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.     

A kinetic model for droplet growth in the transition regime

A variant of the moment expansion method, used in an earlier paper to describe the flow of a gas toward an absorbing sphere, is applied to a more realistic model of a droplet condensing from a supersaturated vapor. In the simplest version a spherical droplet absorbs all incoming vapor molecules, but spontaneously emits molecules with a Maxwellian distribution at the droplet temperature and with the corresponding saturated vapor density. From a solution of the stationary linearized Boltzmann equation with these boundary conditions we obtain expressions for the heat and mass currents toward the sphere as a function of the supersaturation and the temperature difference between the droplet and the vapor at infinity. For small droplet radii the known free flow limit is obtained in a natural way. From the calculated expressions for the heat and mass current we derive evolution equations for the radius and temperature of the droplet. The temperature evolves more rapidly and can thus be eliminated adiabatically; the resulting growth curve for the radius shows a sharp transition from a kinetically controlled regime for small radii to a regime dominated by heat conduction for large radii. The effect of incomplete absorption at the surface is also studied. The actual calculations are carried out for Maxwell molecules, with parameters corresponding to argon at 0.65T _c and 100% supersaturation.     

Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation

Rarefied gas flow behavior is usually described by the Boltzmann equation, the Navier-Stokes system being valid when the gas is less rarefied. Slip boundary conditions for the Navier-Stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level (Boltzmann equation). These slip conditions are explicitly written in terms of asymptotic behavior of some linear half-space problems. The validity of this analysis is established in the simple case of the Couette flow, for which it is proved that the right boundary conditions are obtained.     

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 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%).     

Diffusive Limit of a Kinetic Model for Cometary Flows

A kinetic equation with a relaxation time model for wave-particle collisions is considered. Similarly to the BGK-model of gas dynamics, it involves a projection onto the set of equilibrium distributions, nonlinearly dependent on the moments of the distribution function. Under a diffusive and low Mach number scaling the macroscopic limit is a generalization of the incompressible Navier-Stokes equations, where the momentum equations are coupled to a diffusive equation for an energy distribution function. By a moment approximation, this system can be related to a low Mach number model of fluid mechanics, which already appeared in the literature. Finally, for a linearized version corresponding to Stokes flow an existence result for initial value problems is proved.     

Modeling evaporation effects in conditional moment closure for spray autoignition

Simulations of an n-heptane spray autoigniting under conditions relevant to a diesel engine are performed using two-dimensional, first-order conditional moment closure (CMC) with full treatment of spray terms in the mixture fraction variance and CMC equations. The conditional evaporation term in the CMC equations is closed assuming interphase exchange to occur at the droplet saturation mixture fraction values only. Modeling of the unclosed terms in the mixture fraction variance equation is done accordingly. Comparison with experimental data for a range of ambient oxygen concentrations shows that the ignition delay is overpredicted. The trend of increasing ignition delay with decreasing oxygen concentration, however, is correctly captured. Good agreement is found between the computed and measured flame lift-off height for all conditions investigated. Analysis of source terms in the CMC temperature equation reveals that a convective–reactive balance sets in at the flame base, with spatial diffusion terms being important, but not as important as in lifted jet flames in cold air. Inclusion of droplet terms in the governing equations is found to affect the mixture fraction variance field in the region where evaporation is the strongest, and to slightly increase the ignition delay time due to the cooling associated with the evaporation. Both flame propagation and stabilization mechanisms, however, remain unaffected.     

Thermal explosion in a combustible gas containing fuel droplets

An original physical model of self-ignition in a combustible gas mixture containing liquid fuel droplets is developed. The droplets are small enough for the gas-droplet mixture to be considered as a fine mist such that individual droplet burning is subsumed into a well-stirred, spatially invariant burning approximation. A classical Semenov-type analysis is used to describe the exothermic reaction, and the endothermic terms involve the use of quasi-steady mass transfer/heat balance and the Clausius-Clapeyron evaporative law. The resulting analysis predicts the ignition delay which is a function of the system parameters. Results are given for typical dynamical regimes. The case of different initial temperatures for droplets and gas is highly relevant to gas turbine lean blow-out and re-ignition.     

Thermodynamics of nonstationary and transient effects in a relativistic gas

We show how a system of generalized Fourier and Navier-Stokes equations, containing relaxation terms and couplings between heat flow and viscosity, can be consistently derived from phenomenological thermodynamics and from kinetic theory. The coefficients are given explicitly for a relativistic Boltzmann gas.     

Fluid dynamic limits of kinetic equations. I. Formal derivations

The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog. The moment formalism shows that the limit leading to the incompressible Navier-Stokes equations, like that leading to the compressible Euler equations, is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible Navier-Stokes equations. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution of the classical Boltzmann equation and the Leray solution of the Navier-Stokes equations is discussed.This paper is dedicated to Joel Lebowitz on his 60th-birthday.     

Determination of temperature and concentration of a vapor–gas mixture in a wake of water droplets moving through combustion products

Characteristic temperatures and concentrations of a vapor–gas mixture in a wake of water droplets moving through combustion products (initial temperature 1170 K) were determined using the Ansys Fluent mathematical modeling package. We investigated two variants of motion: motion of two droplets (with sizes from 1 mm to 3 mm), consecutive and parallel, and motion of five staggered droplets. The influence of the relative position of droplets and also of distances between them (varied from 0.01 mm to 5 mm) on temperatures and concentrations of water vapor was established. The distances determine the relation between the evaporation areas and the total volume occupied by a droplet aggregate in the gas medium. The results of modeling for conditions that take into account vaporization on the droplet surface at average constant values of evaporation rate and also with consideration of the change in the latter, depending on the droplet temperature field, are compared. We determined conditions under which the modeling results are comparable for the assumption of a constant vaporization rate and with regard to the dependence of the latter on temperature. The earlier hypothesis on formation of a buffer vapor layer (“thermal protection”) around a droplet, which decreases the thermal flow from the external gas medium, was validated.     

