Study of vortex breakdown by particle tracking velocimetry (PTV)

Using a quasi three-dimensional instantaneous measurement technique, which combines particle tracking velocimetry (PTV) with volume scanning, first quantitative experimental results of the unsteady and asymmetric interior region of vortex breakdown were obtained. The study was carried out in a low speed flow through a cylindrical tube. A vortex was generated by a set of guidevanes and subjected to an adverse pressure gradient causing its breakdown. By scanning a pulsed illuminated planar laser light sheet, a set of meridional and azimuthal cuts of the flow was obtained. With PTV the recorded particle paths in the cuts were processed in order to obtain the instantaneous two-dimensional velocity field, mean streamlines and vorticity distribution. Moreover, the three-dimensional shape of the appearing breakdown, visualized with fluorescent dye, was reconstructed from the cuts. The results revealed that the shape of the bubble nearly equals the streamsurface of the stagnation point. According to the conditions in the water tunnel a single tilted vortex ring at the open rear part of the bubble dominates the interior flow structure of the bubble as first noted by Sarpkaya (1971). The vortical flow is bulged over the bubble, restored and intensified at the lower end. The gathered data lead to the conclusion that the vortex axis remains parallel to the centerline.     

Quasi-cylindrical translation shells

Summary For shells generated by translation of an arbitrary profile along an arc of finite rise and large chord, as compared with the dimensions of , an elastic theory is developed on three-dimensional basis by the method of parametric expansions.

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Sommario Per il guscio generato dalla traslazione di una curva generica lungo un arco di monta finita e corda grande a fronte delle dimensioni di A, è sviluppata su base tridimensionale una teoria elastica col metodo degli sviluppi parametrici.

     

On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

An experimental study on vortex breakdown in a differentially-rotating cylindrical container

The vortex breakdown phenomenon in a closed cylindrical container with a rotating endwall disk was reproduced. Visualizations were performed to capture the prominent flow characteristics. The locations of the stagnation points of breakdown bubbles and the attendant global flow features were in excellent agreement with the preceding observations. Experiments were also carried out in a differentially-rotating cylindrical container in which the top endwall rotates at a relatively high angular velocity _t, and the bottom endwall and the sidewall rotate at a low angular velocity _sb. For a fixed cylinder aspect ratio, and for a given relative rotational Reynolds number based on the angular velocity difference _t–_sb, the flow behavior is examined as |_sb/_t| increases. For a co-rotation (_sb/_t>0), the breakdown bubble is located closer to the bottom endwall disk. However, for a counter-rotation (_sb/_t<0), the bubble is seen closer to the top endwall disk. For sufficiently large values of _sb, the bubble ceases to exist for both cases.     

A visualization study of the interaction of a free vortex with the wake behind an airfoil

The two-dimensional interaction of a single vortex with a thin symmetrical airfoil and its vortex wake has been investigated in a low turbulence wind tunnel having velocity of about 2 m/s in the measuring section. The flow Reynolds number based on the airfoil chord length was 4.5 × 10³. The investigation was carried out using a smoke-wire visualization technique with some support of standard hot-wire measurements. The experiment has proved that under certain conditions the vortex-airfoil-wake interaction leads to the formation of new vortices from the part of the wake positioned closely to the vortex. After the formation, the vortices rotate in the direction opposite to that of the incident vortex.List of symbols c test airfoil chord - C vortex generator airfoil chord - TA test airfoil - TE test airfoil trailing edge - TE _G vortex generator airfoil trailing edge - t dimensionless time-interval measured from the vortex passage by the test airfoil trailing edge: gDt=(T-T- _TE)·U/c - T time-interval measured from the start of VGA rotation - U free stream velocity - U vortex induced velocity fluctuation - VGA vortex generator airfoil - y distance in which the vortex passes the test airfoil - Z vortex circulation coefficient: Z=/(U · c/2) - vortex generator airfoil inclination angle - vortex circulation - vortex strength: =/2     

Two-dimensional transonic vortex flows of an ideal gas

Equations are obtained for two-dimensional transonic adiabatic (nonisoenergetic and nonisoentropic) vortex flows of an ideal gas, using the natural coordinates (=const is the family of streamlines, and =const is the family of lines orthogonal to them). It is not required that the transonic gas flow be close to a uniform sonic flow (the derivation is given without estimates). Solutions are found for equations describing vortex flows inside a Laval nozzle and near the sonic boundary of a free stream.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–109, September–October, 1973.     

