Emissivities and spectral absorptivities of combustion products

This paper is concerned with the determination of the spectral absorptivities K and emissivities of a thermodynamic-equilibrium mixture of combustion products consisting of CO₂, H₂O, CO, OH, HCl, HF, H₂, and NO molecules on the temperature interval 2000–5000° K. The proposed calculation method, based on the use of a graph of the molecular absorptivities reduced to atmospheric pressure, enables K and to be calculated for any gas mixture composed of the above-mentioned molecules.     

Application of dispersion theory to time domain reflectometry in soils

With time domain reflectometry (TDR) two dispersive parameters, the dielectric constant, , and the electrical conductivity, can be measured. Both parameters are nonlinear functions of the volume fractions in soil. Because the volume function of water ( _w) can change widely in the same soil, empirical equations have been derived to describe these relations. In this paper, a theoretical model is proposed based upon the theory of dispersive behaviour. This is compared with the empirical equations. The agreement between the empirical and theoretical aproaches was highly significant: the ( _w) relation of Topp et al. had a coefficient of determination r ² = 0.996 and the (_u) relation of Smith and Tice, for the unfrozen water content, _u, at temperatures below 0°C, had an r ² = 0.997. To obtain ( _w) relations, calibration measurements were performed on two soils: Caledon sand and Guelph silt loam. For both soils, an r ² = 0.983 was obtained between the theoretical model and the measured values. The correct relations are especially important at low water contents, where the interaction between water molecules and soil particles is strong.     

Generation of vortices in a partially filled,rapidly rotating cylinder

We describe a system in which vortices are shed from a cylindrical free surface approximately centered in a rotating flow. Shedding is controlled by the parameter =2 g/ ² d, where g, , d denote gravity, rotation rate and the diameter of the free surface. We find vortex shedding for >0.162 and no vortex shedding for < 0.0847. The range depends on the aspect ratio L/d, where L is the column length, in a nonmonotonic fashion. These results are independent of viscosity and surface tension for small values of these parameters.Now at Martin Marietta, Orlando Aerospace, PO Box 5837, Mail Point 150, Orlando, FL 32855, USA     

Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation

For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu _tt +u _t –u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA inH ₀ ¹ () ×L ²(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA ₀ inH ₀ ¹ () ×L ²(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA ₀ andA .     

Modellvorstellung zur turbulenten Filmkondensation strömenden Dampfes

Zusammenfassung Der Wärmeübergang bei turbulenter Film kondensation strömenden Dampfes an einer waagerechten ebenen Platte wurde mit Hilfe der Analogie zwischen Impuls-und Wärmeaustausch untersucht. Zur Beschreibung des Impulsaustausches im Film wurde ein Vierbereichmodell vorgestellt. Nach diesem Modell wird die wellige Phasengrenze als starre rauhe Wand angesehen. Die Abhängigkeit einer Schubspannungs-Nusseltzahl von der Film-Reynoldszahl und Prandtlzahl wurde berechnet und dargestellt.

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A model for turbulent film condensation of flowing vapour

The heat transfer in turbulent film condensation of flowing vapour on a horizontal flat plate was investigated by means of the analogy between momentum and heat transfer. To describe the momentum transfer in the film a four-region model was presented. With this model the wavy interfacial surface is treated as a stiff rough wall. A shear Nusselt number has been calculated and represented as a function of film Reynolds number and Prandtl number.

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Formelzeichen a Temperaturleitkoeffizient - k Mischungswegkonstante - k _s äquivalente Sandkornrauhigkeit - Nu _x lokale Schubspannungs-Nusseltzahl,Nu _x=x_xv/uw - Pr Prandtlzahl,Pr=v/a - Pr _t turbulente Prandtlzahl,Pr _t =_m/_q - q Wärmestromdichte _q - R Wärmeübergangswiderstand - R_f Wärmeübergangswiderstand des Films - Re _F Reynoldszahl der Filmströmung - T Temperatur - U, V Geschwindigkeitskomponenten des Dampfes in waagerechter und senkrechter Richtung - u, Geschwindigkeitskomponenten des Kondensats in waagerechter und senkrechter Richtung - V Querschwankungsgeschwindigkeit des Kondensats und des Dampfes - u _/gtD Schubspannungsgeschwindigkeit an der Phasengrenze für die Dampfgrenzschicht, uD =(/)^1/2 - u _F Schubspannungsgeschwindigkeit an der Phasengrenze für den Kondensatfilm,u _F =(/)^1/2 - u _w Schubspannungsgeschwindigkeit an der Wand der Kühlplatte,u _w =(w/)^1/2 - y Wandabstand - x Wärmeübergangskoeffizient - gemittelte Kondensatfilmdicke - _s Dicke der zähen Schicht der Filmströmung an der welligen Phasengrenze - ₄ Dicke der zähen Schicht der Filmströmung an der gemittelten glatten Phasengrenze - Wärmeleitzahl - dynamische Viskosität - v kinematische Viskosität - Dichte - Oberflächenspannung - _w Wandschubspannung - Schubspannung an der Phasengrenzfläche - _m turbulente Impulsaustauschgröße - _q turbulente Wärmeaustauschgröße Indizes d Wert des Dampfes - w Wert an der Wand - x lokaler Wert inx - Wert an der Phasengrenze Stoffgrößen ohne Index gelten für das Kondensat     

