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.     

Drag reduction by polymer solutions in a riblet-lined pipe

The flow of 3 to 100 wppm aqueous solutions of a polyethyleneoxide polymer,M _w=6.2×;10⁶, was studied in a 10.2 mm i.d. pipe lined with 0.15 mm V-groove riblets, at diametral Reynolds numbers from 300 to 150000. Measurements in the riblet pipe were accompanied by simultaneous measurements in a smooth pipe of the same diameter placed in tandem. The chosen conditions provided turbulent drag reductions from zero to the asymptotic maximum possible. The onset of polymer-induced drag reduction in the riblet pipe occurred at the same wall shear stress, * _w =0.65 N/m², as that in the smooth pipe. After onset, the polymer solutions in the riblet pipe initially exhibited linear segments on Prandtl-Karman coordinates, akin to those seen in the smooth pipe, with specific slope increment . The maximum drag reduction observed in the riblet pipe was independent of polymer concentration and well below the asymptotic maximum drag reduction observed in the smooth pipe. Polymer solution flows in the riblet pipe exhibited three regimes: (i) Hydraulically smooth, in which riblets induced no drag reduction, amid varying, and considerable, polymer-induced drag reduction; this regime extended to non-dimensional riblet heightsh ⁺<5 in solvent andh ⁺<10 in polymer solutions. (ii) Riblet drag reduction, in which riblet-induced flow enhancementR>0; this regime extended from 5<h ⁺<22 in solvent and from 10<h ⁺<30 in the 3 wppm polymer solution, with respective maximaR=0.6 ath ⁺=14 andR=1.6 ath ⁺=21. Riblet drag reduction decreased with increasing polymer concentration and increasing polymer-induced flow enhancement S. (iii) Riblet drag enhancement, whereinR<0; this regime extended for 22<h ⁺<110 in solvent, withR;–2 forh ⁺>70, and was observed in all polymer solutions at highh ⁺, the more so as polymer-induced drag reduction increased, withR<0 for allS>8. The greatest drag enhancement in polymer solutions,R=–7±1 ath ⁺=55 whereS=20, considerably exceeded that in solvent. Three-dimensional representations of riblet- and polymer-induced drag reductions versus turbulent flow parameters revealed a hitherto unknown dome region, 8<h ⁺<31, 0<S<10, 0<R<1.5, containing a broad maximum at (h ⁺,S,R) = (18, 5, 1.5). The existence of a dome was physically interpreted to suggest that riblets and polymers reduce drag by separate mechanisms.     

Eine Fluoreszenzmethode zur Untersuchung des Transportmechanismus bei der Gasabsorption im Rieselfilm

Zusammenfassung Gewissep h-Indikatoren zeigen bei einem bestimmtenp h-Wert eine stufenförmige Änderung der Fluoreszenzfähigkeit. Diese Eigenschaft wurde für die Untersuchung der Absorption eines sauren oder basischen Gases in einem Rieselfilm ausgenutzt; das Gas geht eine schnelle Neutralisationsreaktion mit der basischen bzw. sauren Rieselflüssigkeit ein, die den Indikator enthält. Der Rieselfilm wird durch ultraviolette Strahlung zur Fluoreszenz angeregt; aus der gemessenen Lichtemission wird der Abstand zwischen der Phasengrenze und der Neutralfläche im Film ermittelt. Es ist möglich, aus den Meßergebnissen die Konzentrationsverteilung in der Flüssigkeit und daraus die Abhängigkeit des effektiven Diffusionskoeffizienten vom Abstand zur Phasengrenze zu berechnen. Letztere bestätigt das verallgemeinerte Diffusionsmodell des Stofftransportes. Blitzaufnahmen zeigen, daß die Absorption nicht gleichmäßig über die Fläche verteilt ist, sondern es bilden sich flecken- oder streifenförmige Muster.

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The fluorescence emission of certainp h-indicators changes according to a step function at a particularp h value. This property has been used to study the absorption of an acidic or alkaline gas into a falling film; the gas undergoes an instantaneous neutralisation reaction with a dilute solution of an alkali or acid containing the indicator. The falling film is exposed to ultraviolet radiation; from the measured intensity of fluorescence the distance between the phase boundary and the neutralisation plane within the film is evaluated. From these measurements one can calculate the concentration distribution in the liquid and consequently the variation of effective diffusivity with distance from the interface. The latter supports the generalised diffusion model of mass transfer. It is evident from flash photographs that absorption is not uniform across the surface but shows characteristic patterns of spots or stripes.

