Experimental research on the laws of propagation of an underexpanded jet in a subsonic counterplan

The fields are obtained of total pressures and velocities along the axis of a jet flowing out into a subsonic counterflow from an isolated conical nozzle with semiaperture angle _a = 10°. The nozzle was located at the vertex of a cone with half-angle = 75° at the vertex and ratio of the radii at the midsection and the nozzle equal to 10. It is established that for an underexpanded jets in sub- and supersonic counterflows the main dimensionless number generalizing the data on the distribution of the gas-dynamic parameters in the jet at various pressure-ratio numbers n, and Mach numbers of the counterflow M and of the jet M_a, is the ratio of the momentum flux density J_a of the jet to the velocity head q: J=J_a/q. Generalizing dependences are obtained for the distribution of the total pressure and the velocity along the axis of the jet, and also for the jet range.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 186–188, May–June, 1986.     

Die swell of filled polymer melts

The Barus effect in polypropylene and polystyrene blended with a variety of fillers at various concentrations was investigated using a capillary extrusion rheometer. If the die swell is defined as the square of the ratio of the extrudate diameterd to the die diameterD, it is found to depend on the apparent shear stress _W . Below a certain value of _w the relation =B _B ^A applies. The die swell, _M , of a filled polymer depends on the type, size and volume fraction of the filler. In particular,A increases as the volume fraction increases and is largest for powders, smaller for flakes and smallest for fibres, whereasB shows the opposite trend but to a lesser extent.     

Tensiometer reaction related to its filter dimensions

A theoretical model for tensiometers is presented. It is based on a new physical considerations: the tensiometer filter is a quasi-saturated porous medium and the transmission fluid in the cavity is in hydrostatic equilibrium and is incompressible. The evolution equations form a complete system which could be used and coupled in a wide number of situations once filter dimensions and geometry have been correctly defined. The model is applied to tensiometer design and leads to new design recommendations. It predicts the existence of two distinct evolution modes for tensiometers. The time constant of the first varies linearly with the ratio of filter thickness to contact area and that of the second varies according to the square of the filter thickness and is independent on the contract area. The model leads to the formulation of an equation for fine-filter tensiometers. This extends Richards and Neal's equation by taking fine-filter geometry and gravity into account.Nomenclature A area of surfaceS _i - A _n ,B _n coefficients defined in Appendix B - C filter capacity - da boundary integration element - g constant gravity vector field - K permeability - L filter thickness - M _f mass of transmission fluid exchanged for a unit variation of the potential - M _mn components of a matrix defined in Appendix B - n porosity - n outward unit vector to filter rim (boundary) - N number of terms (Appendix B) - p pressure - P _i ,p _e internal, external pressure - q volume flux - r variable defined in Appendix B - r _n coefficients defined in Appendix B - S global sensitivity and gauge sensitivity - S _i ,S _e ,S _r filter rim - S _f saturation of filter - t time - U velocity of the transmission fluid in the cavity - V volume of filter - V _c volume of cavity - V _p volume of parasitic fluid - x positional vector - z spatial coordinate Greek Letters _p compressibility of the parasitic fluid - potential - _e potential outside of tensiometer - _i potential inside of tensiometer - ₀ initial potential - _p potential of parasitic fluid - adimensional parameter defined in (5.8) - conductance - dynamic viscosity - pi - density of transmission fluid - _p density of parasitic fluid - temporal parameter and time constant - adimensional temporal coordinate - adimensional spatial coordinate Symbols gradient operator - a·b scalar product ofa andb - a×b vector product ofa andb - partial derivative of with respect to - partial derivative of with respect to - mean geometrical value of _e(t,x) defined in (4.7) - x V x belongs toV     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

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.     

