simulations, we study the transport phenomena in the Knudsen layer near a planar evaporating surface. The MD simulation is first used to validate the assumption regarding the anisotropic velocity distribution of vapor molecules in the Knudsen layer. Based on this assumption, we use the KTG to formulate the temperature and density of vapor at the evaporating surface as a function of the evaporation rate and the mass accommodation coefficient (MAC), and we use these vapor properties as the boundary conditions to find the solution to the BTE for the anisotropic vapor flow in the Knudsen layer. From the study of the evaporation into a vacuum, we show the ratio of the macroscopic speed of vapor to the most probable thermal speed of vapor molecules in the flow direction will always reach the maximum value of sqrt[1.5] at the vacuum boundary. The BTE solutions predict that the maximum evaporation flux from a liquid surface at a given temperature depends on both the MAC and the distance between the evaporating surface and the vacuum boundary. From the study of the evaporation and condensation between two parallel plates, we show the BTE solutions give good predictions of transport phenomena in both the anisotropic vapor flow within the Knudsen layer and the isotropic flow out of the Knudsen layer. All the predictions from the BTE are verified by the MD simulation results.

Lifetime of a Nanodroplet: Kinetic Effects and Regime Transitions.

A transition from a d^{2} to a d law is observed in molecular dynamics (MD) simulations when the diameter (d) of an evaporating droplet reduces to the order of the vapor's mean free path; this cannot be explained by classical theory. This Letter shows that the d law can be predicted within the Navier-Stokes-Fourier (NSF) paradigm if a temperature-jump boundary condition derived from kinetic theory is utilized. The results from this model agree with those from MD in terms of the total lifetime, droplet radius, and temperature, while the classical d^{2} law underpredicts the lifetime of the droplet by a factor of 2. Theories beyond NSF are also employed in order to investigate vapor rarefaction effects within the Knudsen layer adjacent to the interface.

Implementation of Knudsen Layer Phenomena in Rarefied High-Speed Gas Flows

AbstractA numerical investigation of Knudsen layer effects in high-speed flows in a rarefied flow regime has been carried out. The conventional compressible flow computational fluid dynamics (CFD) ...

A numerical method for coupling the BGK model and Euler equations through the linearized Knudsen layer

The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is small. In practice, however, Knudsen layers emerge at the physical boundary, or at the interfaces between the two regimes. We model the Knudsen layer using a half-space kinetic equation, and apply a half-space numerical solver [19,20] to quantify the transition between the kinetic to the fluid regime. A full domain numerical solver is developed with a domain-decomposition approach, where we apply the Euler solver and kinetic solver on the appropriate subdomains and connect them via the half-space solver. In the nonlinear case, linearization is performed upon local Maxwellian. Despite the lack of analytical support, the numerical evidence nevertheless demonstrate that the linearization approach is promising.

Comprehensive assessment of newly-developed slip-jump boundary conditions in high-speed rarefied gas flow simulations

In this paper we numerically evaluate the recently developed Aoki et al. slip and jump conditions in high-speed rarefied gas flows for the first time. These slip and jump conditions are developed to be employed with the Navier–Stokes–Fourier equations. They were derived based on the Boltzmann equation with the first order Chapman–Enskog solution, and the analysis of the Knudsen layer. Four aerodynamic configurations are selected for a comprehensive evaluation of these conditions such as sharp-leading-edge flat plate, vertical plate, wedge and circular cylinder in cross-flow with the Knudsen number varying from 0.004 to 0.07, and argon as the working gas. The simulation results using the Aoki et al. boundary conditions show suitable agreement with the DSMC data for slip velocity and surface gas temperature. The accuracy of these boundary conditions is superior to the conventional Maxwell, Smoluchowski and Le boundary conditions.

Boundary slip phenomena in multicomponent gas mixtures

The slip phenomena in multicomponent gas mixtures are studied. The moment equations, derived from the linearized Boltzmann equation by using the 13-moment approximation of the Grad’s method, are used to obtain the full (containing the contribution from the Knudsen layer) and asymptotic (valid far from the plane physical boundary of the domain) expressions for the nonequilibrium macroscopic parameters of the mixture species. The latter relations are employed to deduce the expressions for the mixture slip velocity and the viscous, thermal, and diffusion slip coefficients by using the modified Maxwell method and the diffuse-specular model of molecule scattering on the wall. The derived relations for the slip coefficients are given in the convenient form expressed in terms of basic transport coefficients, such as partial viscosity and thermal conductivity coefficients, diffusion and thermal diffusion coefficients, and coefficients of molecule momentum accommodation at the wall. The expressions found are used to calculate the slip coefficients for binary (He-Ar) and ternary (He-Ar-Xe) gas mixtures. The numerical results are in good agreement with the data calculated by using other methods.

