Unsteady forces on circular cylinders in a cross-flow

A three-axis piezoelectric load cell was used to measure the local unsteady forces induced on cylinders placed in a cross-flow. In conjunction with this, a single hot-wire was used to traverse the wake at a fixed distance behind the cylinder so that correlations between the induced forces on the cylinder and the wake velocity could be calculated to provide insight into the character of the flow-induced unsteady forces. Experiments were carried out on both two-dimensional and finite-span cylinders at a Reynolds number of 46,000. For the two-dimensional cylinder case, substantial evidence was obtained to demonstrate that the strength of the vortex roll-up along the span was quite uniform. Consequently, the lift-velocity correlation along the span remained unchanged. On the other hand, there was a total lack of correlation between the fluctuating drag and the wake velocity, thus indicating that the drag signal was not quite periodic. In the finite-span cylinder case, the separated flow from the top edge of the cylinder was found to suppress vortex shedding along the span of the cylinder, destroyed its coherence and caused the wake flow to oscillate in the stream direction. This oscillation induced a significant fluctuating drag on the cylinder. Consequently, the fluctuating drag far exceeded the fluctuating lift and the wake velocity was found to correlate well with the drag and not with the lift. This correlation remained intact along the span of the cylinder. Finally, the rms fluctuating lift and drag forces were found to vary along the cylinder span, with the lift increasing and the drag decreasing as the base of the cylinder is approached; thus suggesting that a submerged two-dimensional region exists near the base of the cylinder.List of symbols a span of active element on cylinder - C _D local rms drag coefficient, - C _L local rms lift coefficient, - C _D local mean drag coefficient - (C _D ) _2D spanwise-averaged mean drag coefficient for two dimensional cylinder - d diameter of cylinder (= 10.2 cm) - D fluctuating component of instantaneous drag - D local rms of fluctuating drag - E _D power spectrum of fluctuating drag, defined as - E _L power spectrum of fluctuating lift, defined as - E _U power spectrum of fluctuating streamwise velocity, defined as - f _L dominant frequency of lift spectrum - f _D dominant frequency of drag spectrum - f _u dominant frequency of velocity spectrum - h span of cylinder - H height of test section (= 30.5 cm) - L fluctuating component of instantaneous lift - L local rms of fluctuating lift - R _Du () cross-correlation function of streamwise velocity and local drag - R _Lu () cross-correlation function of streamwise velocity and local lift - Re Reynolds number, - S _L Strouhal number based on f _L , - S _D Strouhal number based on f _D , - S _U Strouhal number based on f _u , - t time - u fluctuating component of instantaneous streamwise velocity - u rms of streamwise fluctuating velocity - u rms of streamwise fluctuating velocity upstream of cylinder - U mean streamwise velocity - U mean stream velocity upstream of cylinder - x streamwise distance measured from axis of cylinder - y transverse distance measured from axis of cylinder - z spanwise distance measured from floor of test section - v kinematic viscosity of air - density of air - time lag in cross-correlation function - _D normalized spectrum of fluctuating drag - _L normalized spectrum of fluctuating lift - _U normalized spectrum of fluctuating streamwise velocity     

Effects of couple-stresses on the dispersion of a soluble matter in a pipe flow of blood

Summary An analysis of the effects of couple-stresses on the effective Taylor diffusion coefficient has been carried out with the help of two non-dimensional parameters based on the concentration of suspensions and , a constant associated with the couple-stresses. It is observed that the concentration distribution increases with increasing or The effective Taylor diffusion coefficient falls rapidly with increasing when is negative.

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Zusammenfassung Der Einfluß der Momentenspannungen auf den effektiven Taylorschen Diffusionskoeffizienten wird untersucht. Dabei treten zwei dimensionslose Parameter and auf: Der erste bezieht sich auf die Suspensionskonzentration, der zweite kennzeichnet die Momentenspannungen. Man findet, daß die Verteilungsgeschwindigkeit mit wachsendem oder zunimmt. Dagegen fällt der Taylorsche Diffusionskoeffizient bei wachsendem stark ab, wenn negativ ist.

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a Tube radius - C Concentration - C _i Body moment vector - C ₀ Concentration at the axis of the tube - C _m Mean concentration - D Molecular diffusion coefficient - d _ij Symmetric part of velocity gradient - F Function of and characterising effective Taylor diffusion coefficient - f _i Body force vector - H A function of and - K ₂ Integration constant - K ^* Effective Taylor diffusion coefficient - k Radius of gyration of a unit cuboid with its sides normal to the spatial axes - I _n Modified Bessel's function ofnth order - L Length of the tube over which the concentration is spread - M Function ofH and - M _ij Couple stress tensor - P Function of - p Fluid pressure - Q Volume rate of the transport of the solute across a section of the tube - r Radial distance from the axis of the tube - T _ij Stress tensor - t Time coordinate - T _ij ^A Antisymmetric part of the stress tensor - u Relative fluid velocity - Average velocity - v _i Velocity vector - Fluid velocity at any point of the tube - v ₀ ⁿ Velocity of Newtonian fluid at the axis of the tube - _i Vorticity vector - x Axial coordinate - x ₁ Relative axial coordinate - z Non-Dimensional radial coordinate - Density - _ij Symmetric part of the stress tensor - µ Viscosity of the fluid - µ _ij Deviatoric part ofM _ij - , Constants associated with couple-stress With 3 figures     

