Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

Mixed convection flow in narrow vertical ducts

The mixed convection flow in a vertical duct is analysed under the assumption that , the ratio of the duct width to the length over which the wall is heated, is small. It is assumed that a fully developed Poiseuille flow has already been set up in the duct before heat from the wall causes this to be changed by the action of the buoyancy forces, as measured by a buoyancy parameter . An analytical solution is derived for the case when the Reynolds numberRe, based on the duct width, is of 0 (1). This is extended to the case whenRe is 0 (^–1) by numerical integrations of the governing equations for a range of values of representing both aiding and opposing flows. The limiting cases, || 1 andR=Re of 0 (1), andR and both large, with of 0 (R ^1/3) are considered further. Finally, the free convection limit, large with R of 0 (1), is discussed.

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Mischkonvektion in engen senkrechten Rohren

Zusammenfassung Mischkonvektion in einem senkrechten Rohr wird unter der Voraussetzung untersucht, daß das Verhältnis der Rohrbreite zur Länge, über welche die Wand beheizt wird, klein ist. Es wird angenommen, daß sich bereits eine voll entwickelte Poiseuille-Strömung in dem Rohr eingestellt hat, bevor Antriebskräfte, gemessen mit dem Auftriebsparameter , aufgrund der Wandbeheizung die Strömung verändern. Es wird eine analytische Lösung für den Fall erhalten, daß die mit der Rohrbreite als charakteristische Länge gebildete Reynolds-ZahlRe konstant ist. Dies wird mittels einer numerischen Integration der wichtigsten Gleichungen auf den FallRe =f (^–1) sowohl für Gleich- als auch für Gegenstrom ausgedehnt. Weiterhin werden die beiden Grenzfälle betrachtet, wenn || 1 undR=Re konstant ist, sowieR und beide groß mit proportionalR ^1/3. Schließlich wird der Grenzfall der freien Konvektion, großes mit konstantem R, diskutiert.

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Nomenclature g acceleration due to gravity - Gr Grashof number - G modified Grashof number - h duct width - l length of the heated section of the duct wall - p pressure - Pr Prandtl number - Q flow rate through the duct - Q ₀ heat transfer on the wally=0 - Q ₁ heat transfer on the wally=1 - Re Reynolds number - R modified Reynolds number - T temperature of the fluid - T ₀ ambient temperature - T applied temperature difference - u, velocity component in thex-direction - v, velocity component in they-direction - x, co-ordinate measuring distance along the duct - y, co-ordinate measuring distance across the duct - buoyancy parameter - ₀ modified buoyancy parameter, ₀=R ^–1/3 - coefficient of thermal expansion - ratio of duct width to heated length, =h/l - (non-dimensional) temperature - _w applied temperature on the wally=0 - kinematic viscosity - density of the fluid - ₀ shear stress on the wally=0 - ₁ shear stress on the wally=1 - stream function     

A noninvasive method for the measurement of flow-induced surface displacement of a compliant surface

A noninvasive optical method is described which allows the measurement of the vertical component of the instantaneous displacement of a surface at one or more points. The method has been used to study the motion of a passive compliant layer responding to the random forcing of a fully developed turbulent boundary layer. However, in principle, the measurement technique described here can be used equally well with any surface capable of scattering light and to which optical access can be gained. The technique relies on the use of electro-optic position-sensitive detectors; this type of transducer produces changes in current which are linearly proportional to the displacement of a spot of light imaged onto the active area of the detector. The system can resolve displacements as small as 2 m for a point 1.8 mm in diameter; the final output signal of the system is found to be linear for displacements up to 200 m, and the overall frequency response is from DC to greater than 1 kHz. As an example of the use of the system, results detailing measurements obtained at both one and two points simultaneously are presented.List of symbols C _t elastic transverse wave speed = (G/)^1/2 - d ⁺ spot diameter normalized by viscous length scale - G frequency average of G() - G() shear storage modulus - G() shear loss modulus - l. viscous length scale = v/u _* - N total number of sampled data values - r separation vector for 2-point measurements = (, ) - rms root-mean-square value - R momentum thickness Reynolds number = U _t8/v - t time - u (y) mean streamwise component of velocity in boundary layer - u _* friction velocity = (t _w/)^1/2 - U free-stream velocity - x, y, z longitudinal, normal and spanwise directions - y _o undisturbed surface position - vertical component of compliant surface displacement - ₉₉ boundary layer thickness for which u(y) = 0.99 U _t8 - _l viscous sublayer thickness 5 l _* - frequency average of G()/ - boundary layer momentum thicknes = - fluid dynamic viscosity - v fluid kinematic viscosity = / - , longitudinal, spanwise components of separation vector r - fluid density - time delay - _w wall shear stress     

