On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

Resonant generation of a solitary wave in a thermocline

The resonant generation of a second-mode internal solitary wave, resulting from a ship internal waves system damping in a thermocline, is studied experimentally. The source of the stationary internal waves was provided by an oblong ellipsoid of revolution towed horizontally and uniformly at the depth of the thermocline center. The ranges of the Reynolds and Froude numbers were 500Re=Ul/v 15000 and 0.3Fi=U/N _max D1.0, respectively. When the body's speed and the linear long-wave second-mode phase speed were equal, an internal solitary wave of the bulge type was observed. The shape of the wave satisfied the Korteweg-de Vries equation. The Urcell parameter was equal to 10.2.List of Symbols L, B, H towing tank length, breadth and height respectively - z vertical coordinate - D characteristic vertical dimension of the body - a minor semiaxis of an ellipsoid - b major semiaxis of an ellipsoid (maximum ellipsoid diameter D=2a) - l length of the body ( =2b) - U velocity of the body - t temperature - g acceleration due to gravity - i fresh water density at ith level - _4° fresh water density for temperature t=4°C - o water density at the center of the thermocline - _i density variation due to the temperature variation at the ith horizon - N Brunt-Väisälä frequency - N _max maximum value of Brunt-Väisälä frequency - Re Reynolds number - Fi internal Froude number - f _n eigenfunction of the boundary-value problem for the nth mode - _n nth mode frequency - k _{n n}th mode horizontal wavenumber - C _n limiting phase speed of a linear nth mode interval wave (= _n/k_n;k_n 0) - Ur Urcell parameter - v fresh water kinematic viscosity - conventional density - half-length of a solitary wave - ₀ solitary wave height - time This work was partially supported by the INTAS (grant no. 94-4057) and by the Russian Foundation of Basic Research under grant no. 94-05-17004-a.A version of this paper was presented at the Second International Conference on Experimental Fluid Mechanics, Torino, Italy, 4–8 July, 1994.     

Diffusion in anisotropic porous media

An experimental system was constructed in order to measure the two distinct components of the effective diffusivity tensor in transversely isotropic, unconsolidated porous media. Measurements were made for porous media consisting of glass spheres, mica particles, and disks made from mylar sheets. Both the particle geometry and the void fraction of the porous media were determined experimentally, and theoretical calculations for the two components of the effective diffusivity tensor were carried out. The comparison between theory and experiment clearly indicates that the void fraction and particle geometry are insufficient to characterize the process of diffusion in anisotropic porous media. Roman Letters A interfacial area between - and -phases for the macroscopic system, m² - A _e area of entrances and exits of the -phase for the macroscopic system, m² - A interfacial area contained within the averaging volume, m² - a characteristic length of a particle, m - b average thickness of a particle, m - c _A concentration of species A, moles/m³ - c _o reference concentration of species A, moles/m³ - c _A intrinsic phase average concentration of species A, moles/m³ - c _a c _A–c _A, spatial deviation concentration of species A, moles/m³ - C c _A/c ₀, dimensionless concentration of species A - binary molecular diffusion coefficient, m²/s - D _eff effective diffusivity tensor, m²/s - D _xx component of the effective diffusivity tensor associated with diffusion parallel to the bedding plane, m²/s - D _yy component of the effective diffusivity tensor associated with diffusion perpendicular to the bedding plane, m²/s - D _eff effective diffusivity for isotropic systems, m₂/s - f vector field that maps c _A on to c _a , m - h depth of the mixing chamber, m     

