Kinetic Boundary Layer Research Articles

Kinetic boundary layers for the Boltzmann equation on discrete velocity lattices

We consider families of discrete velocity models with physical collision invariants, and develop algebraic criteria for well-posedness of the linearized kinetic boundary layer problem. Using the obtained criteria we discuss various hierarchies of symmetric discrete velocity models, and calculate analytical and numerical slip coefficients for the representants of the hierarchies.

The Boltzmann equation and its fluid dynamic limits: Numerical studies

AbstractWe introduce a Boltzmann equation on discrete lattices and demonstrate its applicability for the numerical simulation of flows in the transition regime to fluid dynamics. The application concerns an evaporation condensation where the fluid dynamic flow is ruled by a thin kinetic boundary layer. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Water diffusion through a membrane protein channel: A first passage time approach

Water diffusion through OmpF, a porin in the outer membrane of Escherichia coli, is studied by molecular dynamics simulation. A first passage time approach allows characterizing the diffusive properties of a well-defined region of this channel. A carbon nanotube, which is considerably more homogeneous, serves as a model to validate the methodology. Here we find, in addition to the expected regular behavior, a gradient of the diffusion coefficient at the channel ends, witness of the transition from confinement in the channel to bulk behavior in the connected reservoirs. Moreover, we observe the effect of a kinetic boundary layer, which is the counterpart of the initial ballistic regime in a mean square displacement analysis. The overall diffusive behavior of water in OmpF shows remarkable similarity with that in a homogeneous channel. However, a small fraction of the water molecules appears to be trapped by the protein wall for considerable lengths of time. The distribution of trapping times exhibits a broad power law distribution psi(tau) approximately tau (-2.4), up to tau=10 ns, a bound set by the length of the simulation run. We discuss the effect of this distribution on the dynamic properties of water in OmpF in terms of incomplete sampling of phase space.

Planetary boundary layer energetics simulated from a regional climate model over Europe for present climate and climate change conditions

This paper presents a description of the planetary boundary layer (PBL) for current (1960–1990) and future (2070–2100) climate periods as obtained from a regional climate model (RCM) centered on the Mediterranean basin. Vertically integrated turbulent kinetic energy (TKEZ) and boundary layer height (zi) are used to describe PBL energetics. Present climate shows a TKEZ annual cycle with a clear summer maximum for southern regions, while northern regions of Europe exhibit a smoother or even a lack of cycle. Future climate conditions exhibit a similar behaviour, with an increase in the summer maximum peaks. A detailed analysis of summer surface climate change energetics over land shows an increased Bowen ratio and decreases in the evaporative fraction. The enhanced sensible heat flux responsible for these results causes an energy surplus inside the PBL, resulting in increased convective activity and corresponding TKEZ. These results are consistent with temperature increases obtained by several other model simulations, and also indicate that changes in the turbulent transport from the PBL to the free troposphere can affect atmospheric circulations.

Validation of a Second-Order Slip Model for Dilute Gas Flows

We present an extensive comparison between direct Monte Carlo simulations and the predictions of the Navier-Stokes description coupled to a recently proposed second-order slip model for hard sphere gases. Two one-dimensional, time-dependent channel flows are considered. In both cases excellent agreement is found between molecular simulation and the proposed model well into the transition regime for both the velocity and stress fields. The excellent quantitative agreement extends, approximately, to a Knudsen number based on the channel width of 0.4. The transient nature of the flows suggests that the slip model, despite its (quasi-) steady origins, remains reliable as long as the evolution timescale is long compared to the molecular collision time. Our discussion elucidates the effect of the Knudsen layer, a kinetic “boundary layer” in the vicinity of walls that needs to be superposed to the Navier-Stokes component of the flow in order to obtain the complete flow solution. The existence of this kinetic boundary layer, whose importance grows with increasing Knudsen number and which ultimately renders the Navier-Stokes component insufficient to describe the flow, implies that special care is needed in interpreting slip-flow flowfields.

Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection.

The Prandtl and Rayleigh number dependences of the Reynolds number in turbulent thermal convection following from the unifying theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000); Phys. Rev. Lett. 86, 3316 (2001)] are presented and compared with various recent experimental findings. This dependence Re(Ra,Pr) is more complicated than a simple global power law. For Pr=5.5 and 10(8)<Ra<10(10) the effective or local power law exponent of Re as a function of Ra is definitely less than 0.50, namely, Re approximately Ra(0.45), in agreement with Qiu and Tong's experimental findings [Phys. Rev. E 64, 036304 (2001)]. We also calculated the kinetic boundary layer width. Both in magnitude and in Ra scaling it is consistent with the data.

Scaling in thermal convection: a unifying theory

A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Bénard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [lsim ] 1011) the leading terms are Nu ∼ Ra1/4Pr1/8, Re ∼ Ra1/2Pr−3/4 for Pr [lsim ] 1 and Nu ∼ Ra1/4Pr−1/12, Re ∼ Ra1/2Pr−5/6 for Pr [gsim ] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ∼ Ra1/2Pr1/2, Re ∼ Ra1/2Pr−1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu ∼ Ra1/5Pr1/5, Re ∼ Ra2/5Pr−3/5 for small Pr, a regime with Nu ∼ Ra1/3, Re ∼ Ra4/9Pr−2/3 for larger Pr, and a regime with scaling Nu ∼ Ra3/7Pr−1/7, Re ∼ Ra4/7Pr−6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.

A constructive approach to steady nonlinear kinetic equations

We study steady boundary value problems of nonlinear kinetic theory. Using a continuation argument based on the variation of the Knudsen number we derive a method for the construction of steady solutions of discrete velocity models in a slab. This method is readily transformed into a numerical code. In a preliminary numerical test case the numerical scheme turns out to yield solutions even for Knudsen numbers small enough to calculate with high precision the asymptotic flow field adjacent to a kinetic boundary layer. Thus, we are able to numerically simulate in a simplified situation the transition from a (mesoscopic) kinetic boundary layer to the (macroscopic) far field.

Role of vegetation in generation of mesoscale circulation

A soil-vegetation module is incorporated into a two-dimensional mesoscale model for simulating mesoscale circulations that develop due to changes in surface characteristics. The model was verified by evaluating the diurnal changes of heat fluxes, surface temperature, soil moisture and soil water content with different vegetation covers using a one-dimensional version. Thermally induced mesoscale circulations between vegetated and bare soil areas are simulated with the two-dimensional model using three different types of bare soil adjacent to the vegetated area. The properties of the vegetation breeze, which are similar to that of a sea breeze, are investigated. The intensity of the vegetation breeze circulation is directly related to the characteristics of the bare soil. There is a strong relationship between surface fluxes and the intensity of the vegetation breeze circulation. More soil moisture is transferred to the atmosphere over the vegetated area than over the bare soil area. The effect of vegetation on planetary boundary layer (PBL) structure is presented by comparing the differences of turbulent kinetic energy (TKE), eddy diffusivity and boundary layer height between vegetated area and bare soil area. The effect of bare soil properties on PBL structure is also described.

Kinetic boundary layers in gas mixtures: Systems described by nonlinearly coupled kinetic and hydrodynamic equations and applications to droplet condensation and evaporation