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     

On the prediction of riblet performance with engineering turbulence models

The paper reports the outcome of a numerical study of fully developed flow through a plane channel composed of ribleted surfaces adopting a two-equation turbulence model to describe turbulent mixing. Three families of riblets have been examined: idealized blade-type, V-groove and a novel U-form that, according to computations, achieves a superior performance to that of the commercial V-groove configuration. The maximum drag reduction attained for any particular geometry is broadly in accord with experiment though this optimum occurs for considerably larger riblet heights than measurements indicate. Further explorations bring out a substantial sensitivity in the level of drag reduction to the channel Reynolds number below values of 15 000 as well as to the thickness of the blade riblet. The latter is in accord with the trends of very recent, independent experimental studies.Possible shortcomings in the model of turbulence are discussed particularly with reference to the absence of any turbulence-driven secondary motions when an isotropic turbulent viscosity is adopted. For illustration, results are obtained for the case where a stress transport turbulence model is adopted above the riblet crests, an elaboration that leads to the formation of a plausible secondary motion sweeping high momentum fluid towards the wall close to the riblet and thereby raising momentum transport.Nomenclature c _f Skin friction coefficient - c _f Skin friction coefficient in smooth channel at the same Reynolds number - k Turbulent kinetic energy - K ⁺ k/ _w - h Riblet height - S Riblet width - H Half height of channel - Re Reynolds number = volume flow/unit width/ - Modified turbulent Reynolds number - R _t turbulent Reynolds numberk ²/ - P _k Shear production rate ofk, _t (U _i /x _j + U _j /x _i ) U _i /x _j - dP/dz Streamwise static pressure gradient - U _i Mean velocity vector (tensor notation) - U Friction velocity, _w/ where _w=–H dP/dz - W Mean velocity - W _b Bulk mean velocity through channel - y ⁺ yU /v. Unless otherwise stated, origin is at wall on trough plane of symmetry - Kinematic viscosity - _t Turbulent kinematic viscosity - Turbulence energy dissipation rate - Modified dissipation rate – 2(k ^1/2/x _j )² - Density - _k , Effective turbulent Prandtl numbers for diffusion ofk and      

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

On physical criteria for the selection of wood for soundboards of musical instruments

In order to develop criteria for the physical evaluation of wood for soundboards of musical instruments, measurements were made of dynamic Young's modulusE, static Young's modulusE, internal frictionQ ^–1 in longitudinal direction, and specific gravity for numerous species of broad-leaved wood. From the results obtained, including those of our previous paper on coniferous wood [1], it was found that the suitability of wood for soundboards could be evaluated by the quantity ofQ ^–1/(E/), and that there were very high correlations betweenQ ^–1/(E/) andE/, and betweenE andE, regardless of wood species. Consequently, it becomes possible to select practically any wood suitable for soundboards by using the value ofE/, which can be measured easily, and it was derived that the relation betweenE/ andQ ^–1 of wood could be expressed by an exponential equation regardless of wood species.     

A mixing length model for turbulent flow in constant cross section ducts

In this paper, a study is made of the damping influence of the wall on turbulent fluid flow. By considering the oscillation of the whole of the boundary, van Driest's original hypothesis has been extended to obtain the wall damping factor in flow in a duct of constant cross section. The damping factor is used in conjunction with mixing length expressions to obtain the velocity field. Particular examples considered are plane parallel flow and axisymmetric flow in a pipe and in an annulus.

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Ein Modell für die Mischungslänge von turbulenten Strömungen in Rohren mit konstantem Querschnitt

Zusammenfassung In dieser Arbeit wurde der dämpfende Wandeinfluß in turbulenten Strömungen untersucht. Unter Berücksichtigung der Schwingungen in der gesamten Grenzschicht wurde die ursprüngliche Theorie von van Driest erweitert und ein Dämpfungsfaktor an der Wand in Rohrströmungen mit konstantem Querschnitt ermittelt. Dieser Dämpfungsfaktor diente in Verbindung mit Ausdrücken für die Mischungslänge zur Bestimmung des Geschwindigkeitsfeldes. Ausgewählte Beispiele waren die ebene Parallelströmung sowie die Zylinderströmung in einem Rohr und einem Ringspalt.