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     

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     

Flying hot-wire anemometry

A flying hot-wire arrangement has been developed for the measurement of the velocity characteristics of the flow around airfoils, and particularly in regions where negative values of instantaneous velocity occur. The mechanism and signal processing system are described and appraised by comparing stationary and flying wire measurements obtained in the trailing edge region of a flap at an angle of attack which leads to upper-surface separation.List of symbols time averaged quantity - E voltage from hot wire anemometer - Q, magnitude and direction of cooling velocity viewed from a frame of reference on the probe: Q=U+V _p - q ₁, q ₂, q ₃ cooling velocity fluctuations oriented with respect to the -direction - Q _eff magnitude of effective cooling velocity measured by the hot wire: Q _eff = (Q _N1 ² + h ² Q _N2 ² + k ² Q _T ^{2 1/2} - t time - q _eff fluctuations of the effective cooling velocity - Q _N1, Q _N2, Q _T axial, normal and tangential components of the cooling velocity relative to the hot wire - Q _eff (= 10°) magnitude of the effective cooling velocity with - Q _eff ( = 0°) Q _eff ( = 30°) magnitude of the effective cooling velocity with the wire pitched at 10 ° and 0° to the flow velocity - Q _eff ( = 45°) Q _eff ( = 30°) magnitude of the effective colling velocity with the wire yawed at 45° and 30° to the flow velocity - U, magnitude and direction of flow velocity - u, v, w flow velocity fluctuations (x, y, z) - u ₁, u ₂, u ₃ normalised fluctuations of cooling velocity: u _i=q _iQ for i=1,2,3 - V _p, magnitude and direction of probe velocity - v _p probe velocity fluctuations along the -direction - yaw angle of hot wire relative to the probe axis - angle of mean flow velocity to the probe axis - angle of mean axial cooling component to mean cooling velocity viewed from the wire - pitch angle of probe axis relative to tunnel coordinates (x, y, z) - x, y, z orthogonal coordinate system with the x-direction aligned with the wall (boundary layer) or tunnel centre-line (wake) - x _w, y _w, z _w orthogonal coordinate system with the z _w-direction aligned with the wire and the probe pintels in the x _w- z _w plane     

Free convection from a vertical plate with concentrated and distributed thermal sources

An analysis is developed for the laminar free convection from a vertical plate with uniformly distributed wall heat flux and a concentrated line thermal source embedded at the leading edge. We introduce a parameter=(1 +Q _L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 to describe the relative strength of the two thermal sources; and propose a unified buoyancy parameter=( Ra_L+ Ra_w)^1/5 with=1/(1 +Pr ^–1) to properly scale the dependent and independent variables. The variables are so defined that the resulting nonsimilar boundary-layer equations can describe exactly the buoyancy-induced flow from the dual sources with any relative strength to fluids of any Prandtl number from very small values to infinity. These nonsimilar equations are readily reducible to the self-similar equations of an adiabatic wall plume for=0, and to those of free convection from uniform flux plate for=1. Rigorous finite-difference solutions for fluids of Pr from 0.001 to are obtained over the entire range of from 0 to 1. The effects of both relative source strength and Prandtl number on the velocity profiles, temperature profiles, and the variations of wall temperature, are clearly illustrated.

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Freie Konvektion an einer vertikalen Platte mit einer konzentrierten und einer gleichmäßig verteilten Wärmequelle

Zusammenfassung Für die freie Konvektion an einer vertikalen Platte mit einer gleichmäßig verteilten Wandwärmestromdichte und einer in der Vorderkante eingebetteten linienförmigen Wärmequelle wird eine Berechnungsmethode entwickelt. Zur Beschreibung der relativen Stärke der beiden Wärmequellen führen wir einen Parameter=(1 + Q_L/Q_w)^–1=(1 + Ra_L/Ra_w)^–1 ein und schlagen einen vereinheitlichten Auftriebsparameter=( Ra _L+ Ra _w)^1/5 mit=1/(1 +Pr ^–1 für die Skalierung der abhängigen und unabhängigen Variablen vor. Die Variablen werden so definiert, daß mit den sich ergebenden unabhängigen Grenzschichtgleichungen die von den beiden Wärmequellen beliebiger Stärke verursachte Auftriebsströmung von Fluiden beliebiger Prandtl-Zahl genau beschrieben werden kann. Diese unabhängigen Gleichungen können ohne weiteres auf die selbstähnlichen Gleichungen für den Fall einer lokalen Wärmezufuhr an einer sonst adiabatischen Wand für=0 und jenen der freien konvektion an einer Platte mit einheitlichem Wärmestrom für=1 zurückgeführt werden. Für Fluide mit der Prandtl-Zahl zwischen 0,001 und Unendlich werden nach der strengen finite Differenzen-Methode Lösungen im Bereich von zwischen 0 und 1 erhalten. Der jeweilige Einfluß der relativen Quellenstärke und der Prandtl-Zahl auf die Geschwindigkeits- und Temperaturprofile sowie die Veränderung der Wandtemperatur werden deutlich dargestellt.

