Turbulent heat and mass transfer for Pr ≫ 1 and law of turbulent diffusion damping at a solid wall

The question of determining the law of damping for the turbulent diffusion coefficient at a smooth wall according to data on mass and heat transfer for Pr 1 is discussed. It is proved that the hypothesis that this law is determined by the first member of the Taylor series expansion of , namely, / = yⁿ ₊ is valid in the Pr range from 10³ to 10⁵ only under the assumption that the subsequent terms in the expansion have smaller coefficients. A statistical analysis of electrochemical and other experiments devoted to this problem shows that apparently n = 3, but singularities in the experimental results do not permit making a final conclusion. Requirements on a conclusive experiment are formulated on the basis of the analysis made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–175, March–April, 1977.     

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.     

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.     

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.

     

Extended Korn's inequalities and the associated best possible constants

We are concerned with the coerciveness of the strain energy E(u) (in linear elasticity) associated with a displacement vector u on the Sobolev space H¹ () or its subspaces, a domain in ⁿ representing an isotropic elastic body—certain specific cases are called Korn's inequalities. Sufficient (and necessary) conditions on the Lamé moduli for E(·) to be coercive (or uniformly positive) on such spaces are given, and the associated best possible constants are obtained for some cases.     

The effect of non-steady wall shear on pressure surges in conduits carrying viscous liquids

Summary A study is made of the attenuation of pressure surges in a two-dimension a channel carrying a viscous liquid when a valve at the downstream end is suddenly closed. The analysis differs from previous work in this area by taking into account the transient nature of the wall shear, which in the past has been assumed as equivalent to that existing in steady flow. For large values of the frictional resistance parameter the transient wall shear analysis results in less attenuation than given by the steady wall shear assumption.Nomenclature c /, ft/sec - e base of natural logarithms - F(x, y) integration function, equation (38) - (x) mean value of F(x, y) - g local acceleration of gravity, ft/sec² - h width of conduit, ft - k (2m–1)^{2 2} L/h ² c, equation (35) - k* 12L/h ² c, frictional resistance parameter, equation (46) - L length of conduit, ft - m positive integer - n positive integer - p pressure, lb/ft² - p ₀ constant pressure at inlet of conduit, lb/ft² - P pressure plus elevation head, p+gz, equation (4) - mean value of P over the conduit width h - P ₀ p ₀+gz ₀, lbs/ft² - R frictional resistance coefficient for steady state wall shear, lb sec/ft⁴ - s positive integer; also, condensation, equation (6) - t time, sec - t ct/L, dimensionless time - u x component of fluid velocity, ft/sec - u _m mean velocity in conduit, equation (12), ft/sec - u ₀(y) velocity profile in Poiseuille flow, equation (19), ft/sec - transformed velocity - U initial mean velocity in conduit - x distance along conduit, measured from valve (fig. 1), ft - x x/L, dimensionless distance - y distance normal to conduit wall (fig. 1), ft - y y/h, equation (25) - z elevation, measured from arbitrary datum, ft - z ₀ elevation of constant pressure source, ft - isothermal bulk compression modulus, lbs/ft² - _n , equation (37) - _n (2n–1)/2, equation (36) - viscosity, slugs/ft sec - / = kinematic viscosity, ft²/sec - density of fluid, slugs/ft³ - ₀ density of undisturbed fluid, slugs/ft³ - ø angle between conduit and vertical (fig. 1) The research upon which this paper is based was supported by a grant from the National Science Foundation.     

Über den Einfluß von Querkrümmung und Dichte bei inhomogenen turbulenten Freistrahlen

Zusammenfassung Zur Berechnung turbulenter Strömungen wird das k--Modell im Ansatz für die turbulente Scheinzähigkeit erweitert, so daß es den Querkrümmungs- und Dichteeinfluß auf den turbulenten Transportaustausch erfaßt. Die dabei zu bestimmenden Konstanten werden derart festgelegt, daß die bestmögliche Übereinstimmung zwischen Berechnung und Messung erzielt wird. Die numerische Integration der Grenzschichtgleichungen erfolgt unter Verwendung einer Transformation mit dem Differenzenverfahren vom Hermiteschen Typ. Das erweiterte Modell wird auf rotationssymmetrische Freistrahlen veränderlicher Dichte angewendet und zeigt Übereinstimmung zwischen Rechnung und Experiment.

