Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Multicomponent turbulent boundary layer on a chemically active surface

When a gaseous mixture flows past chemically active surfaces the boundary layer formed on the wetted body may contain a large number of components with different diffusion properties. This leads to the necessity for studying the diffusion of the components in the multicomponent boundary layer.The use of thebinary boundary layer concept in the general case cannot yield satisfactory results, since replacement of the mutual diffusion coefficients D_ij of the various pairs of components by a single diffusion coefficient D in many cases is a rough approximation.In the general case the number of different diffusion coefficients is equal to N(N–1)/2 (N is the number of components). Usually it is possible to identify groups of components with similar molecular weights. Then the number of different diffusion coefficients may be reduced without large error. However, even in the comparatively simple case when it is possible to divide all the components into two groups with similar molecular weights we must take account of three different diffusion coefficients (one diffusion coefficient in each group and also the diffusion coefficient for the components of one group relative to the components of the other group). Only in particular cases when the gaseous mixture consists of only two components with arbitrary molecular weights, or if all the components of the gaseous mixture have similar molecular weights, can we with justification introduce a single diffusion coefficient (if in this case there are no limitations on the direction of the diffusion).Studies have been published covering the laminar multicomponent boundary layer. An analytic method for solving the equations of the laminar multicomponent boundary layer was developed by Tirskii [1]. There are also studies in which concrete results were obtained by numerical methods with the use of computers (for example, [2, 3]).As far as the author knows, for turbulent flow there are studies (for example, [4, 5]) covering flow with chemical reactions only in the case when all the diffusion coefficients are equal (D_ij=D).The present paper presents a method for calculating the turbulent multicomponent boundary layer with account for several different diffusion coefficients.Notation x, y coordinates - u, v velocity components - density - T temperature - h heat content - H enthalpy - c_i mass concentration of the i-th component - c ₁ ⁽¹⁾ element concentrations in solid body - J_i diffusion flux of the i-th component - m molecular weight - dynamic viscosity coefficient - kinematic viscosity coefficient - heat conduction coefficient - c_p specific heat - adiabatic index - D_ij binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - S_t Stanton number - M Mach number - friction - q radiant thermal flux - boundary layer thickness - D rate of displacement of gas-solid interface - degree of gasification - r_ij weight fraction of element i in component j - _ij stoichiometric coefficients - K_i reaction equilibrium constants - l number of components for which I_i0 Indices i, j component number - w quantities for y=0 - * quantities on the edge of the laminar sublayer - ⁽¹⁾ quantities at the solid body - quantities at the outer edge of the boundary layer - molar transport coefficients     

Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

Asymptotic theory of the turbulent boundary layer as a problem of singular perturbations

We consider an asymptotic theory of the turbulent boundary layer [1,2]. In this paper we make an attempt to further develop the mathematical aspects of this theory. We demonstrate the features of this theory by applying it to a problem which is close to the so-called equilibrium turbulent boundary layer with a pressure gradient and blowing.Notation x, y coordinates, parallel and perpendicular to the wall - u velocity component in the x direction - p, ',v pressure, density, and kinematic viscosity coefficient - l' scale of turbulence - tangential stress - u speed at the outer edge of the boundary layer - thickness of the boundary layer - * displacement thickness - ** momentum loss thickness - Cf coefficient of friction - R Reynolds number Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 86–95, September–October, 1971.The author thanks S. S. Kutateladze, A. I. Leont'ev, and G. V. Aronovich for their interest in this effort.     

Numerical Study of Hydrodynamics and Heat and Mass Transfer of a Ducted Gas–Vapor‐Droplet Flow

A computational model has been developed to predict heat and mass transfer and hydrodynamic characteristics of a turbulent gas–vapor–droplet flow. Turbulent characteristics of the gas phase are computed using the k– model of turbulence. It is shown that, with increasing inlet droplet diameter, the rate of heat transfer between the duct surface and the vapor–gas mixture decreases appreciably, whereas the wall friction increases only insignificantly. The predicted values agree fairly well with available experimental and numerical data     

Approximate integration op nonstationary equations of a diffusion or thermal boundary layer

An approximate method is proposed for integrating the nonstationary equations of a diffusion or thermal boundary layer using the known steady solution in the planar or axisymmetric case. It is shown that the proposed method is exact in problems involving mass or heat transfer of reacting drops and bubbles in a laminar flow of a viscous incompressible fluid and also particles moving in an ideal fluid. An integral equation is obtained for the local diffusion or heat flux in the case of abrupt activation of a reaction on the surface of a particle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 87–92, September–October, 1982.     