Response of a viscous annular liquid layer in zero-gravity to different axial excitations of the rigid boundary plates

A cylindrical annular liquid layer between two plates and around a rigid center-core consisting of incompressible and viscous liquid is subjected to different axial excitations, such as one-sided, counter-directional and double-sided unequal excitations. The response of the free liquid surface, the velocity- and pressure-distribution has been determined.

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Zusammenfassung Eine zylindrische Flüssigkeitsschicht bestehend aus inkompressibler und viskoser Flüssigkeit wurde verschiedenen harmonischen Anregungsformen ausgesetzt. Dabei wurden die Fälle einseitiger, doppelseitiger entgegengesetzter und ungleicher doppelseitiger Anregung mit Phase behandelt. Die Vergrößerungsfunktionen für die freie Flüssigkeitsoberfläche, für die Geschwindigkeits- und Druckverteilung wurden bestimmt.

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List of symbols a radius of liquid layer - b radius of inner cylindrical core - (a–b) thickness of layer - e _r , e , k unit vectors in the radial, angular and axial direction resp. - h length of layer - I _m , K _m modified Bessel functions of first and second kind and order m - diameter ratio - p pressure - q 2na/h - q* na/h - r, , z cylindrical coordinates - complex frequency - S sa ²/ - t time - u, w velocity components in the radial- and axial direction - ₀ excitation amplitude - abbreviation - surface tension parameter - surface tension - dynamic viscosity - kinematic viscosity - density of liquid - free liquid surface elevation - dimensionless time - _rz shear stress - reduced forcing frequency - forcing frequency - stream function - _mn natural frequency of non-viscous liquid     

Experimental investigations of fluid flow and heat transfer characteristics of a slot jet impinging on a square cylinder

Experimental investigations on pressure distributions and average heat transfer on square cylinders due to slot jet impingement have been carried out for different parameters such as, slot jet-width, distance of the square cylinder from the nozzle exit, angle of inclination of the cylinder to the jet axis and Reynolds numbers. The minimum value of the pressure coefficient is obtained on the lower face at an angle of inclination of 15° for all distances of the square cylinder from the nozzle exit. At the lowest Reynolds number the maximum average heat transfer rate is obtained at a distance of eight times the jet width from the nozzle exit. An increasing trend of the heat transfer rate is observed for higher Reynolds numbers. The maximum value of the heat transfer rate is obtained between the angles of inclination of 15° and 30° of the square cylinder to the jet axis. A correlation for the average Nusselt number is proposed in terms of the relevant non-dimensional parameters.

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Experimentelle Untersuchung der Strömungs- und Wärmeübergangscharakteristik bei Schlitzdüsenanblasung eines quadratischen Zylinders

Zusammenfassung Druckverteilung und gemittelter Wärmeübergang bei Schlitzanblasung eines quadratischen Zylinders wurden experimentell für folgende Parameter untersucht: Schlitzbreite; Abstand Düsenmündung vom Zylinder; Neigungswinkel des Zylinders zur Strahlachse; Reynoldszahl. Den Minimalwert des Druckkoeffizienten erhält man für alle Abstände an der Unterseite (bei einem Neigungswinkel von 15°). Bei der niedrigsten Reynoldszahl tritt der höchste Wert des gemittelten Wärmestroms in einem Abstand Düsenmündung/Zylinder von 8 Strahlbreiten auf. Mit steigender Reynoldszahl nimmt der Wärmestrom zu. Dessen höchster Wert tritt im Bereich 15 bis 30° des Neigungswinkels zwischen Zylinder und Strahlachse auf. Eine die Meßwerte korrelierende Nusscltbeziehung als Funktion dimensionsloser Parameter wird angegeben.

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Nomenclature A surface area of the square cylinder - a width of the square cylinder - C _p pressure coefficient=(p–p _a )/ - C _pb base pressure coefficient=(p _b –p _a )/ - h _f free convection heat transfer coefficient - average heat transfer coefficient - k thermal conductivity of air - L distance of the axis of the square cylinder from the nozzle exit - l length of the square cylinder - Pr Prandtl number - p static pressure - p _a atmospheric pressure - p _b base pressure on the rear face - Nu _f free convection Nusselt number - average Nusselt number - q heat loss - q _f heat loss due to free convection - Re Reynolds number=u _j W/v _a - T _a ambient air temperature - average surface temperature - u _j average jet velocity at the nozzle exit - W nozzle width - angle of inclination of the square cylinder to the jet axis in degrees - _a kinematic viscosity of air - _a density of air     

Laminar source flow between parallel porous disks with equal suction and injection