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On the influence of transvers-curvature and density in inhomogeneous turbulent free jets

The prediction of turbulent flows based on the k- model is extended to include the influence of transverse-curvature and density on the turbulent transport mechanisms. The empirical constants involved are adjusted such that the best agreement between predictions and experimental results is obtained. Using a transformation the boundary layer equations are solved numerically by means of a finite difference method of Hermitian type. The extended model is applied to predict the axisymmetric jet with variable density. The results of the calculations are in agreement with measurements.

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Bezeichnungen Wirbelabsorptionskoeffizient - c_i Massenkonzentration der Komponente i - c_D, c_L, c, c₁, c₂ Konstanten des Turbulenzmodells - d Düsendurchmesser - E bezogene Dissipationsrate - f bezogene Stromfunktion - f Korrekturfunktion für die turbulente Scheinzähigkeit - j turbulenter Diffusionsstrom - k Turbulenzenergie - k_i Schrittweite in -Richtung - K dimensionslose Turbulenzenergie - L turbulentes Längenmaß - M_i Molmasse der Komponente i - p Druck - allgemeine Gaskonstante - r Querkoordinate - r_0,5 Halbwertsbreite der Geschwindigkeit - r_0,5c Halbwertsbreite der Konzentration - T Temperatur - u Geschwindigkeitskomponente in x-Richtung - v Geschwindigkeitskomponente in r-Richtung - x Längskoordinate - y allgemeine Funktion - Y_i diskreter Wert der Funktion y - Relaxationsfaktor für Iteration - turbulente Dissipationsrate - transformierte r-Koordinate - kinematische Zähigkeit - Exponent - transformierte x-Koordinate - Dichte - _k, Konstanten des Turbulenzmodells - Schubspannung - allgemeine Variable - Stromfunktion - Turbulente Transportgröße Indizes 0 Strahlanfang - m auf der Achse - r mit Berücksichtigung der Krümmung - t turbulent - mit Berücksichtigung der Dichte - im Unendlichen - Schwankungswert oder Ableitung einer Funktion - – Mittelwert Herrn Professor Dr.-Ing. R. Günther zum 70. Geburtstag gewidmet     

Interpolation formulas for maxwell viscosity of certain metals as a function of shear-strain intensity and temperature

The purpose of this study is the construction of interpolation formulas for the dependence of Maxwell viscosity, a quantity which is the reciprocal of shear-strain relaxation time , on shear-strain intensity and temperature for several metals: iron, aluminum, copper, and lead. This function was interpolated in various temperature and deformation velocity ranges in accordance with available experimental data for iron (0 10⁷ sec^–1, 200 ° T 1500 °); aluminum (0 10⁷ sec^–1, 300 ° T 900 °); copper (0 10⁵ sec^–1, 300 ° T 1300 °); lead (0 10⁶ sec^–1, 90 ° T 400 °); temperatures in °K.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 114–118, July–August, 1974.     

Invariant Manifolds for Nonautonomous Systems with Application to One-Step Methods

In this paper we study the existence of invariant manifolds for a special type of nonautonomous systems which arise in the study of discretization methods. According to [10], a one-step scheme of step-size for an autonomous system can be interpreted as the -flow of a perturbed nonautonomous system. The perturbation is rapidly forced in the sense that it is periodic with respect to time with period . Assuming a saddle node for the autonomous system, we prove that these rapidly forced perturbations have center manifolds which exist in a uniform neighborhood and which converge to a center manifold of the autonomous system as tends to zero. Our results are applied to obtain a smooth continuation as well as estimates of the well known center manifolds for one-step schemes. They also form the basis for studying saddle-node homoclinic orbits under discretization.     