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Bezeichnungen A Konzentration der absorbierten Komponente im Rieselfilm - A₀ Eintrittskonzentration der zu absorbierenden Komponente in der Gasphase - A_0z KonzentrationA₀, erforderlich für Fluoreszenzbeginn bei Koordinate z - A* formale Konzentration=A+(B ₀–B)D _B/D _A - B Konzentration des Reaktionspartners im Rieselfilm - B ₀ B am Zulauf - c Konzentration - c _i Indikatorkonzentration - D molekularer Diffusionskoeffizient - D _eff effektiver Diffusionskoeffizient - d Dicke des Rieselfilms (zeitlicher Mittelwert) - g Erdbeschleunigung - I Fluoreszenzintensität (fotoelektrische Anzeige) - I _k Eigenfluoreszenz der Küvette - I _n Nullwert der Fluoreszenz=dc _i i ₀+I _k - i spezifische Fluoreszenzintensität des Indikators beid=1 mm undc _i=1 n - i ₀ i imp h-Bereich schwacher Fluoreszenzfähigkeit - i ₁ i imp h-Bereich starker Fluoreszenzfähigkeit - k beliebiger Faktor - k Geschwindigkeitskonstante für Reaktion 1. Ordnung - k Geschwindigkeitskonstante für Reaktion 2. Ordnung - q absorbierte Mengenstromdichte - t Zeit - w(x) Strömungsgeschwindigkeit in der Flüssigkeit - w _s Strömungsgeschwindigkeit der Flüssigkeitsoberfläche - mittlere Strömungsgeschwindigkeit der Flüssigkeit - x Abstand von der Flüssigkeitsoberfläche - x _N Dicke der stark fluoreszierenden Schicht - z Abstand vom Beginn des Rieselfilms - _g Stoffübergangskoeffizient auf der Gasseite - kinematische Viskosität der Flüssigkeit - Grenzflächenspannung - Re _fl Reynoldszahl der Flüssigkeit= - Re _g Reynoldszahl der Gasströmung Über Teile der vorliegenden Arbeit wurde auf dem Jahrestreffen der Verfahrensingenieure im Oktober 1966 in Hamburg vorgetragen. Eine Zusammenfassung wurde veröffentlicht [1].     

Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

Flow of soft solids squeezed between planar and spherical surfaces

The force to squeeze a Herschel–Bulkley material without slip between two approaching surfaces of various curvature is calculated. The Herschel–Bulkley yield stress requires an infinite force to make plane–plane and plane–concave surfaces touch. However, for plane–convex surfaces this force is finite, which suggests experiments to access the mesoscopic thickness region (1–100 m) of non-Newtonian materials using squeeze flow between a plate and a convex lens. Compared to the plane–parallel surfaces that are used most often for squeeze flow, the dependence of the separation h and approach speed V on the squeezing-time is more complicated. However, when the surfaces become close, a simplification occurs and the near-contact approach speed is found to vary as V h⁰ if the Herschel–Bulkley index is n<1/3, and V h^(3n-1)/(2n) if n 1/3. Using both plane–plane and plane–convex surfaces, concordant measurements are made of the Herschel–Bulkley index n and yield stress ₀ for two soft solids. Good agreement is also found between ₀ measured by the vane and by each squeeze-flow method. However, one of the materials shows a limiting separation and a V(h) behaviour not predicted by theory for h<10 m, possibly owing to an interparticle structure of similar lengthscale.     

Natural convection in rectangular enclosures with adiabatic fins attached on the heated wall

The heat transfer by natural convection in vertical and inclined rectangular enclosures with fins attached to the heated wall is numerically studied using the energy and Navier-Stokes equations with the Boussinesq approximation. The range of study covers 10⁴Ra2×10⁵,A=H/L=2.5 to ,B=l/L=0 to 1,C=h/L=0.25 to 2 andPr=0.72. The inclination angle from the vertical was from 0 to 60 degree. The variation of the local Nusselt numberNu _loc along the enclosure height and the average Nusselt numberNu as a function ofRa are computed. Streamlines and isotherms in the enclosure are produced. The results show thatB is an important parameter affecting the heat transfer through the cold wall of the enclosure. The heat transfer is reduced for decreasingC and it passes from a maximum for an inclination angle. The results show that the heat transfer can generally be reduced using appropriate geometrical parameters in comparison with a similar enclosure without fins.Die Wärmeübertragung bei freier Konvektion in vertikalen und geneigten rechtwinkligen Behältern mit Rippen an den beheizten Wänden wird unter Verwendung der Energie- und Navier-Stokes-Gleichungen sowie der Boussinesq-Approximation numerisch untersucht. Der Bereich der Studie liegt bei 10⁴Ra2·10⁵,A=H/L=2,5 bis ,B=l/L=0 bis 1,C=h/L=0,25 bis 2 undPr=0.72. Der Neigungswinkel der Wand liegt zwischen 0 und 60 Grad. Die Veränderung der lokalen Nusselt-Zahl entlang der Höhe der Behälterwände und die mittlere Nusselt-Zahl in Abhängigkeit derRa-Zahl werden berechnet. Strömungslinien und Isothermen werden im Behälter erzeugt. Die Ergebnisse zeigen, daßB ein wichtiger Parameter für die Wärmeübertragung an der nicht beheizten Wand des Behälters ist. Die übertragene Wärmemenge verringert sich mit abnehmendemC und durchschreitet ein Maximum für eine bestimmte Wandneigung. Die Ergebnisse zeigen, daß im Vergleich zu einer Anordnung ohne Rippen, die Wärmeübertragung bei geeigneten geometrischen Parametern allgemein reduziert werden kann.     