Robustness of Exponential Dichotomies in Infinite-Dimensional Dynamical Systems

In this paper we examine the issue of the robustness, or stability, of an exponential dichotomy, or an exponential trichotomy, in a dynamical system on an Banach space W. These two hyperbolic structures describe long-time dynamical properties of the associated time-varying linearized equation _t +A=B(t) , where the linear operator B(t) is the evaluation of a suitable Fréchet derivative along a given solution in the set K in W. Our main objective is to show, under reasonable conditions, that if B(t)=B(, t) depends continuously on a parameter and there is an exponential dichotomy, or exponential trichotomy, at a value ₀, then there is an exponential dichotomy, or exponential trichotomy, for all near ₀.We present several illustrations indicating the significance of this robustness property.     

Effect of Streamwise Streaky Structures on Turbulization of a Circular Jet

Experimental investigations of the influence of streamwise streaky structures on turbulization of a circular laminar jet are described. The qualitative characteristics of jet evolution are studied by smoke visualization of the flow pattern in the jet and by filming the transverse and longitudinal sections of the jet illuminated by the laser sheet with image stroboscopy. It is shown that the streaky structures can be generated directly at the nozzle exit, and their interaction with the Kelvin–Helmholtz ring vortices leads to emergence of azimuthal beams ( structures) by a mechanism similar to threedimensional distortion of the twodimensional Tollmien–Schlichting wave at the nonlinear stage of the classical transition in nearwall flows. The effect of the jetexhaustion velocity and acoustic action on jet turbulization is considered.     

Interaction of a vortex couple with a free surface

The characteristics of three-dimensional flow structures (scars and striations) resulting from the interaction between a heterostrophic vortex pair in vertical ascent and a clean free surface are described. The flow features at the scar-striation interface (a constellation of whirls or coherent vortical structures) are investigated through the use of flow visualization, a motion analysis system, and the vortex-element method. The results suggest that the striations are a consequence of the short wavelength instability of the vortex pair and the helical instability of the tightly spiralled regions of vorticity. The whirls result from the interaction of striations with the surface vorticity. The whirl-merging is responsible for the reverse energy cascade leading to the formation and longevity of larger vortical structures amidst a rapidly decaying turbulent field.List of symbols A _c Area of a vortex core (Fig. 6b) - AR Aspect ratio of the delta wing model - B base width of delta wing - b ₀ initial separation of the vortex couple - d ₀ depth at which the vortex pair is generated - c average whirl spacing in the x-direction - E energy density - Fr Froude number ( ) - g gravitational acceleration - L length of the scar band - L _ko length of the Kelvin oval - N _w number of whirls in each scar band - P _c Perimeter of a vortex core - q surface velocity vector - r _c core size of the whirl ( = 2A _c/P _c) - Re Reynolds number ( = ) - Rnd a random number - s inboard edge of the scar front (Fig. 6 a) - t time - u velocity in the x-direction - velocity in the y-direction - V _b velocity imposed on a scar by the vortex couple (Fig. 6 a) - V ₀ initial mutual-induction velocity of the vortex couple (=₀/2b ₀) - V _t tangential velocity at the edge of the whirl core - w width of the scar front (Fig. 6 a) - z complex variable - z _k position of the whirl center - half included angle of V-shaped scar band - wave number - _m initial mean circulation of the whirls - ₀ initial circulation of the vortex pair - _w circulation of a whirl - _min minimum survival strength of a whirl - t time step - gDz increment of z - gD change in vorticity - cut-off distance in velocity calculations - critical merging distance - curvature of the surface - wavelength - kinematic viscosity - angular velocity of a whirl core     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

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Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

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Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

Über den Einfluß der veränderlichen Viskosität auf die Stabilität der ebenen Kanalströmung

Zusammenfassung Der Einfluß einer veränderlichen Viskosität auf die Stabilität der ebenen Kanalströmung wird untersucht. Um den Effekt der Viskositätsänderung besonders hervorzuheben, wird ein Materialgesetz ohne Relaxationseigenschaften zugrundegelegt. Außerdem wird nur das Verhalten von ebenen Störungen untersucht. Unter der Ausnutzung der Verwandtschaft der Problemstellung mit dem newtonschen Fall können die Näherungsgleichungen vonC. C. Lin in modifizierter Form übernommen werden. Die Stabilität wird durch die Änderung des Grundprofils infolge der veränderlichen Viskosität und die differentielle Viskosität in der kritischen Schicht bestimmt.

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Summary The influence of shear rate dependent viscosity on the stability of plane channel flow is investigated. In order to demonstrate the effect of the viscosity variation a constitutive model without relaxation properties is choosen. Furthermore only perturbations in the plane of flow are investigated. Since the problem is similar to the newtonian case, the approximate equations ofC. C. Lin can be appropriately modified. The stability depends on the change of the basic profile due to shear rate dependent viscosity and on differential viscosity in the critical layer.