Viscous properties of soil

Dynamic problems connected with the wave propagation in soils not saturated with water and with wave interaction with obstacles and structural elements at the present time are solved on the basis of models in which plastic but not viscous soil properties are taken into account [1–5]. An analysis of experimental data and their comparison with the calculated results [4, 5] confirms that it is permissible to apply the model of an elasticplastic medium to soils in problems concerning the interaction of waves and structures. At the same time plane-wave damping in soils takes place more intensively than would follow from calculations carried out on the basis of models of an elastic-plastic medium. For example, if in a section of a poured sandy soil, taken as the initial section, the maximum stress in the wave is _m=ll kgf/cm² and its duration is 6=8 msec, then at a distance of 25 cm the calculations give _m=9.5 kgf/cm², while the experiment gives _m= 5 kgf/cm². If in the initial section _m= 20 kgf/cm² and =6 msec, then at a distance of 35 cm the calculation gives _m= l7 kgf/cm², while the experiment gives _m= 9 kgf/cm². In the calculations it was assumed that unloading takes place with a constant strain. This deviation of the calculated results from the experiment can be explained, in the first place, by the dependence of the () on the strain rate , which is not taken into account in the model of an elastic-plastic medium. The viscous properties cause additional energy losses and a more intensive damping of the waves. Experimentally the dependence of the () curves on the strain rate has been investigated for many soils [5–8]. The dynamic load on the test sample was produced by a body falling from a height or being accelerated by some method. Below we present test results of viscous soil properties when the test sample is compressed by an air shock wave. Compression curves and approximate numerical values of the coefficient of viscosity are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 68–71, July–August, 1968.The author thanks A. I. Shishikin for his participation in the experiments.     

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     

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.     

Investigations of mean and turbulent flow characteristics of a two dimensional plane diffuser

This paper reports the investigation of mean and turbulent flow characteristics of a two-dimensional plane diffuser. Both experimental and theoretical details are considered. The experimental investigation consists of the measurement of mean velocity profiles, wall static pressure and turbulence stresses. Theoretical study involves the prediction of downstream velocity profiles and the distribution of turbulence kinetic energy using a well tested finite difference procedure. Two models, viz., Prandtl's mixing length hypothesis and k- model of turbulence, have been used and compared. The nondimensional static pressure distribution, the longitudinal pressure gradient, the pressure recovery coefficient, percentage recovery of static pressure, the variation of U _max/U _bar along the length of the diffuser and the blockage factor have been valuated from the predicted results and compared with the experimental data. Further, the predicted and the measured value of kinetic energy of turbulence have also been compared. It is seen that for the prediction of mean flow characteristics and to evaluate the performance of the diffuser, a simple turbulence model like Prandtl's mixing length hypothesis is quite adequate.List of symbols C ₁ , C ₂ ,C turbulence model constants - F _x body force - k kinetic energy of turbulence - l _m mixing length - L length of the diffuser - u, v, w rms value of the fluctuating velocity - u, v, w turbulent component of the velocity - mean velocity in the x direction - _A average velocity at inlet - U _bar average velocity in any cross section - U _max maximum velocity in any cross section - V mean velocity in the y direction - W local width of the diffuser at any cross section - x, y coordinates - dissipation rate of turbulence - _m eddy diffusivity - Von Karman constant - mixing length constant - _l laminar viscosity - _eff effective viscosity - v kinematic viscosity - density - _k effective Schmidt number for k - effective Schmidt number for - stream function - non dimensional stream function     

Das Dispersionsmodell

Zusammenfassung Ein Vergleich im Frequenzbereich zeigt, daß bei der Berechnung der Verweilzeitverteilung mit dem Dispersionsmodell das endlich begrenzte System für Péclet-Zahlen Pe > 10 mit guter Näherung durch ein einseitig unbegrenztes System und für Pe > 50 durch ein beidseitig unbegrenztes System ersetzt werden kann.