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.Nomenclature a distance between disks - A, B, ..., J functions of R _w only - F static pressure - p dimensionless static pressure, p(a ²/ ²) - Q volumetric flow rate of the source - r radial coordinate - r dimensionless radial coordinate, r/a - R radial coordinate of a point in the flow region - R dimensionless radial coordinate of a point in the flow region, R - Re source Reynolds number, Q/2a - R _w wall Reynolds number, Va/ - reduced Reynolds number, Re/r ² - critical Reynolds number - velocity component in radial direction - u dimensionless velocity component in radial direction, a/ - average radial velocity, Q/2a - u dimensionless average radial velocity, Re/r - ratio of radial velocity to average radial velocity, u/u - velocity component in axial direction - v dimensionless velocity component in axial direction, v - V magnitude of suction or injection velocity - z axial coordinate - z dimensionless axial coordinate, z a - viscosity - density - kinematic viscosity, / - shear stress at lower disk - shear stress at upper disk - ₀ dimensionless shear stress at lower disk, - ₁ dimensionless shear stress at upper disk, - dimensionless stream function     

Molecular weight distribution from rheological measurements

Summary A simple and reliable relative method to derive the molecular weight distribution of linear polymers is proposed.It is shown that both the zero-shear viscosity, ₀, and the intrinsic viscosity, [], have a logarithmic dependence on the weight average molecular weight, , and the polydispersity, . The coefficients of these relationships can be determined by applying a multiple regression analysis to a series of samples for which andQ are known.By making use of the two established relationships, the determination of andQ for a given polymer sample reduces to the experimental measurement of its ₀ and [].An analysis has been performed to estimate to what extent experimental errors on ₀ and [] affect the calculated molecular weight distribution.It has been found that only the experimental error on [] contributes heavily to the final error on the polydispersity.

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Zusammenfassung Es wird eine einfache und zuverlässige Relativmethode vorgeschlagen, um die Uneinheitlichkeit linearer Polymere abzuleiten.Es wird gezeigt, daß alle beide, Nullschergradient-viskosität ₀, und Grenzviskositätszahl [], einfach logarithmisch vom Gewichtsmittel des Molekulargewichts , und vom Polymolekularitätsindex , abhängig sind.Die Koeffizienten dieser Beziehungen können mit statistischer Analyse festgesetzt werden, wenn undQ einer Probenreihe bekannt sind.Mit den zwei vorher festgesetzten Beziehungen besteht die Bestimmung von undQ einer gegebenen Polymersprobe nur aus den experimentellen Massen seiner ₀- und []-Werte.Eine Analyse wurde ausgeführt, um die Bedeutung des experimentellen Irrtums über die berechnete Uneinheitlichkeit zu wissen.Es wurde gefunden, daß ein experimenteller Irrtum betreffs [] schwer an endlichem Irrtum der Uneinheitlichkeit teilnimmt.

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With 2 figures and 2 tables     

On the universality of the velocity profiles of a turbulent flow in an axially rotating pipe

If a fluid enters an axially rotating pipe, it receives a tangential component of velocity from the moving wall, and the flow pattern change according to the rotational speed. A flow relaminarization is set up by an increase in the rotational speed of the pipe. It will be shown that the tangential- and the axial velocity distribution adopt a quite universal shape in the case of fully developed flow for a fixed value of a new defined rotation parameter. By taking into account the universal character of the velocity profiles, a formula is derived for describing the velocity distribution in an axially rotating pipe. The resulting velocity profiles are compared with measurements of Reich [10] and generally good agreement is found.Nomenclature b constant, equation (34) - D pipe diameter - l mixing length - l ₀ mixing length in a non-rotating pipe - N rotation rate,N=Re /Re _D - p pressure - R pipe radius - Re _D flow-rate Reynolds number, - Re rotational Reynolds number, Re =v _w D/ - Re* Reynolds number based on the friction velocity, Re*=v*R/ - (Re*)₀ Reynolds number based on the friction velocity in a non-rotating pipe - Ri Richardson number, equation (10) - r coordinate in radial direction - dimensionless coordinate in radial direction, - v _r ,v ,v _z time mean velocity components - v _r ,v ,v _z velocity fluctations - v _w tangential velocity of the pipe wall - v* friction velocity, - axial mean velocity - v _ZM maximum axial velocity - dimensionless radial distance from pipe wall, - y ⁺ dimensionless radial distance from pipe wall - y ₁ ⁺ constant - Z rotation parameter,Z =v _w/v _* =N Re _D /2Re* - _m eddy viscosity - ( _m )₀ eddy viscosity in a non-rotating pipe - coefficient of friction loss - von Karman constant - ₁ constant, equation (31) - density - dynamic viscosity - kinematic viscosity     

Thermal effects in the capillary rheometer

Summary The effect of viscous heating in a capillary rheometer is analysed for a power-law fluid by means of a perturbation expansion based upon a boundary-layer-core structure. This expansion is found to complement the eigenfunction series solution obtained by earlier investigators. A similar analysis is presented for the work-of-expansion effect. These two thermal effects are superimposed together with a third perturbation effect due to the pressure dependence of viscosity.On the basis of the present theory, earlier work in this area is discussed and, in some cases, apparent inaccuracies or inconsistencies are pointed out. A means is indicated for correcting data on the basis of the present theory.