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Nomenclature C _f friction coefficient - C _p specific heat - f reduced stream function - g gravitational acceleration - k thermal conductivity - L width of the plate - Nu local Nusselt number - Pr Prandtl number - q _w wall heat flux - Q _L heat generated by the line source - Q _w heat released by the uniform-flux wall from 0 tox, q _w Lx - Ra _L local Rayleigh number, g T _L ^* x ³/( ) - Ra _w local Rayleigh number,g T _w ^* w ³/( ) - T fluid temperature - T temperature of ambient fluid - T _L ^* characteristic temperature of the line source,Q _{L/(C p} L) - T _w ^* characteristic temperature of the uniform flux wall, =q _w x/k=Q _w /(C _p L) - u velocity component in then-direction - U₀ dimensionless velocity,u/(/x) Ra _L ^2/5 - U ₁ dimensionless velocity,u/(/x) Ra _w ^2/5 - velocity component in they-direction - x coordinate parallel to the plate - y coordinate normal to the plate - thermal diffusivity - thermal expansion coefficient - pseudo-similarity variable,(y/x) - dimensionless temperature, (T–T )/(T _L ^* +T _w ^* ) - ₀ dimensionless temperature, (Ra_l)^1/5 (T–T )/T _L ^* - ₁ dimensionless temperature, (Ra_w)Ra_w)^1/5 (T–T )/T _w ^* - (Ra _L+Ra_w)^1/5 - kinematic viscosity - (1 +Ra _L/Ra_w)^–1=(1 +T _L ^* /T _w ^* )^–1=(1 + Q_L/Q_w)^–1 - density - Pr/(1 +Pr) - _w wall shear stress - stream function     

Effects of viscous dissipation on heat transfer in an oscillating flow past a flat plate

Theoretical investigation has been carried out of laminar thermal boundary layer response to harmonic oscillations in velocity associated with a progressive wave imposed on a steady free stream velocity and convected in the free stream direction. Series solutions are derived both to velocity and temperature field and the resulting equations are solved numerically. The functions affecting the temperature field are shown graphically for different values of Prandtl number. It is observed that there is more reduction in the rate of heat transfer for P _r<1 and a rise in the rate of heat transfer for P _r>1 due to the presence of oscillatory free-stream.Nomenclature u, v velocity components in the x and y direction - x, y Cartesian coordinate axes - t time - U, U ₀ instantaneous value of and mean free stream velocity - density of fluid - kinematic viscosity - T, T _w, T temperature of the fluid, wall and free stream fluid - c _p specific heat at constant pressure - thermal diffusivity - amplitude of free stream velocity - frequency - _p non-dimensional temperature (T–T /T _w–T ) - P _r Prandtl number (c _p/K) - E _c Eckert number (U ₀ ² /c _p(T _w–T )) - a parameter ( ) - ₀ boundary layer thickness of the oscillation of a harmonic oscillation of frequency ( ) - ordinary boundary layer thickness ( ) - time-averaged, time-independent external velocity - A, B, C, D, E, K, L, M, N, P functions used in expansion for u and - Nu Nusselt number (hx/k) - T _w–% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8V4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcadaGcaa% qaaiaadAhacaWG4bGaai4laiqadwfagaqeaaWcbeaakiaacMcaaaa!3CA6!\[(\sqrt {vx/\bar U} )\] - k thermal conductivity     

Green's functions,bi-linear forms,and completeness of the eigenfunctions for the elastostatic strip and wedge

The bi-harmonic Green's functionG(r,r) for the infinite strip region -1y1, -<x<, with the boundary conditionsG=G/y ony=±1, is obtained in integral form. It is shown thatG has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function bi-harmonic in arectangle, and satisfying the same boundary conditions asG, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on sufficient to ensure that is exponentially small asx. In particular it is proved that this is so, solely under the condition that be bounded asx.A corresponding pattern of results is established for the wedge of general angle. The Green's function is obtained in integral form and expressed as a bilinear series of the (Williams) eigenfunctions. These eigenfunctions are proved to be complete for all functions bi-harmonic in anannular sector (and satisfying the same boundary conditions as the Green's function). As an application it is proved that if an elastostatic field exists in a corner region with free-free boundaries, and with either (i) the total strain energy bounded, or (ii) the displacement field bounded, then this field has a unique representation as a sum of those Williams eigenfunctions whichindividually posess the properties (i), (ii).The methods used here extend to all other linear homogeneous boundary conditions for these geometries.On leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A9259 and A9117.     