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The dispersion model. A comparison of approximations

A comparison in the frequency domain shows that for the determination of the residence time distribution with the dispersion model the finitely restricted system may be substituted with good approximation for Peclet numbers Pe > 10 by a one-side unrestricted system and for Pe > 50 by a both-side unrestricted system.

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Bezeichnungen A Rohrquerschnitt - A=A mittlerer Strömungsquerschnitt in der Schüttschicht - Konzentration (Partialdichte) der Bezugskomponente i - Bezugskonzentration nach Gl. (5) - c_i Konzentration (Dichte) der reinen Bezugskomponente i - D axialer Dispersionskoeffizient - E Fehler im Frequenzbereich nach Gl. (36) - G(,) Übertragungsfunktion - G(,i) Frequenzgang - L Länge der Schüttschicht - m Masse - Massenstrom - Péclet-Zahl - s Laplace-Variable - t Zeit - t Impulsbreite - v Strömungsgeschwindigkeit im leeren Rohr - mittlere axiale Strömungsgeschwin digkeit in der Schüttschicht - V=AL Zwischenraumvolumen der Schüttschicht - x Ortskoordinate - (t) Dirac-Stoss - Porosität - dimensionslose Zeit - dimensionslose Konzentration - Laplace-Transformierte der Konzentration - Fourier-Transformierte der Konzentration - dimensionslose Ortskoordinate - =s dimensionslose Laplace-Variable - mittlere Verweilzeit - Kreisfrequenz - = dimensionslose Kreisfrequenz Indices A Ausgang - D Dispersion - E Eingang - i Bezugskomponente - K Konvektion Mitteilung Nr. 44 des Institutes für Mess-und Regel-technik der Eidgenössischen Technischen Hochschule Zürich (Vorsteher: Prof. Dr. P. Profos)     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

Zum Impuls-, Wärme- und Stoffaustausch in turbulenten Strömungen reagierender Binärgemische

Zusammenfassung Die in Teil I vorgestellten Reynolds 'schen Gleichungen und Transportgleichungen werden für Strömungen mit Grenzschichtcharakter angegeben. Weiter werden Integralbedingungen mitgeteilt. Nach einer Diskussion über die Schließung des Gleichungssystems werden Lösungsverfahren besprochen. Dabei wird speziell auf Integralverfahren eingegangen.

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About the transfer of momentum, heat and mass in turbulent flows of binary mixturesPart II: Thin shear flow layers

The Reynolds equations and transport equations given in part I are presented for thin shear flow layers. Integral relations are given. After a discussion of the closure problem methods of solution are described. Specially integral methods are discussed.

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Formelzeichen c Massenkonzentration der Komponente - c_t charakteristische Konzentrationsschwankung - c_o Bezugskonzentration - c spezifische Wärme bei konstantem Druck - c_f Reibungsbeiwert - c_D Dissipationsintegral - c_E Entrainment-Funktion - c Schubspannungsintegral - D binsrer Diffusionskoeffizient - H Formparameter - H₁₂ Formparameter - H₃₂ Formparameter - j Kassendiffusionsstrom - L Bezugslänge - p Druck - p_t charakteristische Druckschwankung - p_o Bezugsdruck - Pr Prandtl-Zahl - q Wärmestrom - q²/2 kinetische Energie der Schwankungsbewegung - Re_L mit L gebildete Reynolds-Zahl - Re mit gebildete Reynolds-Zahl - Re₂ mit ₂ gebildete Reynolds-Zahl - Sc Schmidt-Zahl - T absolute Temperatur - T_t charakteristische TemperaturSchwankung - T_o Bezugstemperatur - u,v,w Geschwindigkeitskomponenten - u_t charakteristische Geschwindigkeitsschwankung - u_o Bezugsgeschwindigkeit - U=/ü dimensionslose. x-Komponente der Geschwindigkeit - x,y,z Komponenten des Ortsvektors Griechische Symbole Grenzschichtdicke - ₁ Verdrängungsdicke - ₂ Impulsverlustdicke - ₃ Energieverlustdicke - _T Enthalpieverlustdicke - _c Konzentrationsverlustdicke - =d/dx Parameter für die Grenzschichtabsch:atzung - turbulente Impulsaustauschgröße - _D turbulente Stoffaustauschgröße - _q turbulente Energieaustauschgröße - Dissipationsfunktion - Wärmeleitfähigkeit - dynamische Viskosität - v=/ kinematische Viskosität - Dichte - Produktionsdichte - Schubspannung Indizes mol molekularer Anteil - tur turbulenter Anteil - res resultierender Anteil - Außenrand der Grenzschicht - w Wand     