Numerical analysis of the cathodic material influence on the arc plasma jet

The cathodic arc discharge is a deposition technique widely used to synthesize hard coatings and thin films. The structure of the plasma generated by the electrical discharge and its interaction with neutral particles was studied using numerical simulations. Typical plasma parameters were characterized considering their spatial and temporal dependence, as well as several cathode materials that are commonly used in these systems. For the evolution of the ion density, it was observed the formation of Knudsen layer, and also a dependence of pressure gradients in the global behavior. With respect to the kinetic energy, it was found a deceleration of ions, which is represented by a shock front produced in the plasma−neutrals interaction. On the other hand, the energy releasing was generated due to the heat transference between electrons and ions. The plasma potential follows a behavior, which is similar to that of the ion density, and it is caused by the dynamics of charged particles which is directly affected by the concentration of neutrals and ions. In general, the physical quantities are directly affected by electrical and thermal conductivity of the cathode material. Our results can be applied to understand the plasma phenomena produced in a cathodic arc discharge

Characteristics of thermal transpiration effect and the hydrogen flow behaviors in the microchannel with semicircular obstacle

The thermal transpiration effect has great potential applications for the hydrogen energy. In this paper, the thermal transpiration effect and the hydrogen flow behaviors are studied in the microchannel with the semicircular obstacles. Firstly, the slip boundary model is used in the simulation of the flow performance in the microchannel. The validity of the model at different Kn is verified by comparing with some previous work. Then, the hydrogen flow characteristics of the thermal transpiration effect with the semicircular obstacle are investigated. The result shows that as the size of the semicircular obstacle increases, the hydrogen flow path of the thermal transpiration effect becomes longer, and the temperature gradient decreases. As the characteristic length of the hydrogen flow decreases, there is an obviously negative influence on the thermal transpiration flow. A deeper analysis shows that the thermal driven flow and the pressure driven flow will produce y-component velocity, which leads to a backflow under the effect of semicircles, and the semicircular obstacles make the Knudsen layer spread to the channel center.

Barodiffusion in Slow Flows of a Gas Mixture

Barodiffusion in slow flows of a gas mixture is studied with an approximation using hydrodynamic equations of motion for the individual mixture components. It is shown that consideration of the viscous momentum transfer and the contribution of Knudsen layers for the mixture flowing in a channel has a considerable effect on the value of the barodiffusion factor. The relations are obtained for the mean diffusion fluxes of components and for the total flux of the mixture in a circular cylindrical capillary; these relations are valid for moderately small Knudsen numbers used for calculation of the diffusion baroeffect and separation effect when the gas mixture flows in a set of capillaries connecting two volumes. The modification of the relations for the barodiffusion factor (and for the diffusion slip coefficient cross-linked with it) allows interpreting the sign alteration of these effects observed experimentally for some gas mixtures at intermediate Knudsen numbers.

Effect of Knudsen Layer on the heat transfer in hypersonic rarefied gas flows

Effect of the Knudsen layer on the surface heat transfer is investigated for the hypersonic flow applications in the slip and early transition flow regime. Knudsen Layer formulation is incorporated within an open source computational fluid dynamics framework, OpenFOAM. Navier-Stokes-Fourier constitutive relations, and first order slip and jump boundary conditions are modified based on the effective mean free path, which is dependent on the geometry of an obstacle and local flow gradients. The modified solver is validated against benchmark cases of low-speed Couette gas flow and hypersonic flow over a flat plate. Investigation is further extended to the high speed flow over a wedge and a circular cylinder covering a wide range of Knudsen numbers within the slip and early transition flow regime. Effect of Knudsen layer on the surface heat transfer is examined, and maximum improvement of 27% is achieved for a wedge case. Moreover, results show excellent agreement with the DSMC data, especially for the cylinder cases with high Knudsen numbers, Kn = 0.05 and 0.25. The simulation results convey that our approach greatly improves the predictive capabilities of the compressible N-S-F solver up to early transition flow regime (Kn ∼ 1). The current work is significant from the perspective of accurate thermal design of hypersonic vehicles.