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Zusammenfassung Es wird der Effekt der Erwärmung einer Potenzflüssigkeit infolge viskoser Reibung in einem Kapillar-Rheometer mittels einer Störungsrechnung untersucht, die auf der Unterteilung der Strömung in eine Grenzschicht und einen Kern basiert. Diese Störungsentwicklung ergänzt eine früher von anderen Autoren gefundene Reihenentwicklung mit Hilfe von Eigenfunktionen. Eine ähnliche Untersuchung wird für die thermische Ausdehnungsarbeit durchgeführt. Diese beiden thermischen Effekte sind zusammen einem dritten Störeffekt superponiert, der von der Druckabhängigkeit der Viskosität herrührt.Aufgrund der vorgelegten Theorie werden verschiedene auf diesem Gebiet früher durchgeführte Arbeiten diskutiert, und es werden in einigen Fällen offensichtliche Ungenauigkeiten und Folgewidrigkeiten aufgedeckt. Schließlich wird eine Methode zur Korrektur von Meßdaten mit Hilfe der vorliegenden Theorie angegeben.

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Nomenclature a tube radius - b ; evaluated atT ₀ andp = 0 when used in perturbation expansion - C _p specific heat - f - f ^* - h defined by eq. [15] - k thermal conductivity - L tube length - m defined by eq. [8] - m ₀ m(T₀, 0) - n power-law index - p pressure - Pe C _p W a/k Peclet number - Pr C _pa/k Prandtl number - Q volumetric flow rate - Q ₀ unperturbed value ofQ in specified-p formulation - r radial coordinate - Re W a/ _a Reynolds number - T temperature - T ₀ inlet temperature - u radial velocity component - u ₀ 0 unperturbed radial velocity - w axial velocity component - w ₀ /W(1 – ) unperturbed axial velocity - W Q/(a ²) average axial velocity - W ₀ Q ₀/(a ²) - z axial coordinate - (3n + 1)/n - ^* ; evaluated atT ₀ andp = 0 when used in perturbation expansion - 4^1-n - ^* - (n + 1)/n - ... shear rate - 4W/a apparent shear rate - p total pressure drop - T _a W ²/k characteristic temperature difference - T _b total bulk-temperature rise - ^* T - r/a - shear viscosity - _a m₀ - (1 –)/ ^1/3 - —p/z - ₀ ... unperturbed value of - z-averaged value of - µ ^{n + 1}/n - z/(a Pe) - _L L/(a Pe) - mass density - _w shear stress at wall - streamfunction - ^*T₀ (absolute temperature scale) - ( )₁ leading-order effect due to viscous heating - ( ) ₁ ^* leading-order effect due to work-of-expansion Note: in specified-p formulation,W gets replaced byW ₀ in definition of Pe, Re, and. With 7 figures and 7 tables     

Wärmeübertragung in einem axial rotierenden,durchströmten Rohr im Bereich des thermischen Einlaufs

Zusammenfassung Der Einfluß der Rotation auf das Temperaturprofil und die Wärmeübergangszahl einer turbulenten Rohrströmung im Bereich des thermischen Einlaufs wird theoretisch untersucht und mit Meßwerten verglichen. Es wird angenommen, daß das Geschwindigkeitsprofil voll ausgebildet ist. Die Rotation hat aufgrund der radial ansteigenden Zentrifugalkräfte einen ausgeprägten Einfluß auf die Unterdrückung der turbulenten Bewegung. Dadurch verschlechtert sich die Wärmeübertragung mit steigender Rotations-Reynoldszahl und die thermische Einlauflänge nimmt beträchtlich zu.

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Heat transfer in an axially rotating pipe in the thermal entrance region. Part 1: Effect of rotation on turbulent pipe flow

The effects of rotation on the temperature distribution and the heat transfer to a fluid flowing inside a tube are examined by analysis in the thermal entrance region. The theoretical results are compared with experimental findings. The flow is assumed to have a fully developed velocity profile. Rotation was found to have a very marked influence on the suppression of the turbulent motion because of radially growing centrifugal forces. Therefore, a remarkable decrease in heat transfer with increasing rotational Reynolds number can be observed. The thermal entrance length increases remarkably with growing rotational Reynolds number.