The secondary flow induced around a sphere in an oscillating stream of elastico-viscous liquid

The present paper is devoted to the theoretical study of the secondary flow induced around a sphere in an oscillating stream of an elastico-viscous liquid. The boundary layer equations are derived following Wang's method and solved by the method of successive approximations. The effect of elasticity of the liquid is to produce a reverse flow in the region close to the surface of the sphere and to shift the entire flow pattern towards the main flow. The resistance on the surface of the sphere and the steady secondary inflow increase with the elasticity of the liquid.Nomenclature a radius of the sphere - b ^ik contravariant components of a tensor - e contravariant components of the rate of strain tensor - F() see (47) - G total nondimensional resistance on the surface of the sphere - g _ik covariant components of the metric tensor - f, g, h secondary flow components introduced in (34) - k ₀ measure of relaxation time minus retardation time (elastico-viscous parameter) - K =k ₀ ²/V ₀ ² , nondimensional parameter characterizing the elasticity of the liquid - n measure of the ratio of the boundary layer thickness and the oscillation amplitude - N, T defined in (44) - p arbitrary isotropic pressure - p _ik covariant components of the stress tensor - p ^ik contravariant components of the stress tensor associated with the change of shape of the material - R =V ₀ a/v, the Reynolds number - S =a/V ₀, the Strouhall number - r, , spherical polar coordinates - u, v, w r, , component of velocity - t time - V(, t) potential velocity distribution around the sphere - V ₀ characteristic velocity - u, v, t, y, P nondimensional quantities defined in (15) - reciprocal of s - density - defined in (32) - defined in (42) - ₀ limiting viscosity for very small changes in deformation velocity - complex conjugate of - oscillation frequency - = ₀/, the kinematic coefficient of viscosity - , defined in (52) - (, y) stream function defined in (45) - =(NT/2n)^1/2 y - /t convective time derivative (1) _ik      

Control of low-speed turbulent separated flow using jet vortex generators

A parametric study has been performed with jet vortex generators to determine their effectiveness in controlling flow separation associated with low-speed turbulent flow over a two-dimensional rearward-facing ramp. Results indicate that flow-separation control can be accomplished, with the level of control achieved being a function of jet speed, jet orientation (with respect to the free-stream direction), and jet location (distance from the separation region in the free-stream direction). Compared to slot blowing, jet vortex generators can provide an equivalent level of flow control over a larger spanwise region (for constant jet flow area and speed).Nomenclature C _p pressure coefficient, 2(P-P_)/V ² - C _Q total flow coefficient, Q/ v - D ₀ jet orifice diameter - Q total volumetric flow rate - R Reynolds number based on momentum thickness - u fluctuating velocity component in the free-stream (x) direction - V free-stream flow speed - VR ratio of jet speed to free-stream flow speed - x coordinate along the wall in the free-stream direction - jet inclination angle (angle between the jet axis and the wall) - jet azimuthal angle (angle between the jet axis and the free-stream direction in a horizontal plane) - boundary-layer thickness - momentum thickness - lateral distance between jet orifices A version of this paper was presented at the 12th Symposium on Turbulence, University of Missouri-Rolla, 24–26 Sept. 1990     

Interaction between a distributed source and the free surface of a viscous fluid

A self-similar solution of the Navier-Stokes equations describing steady-state axisymmetric viscous incompressible fluid flow in a half-space is investigated. The motion is induced by sources or sinks distributed over a vertical axis with a constant density. The horizontal plane bounding the fluid is a free surface. It is found that in the presence of sources a solution of the above type exists and is unique for any value of the Reynolds numberR > 0, but in the case of sinks only on the interval –2 R < 0. The results of calculating the self-similar solutions are presented. The asymptotics of the solutions are found asR 0 andR .Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–65, March–April, 1996.     

Rayleigh-Benard Konvektion in Wasser im Bereich der Dichteanomalie

Zusammenfassung Der Einfluß der Dichteanomalie auf den Wärmetransport in einer von unten mit 0°C gekühlten horizontalen Wasserschicht wurde experimentell untersucht. Nach dem Einsetzen der Konvektion sind zwei Bereiche vorhanden, eine Konvektionsschicht zwischen der Kühlplatte und der 4°C-Isothermen und eine darüberliegende Schicht, in der reine Wärmeleitung herrscht. Die Temperatur der Bereichsgrenze zwischen Konvektions- und Wärmeleitungsschicht steigt mit der Zeit asymptotisch bis auf 8°C an. Die relative Höhe der Konvektionsschicht, die ebenfalls mit der Zeit ansteigt, nähert sich einem konstanten Wert für den Fall, daß die Temperatur der Wasseroberseite konstant gehalten wird. Mit der Höhe der Konvektionsschicht als der charakteristischen Länge lassen sich die Meßwerte sehr gut durch die empirische Beziehung Nu _h =0,073 Ra _h ^0.3 wiedergeben. Der Exponent in dieser Beziehung ist identisch mit dem für normale Fluide, die resultierende Nußelt-Zahl ist dagegen um etwa 22% kleiner.