Effect of the microchannel obstacles on the pressure performance and flow behaviors of the hydrogen Knudsen compressor

The hydrogen Knudsen compressor has potential applications on the hydrogen transmission for the microdevices and systems. In this paper, the numerical model of the hydrogen Knudsen compressor was established, combining the NS continuity equations with the slip boundary conditions. The effect of structures on the performance of the hydrogen Knudsen compressor is studied by generating different obstacles in the microchannels. This paper is mainly concerned on the rectangular and the triangular obstacles, and the influence of the obstacles length and height are investigated, respectively. The Knudsen number distribution and the rarefaction of the hydrogen gas flow are analyzed. Also, the characteristic of the pressure increase for the compressor under different parameters are investigated and discussed. The effect of the structure parameters on the flow velocity distributions are detailed described, as well as the velocity contour and the vortex distributions. Moreover, the variation of the Knudsen layers of the hydrogen gas flow in the hydrogen Knudsen compressor is presented, and the key factor of the Knudsen layers is analyzed and discussed. The results is significantly beneficial for the applications and designs of hydrogen Knudsen compressor.

The general form of transport diffusivity of shale gas in organic-rich nano-slits—A molecular simulation study using Darken approximation

As a potential substitute for the gradually depleted conventional oil and gas resources, shale gas is attracting increasing attention worldwide. Because of the ubiquitous nanopores in shale matrix, studying diffusion behavior of shale gas in nano-scale channels is crucial for its development, which also lays the foundation for the optimization of hydraulic fracturing, horizontal well spacing, etc. This paper focuses on shale gas diffusion behavior in organic-rich nano-slits and establishes a calculation method for generalized transport diffusivity using Darken approximation by molecular simulation (MS). The main conclusions are as follows. (1) For the whole gas, compared to its self-diffusivity which decreases monotonically with pressure, there is a break point for its transport diffusivity in each slit. It’s found that the sequence of magnitudes of transport diffusivities in different pores at low pressures and that at high pressures are contrary. (2) For the bulk district (BD) in 2 nm slit, the self- and transport diffusivities don’t differ much at low pressures and their discrepancy enlarges at high pressures, the deviation of which can be 2.34 times at 41 MPa. (3) The transport diffusivity of the molecules in the Knudsen layer (KL) in 2 nm slit is always larger (1.21–3.76 times) than their self-diffusivity and shows fluctuations at large pressures, other than the self-diffusivity in KL which decreases monotonically with pressure. Furthermore, the deficiencies and directions for improvements of the kinetic self-diffusion model and Knudsen diffusion model are analyzed by the data comparison and the theory background on their derivation and expressions. It is expected that this work will give new insights on the flow capacity of shale gas in nano-scale pores and contribute to a deeper understanding for shale gas evaluation and exploitation.

A unified model for coupling mesoscopic dynamics of keyhole, metal vapor, arc plasma, and weld pool in laser-arc hybrid welding

Although building a model of laser-arc hybrid welding has attracted substantial attention in recent decades, many problems remain. In particular, existing models failed to reproduce the highly transient (down to several tens of nanoseconds) physical interactions between solid, liquid, vapor plume and plasma occurring in the welding process. These interactions control the key physical processes of laser-arc hybrid welding and play dominant roles in process defect formation and final joint quality. Development of a unified model for laser-arc hybrid welding capable of modeling the multiphase physical interactions is faced with three major challenges. The first is how to decouple the highly correlated and mesoscopic interactions between the solid, liquid, vapor plume and plasma phases. The second and third are respectively the numerical treatments of the nonlinear discontinuous sheath layer between the arc plasma and the weld pool and the Knudsen layer between the vapor plume and the weld pool. In this study, we have developed a three-dimensional unified mathematical model that couples the mesoscopic dynamics of the arc plasma, keyhole, metal vapor, and weld pool in the laser-TIG hybrid welding process. To decouple the highly correlated gas, liquid, solid and plasma interactions, we proposed a novel approach in which the electromagnetic field in the workpiece (including solid and liquid phases) and outside the workpiece (gaseous phase and plasma) were continuously modeled globally, but in which the heat transfer and fluid flow are modeled separately within and outside of the workpiece and then coupled through boundary conditions. To treat the nonlinear interface between the arc plasma and weld pool outside the keyhole as well as between the vapor plume/plasma and the weld pool inside the keyhole, we proposed a novel ghost-fluid-based multiple timescale stepping algorithm. The results indicate that our model can be used to predict the time-dependent mesoscopic dynamics of vapor plume, plasma, keyhole and weld pool in a coupled manner. Good agreement was obtained between the numerical predictions and experimental results and literature data.

Kinetic simulation of the non-equilibrium effects at the liquid-vapor interface

Phase change phenomena at microscale is important for novel cooling microsystems with intensive evaporation, so the development of reliable models and simulations are challenging. The vapor behaviors near its condensed phase are simulated using the non-linear S-model kinetic equation. The pressure and temperature jumps obtained numerically are in good agreement with the analytical expressions derived from the appropriate Onsager-Casimir reciprocity relations. The results of the evaporation flux are close to those given by the Hertz-Knudsen-Schrage formula, only when the values of the pressure and temperature at the upper boundary of the Knudsen layer are used. Comparison with recently measured temperature jumps are provided and disagreement with some experiments are discussed.