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Formelzeichen a Temperaturleitzahl - C _n , ,C ₁,C ₃ Konstanten - c _p spezifische Wärme bei konstantem Druck - D Rohrdurchmesser - E Funktion nach Gl. (30) - H _n Eigenfunktionen - l hydrodynamischer Mischungsweg - l _q thermischer Mischungsweg - Massenstrom - N=Re /Re Reynoldszahlenverhältnis - Nu Nusseltzahl - Nu Nusseltzahl für die thermisch voll ausgebildete Strömung - Pr Prandtlzahl - Pr _t turbulente Prandtlzahl - Wärmestromdichte - Re _* Schubspannungsreynoldszahl - R _n Eigenfunktionen - Durchfluß-Reynoldszahl - Re v =D/ Rotations-Reynoldszahl - Ri Richardsonzahl - R Rohrradius - r Koordinate in radialer Richtung - dimensionslose Koordinate in radialer Richtung - T Temperatur - T Temperaturschwankung - T _b bulk temperature - mittlere Axialgeschwindigkeit - v Geschwindigkeit - v Geschwindigkeitsschwankung - turbulenter Wärmestrom - dimensionsloser Wandabstand - =1/6 Konstante - Integrationsvariable - Integrationsvariable - , ₁, ₂, dimensionslose Temperaturen - Wärmeleitzahl - _n Eigenwerte - kinematische Viskosität - Dichte - tangentiale Koordinate - , Hilfsfunktionen Indizes m in der Rohrmitte - r radial - w an der Rohrwand - z axial - 0 am Rohreintritt - 0 ohne Rotation - tangential     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)     

Non-isothermal flow of polymer melts in a curved tube

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions. Br — Brinkman number - c specific heat, J/kg K - De — Dean number - E dimensionless apparent viscosity, eq. (14d) - G dimensionless shear rate, eq. (19) - k parameter of the power-law model, °C^–1, eq. (7) - mass flow rate, kg/s - m ₀ parameter of the power-law model, Pa · s ⁿ , eq. (7) - n parameter of the power-law model, eq. (7) - Nu 2r _p/ — Nusselt number, eqs. (28,31) - p pressure, Pa - Pe — Péclet number - P —(p/)/r _c — pressure gradient, Pa/m - dissipated energy, W, eq. (29) - total energy, W, eq. (30) - r radial coordinate, m - r _c radius of tube-curvature, m, fig. 1 - r _p radius of tube, m, fig. 1 - r _t variable, m, eq. (6) - R dimensionless radial coordinate, eq. (14a) - R _c dimensionlessr _c, eq. (14a) - R _t dimensionlessr _t, eq. (14a) - t temperature, °C - bulk temperature, °C, eq. (27) - t ₀ inlet temperature of the melt, °C - t _w tube wall temperature, °C - T dimensionless temperature, eq. (14c) - T _w dimensionless tube wall temperature - T dimensionless bulk temperature - u ₁ variable, s^–1, eq. (4) - u ₂ variable, s^–1, eq. (5) - U ₁ dimensionlessu ₁, eq. (18) - U ₂ dimensionlessu ₂, eq. (18) - v velocity in-direction, m/s - average velocity of the melt, m/s - V dimensionlessv, eq. (14b) - dimensionless , eq. (15) - z r _c — centre length of the tube, m - Z dimensionlessz, eq. (14e) - heat transfer coefficient, W/m² K - shear rate, s^–1, eq. (8) - — shear rate, s^–1 - apparent viscosity, Pa · s, eq. (7) - ₀ — apparent viscosity, Pa · s - angular coordinate, rad, fig. 1 - thermal conductivity, W/m K - melt density, kg/m³ - axial coordinate, rad, fig. 1 - rate of strain tensor, s^–1, eq. (8) - (—p) pressure drop, Pa     

Instability of Buckley-Leverett flow in a heterogeneous medium

We study the simultaneous one-dimensional flow of water and oil in a heterogeneous medium modelled by the Buckley-Leverett equation. It is shown both by analytical solutions and by numerical experiments that this hyperbolic model is unstable in the following sense: Perturbations in physical parameters in a tiny region of the reservoir may lead to a totally different picture of the flow. This means that simulation results obtained by solving the hyperbolic Buckley-Leverett equation may be unreliable.Symbols and Notation f fractional flow function varying withs andx - value off outsideI - value off insideI - local approximation off around¯x - f ^–,f ⁺ values of - f _j ⁿ value off atS _j ⁿ andx _j - g acceleration due to gravity [ms^–2] - I interval containing a low permeable rock - k dimensionless absolute permeability - k ^* absolute permeability [m²] - k _c ^* characteristic absolute permeability [m²] - k _ro relative oil permeability - k _rw relative water permeability - L ^* characteristic length [m] - L ₁ the space of absolutely integrable functions - L the space of bounded functions - P _c dimensionless capillary pressure function - P _c ^* capillary pressure function [Pa] - P _c ^* characteristic pressure [Pa] - S similarity solution - S _j ⁿ numerical approximation tos(x_j, t_n) - S ₁, S₂,S ₃ constant values ofs - s water saturation - value ofs at - s ^L left state ofs (wrt. ) - s ^R right state ofs (wrt. ) - s s for a fixed value of in Section 3 - T value oft - t dimensionless time coordinate - t ^* time coordinate [s] - t _c ^* characteristic time [s] - t _n temporal grid point,t _n=n t - v ^* total filtration (Darcy) velocity [ms^–1] - W, , v dimensionless numbers defined by Equations (4), (5) and (6) - x dimensionless spatial coordinate [m] - x ^* spatial coordinate [m] - x _j spatial grid piont,x _j=j x - discontinuity curve in (x, t) space - right limiting value of¯x - left limiting value of¯x - angle between flow direction and horizontal direction - t temporal grid spacing - x spatial grid spacing - length ofI - parameter measuring the capillary effects - argument ofS - _o dimensionless dynamic oil viscosity - _w dimensionless dynamic water viscosity - _c ^* characteristic viscosity [kg m^–1s^–1] - _o ^* dynamic oil viscosity [kg m^–1s^–1] - _w ^* dynamic water viscosity [k gm^–1s^–1] - _o dimensionless density of oil - _w dimensionless density of water - _c ^* characteristic density [kgm^–3] - _o ^* density of oil [kgm^–3] - _w ^* density of water [kgm^–3] - porosity - dimensionless diffusion function varying withs andx - ^* dimensionless function varying with s andx ^* [kg^–1m³s] - _j ⁿ value of atS _j ⁿ andx _j This research has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).     