A class of nonlinear,completely integrable abstract wave equations

IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u+Lu+u _H ² u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u) are relatively compact inD(L ^1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball B=(u ₀,u ₀ )D(L ^1/2)×H; L^1/2 u _{0 H} ² +u ² < for sufficiently small.     

Zur Variationsformulierung nichtlinearer Randwertprobleme

Übersicht Es werden verschiedene Bedingungen aufgestellt, die es erlauben, die durch die beiden (Systeme von) nichtlinearen DifferentialgleichungenA (u, ) = q, B (u, ) = und Randbedingungen zusammen mit den nichtlinearen algebraischen Relationenq = C(u, ), = D(u, ) beschriebene Aufgabe durch äquivalente Variationsprobleme zu ersetzen. Dabei zeigt sich ein enger Zusammenhang mit den in der Festkörpermechanik wohlbekannten Prinzipien der virtuellen Verschiebungen und der virtuellen Kräfte. Die auf systematischem Weg konstruierten Variationsfunktionale enthalten viele in der Physik bekannte Funktionale als Sonderfälle, insbesondere jene, die in der Elastomechanik nach Green, Castigliano, Hellinger, Reißner, Hu und Washizu benannt werden.

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Summary In this paper there are established various conditions which allow a variational formulation of the problem described by the two (systems of) nonlinear differential equationsA(u, ) = q, B(u, ) = and boundary conditions together with the nonlinear algebraic relationsq = C(u, ), = D(u, ). Besides a close relationship is revealed to the principles of virtual displacements and virtual forces which are wellknown in solid mechanics. The systematically constructed variational functional contain many functionals in physics as special cases, mainly those of Green, Castigliano, Hellinger, Reißner, Hu and Washizu in elastomechanics.

     

Differential interferometry for studying hypersonic flows

The purpose of this article is to describe an optical technique based on differential interferometry with strongly phase-shifted beams using a white light source and a Wollaston prism. This technique is recommonded particularly for measuring very small index variations. It has been used for analyzing hypersonic flows around slender axisymmetrical bodies. The radial gas density distributions obtained in the shock layers were compared with the analytical solutions developed by Merlen and Andriamanalina (1992) and with Jones' tabulated computations (1969).List of Symbols m exponent of the obstacle's power law - R, R radii of the shock and of the obstacle, respectively - R _c radius of curvature of the spherical mirror - r radial coordinate - L obstacle length - L _m distance from model to spherical mirror - x, y cartesian coordinates with origin at obstacle nose - geometric angle of incidence - birefringence angle of the Wollaston prism, =() - wavelength - relative thickness of the obstacle - _c cone apex semi-angle - Y distance between the two partial beams at the level of the test section - n refractive index of the medium - E optical thickness - e test section width - _y light deviation along the y axis - h length of the path traveled by one of the two beams through the shock layer - gas density - _s gas density under standard conditions - freestream gas density - _min minimum detectable phase difference     

On a generalized nonlinearK-ɛ model for turbulence that models relaxation effects

In this paper, based on a similarity that exists between the constitutive relations for turbulent mean flow of a Newtonian fluid and that for the laminar flow of a non-Newtonian fluid, and making use of extended thermodynamics, we develop a generalized nonlinearK- model, whose approximate form includes the standardK- model and the nonlinearK- model of Speziale (1987) as special cases. Our nonlinearK- model, which is frame indifferent, can predict relaxation of the Reynolds stress, unlike most standardK- models. Also, our model is in keeping with that of Yakhotet al. (1992). Most interestingly, the linearized form of our model bears a striking resemblance to the model due to Yoshizawa and Nisizima (1993); however, it has been obtained from a totally different perspective.     