Free convection effects on the oscillatory flow of an electrically conducting rarefied gas past an infinite, vertical plate with constant suction

An analysis of a two-dimensional, unsteady flow of an electrically conducting, viscous, incompressible rarefied gas past an infinite vertical porous plate is carried out under the following assumptions: (i) the suction velocity normal to the plate is constant (ii) the free stream velocity oscillates in time about a constant mean (iii) the plate temperature is constant (iv) the difference between the temperature of the plate and the free stream is moderately large causing the free convection currents (v) first order velocity-slip and the temperature jump boundary conditions (vi) transverse magnetic field (vii) induced magnetic field is negligible.Approximate solutions to the coupled, non-linear equations governing the flow are derived for the mean velocity, mean temperature, mean-skin-friction, mean rate of heat transfer, transient velocity and temperature, fluctuating parts of the velocity profiles, the amplitude and the phase of the skin-friction and the rate of heat-transfer. They are shown graphically followed by a discussion. The effects of ±G (Grashof number), ±E (Eckert number), M (Magnetic field parameter), h ₁ (rarefaction parameter), h ₂ (temperature jump coefficient), (frequency) are discussed for heating (G<0) or cooling (G>0) of the plate by the free convection currents.Nomenclature |B| amplitude of skin-friction - B ₀ applied magnetic field - c _p specified heat at constant pressure - E Eckert number - f ₁ Maxwell's reflection coefficient - f ₂ thermal accommodation coefficient - g _x acceleration due to gravity - G Grashof number - h ₁ rarefaction parameter (L ₁ v ₀/) - h ₂ non-dimensional temperature jump coefficient (L ₂ v ₀/) - k thermal conductivity - K _n Knudsen number - L mean free path - L ₁ (2–f ₁)L/f ₁ - L ₂ - l ₁ characteristic length - M magnetic field parameter - M _r, M _i fluctuating parts of velocity - m - P Prandtl number - p pressure - q rate of heat transfer - q _m mean rate of heat transfer - |Q| amplitude of rate of heat transfer - R suction Reynolds number - T temperature of fluid - T _w temperature of the plate - T temperature of the fluid in free stream - t time - t dimensionless time - U free stream velocity - U dimensionless free stream velocity - U mean of U(t) - u, v velocity components in x, y directions - u dimensionless velocity in x direction - u ₀ mean velocity - u ₁ fluctuating part of velocity - v ₀ suction velocity - x, y coordinate system - x, y dimensionless coordinates - frequency of the free stream oscillations - dimensionless frequency - dimensionless temperature - ₁ fluctuating part of temperature - phase angle of skin-friction - phase angle of rate of heat transfer - density of the fluid in the boundary layer - density of the fluid in the free stream - viscosity - kinematic viscosity - electrical conductivity of the fluid - small positive constant - skin-friction - _m mean skin-friction - specific heat ratio - ₁ coefficient of volume expansion     

Vibration of rectangular orthotropic plates

Summary An approximate analytical procedure has been given to solve the problem of a vibrating rectangular orthotropic plate, with various combinations of simply supported and clamped boundary conditions. Numerical results have been given for the case of a clamped square plate.Nomenclature 2a, 2b sides of the rectangular plate - h plate thickness - E _x , E _y , E, G elastic constants of te orthotropic material - D _x E _x h ³/12 - D _y E _y h ³/12 - H _xy Eh ³/12+Gh ³/6 D _x , D _y and H _xy are rigidity constants of the orthotropic plate - mass of the plate per unit area - Poisson's ratio - W deflection of the plate - p circular frequency - b/a ratio - X _m , Y _n characteristic functions of the vibrating beam problem - p ² a ² b ²/H _xy the frequency parameter.     