Zur energetischen Optimierung von Schleppströmungspumpen mit durchgehend konstantem Arbeitsspalt für Medien mit nicht-newtonschem Fließverhalten

Zusammenfassung Die zum kontinuierlichen Austragen und Ausformen von strukturviskosen und anderen nicht-newtonschen Medien dienenden Schleppströmungspumpen lassen sich bei vorgegebenem Volumendurchsatz und Betriebsdruckp durch Anpassung des Arbeitsspaltesh und der Arbeitsdrehzahln energetisch optimal auslegen und betreiben. Die entsprechende Kennzahl ist der als Quotient aus Austriebs-Leistung p und AntriebsleistungP definierte Pumpwirkungsgrad . — Die optimalen (h, n, )-Werte werden unter der Voraussetzung berechnet, daß sich das Fließverhalten des geförderten Mediums durch einen Polynomansatz nachRabinowitsch beschreiben läßt. Dabei ergibt sich für die optimalen-Werte ein Bereich zwischen etwa 20% und 33%. Rheologische Ansätze mit einer auf eine mittlere Schergeschwindigkeit bezogenen konstanten scheinbaren Viskosität, welche in jedem Fall auf den für newtonsche Medien charakteristischen Idealwert=33% führen, sind hiernach für strukturviskose und andere nicht-newtonsche Medien unzulässig.

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Summary Drag-flow pumps, as used for the continuous extrusion of non-Newtonian fluids, can be operated with minimum drive powerP at a given volume throughput and pressurep, if the radial dimensionh of the drag channel and working speedn are optimized. The key number of this optimization is the efficiency . — Appropriate (h, n, )-values are calculated on the basis of the rheological equation proposed byRabinowitsch. The optimum range of-values is found to be between 20% and 33%, whilst former calculations with an average apparent viscosity resulted in _opt = 33% generally. Obviously, here is one of the causes of discrepancy between theoretical and actual efficiencies of such pumps.

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Symbole a Stoffkennzahl, Gl. [3] - b Breite des Schleppspalts, Abb. 2 - c Stoffkennzahl, Gl. [3] - C ₁ Integrationskonstante, Gl. [5] - C ₂ Integrationskonstante, Gl. [8] - d Durchmesser des rotierenden Elements, Abb. 1 - e spezifische Antriebsleistung, Gl. [18] - h Höhe (Radialmaß) des Schleppspalts, Abb. 1 - k Anzahl der Schleppspalte - m Fließexponent im Potenzansatz - Massedurchsatz - M _d Drehmoment - n Umdrehungsgeschwindigkeit, Arbeitsdrehzahl des rotierenden Elements, Abb. 1 - p Betriebsdruck - p Druckgradient, Gl. [6] - P aufgenommene Antriebsleistung - r radiale Koordinate - r _i=d/2 – h Innenradius des rotierenden Elements - r _a=d/2 Außenradius des rotierenden Elements - s zirkulare Länge des Schleppspalts - t (mittlere) Verweilzeit des Mediums im Schleppspalt - T Temperatur - v lokale zirkulare Geschwindigkeit - v ₀ Umfangsgeschwindigkeit des rotierenden Elements, Abb. 1 - V Volumen des Schleppspalts - Volumendurchsatz der Schleppströmungspumpe - Volumendurchsatz der Druck(gradienten)strömung - Volumendurchsatz der Schleppströmung - dimensionslose Kennzahl, Gl. [22] - Schergeschwindigkeit, Gl. [2] - dimensionsloser Pumpwirkungsgrad, Gl. [1] - µ Scherviskosität - Dichte - Schubspannung, Gl. [2] - zirkulare Koordinate - Fluidität im Potenzansatz - Winkelgeschwindigkeit Erweiterte Fassung eines Vortrages anläßlich des 5. Stuttgarter Kunststoff-Kolloquiums vom 2. März 1977.Mit 14 Abbildungen     

Production of temperature fluctuations in grid turbulence: Wiskind's experiment revisited