Particle mixing and diffusion in the turbulent wake of cylinder arrays

The collection of 10 DOP droplets from an aerosol flow around an array of five circular cylinders in a turbulent air stream was measured. Axial and angled alignments of the target array, normal to the flow, were examined. Cylinder Reynolds numbers ranged from 1,300 to 5,100, and effective Stokes numbers from 0.50 to 1.58. A continuous line source turbulent diffusion model was used with the centerline velocity and concentration distributions to derive the eddy diffusion coefficients for momentum and particle concentration in the near wake of the cyinders. Angled collection results showed an optimal target alignment at the edge of the momentum wake of the lead cylinder. Further examination with an LDA system suggests the peak in collection is due to a particle velocity surplus in the wake.List of symbols A _k projected area of cylinder k - A _t cross sectional area of flow - b wake width - C particle concentration - C ₁ concentration deficit - C _D.n drag coefficient of nth cylinder - d diameter - vortex shedding frequency - L _c length of cylinder - n cylinder rank - Q strength of line source - R _L Lagrangian time autocorrelation coefficient - Re Reynolds number (U d/) - T _L Lagrangian integral time scale - U x-directed velocity - U ₁ velocity deficit - u fluctuating axial velocity component - v fluctuating lateral velocity component - V drop volume - x intercylinder spacing - y lateral displacement - mixing length proportionality constant - A measured droplet diameter - eddy diffusivity - ^* corrected collection efficiency - contact angle of droplet on surface - viscosity of air - density - time - non-Stokesian drag correction factor - c cylinder - g gas - M momentum - p particles - theoretical (Stokes efficiency) - aneth freestream value - 1, n cylinder rank 1 or n     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

Thin-Walled Beams: the Case of the Rectangular Cross-Section

In this paper we present an asymptotic analysis of the three-dimensional problem for a thin linearly elastic cantilever =×(0,l) with rectangular cross-section of sides and ², as goes to zero. Under suitable assumptions on the given loads, we show that the three-dimensional problem converges in a variational sense to the classical one-dimensional model for extension, flexure and torsion of thin-walled beams. Mathematics Subject Classifications (2000) 474K20, 74B10, 49J45.     

Asymptotic analysis of equilibrium states for rotating turbulent flows

The equilibrium states of homogeneous turbulence simultaneously subjected to a mean velocity gradient and a rotation are examined by using asymptotic analysis. The present work is concerned with the asymptotic behavior of quantities such as the turbulent kinetic energy and its dissipation rate associated with the fixed point (/kS)=0, whereS is the shear rate. The classical form of the model transport equation for (Hanjalic and Launder, 1972) is used. The present analysis shows that, asymptotically, the turbulent kinetic energy (a) undergoes a power-law decay with time for (P/)<1, (b) is independent of time for (P/)=1, (c) undergoes a power-law growth with time for 1<(P/)<(C ₂–1), and (d) is represented by an exponential law versus time for (P/)=(C ₂–1)/(C ₁–1) and (/kS)>0 whereP is the production rate. For the commonly used second-order models the equilibrium solutions forP/,II, andIII (whereII andIII are respectively the second and third invariants of the anisotropy tensor) depend on the rotation number when (P/kS)=(/kS)=0. The variation of (P/kS) andII versusR given by the second-order model of Yakhot and Orzag are compared with results of Rapid Distortion Theory corrected for decay (Townsend, 1970).     

Effect of free-stream turbulence on surface friction in a turbulent boundary layer

The combined effect of the turbulence intensity , the turbulence scaleL, and the Reynolds number Re^** on the surface friction coefficientc _f in a turbulent boundary layer is studied. The dependence of the relative friction increment on the equivalent turbulence level _cq, which takes into account the simultaneous variation in ,L and Re^**, is determined. The threshold value _cq ^* below which the value ofc _f does not depend on _cq is found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 65–75, March–April, 1995.     

The Gibbs-Thompson relation within the gradient theory of phase transitions

This paper discusses the asymptotic behavior as 0+ of the chemical potentials associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ²¦¦²+ W(), where is the density. The main result is that is asymptotically equal to E/d+o(), with E the interfacial energy, per unit surface area, of the interface between phases, the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Numerical solution of two-dimensional problem of gas flow from a chamber with flat walls

A study is made of the classical problem of steady flow of ideal gas from an infinite two-dimensional chamber with straight wall generator making angle _w with the symmetry plane of the flow. Specification of _w, the pressure ratio _a-P_a/P_o (P_o is the stagnation pressure of the gas in the chamber, P is the pressure in the ambient medium), and the specific-heat ratio completely determines the flow of gas from the chamber.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–181, July–August, 1988.I thank A. N. Kraiko for his interest in the work and helpful discussions of the results.     