Critical heat fluxes in boiling of liquid nitrogen underheated to saturation point in forced-flow conditions

This paper gives the results of experimental determinations of the critical heat fluxes in the boiling of Liquid nitrogen in forced-flow conditions in the mass velocity range 2 · 10³-40 · 10³ kg/m² · sec, pressure range 29 · 10⁴–245 · 10⁴ N/m², and at underheatings corresponding to the onset of normal boiling crises.Notation q₀ critical heat flux - r heat of vaporization - i enthalpy of flow corresponding to saturation point - i enthalpy of flow corresponding to liquid temperature - surface tension - density of liquid - density of saturated vapor - C _f friction factor - W_g mass velocity - Fr_* Froude number - g acceleration due to gravity     

The effect of ablation products on heat exchange in the disintegration of graphite in a radiating air plasma

In this article we consider the problem of the stationary hypersonic flow of a viscous, heat-conducting stream of radiating high-temperature air past the forward critical point of a blunt body made of graphite in the region between a passing shock wave and the surface of the body.We investigated the radiative and convective heat exchange taking place at an impenetrable surface, as well as such heat exchange when air is injected.We obtained the characteristics of the graphite ablation on the assumption that there is radiative transfer of heat by the graphite-disintegration products.The diffusion was calculated on the basis of a binary model, i.e., it was assumed that the mixture consists of two components: the advancing air and the disintegration products; the chemical reactions in the boundary layer were assumed to be frozen, and on the outer edge of the boundary layer, reaching out as far as the shock wave, they were assumed to be equilibrium reactions.The condition of the gas on the disintegrating surface was also determined on the assumption that there was chemical equilibrium, and the saturation pressure of the vapors was assumed to be equal to the stagnation pressure.It should be noted that the effect of ablation on heat exchange has been considered in [1–3]. Anfimov and Shari [1] assumed that pure air was injected and their calculations were carried out up to stagnation temperatures of 15·10³° K. Hoshisaki and Zasher [2] gave an analysis of a small number of variants, where the values of the absorption cross sections of the injected components used in their calculations were independent of temperature; and the discussion in [3] was limited to the case of low injection velocities.Notation c_p specific heat at constant pressure - c_i mass concentration of the i-th component - D₁₂ coefficient of binary diffusion - f dimensionless stream function - h static enthalpy of the mixture - h_i enthalpy per unit mass of the i-th component - k thermal conductivity - m_w total rate of mass loss through the surface - p stagnation pressure - q heat flux - T absolute temperature - u velocity component along the x axis - v velocity component along the y axis - optical thickness - coefficient of absorption - Lees-Dorodnitsyn self-similar variable - viscosity coefficient - absorption cross section - density - Stefan-Boltzmann constant - wavelength - angle between the normal to the surface and the incident intensity ray Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–31, January–February, 1971.     

Unsteady forced convection heat/mass transfer from a flat plate

Numerical methods are used to investigate the transient, forced convection heat/mass transfer from a finite flat plate to a steady stream of viscous, incompressible fluid. The temperature/concentration inside the plate is considered uniform. The heat/mass balance equations were solved in elliptic cylindrical coordinates by a finite difference implicit ADI method. These solutions span the parameter ranges 10 Re 400 and 0.1 Pr 10. The computations were focused on the influence of the product (aspect ratio) × (volume heat capacity ratio/Henry number) on the heat/mass transfer rate. The occurrence on the plates surface of heat/mass wake phenomena was also studied.     

Nonsimilar Solutions for Mixed Convection in Non‐Newtonian Fluids Along a Vertical Plate in a Porous Medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in powerlaw type nonNewtonian fluids along a vertical plate with powerlaw wall temperature distribution. The mixed convection regime is divided into two regions, namely,the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

Anomalous transport phenomena in rapid external flows of viscoelastic fluids

Summary The concept of an elastic boundary layer is proposed to explain certain anomalous transport phenomena which occur during rapid external flows of viscoelastic fluids past immersed objects. Reported experimental observations are interpreted by using models based on this concept. Particularly, data on velocity independent drag and heat transfer coefficients for flow of dilute polymer solutions past tiny cylinders are satisfactorily correlated.