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Nomenclature A, A^*(=Au/v) Parameter defined in text - b, b^*(=bu/v) semi-width of parallel plate channel - c(= 1/A) parameter defined in text - E[, /2] complete elliptic integral of the second kind - d damping factor - F, G, H functions - l, l^*(=/v) mixing length - M_O, _O functions - r, r^*(=ru/v) radius - A real part of function - R, S, T, U functions - u, u^*(=u/u) velocity in flow direction Z - friction velocity - x, y, z co-ordinates (z in flow direction) - y^*(=yu/v) non-dimensional wall distance - fluid density - , _eff kinematic viscosity, effective kinematic viscosity - phase angle, or polar coordinate angle - shear stress - (=r/r_W) radius ratio - angular velocity Suffixes w wall value - far from a wall     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Boundary layer stability on a rotating disk with corotation of the surrounding fluid

The stability of the steady self-similar flow in the boundary layer on a rotating disk of infinite radius with corotation of the surrounding fluid is analyzed by the normal mode method. The spectral problem for infinitesimal three-dimensional disturbances is solved by a collocation method with expansion of the amplitude functions in Chebyshev polynomials. It is established that for all values of the parameter 0, equal to the ratio of the angular velocities of the fluid and the disk, the lower critical Reynolds number is determined byA-type, waves, whose development is governed by the parallel instability mechanism typical of an Ekman layer. TheB-type instability, associated with the presence of an inflection point on the velocity profile, disappears when 4. The neutral surfaces are calculated for Karman flow (=0) and Bödewadt flow (). It is found that in Karman flowA-type waves may grow at values of the Reynolds number several times smaller than the critical Reynolds number for spiral vortices. The results of the analysis are compared with the available experimental data.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 69–77, September–October, 1992.     

Free convection heat transfer from an isothermal plate with arbitrary inclination

This paper presents a new formulation for the laminar free convection from an arbitrarily inclined isothermal plate to fluids of any Prandtl number between 0.001 and infinity. A novel inclination parameter is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed equations. Moreover, the self-similar equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting=0 and=1, respectively. Heated upward-facing plates with positive and negative inclination angles are investigated. A very accurate correlation equation of the local Nusselt number is developed for arbitrary inclination angle and for 0.001 Pr .

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Wärmeübertragung bei freier Konvektion an einer isothermen Platte mit beliebiger Neigung

Zusammenfasssung Diese Untersuchung stellt eine neue Formulierung der laminaren freien Konvektion von Flüssigkeiten mit einer Prandtl-Zahl zwischen 0,001 und unendlich an einer beliebig schräggestellten isothermen Platte dar. Ein neuer Neigungsparameter wird eingeführt, so daß alle Fälle der horizontalen, geneigten oder vertikalen Platte von einem einzigen Satz transformierter Gleichungen beschrieben werden können. Die unabhängigen Gleichungen für die beiden Fälle der horizontalen and vertikalen Platte wurden für=0 und=1 aus den transformierten Gleichungen wieder abgeleitet. Es wurden erwärmte aufwärtsgerichtete Platten mit positiven und negativen Neigungswinkeln untersucht. Eine sehr genaue Gleichung wurde für die lokale Nusselt-Zahl bei beliebigen Neigungswinkeln und für 0,001 Pr entwickelt.

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Nomenclature C _p specific heat - f reduced stream function - g gravitational acceleration - Gr local Grashof number,g(T _w –T _w ) x³/v² - h local heat transfer coefficient - k thermal conductivity - n constant exponent - Nu local Nusselt number,hx/k - p pressure - Pr Prandtl number, v/ - Ra local Rayleigh number,g(T _w –T )J x³/v - T fluid temperature - T _w wall temperature - T temperature of ambient fluid - u velocity component in x-direction - v velocity component in y-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra¦sin¦)^1/4/( Ra cos()^1/5 - pseudo-similarity variable, (y/) - dimensionless temperature, (T–T )/(T _w–T ) - ( Ra cos)^1/5+(Rasin)^1/4 - v kinematic viscosity - 1/[1 +(Ra cos)^1/5/( Ra¦sin)^1/4] - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of plate inclination measured from the horizontal - stream function - dimensionless dynamic pressure     