Instantaneous density measurement in two-dimensional gas flow by high speed differential interferometry

The aim of this paper is to present an experimental set-up using a Wollaston prism differential interferometer producing up to twenty successive short exposure white light interferograms at a high framing rate. It is shown that, through optical component calibration, the interferograms can be analysed to yield the instantaneous density field. This method has been successfully tested in the two-dimensional unsteady flow generated by the interaction of a mixing layer and a cavity.List of symbols h height of the downstream edge of the cavity - H height of backward facing step - M Mach number - t time - t time interval between two successive frames - N frequency - double-prism median plane - birefringence angle - _p pressure fluctuation - C _p pressure coefficient - biprism abscissa corresponding to any colour - ₀ biprism reference abscissa corresponding to background colour - _y deviation of light rays - R radius of curvature of spherical mirror - L virtual distance from the middle of the test section to the spherical mirror - E optical thickness - E _e optical thickness corresponding to background colour - d _E difference of optical thickness - d _x abscissa difference - gas density - ₀ stagnation gas density - _e gas density of background colour     

The bipore model in solid-liquid extraction: the batch process

This paper presents the exact analytical solution for the general case of transient mass transfer between a solid with a biporous structure (with a micro and a macroporosity) and the entouring finite fluid. The transport inside the solid is by molecular diffusion and outside of it the convective film resistance is included. A general expression is given which is valid for the infinite plate, for the infinite cylinder and for the sphere. The standard monopore case is obtained as a particular solution.

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Das Bipor-Modell in der fest-flüssig Extraktion: Das diskontinuierliche Verfahren

Zusammenfassung Es wird die exakte analytische Lösung für den allgemeinen Fall der instationären Stoffübertragung zwischen einem Festkörper mit biporöser Struktur (bestehend aus einer Mikro- und einer Makroporosität) und dem äußeren Fluid vorgestellt. Der Transport in dem Feststoff erfolgt mittels molekularer Diffusion. Außerhalb der Feststoffpartikel wird der konvektive Filmwiderstand berücksichtigt. Eine allgemeine Formel wird angegeben, die für die unendliche Platte, für den unendlichen Zylinder und für die Kugel anwendbar ist. Die Lösung für das übliche monopore Modell ergibt sich als Sonderfall.

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Nomenclature c _a concentration of liquid in micropores - c _b concentration of liquid in macropores - ¯c average concentration in the particle - c₁ initial value of ¯c - c _e concentration in liquid outside the particle - c _e1 initial value ofc _e - D _a ,D _b effective diffusivity in the micro resp. in the macro structure limit ofE for infinite time - f _n form-function defined in Eqs. (20), (21) and (22) - F _n function defined in Eq. (33) - f, g,h Laplace transforms ofc _a ^* ,c _b/* and ¯c^* resp. - I ₀ ,I ₁ modified Bessel functions of the first kind, order zero and first order resp. - J ₀ ,J ₁ Bessel functions of the first kind, order zero and first order resp. - k _c mass transfer coefficient - M _p mass of the solid particles - n numerical form constant, 1 for the plate, 2 for the cylinder and 3 for the sphere - N function defined in Eq. (19) - R _a radius of the microporus spheres - R _b size of the particle (for the plate2R _b is its thickness, for cylinder and sphere: the radius) - r radial coordinate inside the microporous sphere - r ^* =r/R _a adimensional forrt time - t ^* -t/ _b adimensional for time (Fourier Number) - V volume of fluid phase (exterior to solid) - x position coordinate inside the solid particle - x ^* =x/R _b adimensional forx - =(M_pp)/(V_p) volume of fluid inside the particles divided by volume of fluid outside - y=(R _b k _c )/D _b adimensional for the mass transfer coefficient - _a mircoporosity - _b microporosity - _p = _b + (1 _b ) _a total porosity of the particle - =_p/_b – 1=(1 -_b @#@) ( _a / _b ) adimensional parameter, characteristic for the biporous structure - _p density of particle - _a =R _a/2 / (D _a / _a) characteristic (micro) time - _b =R _b/2 / (D _b / _b) characteristic (macro) time - = _a / _b adimensional parameter, characteristic for the biporous particle