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).     

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     

Application of Rayleigh-Ritz method to forced convection turbulent heat transfer

An analytical model to predict heat transfer rates to an incompressible fluid in turbulent flow, with fully developed velocity profile, between a heated plate and a parallel, insulated plate is developed. The model employs van Driest's mixing length expression near the wall, a constant eddy diffusivitiy in the core and a constant turbulent Prandtl number. An approximate solution obtained by employing Rayleigh-Ritz method is shown to compare well with the exact solution obtained by numerical integration of the differential equations. The results are compared with the available experimental data and analytical solutions.

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Anwendung der Rayleigh-Ritz-Methode auf die Wärmeübertragung bei erzwungener turbulenter Strömung

Zusammenfassung Es wird ein analytisches Modell zur Berechnung der Wärmeübertragung an ein inkompressibles Fluid in turbulenter Strömung mit voll ausgebildetem Geschwindigkeitsprofil zwischen einer beheizten Platte und einer dazu parallelen isolierten Platte angegeben. Das Modell verwendet van Driest's Ausdruck für die wandnahe Mischungslänge, eine konstante Wirbeldiffusivität im Kern und eine konstante turbulente PrandtlZahl. Eine Näherungslösung nach der Rayleigh-Ritz-Methode läßt sich gut mit der exakten Lösung vergleichen, die durch numerische Integration der Differentialgleichungen erhalten wurde. Die Ergebnisse werden mit verfügbaren Versuchswerten und analytischen Lösungen verglichen.

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Nomenclature A⁺ dimensionless constant in van Driest formula - a⁺ dimensionless distance from the wall after which the eddy diffusivity of momentum is constant - b half-gap of passage - b⁺ dimensionless half-gap=bu^*/ - C_f skin friction coefficient - C_p constant pressure specific heat - d hydraulic mean diameter defined as 4xarea/perimeter=4b - h convective heat transfer coefficient - K⁺ dimensionless constant in van Driest formula - k fluid thermal conductivity - m mass flow rate of fluid - Nu Nusselt number hd/k - P pressure - Pr Prandtl number=/ - Pr_t turbulent Prandtl number=_m/ - q_w heat flux at wall - Re Reynolds number=v_md/ - T Temperature - u⁺ dimensionless velocity=V_x/u^* - u^* friction velocity= - V_x axial velocity - x axial distance from the entrance - x⁺ dimensionless distance=x/d - y distance from the heated wall - y⁺ dimensionless distance=yu^*/ Greek Symbols thermal molecular diffusivity - function equal to (_H+)/ - boundary layer thickness - _H eddy diffusivity of heat - _m eddy diffusivity of momentum - _m0 uniform eddy diffusivity of momentum in the core - dimensionless temperature - T-T_i/q_wd/k uniform heat flux - T-T_w/T_i-T_w uniform temperature - fluid kinematic viscosity - fluid density - fluid shearing stress - bulk mean temperature—fully developed region - fully developed transverse temperature profile Suffixes 1 fully developed - 2 in the entrance region - i at the inlet - m bulk mean value - w at the heated wall