Some flow characteristics of lobed forced mixers at higher velocity ratios

The flow characteristics of two types of lobed forced mixers, the unscalloped and the scalloped mixers, have been examined at velocity ratios higher than unity, in relation to the variation of mass flux uniformity, the decay of the streamwise vorticity, the variation of turbulent kinetic energy and the growth of the shear layer with distance from the trailing edge. Three trailing edge configurations have also been considered for each type of mixer, namely a square wave, a semi-circular wave and a triangular wave. The analysis showed that the strength of the streamwise vorticity shed at the trailing edge and the subsequent decaying rate with downstream distance are found to be very important in studying the mixing effectiveness of the lobed mixers.List of Symbols C _I normalized streamwise circulation, _s/U _r h tan - _s streamwise circulation - k turbulent kinetic energy = 1/2(u²+v²+w²) - Re Reynolds number, U _r /=2.27×10⁴ - h (=) Lobe height, 33 mm - U ₁, U ₂ mean velocity of the slow and fast streams - U _r reference mean velocity, (U ₁ + U ₂)/2 = 10 m/s - U, u streamwise mean and the corresponding rms velocities - V, v horizontal mean and the corresponding rms velocities - W, w vertical mean and the corresponding rms velocities - x,y, z streamwise, horizontal and vertical directions - A _wake cross-sectional area bounded by the wake region. The wake region boundary is defined at the region bounded by one half of a lobe along the y/ direction and at the locations along the z/ direction where U ₂/U _r2 and U ₁/U _r1<0.95. - nominal lobe wavelength, 33 mm - half of the included divergent angle of the penetration region, 22° - U uniformity factor - momentum thickness Financial supports from the Applied Research Grant is gratefully acknowledged. The contribution of Mr. J. K. L. Teh, Dr. J. H. Yeo and Mr. T. H. Yip to the work presented here are sincerely appreciated.     

Pseudoähnlichkeitslösung des Rayleighschen Problems für reinviskose nicht-newtonsche Flüssigkeiten

Zusammenfassung Die exakte Ähnlichkeitslösung des Problems der nichtstationären Strömung einer hypothetischen Potenzflüssigkeit in der Umgebung einer ruckartig beschleunigten Platte (Rayleighsches Problem) wird zur Konstruktion einer Näherungslösung des analogen rheodynamischen Problems für reinviskose nicht-newtonsche Flüssigkeiten benutzt. Die Überprüfung der Genauigkeit der genäherten Pseudoähnlichkeitslösung basiert auf Berechnungen der Residuen der integralen Bilanzen des Impulses und der mechanischen Energie. Numerische Ergebnisse dieses Problems werden für das Powell-Eyringsche Modell der Viskositätsfunktion angegeben.

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Summary The exact similarity solution of the problem of the unsteady flow of a hypothetic power-law liquid near a suddenly started plate (Rayleigh's problem) is employed for the construction of an approximative solution of the same problem for arbitrary purely viscous non-Newtonian liquids. The testing of accuracy of this approximative pseudosimilarity solution is based on calculation of residua in the macroscopic balances of momentum and mechanical energy. Numerical results are reported for the Powell-Eyring model of the viscosity function.

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Symbole a Parameter der Ähnlichkeitslösung, definiert durch die Gln. [21] und [22],a =C ₀/B ₀ - B ₀ Parameter der Ähnlichkeitslösung, definiert durch Gl. [22] - B ₁ Parameter der Ähnlichkeitslösung, definiert durch Gl. [37] - C ₀ Parameter der Ähnlichkeitslösung, definiert durch Gl. [21] - D Differentialoperator, Gl. [10a, b] - Ey Kennzahl der rheologischen Ähnlichkeit für das Powell-Eyringsche Modell der Viskositätsfunktion, Gl. [43]G = / _I G _w = _w/ _I - K Konsistenzkoeffizient, Parameter des Potenzmodells der Viskositätsfunktion [7] - K scheinbarer Konsistenzkoeffizient, Gl. [23b] - n Fließindex des Potenzmodells der Viskositätsfunktion [7] - n scheinbarer Fließindex - n Index der logarithmischen Konvexität der Viskositätsfunktion - r unabhängige Veränderliche in der Ähnlichkeitslösung des Rayleighschen Problems, Gl. [17] - r unabhängige Veränderliche der Pseudoähnlichkeitslösung, Gl. [26]S = / _I S _w = _w/_I - t Zeit - T normierte Zeitvariable, Gl. [9c] - u ₀ Geschwindigkeit der Platte - U (Y, T) Pseudoähnlichkeitsnäherung des FeldesV (Y, T) - V (Y, T) normiertes Geschwindigkeitsfeld, Gl. [9a — c] - Y normierte Entfernung von der Platte, Gl. [9b] - W(r; n) Ähnlichkeitsdarstellung des Geschwindigkeitsfeldes, Gl. [20a, b] - Schergeschwindigkeit, Gl. [4] - _I Schergeschwindigkeits-Parameter der Viskositätsfunktion (Stoffkonstante) - _w momentane Schergeschwindigkeit an der Wand - _M normiertes Residuum der Impulsbilanz - _E normiertes Residuum der Bilanz der mechanischen Energie - dimensionsloser Parameter, definiert durch die Gln. [13] und [14] - dimensionsloser Parameter, definiert durch Gl. [23c] - Dichte - ₁₂, Schubspannung bei einer viskometrischen Strömung - [] Viskositätsfunktion - _I Schubspannungs-Parameter der Viskositätsfunktion (Stoffkonstante) - _w momentane Schubspannung an der Wand Mit 2 Abbildungen     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