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Rayleigh-benard convection in water with maximum density effects

Maximum density effects on the heat transfer through a horizontal water layer cooled from below with 0°C have been experimentally studied. After onset of convection two different regions are observed, a convection layer between the lower cold wall and the 4°C isotherm and a superimposed conduction layer. The temperature of the interface between the convection and conduction layer increases with time and approaches asymptotically 8°C. The relative height of the convection layer which increases with time also approaches a constant value for the case that the temperature of the upper boundary is kept constant. Using the height of the convection layer as the characteristic length scale the measured data follow very closely the empirical relation Nu_h=0.073 Ra _h ^0.3 . The exponent in this relation is identical with that for fluids without maximum density but the Nusselt-number is about 22% lower.

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Bezeichnungen a Temperaturleitfähigkeit - b=c Wärmeeindringkoeffizient - c spezifische Wärmekapazität - H Gesamthöhe der Wasserschicht - h Höhe der Konvektionsschicht - N Zahl der Konvektionszellen pro m² - Nu _h =h/ Nußelt-Zahl, aufh bezogen - Nu _H =H Nußelt-Zahl, aufH bezogen - q Wärmestromdichte - Ra=gh/av( ₂–₁) Rayleigh-Zahl - t Zeit - z Höhenkoordinate - Wärmeübergangskoeffizient - isobarer Volumenausdehnungskoeffi zient - Temperatur in °C - Wärmeleitfähigkeit - v kinematische Viskosität - Dichte Indizes h auf die Höheh der Konvektionsschicht bezogen - H auf die GesamthöheH bezogen - i Grenze zwischen Konvektionsund Wärmeleitungsbereich, bei der Höhez=h - 0 Anfangstemperatur zum Zeitpunkt - t O - 1 Kühlplatte,z=0 - 2 Heizplatte,z=H - Konvektion - Wärmeleitung Herrn Prof. Dr.-Ing. U. Grigull zum 70. Geburtstag gewidmet     

Dielectric properties of the composite system polyurethane — sodium chloride

Summary The dielectric properties of the composite system polyurethane-sodium chloride have been measured at frequencies between 10^–4 Hz and 3 · 10⁵ Hz in the temperature range from –150 °C up to +90 dgC. Three dielectric loss mechanisms have been found; they are indicated by ₁, ₂ and. The activation energy of the ₁-transition is 35 kcal/mole, that of the-transition 6.7 kcal/mole. The ₂-loss peak was only observed at frequencies of 10³ Hz and higher, forming one broad peak with the ₁-loss peak at lower frequencies. In the composite materials, the- and ₂-loss peaks measured at fixed frequencies were found at the same temperature. The ₂-loss peak, however, was shifted to a lower temperature, due to the sodium chloride filler. Comparison of experimental data of and tan with theoretical predictions concerning the dielectric properties of composite systems showed only partial agreement. The difference mainly consisted in. the temperature shift in the tan-peak of the ₁-transition.

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Zusammenfassung Die dielektrischen Eigenschaften des Verbundssystems Kochsalz-Polyurethankautschuk wurden im Frequenzgebiet zwischen 10^–4 Hz und 3.10⁵ Hz und im Temperaturbereich von –150 °C bis +90 °C gemessen. Es wurden drei dielektrische Verlustmaxima gefunden, die mit ₁, ₂ und angedeutet werden. Die Aktivierungsenergie des ₁-Überganges beträgt 35 kcal/Mol, die des-Überganges 6.7 kcal/Mol. Das ₂-Maximum konnte nur bei Frequenzen höher als 10³Hz vom ₁-Maximum gesondert erfaßt werden. Die Lage der ₂- und-Maxima war vom Füllgrad unabhängig. Das ₁-Maximum verschiebt sich mit steigendem Füllgrad zu niedrigeren Temperaturen. Die gemessenen Werte von und tan stimmen nur teilweise mit den Aussagen einer Theorie der dielektrischen Eigenschaften von Mischkörpern überein.