We consider a mixture of heavy vapor molecules and a light carrier gas surrounding a liquid droplet. The vapor is described by a variant of the Klein-Kramers equation, a kinetic equation for Brownian particles moving in a spatially inhomogeneous background; the gas is described by the Navier-Stokes equations; the droplet acts as a heat source due to the released heat of condensation. The exchange of momentum and energy between the constituents of the mixture is taken into account by force terms in the kinetic equation and source terms in the Navier-Stokes equations. These are chosen to obtain maximal agreement with the irreversible thermodynamics of a gas mixture. The structure of the kinetic boundary layer around the sphere is then determined from the self-consistent solution of this set of coupled equations with appropriate boundary conditions at the surface of the sphere. For this purpose the kinetic equation is rewritten as a set of coupled moment equations. A complete set of solutions of these moment equations is constructed by numerical integration inward from the region far away from the droplet, where the background inhomogeneities are small. A technique developed in an earlier paper is used to deal with the severe numerical instability of the moment equations. The solutions so obtained for given temperature and pressure profiles in the gas are then combined linearly in such a way that they obey the boundary conditions at the droplet surface; from this solution source terms for the Navier-Stokes equation of the gas are constructed and used to determine improved temperature and pressure profiles for the background gas. For not too large temperature differences between the droplet and the gas at infinity, self-consistency is reached after a few iterations. The method is applied to the condensation of droplets from a supersaturated vapor, where small but significant corrections to an earlier, not fully consistent version of the theory are found, as well as to strong evaporation of droplets under the influence of an external heat source, where corrections of up to 40 % are obtained.

The structure of the stationary kinetic boundary layer for the linear BGK equation

Recently, very accurate numerical methods for solving kinetic boundary layer problems for linear kinetic equations have been developed. To test such methods, detailed information about exactly solvable models can be very helpful. To this purpose the authors present results for the stationary, 1D linear BGK equation, obtained by the singular eigenfunction method, that extend results already known in the literature, both in scope and in precision. Special attention is paid to the nature of the singularity in the distribution function for small velocities near the boundary; this singularity cannot be reproduced exactly by the numerical methods available. Explicit expressions are presented for the Milne problem and for the albedo problem with input particles of a single velocity. They compare the results with those of a recently developed variant of the two-stream moment method. They find excellent agreement, except close to the singularity; for many quantities of interest, the accuracy obtainable by the novel two-stream moment method cannot be reproduced with comparable numerical effort by evaluation of the exact solution.

An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations

The authors develop a method for solving 1D kinetic boundary layer problems for linear equations that uses separate Hermite moment expansions in the velocity variable for particles moving towards and away from a plane wall. This so-called two-stream method is tested for two especially simple kinetic equations, the linear BGK equation and the Klein-Kramers equation, the kinetic equation for Brownian particles. For both of these equations, extensive exact information is available for two simple boundary layer problems, the Milne and the albedo problem. A comparison of these exact results with those obtained with this version of the two-stream moment method shows that the accuracy obtainable by this method exceeds that of earlier methods by several orders of magnitude. In particular, the method allows them to obtain accurate results not only for the moments of the distribution function, but also for the distribution function itself.

The influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.

The analytic structure of the stationary kinetic boundary layer for Brownian particles near an absorbing wall

A method for solving kinetic boundary layer problems for the (1D) Klein-Kramers equation, the kinetic equation for Brownian particles, was proposed a few years ago by Marshall and Watson (1987) for the special case of particles moving under the influence of a constant or zero external force. In this paper the authors apply this method to a number of classical stationary boundary layer problems for completely or partially absorbing walls. These are the Milne problem, in which a current flows towards the wall from infinity, and the albedo problem, in which particles are injected into the system at the wall with a prescribed velocity distribution. The solutions are known to be non-analytic at the wall for zero normal velocity; the authors pay particular attention to the nature of this singularity for several special cases. The results are compared with results obtained by approximate methods.

Heat transfer between a small sphere and a dilute gas; the influence of the kinetic boundary layer

The transfer of heat between an object not too large compared to a mean free path and a gas surrounding it is influenced considerably by the structure of the kinetic boundary layer around the object. We calculate this effect for a spherical object in a dilute gas by applying a recently developed variant of the moment method to solve the stationary linearized Boltzmann equation for the gas surrounding the sphere, From the solution we determine the temperature jump coefficient, which occurs in the boundary condition to be used for the heat conduction equation at the surface of the sphere. We study the dependence of this quantity on the radius and on the thermal accomodation coefficient. We find that typical boundary layer effects become less important as the accomodation coefficient decreases, and propose a simple approximate formula, which describes the results for large spheres to within about half a percent.