Shear-induced aggregation and break up of fibril clusters close to the percolation concentration

To probe the behaviour of fibrillar assemblies of ovalbumin under oscillatory shear, close to the percolation concentration, c_p (7.5%), rheo-optical measurements and Fourier transform rheology were performed. Different results were found close to c_p (7.3%), compared to slightly further away from c_p (6.9 and 7.1%). For 6.9 and 7.1%, a decrease in complex viscosity, and a linear increase in birefringence, n, with increasing strain was observed, indicating deformation and orientation of the fibril clusters. For 7.3%, a decrease in complex viscosity was followed by an increase in complex viscosity with increasing strain, which coincided with a strong increase in n, dichroism, n, and the intensity of the normalized third harmonic (I₃/I₁). This regime was followed by a second decrease in complex viscosity, where n,n and I₃/I₁ decreased. In the first regime where the viscosity was decreasing with increasing strain, deformation and orientation of existing clusters takes place. At higher oscillatory shear, a larger deformation occurs and larger structures are formed, which is most likely aggregation of the clusters. Finally, at even higher strains, the clusters break up again. An increase in complex viscosity, n, n and I₃/I₁ was observed when a second strain sweep was performed 30 min after the first. This indicates that the shear-induced cluster formation and break up are not completely reversible, and the initial cluster size distribution is not recovered after cessation of flow.     

Laminar flow of non-Newtonian liquid films inside a vertical pipe

Summary Equations describing the laminar, gravitationally accelerated flow of power-law liquid films flowing inside of a vertical pipe have been studied mathematically. The flow depends on the values of the flow behavior index, the initial film thickness, the magnitude of the initial uniform velocity, and the distance traveled by the fluid along the vertical wall. Values for the film thickness, boundary-layer thickness as well as entrance length are obtained numerically. The numerical solutions show the magnitude of deviations from the flow of Newtonian fluids that can be expected with power-law fluid characteristics.

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Zusammenfassung Es werden die Gleichungen, welche eine laminare schwerkraftbeschleunigte Filmströmung von Ostwald-deWaele-Flüssigkeiten längs der Innenseite eines senkrechten Rohrs beschreiben, mathematisch analysiert. Die Strömung hängt vom Fließindex, der anfänglichen Filmdicke, der Größe der gleichförmigen Anfangsgeschwindigkeit und der Laufstrecke ab. Die Werte von Filmdicke und Eintrittslänge werden numerisch bestimmt. Diese Lösungen zeigen die Größenordnung der Abweichungen von der Strömung newtonscher Flüssigkeiten, die bei Ostwald-deWaele-Flüssigkeiten erwartet werden können.

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G Dimensionless parameter, (3U ₀) ⁿ m/(gh ₀ ⁿ⁺¹ ) - g Gravitational acceleration, ft/sec² - h Liquid film thickness, ft;h ₀ initial value; - k Fluid consistency index, lb _f ^sec nft^–2 - m k/, lb _f lb _m ^-1 sec ⁿ ft - n Flow behavior index, dimensionless - R Radius of the considered curved surface, ft - r Radial distance in cylindrical coordinates, ft - U ₀ Initial velocity, ft/sec - U _s Free stream velocity, ft/sec - V _r Velocity component inr-direction in cylindrical coordinates, ft/sec - V _z Velocity component inz-direction in cylindrical coordinates, ft/sec - z Coordinate axis, distance from original in direction of flow, ft; , dimensionless - z ^* Entrance length, ft; , dimensionless - Boundary-layer thickness, ft; ; ^* = at end of entrance region; - Defined by Eq. [8], dimensionless; ^* = when - Dimensionless parameter, (R –r)/ - Fluid density, lb _m /ft³ - _rz Shear stress inz-direction on surface normal tor, lb _f /ft² With 4 figures and 1 table     

The effect of canopy roughness density on the constitutive components of the dispersive stresses

How to represent the effects of variable canopy morphology on turbulence remains a fundamental challenge yet to be confronted. Planar averaging over some minimal area can be applied to average-out this sort of spatial variability in the time-averaged mean momentum balance. Because of the multiply connected air-spaces, spatial averaging gives rise to covariance or dispersive stress terms that are produced by the spatial correlations of the time-averaged quantities. These terms are “unclosed” and require parameterization, which to date remains lacking due to the absence of data. Here, flume experiments were conducted to quantify the magnitude and sign of the dispersive stresses for a cylindric canopy where the rod density was varied but the individual rod dimensions (rod height h _c and rod diameter d _r) remained the same. Quadrant analysis was used to explore the genesis of their spatial coherency inside the canopy for a wide range of rod densities. When compared to the conventional turbulent stresses, these dispersive stresses can be significant in the lowest layers of sparse canopies. For dense canopies, the dispersive terms remain negligible when compared to the conventional momentum fluxes at all the canopy levels consistent with previous experiments in vegetated and urban canopies. It was also shown that the spatial locations contributing most to the dispersive terms were in the immediate vicinity downstream of the rods. In the deeper layers of sparse canopies, these positions contributed large and negative stresses, but in the upper levels of the canopy, they contributed large but positive stresses. Because the longitudinal velocity spatial perturbation behind the rods is negative, the switch in sign in these stresses was connected with the sign of the vertical velocity spatial perturbation Simplified scaling arguments, using a reduced mean continuity equation and the vertical mean momentum balance for the flow field near the rods, offer clues as to why in much of the lower canopy levels (about 0.75 h _c ) while in the upper canopy levels.     