     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable     

Binary laminar boundary layer on a vertical surface in the presence of combined free and forced convection

Heat and mass transfer at a vertical surface is examined in the case of combined free and forced convection. The boundary layer equations, transformed to ordinary differential equations, contain a parameter that determines the effect of free convection on the forced motion. Criteria are offered for differentiating the free-convection, forced-convection, and combined regimes.Notation x, y coordinates - u, v velocity components - g acceleration of gravity - T temperature - kinematic viscosity - coefficient of thermal expansion - a thermal diffusivity - ₁ partial vapor density - D diffusion coefficient - W₂ mass velocity of air - independent variable - _w shear stress at wall - thermal conductivity - r latent heat of phase transition - , dimensionless temperature and partial vapor density - m^* the complex (m ₁–m _1w )/(1–m(_1w ) - c_p specific heat at constant pressure - G Grashof number - R Reynolds number - P Prandtl number - S Schmidt number     

Design of a Beam with Controllable Force Elements

A method for solving the problem of design of an intellectual structure formulated for the pair optimal position of actuators, optimal control of actuators is developed. In the method proposed, physical and logical objects are treated as equivalent.     

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–)     

Parameter identification in a soil with constant diffusivity

We consider infiltration into a soil that is assumed to have hydraulic conductivity of the form K = K = K_se^h and water content of the form = K – _r. Here h denotes capillary pressure head while K_s, , and _r represent soil specific parameters. These assumptions linearize the flow equation and permit a closed form solution that displays the roles of all the parameters appearing in the hydraulic function K and . We assume K_s and _r to be known. A measurement of diffusivity fixes the product of and resulting in a parameter identification problem for one parameter. We show that this parameter identification problem, in some cases, has a unique solution. We also show that, in some cases, this parameter identification problem can have multiple solutions, or no solution. In addition it is shown that solutions to the parameter identification problem can be very sensitive to small changes in the problem data.     

Pressure distribution near wedges for separated supersonic turbulent flow

Summary A theoretical analysis of the pressure distribution in the vicinity of a wedge for separated turbulent flow is made. The solution is based on Vasiliu's analysis of the pressure distribution for step-induced separation using the Crocco-Lees mixing coefficient and Chapman's dividing streamline model. Theoretical results are compared with experimental data by Sterrett and Emery for Mach 5.8 and wedge angles of 28° and 34.17°.Nomenclature b mixing coefficient distribution factor - C _p pressure coefficient - F() defined by equation (3) - f ₁() defined by equation (5) - f() defined by equation (10) - I ₁ momentum integral, reference 4 - K mixing coefficient, defined by equation (4) - K _j jet flow parameter, reference 4 - K ₀ value of K at separation - K _0r value of K in the reattachment zone - () defined by equation (11) - M Mach number - M free stream Mach number - P pressure - P _S pressure at the separation point - P free stream pressure - r _S defined as P _S/P - X distance from the separation point - X _n distance from separation to reattachment point - X _W distance from separation point to wedge corner - wedge angle - specific heats ratio - mixing layer thickness - _j mixing layer thickness in jet flow solution - _j ^* displacement thickness in jet flow solution - _S boundary layer thickness at separation - dimensionless coordinate, defined as X/ _S - _n value of at the reattachment point - deflection angle of flow outside the mixing layer - jet flow parameter, reference 4 - dimensionless pressure, defined as P/P _S - [ _c ]_max jet flow parameter, reference 4 - _c jet spread factor, reference 4     

The rheology of Hydroxy-propylguar (HPG) solutions and its influence on the flow through a porous medium and turbulent tube flow,respectively (Part 1)

The rheology of aqueous HPG solutions in the range 100 wppm to 5000 wppm is investigated. The flow through a porous medium and turbulent tube flow, respectively, of these solutions is studied as well. Especially with respect to the higher concentrations, the data correlate nicely only after the effect of shear is extracted, i.e., after the variable viscosity is taken into account. This is accomplished by working with an apparent viscosity _c , defined such that, the Hagen Poiseuille law (with _c ) holds in laminar tube flow.     