Rheology of molten polymers

Summary TheCross equation describes the flow of pseudoplastic liquids in terms of an upper and a lower Newtonian viscosity corresponding to infinite and zero shear, and ₀, and of a third material constant related to the mechanism of rupture of linkages between particles in the intermediate, non-Newtonian flow regime, Calculation of of bulk polymers is important, since it cannot be determined experimentally. The equation was applied to the melt flow data of two low density polyethylenes at three temperatures.Using data in the non-Newtonian region covering 3 decades of shear rate to extrapolate to the zero-shear viscosity resulted in errors amounting to about onethird of the measured ₀ values. The extrapolated upper Newtonian viscosity was found to be independent of temperature within the precision of the data, indicating that it has a small activation energy.The ₀ values were from 100 to 1,400 times larger than the values at the corresponding temperatures.The values of were large compared to the values found for colloidal dispersions and polymer solutions, but decreased with increasing temperature. This shows that shear is the main factor in reducing chain entanglements, but that the contribution of Brownian motion becomes greater at higher temperatures.

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Zusammenfassung Die Gleichung vonCross beschreibt das Fließverhalten von pseudoplastischen Flüssigkeiten durch drei Konstante: Die obereNewtonsche Viskosität (bei sehr hohen Schergeschwindigkeiten), die untereNewtonsche Viskosität ₀ (bei Scherspannung Null), und eine Materialkonstante, die vom Brechen der Bindungen zwischen Partikeln im nicht-Newtonschen Fließbereich abhängt. Die Berechnung von ist wichtig für unverdünnte Polymere, wo man sie nicht messen kann.Die Gleichung wurde auf das Fließverhalten der Schmelzen von zwei handelsüblichen Hochdruckpolyäthylenen bei drei Temperaturen angewandt. Die Werte von ₀, durch Extrapolation von gemessenen scheinbaren Viskositäten im Schergeschwindigkeitsbereich von 10 bis 4000 sec^–1 errechnet, wichen bis 30% von den gemessenen ₀-Werten ab. Die Aktivierungsenergie der war so klein, daß die-Werte bei den drei Temperaturen innerhalb der Genauigkeit der Extrapolation anscheinend gleich waren. Die ₀-Werte waren 100 bis 1400 mal größer als die-Werte.Im Verhältnis zu kolloidalen Dispersionen und verdünnten Polymerlösungen war das der Schmelzen groß, nahm aber mit steigender Temperatur ab. Deshalb wird die Verhakung der Molekülketten hauptsächlich durch Scherbeanspruchung vermindert, aber der Beitrag derBrownschen Bewegung nimmt mit steigender Temperatur zu.

     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

Reflection, transmission and excitation of SH-surface waves by a discontinuity in mass-loading on a semi-infinite elastic medium

The scattering of an SH-wave by a discontinuity in mass-loading on a semi-infinite elastic medium is investigated theoretically. The incident wave is either a plane body wave or a plane SH-surface wave. The problem is reduced to a Wiener-Hopf problem for the scattered wave. In this problem the amplitude spectral density of the particle displacement occurs as unknown function. Special attention is given to the numerical values of the surface wave contributions to the scattered field.Nomenclature x ₁, x ₂, x ₃ Cartesian coordinates - , polar coordinates in x ₁, x ₃-plane - volume mass density - surface mass density of mass-loading - , Lamé constants - U scalar wave function, defined by (2.1) - c _S speed of propagation of uniform shear waves in bulk medium (c _S=(/)^1/2) - angular frequency - t time - k _S wave number of uniform shear waves (k _S=/c _S) - reduced specific acoustic impedance of mass-loading (=k _S /) - k _m wave number of SH-surface wave (k _m=k _S(1+ ²)^1/2) - _1,2,3 partial differentiation with respect to x _1,2,3 - ⁱ angle between x ₃-axis and direction of propagation of incident body wave - ⁱ wave number in horizontal direction of incident body wave ( ⁱ=k _S sin( ⁱ)) - ⁱ wave number in vertical direction of incident body wave ( ⁱ=k _S cos( ⁱ)) - C _1,2 complex amplitude of surface wave excited by a body wave - R reflection factor of surface wave, when surface wave is incident - T transmission factor of surface wave, when surface wave is incident - S particle displacement vector The research presented in this paper has been carried out with partial financial support from the Delfts Hogeschoolfonds.     