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Zusammenfassung Es wird das Konzept einer elastischen Grenzschicht entworfen, um gewisse anomale Transportphänomene zu erklären, welche bei schnellen Strömungen viskoelastischer Flüssigkeiten um eingetauchte Körper auftreten. Die berichteten experimentellen Beobachtungen werden mit Hilfe von Modellen interpretiert, die auf diesem Konzept basieren. Insbesondere werden Daten über geschwindigkeitsunabhängige Widerstands- und Wärmeübertragungs-Koeffizienten bei der Strömung verdünnter Polymerlösungen um dünne Zylinder befriedigend korreliert.

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A, B numerical constants - A ₁,A ₂ surface areas - C _D drag coefficient - D cylinder diameter - F hoop force - h heat transfer coefficient - k thermal conductivity - M molecular weight - Nu Nusselt number - R gas constant - T absolute temperature - u x-component of the velocity - U free stream velocity - x, y Cartesian coordinates - shear rate - boundary layer thickness - ₀ elastic boundary layer thickness - relaxation time - µ viscosity - v kinematic viscosity - [] intrinsic viscosity - density - normal stress difference - shear stress With 3 figures     

Thermal free convection from a solid sphere

Summary Thermal free convection from a sphere has been studied by melting solid benzene spheres in excess liquid benzene (Pr=8,3; 10⁸<Gr<10⁹). Overall heat transfer as well as local heat transfer were investigated. For the effect of cold liquid produced by the melting a correction has been applied. Results are compared with those obtained by other workers who used alternative experimental methods.Nomenclature coefficient of heat transfer - d characteristic length, here diameter of sphere - thermal conductivity - g acceleration of free fall - cubic expansion coefficient - T temperature difference between wall and fluid at infinity - kinematic viscosity - density - c specific heat capacity - a thermal diffusivity (=/c) - D diffusion coefficient - Nu dimensionless Nusselt number (=d/) - Nu* the analogous number for mass transfer (=kd/D) - mean value of Nusselt number - Gr dimensionless Grashof number (=gd ³T/ ²) - Gr* the analogous number for mass transfer (=gd ³x/ ²) - Pr dimensionless Prandtl number (=/a) - Sc dimensionless Schmidt number (=/D)     

Determination of heat fluxes in the neighborhood of a double curvature stagnation point in a flow of dissociating gas of arbitrary chemical composition

Numerical solutions are obtained for the equations of a uniform compressible boundary layer with variable physical properties in the vicinity of a stagnation point with different principal curvatures in the presence of an injected gas with the same properties as the incident flow. The results of the numerical solutions are approximated for the heat flux in the form of a relation that depends on the variation of the product of viscosity and density across the boundary layer and on the ratio of the principal radii of curvature.Using the concepts of effective diffusion coefficients in a multicomponent boundary layer, previously introduced by the author in [1], and the generalized analogy between heat and mass transfer in the presence of injection, together with the numerical solutions obtained, it is always possible, even without additional solutions of the boundary-layer equations, to derive final formulas for the heat fluxes in a flow of dissociating gas of arbitrary chemical composition, provided that we make the fundamental assumption that all recombination reactions take place at the surface.By way of example, formulas are given for the heat transfer to the surface of a body from dissociating air, regarded as a five-component mixture of the gases O, N, NO, O₂, N₂, and from a dissociating mixture of carbon dioxide and molecular nitrogen of arbitrary composition, regarded as an eleven-component mixture of the gases O, N, C, NO, C₂, O₂, N₂, CO, CN, C₃, CO₂.In the process of obtaining and analyzing these solutions it was found that, in computing the heat flux, a multicomponent mixture can be replaced with an effective binary mixture with a single diffusion coefficient only when the former can be divided into two groups of components with different (but similar) diffusion properties. In this case the concentrations of one group at the surface must be zero, while the diffusion flows of the second group at the surface are expressible, using the laws of mass conservation of the chemical elements, in terms of the diffusion flows of the first. Then the single effective diffusion coefficient is the binary diffusion coefficient D(A,M), where A relates to one group of components and M to the other.In view of the small amount of NO(c(NO) < 0.05), the diffusion transport of energy in dissociated air maybe described with the aid of a single binary diffusion coefficient D(A, M)(A=O, N, M=O₂, N₂, NO). However even in the case of complete dissociation into O and C atoms at the outer edge of the boundary layer, the diffusion transport of energy in dissociated carbon dioxide can not be described accurately enough by means of a model of a binary mixture with a single diffusion coefficient, since the diffusion properties of the O and C atoms are distinctly different.     