Some measurements in a binary gas jet

Simultaneous measurements of species volume concentration and velocities in a helium/air binary gas jet with a jet Reynolds number of 4,300 and a jet-to-ambient fluid density ratio of 0.64 were carried out using a laser/hot-wire technique. From the measurements, the turbulent axial and radial mass fluxes were evaluated together with the means, variances and spatial gradients of the mixture density and velocity. In the jet near field (up to ten diameters downstream of the jet exit), detailed measurements of u/ ₀ U ₀, v/ ₀ U₀, u v/ ₀ U ₀ ² , u ² / ₀ U ₀ ² and v ² / ₀ U ₀ ² reveal that the first three terms are of the same order of magnitude, while the last two are at least one order of magnitude smaller than the first three. Therefore, the binary gas jet in the near field cannot be approximated by a set of Reynolds-averaged boundary-layer equations. Both the mean and turbulent velocity and density fields achieve self-preservation around 24 diameters. Jet growth and centerline decay measurements are consistent with existing data on binary gas jets and the growth rate of the velocity field is slightly slower than that of the scalar field. Finally, the turbulent axial mass flux is found to follow gradient diffusion relation near the center of the jet, but the relation is not valid in other regions where the flow is intermittent.     

Wind-driven nonlinear oscillations of cables

The objective of this paper is the study of the dynamics of damped cable systems, which are suspended in space, and their resonance characteristics. Of interest is the study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a first-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables are acted upon by their own weight in the perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes are used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations.In order to investigate the behavior of the cable motion in detail, the linear and the nonlinear analyses are discussed separately. The first part of this paper deals with the solution to the self adjoint boundary-value problem for small-amplitude vibrations and the determination of mode shapes and natural frequencies. The second problem dealt with in this paper is the determination of the phenomena produced by the primary resonance of the system. The method of multiple time scales is used to develop solutions for the resulting multi-dimensional dynamical system with quadratic nonlinearity.Numerical results for the steady state response amplitude, and their variation with external excitation and external detuning for various values of internal detuning parameters are obtained. Saturation and jump phenomena are also observed. The jump phenomenon occurs when there are multi-valued solutions and there exists a variation of kinetic energy among solutions.Notation A=diag(a _i ,i=1, 2, 3) amplitude matrix (diagonal) - A _n,A undeformed area, deformed area - B span of hanging cables - D sag for static conditions - E Young's modulus - vector of external force - diagonal matrix - symmetric coefficient matrix - H ^* =HR I unit matrix - diagonal matrix - L original length of cables before hanging - M the symmetric stiffness matrix - N integer - P damping constant matrix (diagonal) - R linear mode shape matrix (diagonal) - S sway of hanging cables - T tension of cables - T _o tension of cables for static conditions - T _o(0) tension of the lowest point for static conditions - V eigenfunction matrix - b=y ^T R coefficient vector - b - c,c ₁,c ₂,c ₃ vector, and the components in thex ₁,x ₂,x ₃ directions respectively, in terms of cosine functions. - e, e _o strain, and static strain of elongation - e ₁ time-dependent perturbation ine - f wind force in the sway direction - f, f ₀,f ₁ vector of external force - g gravity constant - h time-dependent amplitude vector - m mass density per unit length of the undeformed cable - r=(R ₁,R ₂,R ₃) ^T vector of modal shapes - s undeformed arc length - t time - u ₁ linear scalar in z - u ₂ quadratic scalar in z - v ₁,v ₂,v ₃ eigenfunctions inx ₁,x ₂, andx ₃ directions, respectively - x=(x ₁,x ₂,x ₃) ^T Cartesian position vector and components - y=(y ₁,y ₂,y ₃) ^T static position vector and components - error vector - matrix operator - =diag[₁, ₂, ₃] internal frequency matrix and components - excitation frequency - global matrix of coordinate functions - T _o(0)/mgL - mgL/EA _o - yy ^T - s/L - = diag[₁, ₂, ₃] phase angle matrix and components of characteristic modes - phase angle of excitation force - ₁, ₂ time-dependent amplitude vectors in timet _o and timet ₁ - _ij,i=1, 2...N,j=1, 2, 3 theith coordinate function of thejth component - _i = diag[_i1, _i2, _i3] theith matrix of coordinate functions - global vector of modal amplitudes - ₁ external detuning parameter - _i,i=2, 3 internal detuning parameter - _i,i=1, 2, 3 phase angles     