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Liste der wichtigsten Symbole A Dimensionslose Kennzahl: - b Stoffkonstante - h Halbe Kanalhöhe - Druckgradient - Re Reynoldszahl - Re _k Kritische Reynoldszahl - Re _k Kritische Reynoldszahl für ein newtonsches Fluid mit der Viskosität - u _g(y) Grundgeschwindigkeitsprofil - U _M Maximale Geschwindigkeit - Viskosität - Viskosität im zweiten newtonschen Bereich - _D Differentielle Viskosität - Stoffkonstante - _k Kritischer Druckgradient _k = –(dp/dx)_k - _k Kritischer Druckgradient für ein newtonsches Fluid mit der Viskosität - Dichte des Fluids Mit 8 Abbildungen     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

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     

Factors influencing non-circular ring vortex motion

Non-circular ring vortices are innately unstable, giving rise to a range of new phenomena. Here we report on our and Heertsch's [1] experiments in which vortices were generated at rectangular holes and nozzles with aspect ratios 2<<20. Different piston histories were also used. For forestrokes alone we were able to confirm the typical non-splitting motion of the primary vortex. On introducing a backstroke following the forestroke even for values of as low as 2 — values which should not give rise to splitting vortices — vortices could be made to split into 2, 3 or 4 secondary vortices. For cases where they rejoined the process was significantly different to that predicted by theory [2]. For 3 for a nozzle geometry the splitting angle is extremely sensitive to the stroke (length) so long as splitting takes place, whereas for 9>>5 the splitting angle tends to become independent of the stroke. This sensitivity on the stroke is reduced for vortices generated at a hole geometry. For all cases investigated here the splitting angle seems to be relatively insensitive to the Reynolds number. Vortices generated at hole geometries also tend to be less stable than those generated at tube geometries. Finally, the dependence of the splitting angle on the stroke length only scales with the nozzle breadth for 7>>5.

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Sommario Vortici ad anello non circolari sono intrinsecamente instabili e danno luogo ad una gamma di nuovi fenomeni. In questo articolo vengono riportati gli esperimenti degli Autori e di Heertsch in cui sono generati vortici in ugelli e fori rettangolari con rapporti geometrici =2÷20. Sono state anche usate differenti storie del moto del pistone. Nel caso in cui si usi solo la corsa in avanti si è stati capaci di confermare il moto tipico del vortice primario senza divisione del vortice stesso. Introducendo una corsa inversa, subito dopo la corsa in avanti, persino per pari circa a 2 — valore in cui il vortice non si dovrebbe dividere — i vortici si potevano dividere in due, tre o quattro vortici secondari. Nei casi in cui si verificava la riconnessione, l'evoluzione del processo era molto differente rispetto alla teoria. Per <3, per una data geometria dell'ugello, l'angolo di separazione è estremamente sensibile alla lunghezza della corsa, mentre per =5÷9 l'angolo di separazione tende a diventare indipendente dalla corsa. Questa sensibilità è ridotta per vortici generati in fori. In tutti i casi l'angolo di separazione sembra abbastanza indipendente dal numero di Reynolds. Vortici generati in corrispondenza di fori tendono ad essere meno stabili di quelli generati in ugelli. Infine, la dipendenza dall'angolo di separazione sulla lunghezza della corsa scala con l'ampiezza dell'ugello solamente per =5÷7.

     

Asymptotic theory of vortex breakdown

Axisymmetric breakdown of a vortex in a steady flow of incompressible fluid is investigated by means of an asymptotic analys is of the Navier-Stokes equations for large Reynolds number (R). A criterion for vortex breakdownis formulated. The region of slow motion resulting from vortex breakdown is shown to be bounded by an ellipsoid with length of order R^–1/4 and thickness of the order R^–3/8.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 78–90, May–June, 1993.     

Interaction of an oscillating vortex with a turbulent boundary layer

An oscillating vortex embedded within a turbulent boundary layer was generated experimentally by forcing a periodic lateral translation of a half-delta wing vortex generator. The objective of the experiment was to investigate the possibility that a natural oscillation, or meander, might be responsible for flattened vortex cores observed in previous work, which could also have contaminated previous turbulence measurements. The effect of this forced oscillation was characterized by comparison of measurements of the mean velocities and Reynolds stresses at two streamwise stations, for cases with and without forcing. The Reynolds stresses, especially w, were affected significantly by the forced oscillation, mainly through contributions from the individual production terms, provided the vortex was not too diffuse.List of Symbols a amplitude of forced vortex motion - f frequency of forced vortex generator motion - l vortex generator root chord - L flow length scale - R _Y , R _Z vortex core radial dimensions in vertical and spanwise directions, respectively - R_r vortex circulation Reynolds number R = / - u, v, w instantaneous velocity components in X, Y, Z directions - U, V, W mean velocities; shorthand notation for u, , w - X, Y, Z right-hand Cartesian streamwise, vertical, and spanwise coordinate directions - boundary-layer thickness - overall circulation - air kinematic viscosity - _x streamwise vorticity, _X = W/Y–V/d+t6Z - ( )₀ reference value (measured at X = 10 cm) - ( )c refers to vortex center - ( ) _max maximum value for a particular crossflow plane - ( ) (overbar) time average - ( ) (prime) fluctuating component, e.g., u=U+u     