Heat transfer in a thin layer of a viscous heavy noncompressible liquid under wavy flow conditions

The article describes a method for calculating the flow of heat through a wavy boundary separating a layer of liquid from a layer of gas, under the assumption that the viscosity and heat-transfer coefficients are constant, and that a constant temperature of the fixed wall and a constant temperature of the gas flow are given. A study is made of the equations of motion and thermal conductivity (without taking the dissipation energy into account) in the approximations of the theory of the boundary layer; the left-hand sides of these equations are replaced by their averaged values over the layer. These equations, after linearization, are used to determine the velocity and temperature distributions. The qualitative aspect of heat transfer in a thin layer of viscous liquid, under regular-wavy flow conditions, is examined. Particular attention is paid to the effect of the surface tension coefficient on the flow of heat through the interface.Notation x, y coordinates of a liquid particle - t time - v and u coordinates of the velocity vector of the liquid - p pressure in the liquid - c_v, , T,, andv heat capacity, thermal conductivity coefficient, temperature, density, and viscosity of the liquid, respectively - g acceleration due to gravity - surface-tension coefficient - c phase velocity of the waves at the interface - T_w wall temperature - h₀ thickness of the liquid layer - u₀ velocity of the liquid over the layer Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 147–151, July–August, 1970.     

Generalization of the experimental data on extended crisis in heat transfer in the boiling of a liquid

A generalized formula is given for the critical heat flux, and it is shown that crises of this type are most characteristic of the boiling of organic liquids at high temperatures.Notation q_* critical heat flux - q heat flux - W mean flow speed of liquid in crisis section; - W_g mass flow rate - r latent heat of evaporation - coefficient of surface tension - -@#@ density of dry saturated vapor - density of liquid on saturation line - i enthalpy of liquid on saturation line - i mean enthalpy of liquid in crisis cross section - cf coefficient of friction - g acceleration due to gravity - P static pressure in crisis cross section - T saturation temperature - T_* temperature of surface of tube - mean density of liquid in crisis cross section I am indebted to I. N. Svorkova for assistance.I am also indebted to S. S. Kutateladze and A. I. Leont'ev for discussions and valuable comments.     

On hydromagnetic fluctuating flow past an infinite porous plate in slip flow regime

The effect of a uniform external magnetic field on the laminar, incompressible rarefied gas flow along an infinite porous flat plate is studied under the following conditions: 1) there is uniform suction, 2) the external flow velocity varies periodically with time in magnitude but not in direction, 3) the magnetic Reynolds number is small and 4) the current occurs under slip flow boundary conditions. Expressions for the velocity and temperature fields in the boundary layer are obtained. The response of skin friction, and heat transfer to the fluctuating stream is studied for variations in the rarefaction parameter h ₁, the magnetic field parameter M, and the frequency of the fluctuating stream.Nomenclature c _p specific heat of the gas - f ₁ Maxwells reflection coefficient - f ₂ thermal accommodation coefficient - G as defined in (36) - h ₁ rarefaction parameter (L ₁ v ₀/) - h ₂ nondimensional temperature jump coefficient (L ₂ v ₀/) - H amplitude of the skin friction - k thermal conductivity - K _n Knudsen number - L mean free path - L ₁ (2–f ₁/f ₁) L - L ₂ - M magnetic field parameter ( ₀ B ₀ ² /v ₀ ² ) - m 1/2[1+(1+4M+4i)^1/2], m _r+im _i - n ₁ 1/2[1+(1+4M)^1/2] - q heat flux - R suction Reynolds number - T temperature - x, y coordinates along and perpendicular to the plates - u, v velocity components along x, y-directions - density - kinematic viscosity - ₀ electrical conductivity - Prandtl number - frequency of the fluctuating stream - nondimensional frequency parameter (/v ₀ ² ) - nondimensional distance from wall (v ₀ y/) - phase lead - U _{0 0} mean velocity in the boundary layer - U _{0 1}, U _{0 2} amplitude of the velocity fluctuation in the boundary layer - specific heat ratio     