The kinetic boundary layer for the linearized Boltzmann equation around an absorbing sphere

We construct approximate solutions of the linearized Boltzmann equation for a gas outside of a completely absorbing sphere, a simple model for a liquid droplet growing in a supersaturated vapor. The solutions are linear combinations of two Chapman-Enskog-type solutions, which carry heat and particle currents, and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path. To construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere, we expand the solution in Burnett functions and truncate the resulting system of equations for the expansion coefficients. For one particular truncation prescription, which generalizes Grad's 13-moment scheme, good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere; the results for small radii extrapolate smoothly toward the known limit of zero radius. We present results for the reaction rate (the particle current arriving at the sphere divided by the density at infinity) and for the density and temperature profiles in the boundary layer. The explicit calculations are carried out for Maxwell molecules, but the method appears to be suitable for more general intermolecular potentials.

The kinetic boundary layer around an absorbing sphere and the growth of small droplets

Deviations from the classical Smoluchowski expression for the growth rate of a droplet in a supersaturated vapor can be expected when the droplet radius is not large compared to the mean free path of a vapor molecule. The growth rate then depends significantly on the structure of the kinetic boundary layer around a sphere. We consider this kinetic boundary layer for a dilute system of Brownian particles. For this system a large class of boundary layer problems for a planar wall have been solved. We show how the spherical boundary layer can be treated by a perturbation expansion in the reciprocal droplet radius. In each order one has to solve a finite number ofplanar boundary layer problems. The first two corrections to the planar problem are calculated explicitly. For radii down to about two velocity persistence lengths (the analog of the mean free path for a Brownian particle) the successive approximations for the growth rate agree to within a few percent. A reasonable estimate of the growth rate for all radii can be obtained by extrapolating toward the exactly known value at zero radius. Kinetic boundary layer effects increase the time needed for growth from 0 to 10 (or 2 1/2) velocity persistence lengths by roughly 35% (or 175%).

On the approximation of diffusion memory functions by time correlation functions in inhomogeneous systems

Approximations commonly used to relate memory functions to time correlation functions governed by Newtonian dynamics are shown to be incorrect for strongly inhomogeneous systems. An exact expression for the space dependent diffusion coefficient is given in terms of time correlation functions whose dynamics are governed by Newton’s equations. The resulting expression is evaluated using both the Bhatnagar–Gross–Krook kinetic equation and the Fokker–Planck equation to calculate the required time correlation functions for a system containing a specularly reflecting wall and for a system having a harmonic potential. For inhomogeneous systems, the memory function may not be approximated by the corresponding time correlation function. In addition, certain velocity correlations near a specularly reflecting wall are shown to contain a t−1/2 long time tail and a kinetic boundary layer whose thickness grows as t1/2; these features are absent from the memory function.

Kinetic boundary layer treatment in the problem of a rarefied gas flow through a perforated wall by the method of moments

Gas flow from one chamber to another through a plane wall perforated with cylindrical channels is investigated. Gas flow close to the surface of a wall is studied by the methods of kinetic gas theory. The molecule velocity distribution function is found from the Boltzmann equation as a result of its solution by the method of moments. Intermolecular collisions in the channels are supposed to be absent. Molecules collide in the Knudsen layer near the perforated wall. This leads to the formation of a uniform flow far from the wall.

Navier–Stokes boundary conditions of a gas mixture

The Navier–Stokes velocity and density boundary conditions of a multicomponent gas mixture in front of a wall are considered. The following gas–wall interactions are taken into account: partially inelastic interactions between the gas molecules and the wall (accommodation), conversion among the different molecule species (ionization, etc.), and gas flow through the wall. Since the influence of the kinetic boundary layer on the gas continuum is negligible, the moments in the boundary conditions for the Navier–Stokes equations are calculated with the first-order Chapman–Enskog solution (Maxwell’s method).