Measurements have been made in nearly-isotropic grid turbulence on which is superimposed a linearly-varying transverse temperature distribution. The mean-square temperature fluctuations, , increase indefinitely with streamwise distance, in accordance with theoretical predictions, and consistent with an excess of production over dissipation some 50% greater than values recorded in previous experiments. This high level of production has the effect of reducing the ratio,r, of the time scales of the fluctuating velocity and temperature fields. The results have been used to estimate the coefficient,C, in Monin's return-to-isotropy model for the slow part of the pressure terms in the temperature-flux equations. An empirical expression by Shih and Lumley is consistent with the results of earlier experiments in whichr 1.5, C 3.0, but not with the present data where r 0.5, C 1.6. Monin's model is improved when it incorporates both time scales.List of symbols C coefficient in Monin model, Eq. (5) - M grid mesh length - m exponent in power law for temperature variance, x ^m - n turbulence-energy decay exponent,q ² x ^-n - p production rate of - p pressure - q ² - R microscale Reynolds number - r time-scale ratiot/t - T mean temperature - U mean velocity - mean-square velocity fluctuations (turbulent energy components) - turbulent temperature flux - x, y, z spatial coordinates - temperature gradient dT/dy - thermal diffusivity - dissipation rate ofq ²/2 - dissipation rate of - Taylor microscale (₂=5q₂/) - temperature microscale - _v temperature-flux correlation coefficient, /v - dimensionless distance from the grid,x/M     

Investigation of the flowfield immediately upstream of a hot film probe

The laminar flowfield in a rectangular channel immediately upstream of a hot film gradient probe with two parallel films was investigated in the range of Reynolds number Re _pr= 6 to 95, with the Reynolds number based on the probe diameter and the local flow velocity. For this study a photochromic dye flow visualization technique was used. The results show that the smaller the Reynolds number Re _prthe larger the influence of the probe is upon the flowfield. No distinct influence of the probe location relative to the channel walls on the flow deceleration process immediately upstream of the probe was observed.List of symbols a distance between the hot films - d _h hydraulic diameter - d _pr diameter of the probe body - Reynolds number based on hydraulic diameter and mean flow velocity - Reynolds number based on probe diameter and the undisturbed flow velocity at the centerline of probe - u flow velocity in x-direction - u ₀ undisturbed velocity in the center of the channel - undisturbed mean flow velocity - u(x,y) velocity at position (x,y) - averaged velocity gradient - x coordinate in main flow direction - y coordinate normal to the larger wall of the rectangular channel - z coordinate normal to x and y - v kinematic viscosity     

Time averaged temperature distribution in a cylindrical resistor with alternating current

The non-uniform heat generation in a cylindrical resistor crossed by an alternating electric current is considered. The time averaged and dimensionless temperature distribution in the resistor is analytically evaluated. Two dimensionless functions are reported in tables which allow one to determine the time averaged temperature field for arbitrarily chosen values of the physical properties and of the radius of the resistor, of the electric current frequency, of the Biot number and of either the power generated per unit length or the effective electric current.

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Zeitliche Temperaturverteilung in einem zylinderförmigen Wechselstromwiderstand

Zusammenfassung Es wird ungleichförmige Wärmeerzeugung in einem mit Wechselstrom belasteten Widerstand unterstellt, woraus sich die darin einstellende, zeitlich gemittelte, dimensionslose Temperaturverteilung analytisch berechnen läßt. Zwei tabellierte dimensionslose Funktionen gestatten die Bestimmung dieser Temperaturverteilung für beliebige Werte der Stoff- und Feldparameter, des Widerstandhalbmessers, der elektrischen Frequenz, der Biot-Zahl, sowie der erzeugten Leistung pro Längeneinheit oder des effektiven Stroms.

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Nomenclature A intregration constant introduced in Eq. (15) - Bi Biot numberhr ₀/ - c speed of light in empty space - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E amplitude of the electric field oscillations - electric permittivity - f function ofs and defined in Eq. (22) - function of defined in Eq. (45) - g function ofs and defined in Eq. (34) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=–1 - I electric current - I _eff effective electric currentI _eff=I/2 - Im imaginary part of a complex number - J current density - J _n Bessel function of first kind and ordern - thermal conductivity - magnetic permeability - ₀ magnetic permeability of free space - q _g power generated per unit volume - time average of the power generated per unit volume - Q time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - mass density - s dimensionless radial coordinates=r/r ₀ - s,s integration variables - electric conductivity - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - dimensionless temperature defined in Eq. (27) - x position vector - x arbitrary real variable - x integration variable - Y ₀ Bessel function of second kind and order 0 - z axial coordinate - z unit vector parallel to the axis of the cylinder - angular frequency - dimensionless parameter =r₀ - · modulus of a complex number - equal by definition     

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)     

Eine stationäre Meßmethode zur Bestimmung der Relaxationszeit viskoelastischer Fluide bei räumlicher Beanspruchung