Über eine Anwendung der Hillschen Minimalbedingung in der Plastizitätstheorie

Zusammenfassung Auf dem gezeigten Weg wurden die Spannungen _r , , _z berechnet, wobei an Stelle der Veränderlichen r und die dimensionlosen Größen x _i = r _i /, x=r/ und x _a = r/ in die Rechnung eingeführt wurden. Die Funktion (r, ) wurde dann für den Bereich 0,45x_i1,0, 1x_a 2 tabuliert. Hierbei zeigte sich, daß der Rechenaufwand bei der Durchrechnung eines Einzelbeispiels nach der Charakteristikenmethode wesentlich geringer ist. Bei der Anlage von Zahlentafeln zur Berechnung von Spannungen für beliebige Durchmesserverhältnisse ergab sich, daß der aufgezeigte Wege zu geringerem Rechenaufwand führt. Für das Beispiel r _i /r _a=1/2 wurden die Rohraufweitungen bestimmt und diese Werte noch durch praktische Versuche nachgeprüft. Hierbei ergab sich, daß die theoretisch bestimmten Rohraufweitungen in dem Streubereich der gemessenen Rohraufweitungen lagen, wobei Messungen an drei Rohren aus demselben Material und demselben Rohrverhältnis durchgeführt wurden. Insbesondere stimmten die theoretischen Rohraufweitungen auch mit den gemessenen Rohraufweitungen überein, wenn das Rohr entlastet wurde und die Restdeformationen bestimmt wurden.Daraus kann geschlossen werden, daß durch die berücksichtigte lineare Verfestigung die tatsächlichen Verhältnisse außerordentlich gut erfaßt werden.Der sogenannte Platzdruek eines Rohres kann auf rein rechnerischem Weg nicht erfaßt werden, da für =r_a die geometrische Gestalt des Rohres instabil wird. Bei den Versuchen zeigt sich, wenn der Innendruck über p _i ( =r **** _a ) gesteigert wird, daß das Rohr schon bei geringen Überschreitungen aufzubauchen beginnt.Meinem Lehrer Herrn Prof. Dr. Dr. R. Grammel zum 65. Geburtstag gewidmet.     

An experimental study on the two-dimensional opposed wall jet in a turbulent boundary layer: Change in the flow pattern with velocity ratio

Results of the measurement of flow properties in a two-dimensional turbulent wall jet which is injected into the turbulent boundary layer in the direction opposite to that of the boundary layer flow are presented by varying the ratio of the jet issuing velocity to the mainstream velocity . This flow includes the flow separation and the recirculating flow, and so it requires the magnitude and direction of instantaneous velocity be measured. In the present work, a tandem hot-wire probe is manufactured and its characteristics are examined experimentally. With the use of this probe the change in the penetration length of the jet with the velocity ratio is investigated. It is clarified that two regimes of flow patterns exist: in the low velocity ratio the penetration length of the jet changes little with , and in the high velocity ratio it goes far from the nozzle with increasing . Streamlines, turbulence intensity contours and static pressure contours are presented in the two typical velocity ratios corresponding to each of two flow patterns, and they are compared.List of symbols b ₀ nozzle width (Fig. 1) - , e mean and fluctuating output voltages of hot-wire anemometer - p, p mean static pressure, p = p – p _o - p ₀ static pressure in undisturbed mainstream - p _w , p _w mean wall pressure, p _w = p _w – p _o - U ₀ mainstream velocity - U _j jet velocity at the nozzle exit - , u mean and fluctuating velocity components in x-direction - u _e effective cooling velocity - x distance along the wall from nozzle exit - x _pmax, x _pmin positions where the wall pressure has maximum and minimum values respectively - x _s penetration length of jet - y distance from the wall - forward flow fraction - ₁, ₂ yaw and pitch angles of flow direction (Fig. 4) - velocity ratio, = U _j /U _o - density of the fluid - nondimensional stream function The authors wish to express their appreciation to Prof. Toshio Tanaka of Gifu University for his advice in the course of the experiment. They also would like to thank the Research Foundation for the Electrotechnology of Chubu which partly supported this work.