An experimental investigation of separating flow on a convex surface

The mean and turbulent characteristics of an incompressible turbulent boundary layer developing on a convex surface under the influence of an adverse pressure gradient are presented in this paper.The turbulence quantities measured include all the components of Reynolds stresses, auto-correlation functions and power spectra of the three components of turbulence. The results indicate the comparative influence of the convex curvature and adverse pressure gradient which are simultaneously acting on the flow. The investigation provides extensive experimental information which is much needed for a better understanding of turbulent shear flows.Nomenclature a, b constants in equation for velocity defect profile (Fig. 6) - c _f skin-friction coefficient (= _w/F _1/2 U ₁ ² ) - E(k ₁) one-dimensional wave number spectra - f frequency in Hz - G Clauser's equilibrium parameter = (H–1)/H(c _f /2) - H shape parameter (= ₁/ ₂) - k ₁ wave number (=2f/U) - L _u, L _v, L _w length scales of u, v and w fluctuations - p _s static pressure on the measurement surface - p _w reference tunnel wall static pressure - q ² total turbulent kinetic energy - R radius of curvature of the convex surface - R() auto-correlation function - T _u, T _v, T _w time scales of u, v and w fluctuations - U local mean velocity - U ₁ local free stream velocity - U _* friction velocity - u, v, w velocity fluctuations in x, y and z directions respectively - X streamwise coordinate measured along the surface from A (Fig. 1b) - x streamwise coordinate measured along the surface reckoned from station 9 - y coordinate normal to the surface - z spanwise coordinate - ₁/ _w · dp/dx - - boundary layer thickness - ₁ displacement thickness - ₂ momentum thickness - ₃ energy thickness - kinematic viscosity - density - time delay - _w wall shear stress     

Evaluation of the performance characteristics of a thermal transient anemometer

The Goertler instability of a hypersonic boundary layer and its influence on the wall heat transfer are experimentally analyzed. Measurements, made in a wind tunnel by means of a computerized infrared (IR) imaging system, refer to the flow over two-dimensional concave walls. Wall temperature maps (that are interpreted as surface flow visualizations) and spanwise heat transfer fluctuations are presented. Measured vortices wavelengths are correlated to non-dimensional parameters and compared with numerical predictions from the literature.List of symbols c _p Specific heat coefficient at constant pressure of the free stream - F Input (true) image - F ₀ Fourier number - Restored image - G Recorded (degraded) image - G Goertler number based on the boundary layer thickness, as defined by Eq. (3) - H System transfer function - M Mach number - Pr Prandtl number - p ₀ Stagnation pressure - Exchanged net heat flux - Convective heat flux - Radiative heat flux - r Recovery factor - Re _m Unit Reynolds number - Re _x Local Reynolds number based on the distance from the leading edge - Re Local Reynolds number based on the boundary layer thickness - Curvature radius - St Stanton number, as defined by Eq. (7) - T _aw Adiabatic wall temperature - T _w Wall temperature - T ₀ Stagnation temperature - t Time - V Free stream velocity - x Streamwise spatial coordinate - y Normal-to-wall spatial coordinate - z Spanwise spatial coordinate - Thermal diffusivity coefficient - Disturbance wavenumber - Non dimensional wavenumber - Boundary layer thickness - Goertler number based on the vortices wavelength - Vortices wavelength - Free stream density - Disturbance total amplification, as defined by Eq. (3) - Disturbance (spatial) growth rate - Non-dimensional growth rate - Perturbation amplitude of a generic quantity - Perturbation amount     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