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

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

Flow visualization of compressible vortex structures using density gradient techniques

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.Nomenclature a Unobstructed height of schlieren light source in cutoff plane, m - c Blade chord, m - f Focal length of schlieren focusing mirror, m - C _T Rotor thrust coefficient, T/( ² R ⁴) - I Image screen illumination, Lm/m ² - l Distance from vortex to shadowgraph screen, m - n _b Number of blades - p Pressure,N/m ² - p Ambient pressure, N/m ² - r, , z Cylindrical coordinate system - r _c Vortex core radius, m - Non-dimensional radial coordinate, (r/r _c ) - R Rotor radius, m - Tangential velocity, m/s - Specific heat ratio of air - Circulation (strength of vortex), m ²/s - Non-dimensional quantity, ² 8²p r _c ² - Refractive index of fluid medium - ₀ Refractive index of fluid medium at reference conditions - Gladstone-Dale constant, m ³/kg - Density, kg/m ³ - Density at ambient conditions, kg/m ³ - Non-dimensional density, (/ ) - Rotor solidity, (n _b c/ R) - Rotor rotational frequency, rad/s     

Stabilität einer in zylindrischen Gleitlagern laufenden,unwuchtfreien Welle

Zusammenfassung Das Problem der sogenannten Lagerinstabilität wurde rechnerisch untersucht. Als Vorbereitung wurde der Schmiermitteldruck und die Feder- und Dämpfungskonstanten des Schmierfilms berechnet. Die lineare Stabilitätsrechnung ergab, daß die Größe des Breitenverhältnisses die Stabilitätsgrenzkurve nur wenig, die Anfahrkurve jedoch stark beeinflußt. Wie man aus dem Verlauf der Grenzkurven ersieht, muß man bei der Stabilisierung eines instabilen Gleichgewichtes zwei Fälle unterscheiden: Beim leicht belasteten Lager ist das Lagerspiel r zu verkleinern und die Eigenfrequenz _k der Welle zu erhöhen, beim schwer belasteten Lager ist die Zähigkeit und die Lagerbreite B zu verkleinern und das Lagerspiel r zu vergrößern. Die nichtlineare Bahnberechnung ergab: Auch wenn nach linearer Rechnung das Gleichgewicht stabil ist, kann bei höheren Drehzahlen eine hinreichend große Anfangsstörung Instabilität hervorrufen. Die Bahnkurven werden bei Instabilität mit wachsender Amplitude mehr und mehr kreisförmig. Dabei stellt sich bei Drehzahlen unter 2 _k die Halbdrehfrequenzschwingung mit einer endlichen Rotoramplitude ein, während bei Drehzahlen über 2 _k die Eigenfrequenzschwingung entsteht, deren Amplitude unaufhörlich anwächst. Dieses Anwachsen beruht auf einer durch den Schmiermitteldruck bedingten Energiezufuhr. Soll also in Wirklichkeit die Amplitude endlich bleiben, so muß eine äußere Dämpfung auf die Welle wirken. Für einen einfachen Dämpfungsansatz wurde daher abschließend die stationäre, kreisförmige Wellenschwingung berechnet, wobei sich wiederum die Halbdrehfrequenz- und die Eigenfrequenzschwingung ergaben. Für große Werte des Parameters / nimmt danach die Rotoramplitude mit der Drehzahl zu, für kleine /-Werte nimmt sie dagegen nach Überschreiten eines Maximums wieder ab. In diesem Falle läßt sich die Rotoramplitude auch durch ein Verkleinern des Parameters / vermindern. Die bisher bekannten Versuchsergebnisse scheinen die Rechenergebnisse zu bestätigen.Gekürzte Fassung der Dissertation an der T.H. Karlsruhe, 1962; Referent: Prof. Dr.-Ing. K. Kollmann; Korreferent: Prof. Dr. K. Nickel und Prof. Dr. F. Weidenhammer; insbesondere Herrn Professor Kollmann danke ich herzlich für die Anregung und großzügige Förderung der Arbeit.     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.