Continuity and momentum equations for moist atmospheres

The moist atmosphere with occurring precipitation is considered to be a multiphase fluid composed of dry air, water vapor and hydrometeors. These compositions move with different velocities: they take a macroscopic motion with a reference velocity and a relative motion with a velocity deviated from the reference velocity. The reference velocity can be chosen as the velocities of dry air, a gas mixture and the total air mixture. The budget equations of continuity and momentum are formulated in the three reference-velocity frames. It is shown that the resulting equations are dependent on the chosen reference velocity. The diffusive flux due to compositions moving with velocities deviated from the reference velocity and the internal sources due to the phase transitions of water substances result in additional source terms in continuity and momentum equations. A continuity equation of the total mass is conserved and free of diffusive flux divergence if the reference velocity is referred to the velocity of the total air mixture. However, continuity equations in the dry-air and gasmixture frames are not conserved due to the mass diffusive flux divergence. The diffusive flux introduces additional source terms in the momentum equation. In the dry-air frame, the diffusive flux of water substances and the phase transitions of water substances contribute to the change of the total momentum. The additional sources of total momentum in the frame of a gas mixture are associated with the diffusive flux of hydrometeors, the phase transitions of hydrometeors and the gasmixture diffusive flux. In the frame of total air mixture, the contribution to the total momentum comes from the diffusive flux of all atmospheric compositions instead of the phase transitions. The continuity and momentum equations derived here are more complicated than the traditional model equations. With increasing computing power, it becomes possible to simulate atmospheric processes with these sophisticated equations. It is helpful to the improvement of precipitation forecast.     

Vaporization and combustion in three-dimensional droplet arrays

The quasi-steady vaporization and combustion of multiple-droplet arrays is studied numerically. Utilizing the Shvab–Zeldovich formulation, a transformation of the governing equations to a three-dimensional Laplace’s equation is performed, and the solution to Laplace’s equation is obtained numerically to find the effects of droplet interactions in symmetric, multiple-droplet arrays. Vaporization rates, flame surface shapes, and flame locations are found for different droplet array configurations and fuels. The number of droplets, the droplet arrangement within the arrays, and the droplet spacing within the arrays are varied to determine the effects of these parameters. Computations are performed for uniformly spaced three-dimensional arrays of up to 216 droplets, with center-to-center spacing ranging from 3 to 25 droplet radii. As a result of the droplet interactions, the number of droplets and relative droplet spacing significantly affect the vaporization rate of individual droplets within the array, and consequently the flame shape and location. For small droplet spacing, the individual droplet vaporization rate decreases below that obtained for an isolated droplet by several orders of magnitude. A similarity parameter which correlates vaporization rates with array size and spacing is identified. Individual droplet flames, internal group combustion, and external group combustion can be observed depending on the droplet geometry and boundary conditions.     

Force multipole moments for a spherically symmetric particle in solution

We present a general theorem for the force multipole moments of arbitrary order induced in a spherically symmetric particle immersed in a fluid whose motion satisfies the linear Navier-Stokes equation for steady incompressible viscous flow. The multipole moments are expressed in terms of the unperturbed fluid velocity field. It is shown that for a particle with a finite extension only a few terms give rise to fluid perturbations which are not confined to the interior of the particle. We give explicit results for a polymer satisfying the Debye-Bueche-Brinkman equations and for a hard sphere with mixed slip-stick boundary conditions.     

Acoustic-wave reflection from a gas−droplet–mixture boundary

The problem on acoustic-wave reflection from the boundary of a gas?droplet mixture is considered. For the interface between the pure gas and the gas?droplet mixture at a wave inclined incidence, a nonmonotonic dependence of the reflectance on the volume content of droplets is established. The values of the critical wave-incidence angle and the volume content of inclusions at which the reflectance vanishes are found and illustrated. The possibility of occurrence of the lowest reflectance in dependence on the frequency of perturbations in a certain interval of the wave-incidence angle related mainly to the difference in the densities of the gas-droplet mixture and the pure gas is shown.     

The non-linear Enskog-Boltzmann equation

Some implications of a modified form of the non-linear Enskog equation for a hard sphere fluid are investigated. It leads to different Navier-Stokes equations in a multicomponent mixture, and its linearized form describes the space and time dependence fluctuations.     

Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space.¶ II. Construction of the Navier-Stokes Solution

This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem. Received: 5 September 1996 / Accepted: 14 July 1997     

Modeling the Water Droplet Evaporation Processes with Regard to Convection,Conduction and Thermal Radiation

A predictive model was developed for investigation of high-temperature heating and evaporation of water droplets. The model takes into account the basic interrelated processes of heat transfer and phase transitions. Typical velocity and temperature profiles were found in the high-temperature gas–water droplet system with external gas medium temperature varied from 100 to 800°C. Various formulations of the problem, significantly different in the type of considered processes and factors, are considered.We analyzed temperature conditions of heating and evaporation of water droplets, which allow the use of simplified models and which need consideration of all complex interrelated processes of heat and mass transfer (including convection, conduction and radiant heat transfer in droplets, and also in the surface vapor–gas layer).