Influence of vorticity on the heat transfer to blunt bodies in hypersonic flight

When blunt bodies are in hypersonic flight, a high-entropy layer of gas with nonzero vorticity is formed near their surface. The transverse gradients of the entropy, density, and gas velocity in the layer are high, which makes it necessary to take into account its absorption by the boundary layer of finite thickness . This vortex interaction is usually accompanied by an increase in the heat flux q and the frictional stress on the wall compared with their values as calculated in accordance with the classical scheme of a thin boundary layer, when the parameters on the outer edge of the boundary layer are set equal to the inviscid parameters on the body. This effect has been investigated on the side surface of slender (with angle 1 to the undisturbed flow) blunt bodies in a hypersonic stream [1–3]. It is shown in the present paper that the effect can have a stronger and even qualitative influence on the flow over blunt bodies with 1 if the radius of curvature R_s of the detached shock wave on the axis is small compared with the midsection radius R of the body. It is shown that the distributions of the heat fluxes with allowance for the vorticity of the inviscid shock layer are similar in the case of slightly blunt (r₀/R 0) cones with half-angles less than a critical _*.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 50–57, March–April, 1981.     

Experimentelle Untersuchung des Einflusses der Nebelbildung auf den Wärme- und Stofftransport an einer senkrechten gekühlten Platte bei freier Konvektion

Zusammenfassung Experimentell untersucht wurde der Einfluß der Bildung von Nebel innerhalb der Grenzschicht auf den Wärme- und Stofftransport an einer senkrechten gekühlten Platte in feuchter Luft bei freier Konvektion. Gemessen wurde das Temperaturfeld, Wärme- und Stoffübergangsraten sowie die Dicke und Struktur der Nebelschicht. Die Bildung von Nebel steigert den Wärmetransport an die Wand und behindert den Stofftransport erheblich.

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Experimental investigation of the influence of fog formation on the free convective heat and mass transfer at a vertical cooled plate

The influence of fog formation within the boundary layer on free convective heat and mass transfer at a vertical cooled plate in humid air was studied experimentally. Temperature field, heat and mass transfer rates as well as thickness and structure of the fog layer were measured. Caused by fog formation, heat transfer at the wall is increased and mass transfer is decreased considerably.

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Formelzeichen c Massenkonzentration - c _{, rel} relative Dampfkonzentration in der Umgebung - d Dicke - Gr_x örtliche Grashofzahl - L Modellänge - m Massenstromdichte - n Brechungsindex - n dn/dy Brechzahlgradient - Nu_x örtliche Nusseltzahl - p Druck - q Wärrnestromdichte - r spezifische Refraktivität - R spezielle Gaskonstante - S Streifenordnung - t Zeit - T Temperatur - x, y, z Ortskoordinaten - _T Wärmeübergangskoeffizient - _Hg Lichtwellenlänge - Dichte Indices D Dampf - L Luft - RF Reif - Tr Tropfen, Nebel - W an der Wand - in der Umgebung Herrn Prof. Dr.-Ing. U. Grigull zum 75. Geburtstag gewidmet     

Some refinements concerning the boundary conditions at the macroscopic level

The study of boundary effects initiated in a previous paper is continued. New assumptions regarding the geometrical structure of the boundary surface are introduced. Under these assumptions, it is shown that macroscopic Neumann conditions do not generally affect the determination of the macroscopic field in the case of the transport process considered — heat conduction. For this type of boundary condition, the boundary effect is generally confined within a thin layer near the boundary. When heat sources are taken into account within the porous domain, the result is different. In this case, making use of a Neumann boundary condition, expressed in terms of macroscopic variables, amounts to introducing an extra flux. Under normal circumstances, however, this additional flux is negligible.Roman Letters A cross-sectional area of a unit cell - A _e cross-sectional area of a unit cell at the boundary surface - A _sf interfacial area of the s-f interface contained within the averaging volume - surface area per unit volume (A _sf/ ) - A _sf interfacial area of the s-f interface contained within the macroscopic system - g closure vector - h closure vector - k heat transfer coefficient at the s-f interface - K_eff effective thermal conductivity tensor - _x unit cell length - n unit vector - n_e outwardly directed unit normal vector at the boundary - n_sf outwardly directed unit normal vector for thes-phase at f-s interface - q heat flux density - T ^* macroscopic temperature defined by the macroscopic problem - s closure variable - V volume of the macroscopic system - V boundary surface of the macroscopic domain - V ₁ macroscopic sub-surface of the boundary surface - x local coordinate Greek Letters _s,_f volume fraction - _s, gl_f microscopic thermal conductivities - true microscopic temperature - ^* microscopic temperature corresponding toT ^* - microscopic error temperature - vector defined by Equation (34) - < > spatial average