The Preston tube in adiabatic compressible flow

A new procedure for the reduction of Preston tube data is introduced, based on the van Driest transformation. It appears to give results agreeing with the better calibration experiments, although a significant assumption in its derivation is violated.List of Symbols M _s Mach number sensed by Preston tube - M Friction Mach number (=u/wall sound speed) - R Gas constant - T _w Wall temperature - d Diameter of Preston tube - h Height of effective centre of Preston tube - p Preston tube pressure difference reading - p _i Equivalent incompressible Preston tube reading - p _w Wall pressure - r Recovery factor (=0.896) - u Friction velocity (=[_w/wall density]^1/2) - Empirical constant allowing for departure from Crocco temperature-velocity correlation (=0.975) - Specific heat ratio - Fluid kinematic viscosity - _w Wall shear-stress     

Elongational flow behavior of viscoelastic liquids: Modelling bubble dynamics with viscoelastic constitutive relations

Summary Earlier parts of this series have described a technique based on the collapse of single bubbles in the fluids for studying the elongational rheology of viscoelastic solutions and melts of moderate viscosities ( ₀ > 10²p) at relatively high strain rates . The present paper describes the modelling of bubble collapse with both rate and integral type constitutive relations using a body coordinate system. Predictions of the stress at the bubble wall as a function of time during collapse from a BKZ model and a modified corotational Maxwell model compared favorably with experimental data for two polymer solutions, 1% polyacrylamide in water/glycerine and 2% hydroxypropyl cellulose in water.

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Zusammenfassung In vorangehenden Veröffentlichungen dieser Reihe wurde eine Methode beschrieben, mit Hilfe derer man aus dem Zerfall von einzelnen Blasen in einer Flüssigkeit auf die Dehn-Rheologie viskoelastischer Lösungen und Schmelzen mittlerer Viskosität ( ₀ > 10² P) bei relativ hohen Dehngeschwindigkeiten schließen kann. Die vorliegende Untersuchung beschreibt Modelle des Blasenzerfalls mit Hilfe von Stoffgleichungen sowohl vom rate- als auch vom Integral-Typ, wobei ein körperfestes Koordinatensystem benutzt wird. Die Voraussagen der Spannung an der Blasenwand als Funktion der Zeit während des Zerfalls bei Verwendung eines BKZ- und eines modifizierten korotatorischen Maxwell-Modells zeigen eine recht gute Übereinstimmung mit experimentellen Werten, die an zwei Polymerlösungen, nämlich einer 1%igen Polyacrylamid-Lösung in einer Wasser-Glycerin-Mischung und einer 2%igen wäßrigen Hydropropylcellulose, erhalten worden sind.

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Nomenclature a material constant - b material constant - g metric tensor, space coordinates - m material constant - n material constant - p pressure - P _G pressure within bubble - P _R pressure outside bubble at the wall - P pressure far away from the bubble - R bubble radius - dR/dt - R ₀ initial bubble radius - t time - u velocity - U potential function - Y R/R ₀ Greek symbols covariant body metric tensor - surface tension - rate of deformation matrix, II -second invariant of - strain rate - ₀ zero shear rate viscosity - _e elongational viscosity - _ef effective viscosity - ¹, ², ³ coordinates in body system - ^{1 1}/R ₀ ³ - body stress tensor - density - space stress tensor - relaxation time - _ef effective relaxation time - bubble pressure function, defined in eq. [19] - vorticity tensor With 11 figures and 1 table     

Stress analysis of the cylinder subjected to arbitrarily distributed axisymmetrical loads

Summary Stress analysis has been carried out for a finite cylinder subjected to arbitrarily distributed axisymmetrical surface loads. Direct stress _x in the axial direction is assumed to be of the form _x = ₀+r ₁ +r ₂ where ₀ to ₂ are functions of x. Using the equations of equilibrium and compatibility the other direct stresses and the shearing stress are expressed by ₁ and ₂. Fundamental equations governing ₁ and ₂ are introduced using the variational principle of complementary energy. From the results of the present analysis it is evident that the boundary conditions can be satisfied completely even for the case where the external forces are specified in complicated form, and that more accurate solutions can easily be obtained by introducing additional terms in _x.