Experimental investigation of impinging jet arrays

We report on measurements of the velocity field and turbulence fluctuations in a hexagonal array of circular jets, impinging normally on a plane wall, using particle image velocimetry (PIV). Results for mean velocity and turbulent stresses are presented in various horizontal and vertical planes. From the measurements, we have identified some major features of impinging jet arrays and we discuss their mutual interaction, collision on the plate, and consequent backwash, which generate recirculating motion between the jets. The length of the jet core, the production of turbulence kinetic energy, and the model of the exhaust mechanisms for spent fluid are also discussed. The measurements indicated that the interaction between the self-induced cross flow and the wall jets resulted in the formation of a system of horseshoe-type vortices that circumscribe the outer jets of the array. The instantaneous snapshots of the velocity field reveal some interesting features of the flow dynamics, indicating a breakdown of some of the jets before reaching the plate, which may have consequences on the distribution of the instantaneous heat transfer.List of symbols D_m Nozzle diameter in multiple jet array nozzle plate (m) - D_s Pipe diameter in single jet rig (m) - H Distance between nozzle and impingement plate (m) - k Turbulent kinetic energy (m²/s²) - L Pipe length (m) - P_k Production of turbulent kinetic energy (m²/s³) - P_uu , P_vv Normal components of P_k (m²/s³) - P_uv Shear component of P_k (m²/s³) - s Pitch (m) - U_bulk Surface-averaged exit velocity (single jet) (m/s) - U_CL Center line jet exit velocity (jet array), m/s - u, v Mean velocity components in x and y directions (m/s) - u, v, w Instantaneous velocity in x, y, and z directions (m/s) - u, v, w Velocity fluctuation in x, y, and z directions (m/s) - u², v², w² Reynolds normal stress components (m²/s²) - uv Reynolds shear stress component (m²/s²) - x, z Coordinates parallel to impingement plate (m) - y Coordinate perpendicular to impingement plate (m)     

Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping (1/2 order subharmonic oscillations and entrainment)

Nonlinear forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping are studied. In particular, entrainment phenomena at the critical speeds of 1/2 order subharmonic oscillations of forward and backward whirling modes are investigated. A self-excited oscillation appears in the wide range above the major critical speed. The amplitude of this oscillation reaches a limit value and then a self-sustained oscillation occurs. In the vicinity of a 1/2 order subharmonic oscillation of a forward whirling mode, a self-excited oscillation is entrained by a subharmonic oscillation. In the vicinity of a 1/2 order subharmonic oscillation of a backward whirling mode, either a self-excited oscillation or a subharmonic oscillation occurs.Experiments were made by an elastic rotating shaft with a disc. Nonlinearity in its restoring force was due to an angular clearance of a bearing and internal damping was due to friction between the shaft and an inner ring of the bearing. A self-excited oscillation was observed in the range above the major critical speed and this self-excited oscillation was entrained by a 1/2 order subharmonic oscillation of a forward whirling mode.Nomenclature O–xyz rectangular coordinate system - , _x, _y inclination angle of a shaft and its projections on the xz- and yz-planes - _x, _y inclination angles in rotating coordinates - , polar coordinates - I _p polar moment of inertia of a rotor - I diametral moment of inertia of a rotor - i _p ratio of I _p to I - dynamic unbalance of a rotor - rotating speed (angular velocity) - F magnitude of a dynamic unbalance force, F = (1 – i _p )² - c external damping coefficient - h internal damping coefficient - t time - D _x , D _y internal damping terms in stationary coordinates - D _x , D _y internal damping terms in rotating coordinates - N _x , N _y nonlinear terms in restoring forces     

The psychrometric problem for the evaporation of NH3 in NH3/H2 atmosphere in neutral gas absorption refrigeration units for pressures 17.5 to 27.5 bar

The present work considers the thermodynamic and transport properties of the real NH₃/H₂ gas mixture as a function of the state variables temperature, mass fraction and pressure. The relevant properties are presented in the form of analytical expressions, valid in the pressure range of 17.5 to 27.5 bar. The psychrometric problem is used to determine the mass fraction of the NH₃/H₂ gas mixture with the dry and wet bulb temperatures as input variables.

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Das Psychrometrische Problem der Verdampfung von NH₃ in NH₃/H₂ Atmosphäre bei Neutralgasabsorptionskälteanlagen für Drücke von 17,5 bis 27,5 bar

Zusammenfassung In der vorliegenden Arbeit werden die thermodynamischen und Transporteigenschaften des reellen NH₃/H₂ Gasgemisches als Funktion der Zustandsvariabeln Temperatur, Massenkonzentration und Druck betrachtet. Die verschiedenen Eigenschaften sind in Form von analytischen Ausdrücken dargestellt und gelten in dem Druckbereich von 17,5 bis 27,5 bar. Das Psychrometrische Problem wurde benutzt um mit Hilfe der Trocken- und Feuchtkugeltemperatur, als Eintrittsparameter, die Massenkonzentration des NH₃/H₂-Gasgemisches zu bestimmen.