Dehnströmungen mit verdünnten Polymerlösungen: Ein theoretisches Modell und seine experimentelle Verifikation

Zusammenfassung Es werden theoretische Überlegungen zusammenfassend dargestellt, welche die Streckung und Ausrichtung von flexiblen Makromolekülen in stationären einfachen Dehnströmungen beschreiben. Die Makromoleküle werden hierbei als EDNE-(endlich dehnbare, nichtlinear elastische) Hanteln modelliert. Für den Fall niedriger bzw. hoher Dehnungsraten werden Dehnviskositätsgleichungen für Strömungen mit verdünnten Polymerlösungen angegeben.Die Arbeit vergleicht die abgeleiteten theoretischen Gleichungen mit experimentellen Ergebnissen, welche für Porenströmungen erhalten wurden; Porenströmungen weisen Dehnströmungen auf. Anhand der durchgeführten experimentellen Untersuchungen, in denen alle die den Druckverlust maßgebend beeinflussenden strömungsmechanischen und physikalisch-chemischen Parameter variiert wurden, kann gezeigt werden, daß sich die aufgezeigten theoretischen Zusammenhänge quantitativ bestätigen lassen.Schlüsselwörter Dehnströmung, Makromolekülmodell, Porenströmung, EDNE-Hantelmodell, Polymerlösung

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Summary The present paper summarizes theoretical considerations regarding the elongation of flexible macromolecules in simple steady elongational flows. The macromolecules are treated as FENE(finite extensible, nonlinear elastic)-dumbbells. Equations for extensional viscosity are given for flows of dilute polymer solutions applicable at low and high elongation rates.The present paper compares the derived theoretical relationships with experimental results. These results were obtained in porous media flows, which exhibit strong elongational rates. It can be shown on the basis of the experimental investigations, that all fluid mechanic and physico-chemical parameters that influence the measured pressure losses responded as predicted by the theory.

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a Mark-Houwink-Exponent - A Avogadro-Konstante - b Verhältnis von Molekülzeitkonstanten - c Polymergewichtskonzentration - d Kugeldurchmesser der Schüttung - D Diffusionskonstante - De Deborahzahl - f Reibungsbeiwert der Porenströmung - F Kraftvektor des Hantelmodells - g Erdbeschleunigung - H Hookesche Federkonstante des Makromoleküls - k Boltzmann-Konstante - k _1,2,3 empirische Konstanten - K Mark-Houwink-Konstante - l ₀ Länge des Monomeren - L Länge des statistischen Fadenelementes - L ₀ Maximallänge des gestreckten Polymermoleküls - L Bezugslänge für den Druckverlust der Porenströmung - m Masse des statistischen Fadenelementes - m ₀ Masse des Monomeren - Molarität - M Molekulargewicht des Polymeren - n Porosität der Kugelschüttung - n ₀ Hantelkonzentration - N Anzahl der statistischen Fadenelemente - p Druckverlust der Porenströmung - P Polymerisationsgrad - R Endpunktabstand des Makromoleküls - R ₀ maximaler Endpunktabstand des gestreckten Moleküls - mittlerer Endpunktabstand des Moleküls - Orientierungsvektor des Hantelmodells - Re Reynoldszahl der Porenströmung - t Zeit - T Temperatur - mittlere Filtergeschwindigkeit der Porenströmung - v Strömungsfeld - Aufweitungsparameter - Bindungswinkel zweier Kohlenstoffatome - Dehnungsrate - Stokesscher Reibungsfaktor - dynamische Viskosität - ^* reduzierte Viskosität - [] Grenzviskositätszahl - Dehnviskosität - ^* reduzierte Dehnviskosität - Widerstandskennzahl der Porenströmung - v kinematische Viskosität - Dichte des Fluids - _H Hookesche Relaxationszeit des EDNE-Hantelmodells - _H,e Hookesche Relaxationszeit des linear elastischen Hantelmodells - _R Relaxationszeit des starren Hantelmodells - _zz , _yy Normalspannungen - Volumenkonzentration - _fl. dimensionsloser Faktor des Strömungsfeldes - ₀ Konstante der Flory-Fox-Gleichung - Verteilungsfunktion des Hantelmodells - _eq. Gleichgewichtsverteilungsfunktion - a aufgeweitet - e effektiv - max maximal - p polymer - s solvent, Lösungsmittel - Theta-Zustand Mit 12 Abbildungen und 2 Tabellen     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

Attractors for the Two-Dimensional Navier–Stokes-α Model: An α-Dependence Study

The two-dimensional Navier–Stokes- model is considered on the torus and on the sphere. Upper and lower bounds for the dimension of the global attractors are given. The dependence of the dimension of the global attractors on is studied. Special attention is given for the limiting cases when 0, that is, when the Navier–Stokes- model tends to the Navier–Stokes equations, and to the case when .