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Spannungsanalyse für den Zylinder unter axialsymmetrischer Last in beliebiger Verteilung

Übersicht Für einen endlichen Zylinder unter axialsymmetrischer Oberflächenlast in beliebiger Verteilung werden die Spannungen ermittelt. Die Normalspannung in Axialrichtung wird in der Form _x = ₀+r ₁ +r ₂ angesetzt mit ₀, ₁, ₂ als Funktionen von x. Mit Hilfe der Gleichgewichtsund Verträglichkeitsbedingungen werden die anderen Normalspannungen und die Schubspannung durch ₁ und ₂ ausgedrückt. Über das Variationsprinzip für die Komplementärenergie werden die grundlegenden Gleichungen für ₁ und ₂ eingeführt. Die Ergebnisse zeigen, daß die Randbedingungen selbst für komplizierte Belastungsarten vollständig erfüllbar sind und mit zusätzlichen Termen in _x mühelos noch genauere Lösungen bestimmt werden können.

     

Shear softening and thixotropic properties of wheat flour doughs in dynamic testing at high shear strain

Shear softening and thixotropic properties of wheat flour doughs are demonstrated in dynamic testing with a constant stress rheometer. This behaviour appears beyond the strictly linear domain (strain amplitude ₀ 0.2%),G,G and |^*| decreasing with ₀, the strain response to a sine stress wave yet retaining a sinusoidal shape. It is also shown thatG recovers progressively in function of rest time. In this domain, as well as in the strictly linear domain, the Cox-Merz rule did not apply but() and | ^*())| may be superimposed by using a shift factor, its value decreasing in the former domain when ₀ increases. Beyond a strain amplitude of about 10–20%, the strain response is progressively distorted and the shear softening effects become irreversible following rest.     

Modellvorstellung zur turbulenten Filmkondensation strömenden Dampfes

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

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A model for turbulent film condensation of flowing vapour

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

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

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

A turbulence model for the heat transfer near stagnation point of a circular cylinder

A one-equation low-Reynolds number turbulence model has been applied successfully to the flow and heat transfer over a circular cylinder in turbulent cross flow. The turbulence length-scale was found to be equal 3.7y up to a distance 0.05 and then constant equal to 0.185 up to the edge of the boundary layer (wherey is the distance from the surface and is the boundary layer thickness).The model predictions for heat transfer coefficient, skin friction factor, velocity and kinetic energy profiles were in good agreement with the data. The model was applied for Re 250,000 and Tu0.07.Nomenclature µ,C _D Constants in the turbulence kinetic energy equation - C ₁,C ₂ Constants in the turbulence length-scale equation - Skin friction coefficient atx - D Cylinder diameter - F Dimensionless flow streamwise velocityu/u _e - k Turbulence kinetic energy =1/2 the sum of the squared three fluctuating velocities - K Dimensionless turbulence kinetic energyk/u _e ^/2 - I Dimensionless temperature (T–T _w )/(T –T _w ) - l Turbulence length-scale - l _e Turbulence length-scale at outer region - Nu _D Nusselt number - p Pressure - Pr Prandtl number - Pr _t Turbulent Prandtl number - Pr _k Constant in the turbulence kinetic energy equation - R Cylinder radius - Re _D Reynolds number u D/µ - Re _x Reynolds number u x/µ - R _K Reynolds number of turbulence - T Mean temperature - T Mean temperature at ambient - T _s Mean temperature at surface - Tu Cross flow turbulence intensity, - u Mean flow streamwise velocity - u Fluctuating streamwise velocity - u _e Mean flow velocity at far field distance - u Mean flow velocity at ambient - u* Friction velocity - v Mean velocity normal to surface - V Dimensionless mean velocity normal to surface - x,x ₁ Distance along the surface - y Distance normal to surface - Dimensionless pressure gradient parameter - Boundary layer thickness atu=0.9995u _e - Transformed coordinate iny direction - Fluid molecular viscosity - _t Turbulent viscosity - _eff + _t - µ Fluid molecular viscosity at ambient - Kinematic viscosity/ - Density - Density at ambient - _w Wall shear stress - _w,0 Wall shear stress at zero free stream turbulence