Modeling of Knudsen Layer Effects in the Micro-Scale Backward-Facing Step in the Slip Flow Regime

The effect of the Knudsen layer in the thermal micro-scale gas flows has been investigated. The effective mean free path model has been implemented in the open source computational fluid dynamics (CFD) code, to extend its applicability up to slip and early transition flow regime. The conventional Navier-Stokes constitutive relations and the first-order non-equilibrium boundary conditions are modified based on the effective mean free path, which depends on the distance from the solid surface. The predictive capability of the standard ‘Maxwell velocity slip—Smoluchwoski temperature jump’ and hybrid boundary conditions ‘Langmuir Maxwell velocity slip—Langmuir Smoluchwoski temperature jump’ in conjunction with the Knudsen layer formulation has been evaluated in the present work. Simulations are carried out over a nano-/micro-scale backward facing step geometry in which flow experiences adverse pressure gradient, separation and re-attachment. Results are validated against the direct simulation Monte Carlo (DSMC) data, and have shown significant improvement over the existing CFD solvers. Non-equilibrium effects on the velocity and temperature of gas on the surface of the backward facing step channel are studied by varying the flow Knudsen number, inlet flow temperature, and wall temperature. Results show that the modified solver with hybrid Langmuir based boundary conditions gives the best predictions when the Knudsen layer is incorporated, and the standard Maxwell-Smoluchowski can accurately capture momentum and the thermal Knudsen layer when the temperature of the wall is higher than the fluid flow.

Resolving Knudsen layer by high-order moment expansion

We model the Knudsen layer in Kramers’ problem by the linearized high-order hyperbolic moment system. Thanks to the hyperbolicity of the moment system, its boundary conditions are properly reduced from the kinetic boundary condition. For the Kramers’ problem, we present the analytical solutions of the linearized moment systems. The velocity profile in the Knudsen layer is captured with improved accuracy for a wide range of accommodation coefficients. With the order of the moment system increasing, the velocity profile approaches to that of the linearized Boltzmann–BGK equation.

О бародиффузии при медленных течениях газовой смеси

AbstractBarodiffusion in slow flows of a gas mixture is studied with an approximation using hydrodynamic equations of motion for the individual mixture components. It is shown that consideration of the viscous momentum transfer and the contribution of Knudsen layers for the mixture flowing in a channel has a considerable effect on the value of the barodiffusion factor. The relations are obtained for the mean diffusion fluxes of components and for the total flux of the mixture in a circular cylindrical capillary; these relations are valid for moderately small Knudsen numbers used for calculation of the diffusion baroeffect and separation effect when the gas mixture flows in a set of capillaries connecting two volumes. The modification of the relations for the barodiffusion factor (and for the diffusion slip coefficient cross-linked with it) allows interpreting the sign alteration of these effects observed experimentally for some gas mixtures at intermediate Knudsen numbers.

Diffusive tunneling in an isobaric but non-isothermal fuel-pusher mixture

The hydrodynamic mix of fusion fuel and inert pusher can simultaneously generate smaller fuel pockets and finer pusher layers that separate them. Smaller fuel pockets have greater local Knudsen numbers, which tend to exacerbate the Knudsen layer reactivity reduction. A thinner pusher layer separating the neighboring fuel pockets, on the other hand, can enable the diffusive tunneling of Gamow fuel ions through the pusher layer and hence alleviate the Knudsen layer reactivity degradation. Here, the diffusive tunneling phenomenon describes a random walk process by which the Gamow fuel ions from one fuel pocket can traverse the inert pusher layer to join a neighboring fuel pocket without losing much of their energy. This is made possible by the much slower collisional slowing down rate compared with the pitch angle scattering rate of light fuel ions with heavier pusher ions. In an isobaric target mixture where fuel and pusher segments can have distinct temperatures, due to their different compressibilities, the temperature effect on the critical pusher layer areal density below which diffusive tunneling can occur, which is a property of the hydrodynamic mix, is understood by computing the ion charge state distribution using a collisional radiative model. This information is fed into the collisionality evaluation, enabling a parametric scan of the diffusive tunneling physics in terms of the target pressure, fuel, and pusher temperatures. It is found that when the gold pusher layer has a temperature above 1 keV, the variation of the pusher temperature has little effect on the critical areal mass density below which diffusive tunneling can occur. If the pusher layer is 1 keV or below, the critical areal mass density rises sharply, indicating that for a stronger fuel-pusher temperature disparity, the onset of diffusive tunneling will be at an earlier stage of the hydrodynamic mix when the fuel-pusher mixing structures are of less reduced size.