Zusammenfassung Es wird gezeigt, daß bei Kenntnis der Fließkurve viskoelastischer Flüssigkeiten allein aus der Drehmomentkennlinie des stationär betriebenen Kugel-Kugel-Rheometers eine Relaxationszeit der räumlichen Beanspruchung bestimmt werden kann. Ausgehend von derColeman-Nollschen Entwicklungsschreibweise der rheologischen Zustandsfunktion wird das Geschwindigkeitsfeld als Potenzreihe der Kreisfrequenz bis zur 3. Ordnung bestimmt und zur Drehmomentbeziehung integriert.Messungen an einigen Versuchssubstanzen bestätigen die Tauglichkeit der entwickelten Methode.Häufig verwendete Formelzeichen –a N/m² isotroper Druckanteil - m/s Geschwindigkeitsvektor - e ₁₄ Integrationskonstanten - f _i() Geometriefunktionen - m vektorielle Feldfunktion - ms vektorielle Feldfunktion - ms² vektorielle Feldfunktion - k _i() Geometriefunktionen - t ₀ s Relaxationszeit der räumlichen Beanspruchung - m/s Geschwindigkeitsvektor erster Ordnung - m/s Geschwindigkeitsvektor zweiter Ordnung - m/s Geschwindigkeitsvektor dritter Ordnung - D 1/s Deformationsgeschwindigkeitstensor - 1/s², 1/s³ korotationale, zeitliche Ableitung vonD - 1 Einheitstensor - M Nm Antriebsmoment der rotierenden Kugel - M _i Nm Teilmomente - R m Kugelradius - R _G m Hohlkugelradius - S N/m² Spannungstensor - W 1/s Rotationsgeschwindigkeitstensor - ₁ N s/m² Stoffparameter 1. Ordnung - ₂, ₃ N s²/m² Stoffparameter 2. Ordnung - ₄, ₅, ₆ N s³/m² Stoffparameter 3. Ordnung - RadienverhältnisR/R _G - ₀ N s/m² Anfangsviskosität - kg/m³ Dichte der Flüssigkeit - 1/s Kreisfrequenz der rotierenden Kugel Vorgetragen auf dem 6. Internationalen Rheologie-Kongreß in Lyon-Frankreich vom 4.–8. September 1972.Jetzt: BASF-AG, LudwigshafenMit 4 Abbildungen     

Wärmeübertragung in einem axial rotierenden,durchströmten Rohr im Bereich des thermischen Einlaufs

Zusammenfassung Der Einfluß der Rotation auf das Temperaturprofil und die Wärmeübergangszahl einer laminaren Rohrströmung im Bereich des thermischen Einlaufs wird theoretisch untersucht. Es wird angenommen, daß das Geschwindigkeitsprofil voll ausgebildet ist. Die Rotation hat einen destabilisierenden Einfluß auf die Laminarströmung, die umschlägt und turbulent wird. Aufgrund der Anfachung der Turbulenz durch die Rotation verbessert sich die Wärmeübertragung mit steigender Rotations-Reynoldszahl und die thermische Einlauflänge nimmt beträchtlich ab.

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Heat transfer in an axially rotating pipe in the thermal entrance region. Part 2: Effect of rotation on laminar pipe flow

The effects of tube rotation on the temperature distribution and the heat transfer to a fluid flowing inside a tube are examined by analysis in the thermal entrance region. The flow is assumed to be hydrodynamically fully developed. The rotation has a destabilizing effect on the laminar pipe flow, causing a transition to turbulent flow. Therefore, a remarkable increase in heat transfer with increasing rotational Reynolds number can be observed. The thermal entrance length decreases remarkably with growing rotational Reynolds number.

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Formelzeichen a Temperaturleitzahl - C _n , ,C ₁,C ₃ Konstanten - c _p spezifische Wärme bei konstantem Druck - D Rohrdurchmesser - E Funktion nach Gl. (16) - H _n Eigenfunktionen - l hydrodynamischer Mischungsweg - l _q thermischer Mischungsweg - N=Re /Re Reynoldszahlenverhältnis - Nu Nusseltzahl - Nu Nusseltzahl für die thermisch voll ausgebildete Strömung - Pr Prandtlzahl - Pr _t turbulente Prandtlzahl - Wärmestromdichte - Re _* Schubspannungsreynoldszahl - R _s Eigenfunktionen - Durchfluß-Reynoldszahl - Re =v D/v Rotations-Reynoldszahl - R Rohrradius - r Koordinate in radialer Richtung - Dimensionslose Koordinate in radialer Richtung - T Temperatur - T _b bulk temperature - mittlere Axialgeschwindigkeit - v Geschwindigkeit - dimensionsloser Wandabstand - Integrationsvariable - Integrationsvariable - , dimensionslose Temperaturen - Wärmeleitzahl - _p Eigenwerte - kinematische Viskosität - Dichte - tangentiale Koordinate - , Hilfsfunktionen Indizes m in der Rohrmitte - r radial - w an der Rohrwand - z axial - 0 am Rohreintritt - 0 ohne Rotation - tangential