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Abbreviations

Nomenclature A ₁,...,A₃/A₁,...,A₄/A₁,...,A₄ Enthalpy coefficients [Eq. (2)] - a _ij, a_ij, a_ij: i = 1, 4,j = 1, 4 enthalpy coefficients [Eqs. (2a), (2b)–(2j),(2k)] - B ₁,B₂,..., B₅, B₆ coefficients of the Eq. (3) concerning the mass fraction of the - NH₃/H₂ gas mixture at saturated state - b _ij: i=1, 6,j = 1, 4 coefficients of the Eqs. (3a)-(3f) - B ₁₁ cm³/mole second virial coefficient of H₂ - B ₂₂ cm³/mole second virial coefficient of NH₃ - B ₁₂ cm³/mole mixture property (mixture second virial coefficient) - C ₁,..., C₃/C₁,...,C ₃/C₁,...,C₄ specific volume coefficients [Eq.(4) - c _ij, c_ij, c_ij: i = 1, 4,j = 1,4 specific volume coefficients [Eqs. (4a)-(4j)] - c _p kJ/kg grd specific heat capacity for constant pressure - D ₁₂ cm²/sec diffusion coefficient of the real - NH₃/H₂ gas mixture - D ₁,...,D₄/D₁,...,D₄/D₁,...,D₄ coefficients [Eq. (7)] concerning the thermal conductivity of the - NH₃/H₂ gas mixture - d _ij,d_ij, d_ij: i = 1, 4,j=1,4 thermal conductivity coefficients [Eqs. (7a)-(7l)] - f _G mass fraction of the real NH₃/H₂ gas mixture - > f _sat,f _s mass fraction at saturation state - h kJ/kg specific enthalpy of the NH₃/H₂ gas mixture - H kJ/kmole molar enthalpy - H ₁₁ kJ/kmole molar enthalpy of the H₂ - H ₂₂ kJ/kmole molar enthalpy of the NH₃ - h _G kJ/kg specific enthalpy of the NH₃/H₂ gas mixture at bulk state conditions - h _s kJ/kg specific enthalpy of the NH₃/H₂ gas mixture at interface state conditions - n moles of the mixture - N _pr Prandtl number - Schmidt number - Lewis number - p bar pressure (total pressure of the NH₃/H₂ gas mixture) - r coefficient of the thermal conductivity of the NH₃/H₂ gas mixture - T °C, K temperature - m³/kg specific volume - mole fraction - ₁ moles H₂/moles mixture - ₂ moles NH₃/moles mixture Greek letters , _M kJ/m h grd thermal conductivity of the real NH₃/H₂ gas mixture - ₁ kJ/m h grd thermal conductivity of the H₂ - ₂ kJ/m h grd thermal conductivity of the NH₃ - P dynamic viscosity of the real NH₃/H₂ gas mixture - kg/m₃ density of the real NH₃/H₂ gas mixture     

Bursting and structure of the turbulence in an internal flow manipulated by riblets

The buffer layer of an internal flow manipulated by riblets is investigated. The distributions of the ejection and bursting frequency from the beginning to the middle part of the buffer layer, together with high moments of the fluctuating streamwise velocity,u, and its time derivative are reported. The profiles of the ejection and bursting frequency are determined and compared using three single point detection schemes. The effect of the riblets on the bursting mechanism is found confined in a localized region in the buffer layer. The multiple ejection bursts are more affected than the single ejection bursts. The skewness and flatness factors of theu signal are larger in the manipulated layer than in the standard boundary layer. That, also holds true for the flatness factor of the time derivative, but the Taylor and Liepman scales are not affected. The spectrum of theu signal is altered at the beginning part of the viscous sublayer.Nomenclature u Friction velocity - Viscosity - l _v ;f _v wall scalesv/u ;u ² /v - y Vertical distance to the wall - z Spanwise extent - (⁺) Variable normalized with wall scales - u Velocity;u=Turbulence intensity - h, s Height and width of the riblets - f _e Ejection frequency - f _b Bursting frequency - f _BME Frequency of the Bursts with Multiple Ejection - f _BSE Frequency of Single Ejection Bursts - S andS _du/dt Skewness factor ofu and its time derivative - F _u andF _du/dt Flatness factor ofu and its time derivative Abbreviations SBL Standard (non-manipulated) Boundary Layer - MBL Manipulated Boundary Layer - BME Bursts with Multiple Ejections - BSE Bursts with Single Ejections - VITA Variable Interval Time Averaging technique - u–l u-level technique - mu Modifiedu-level technique