Performance prediction of dual cylindrical wave laser devices for flow measurements

The dual cylindrical wave system is a variant of laser Doppler velocimetry, in which two cylindrical waves of laser light are used to illuminate a moving particle. This instrument is being used for local measurement of the unsteady skin friction in turbulent boundary layers, as well as droplet sizing in spray flows. In the present work, performance of these new devices is examined using the electromagnetic theory of light. Various requirements for the design and operation of these instruments have been further elaborated and extended. The accuracy of the previous experimental results has also been considered. The optics-related errors are shown to be negligible in the measurements of streamwise as well as spanwise wall velocity gradients. However, rigorous simulations appear to be essential for proper calibration of the particle sizing device.List of symbols A, B, C three particle positions - a half-width of an optical slit - a _gm amplitude of a plane wave in the spectrum of a cylindrical wave - d _f fringe spacing - d _p particle diameter - E amplitude of the electric oscillation in the optical field - E _c combined electric field of two cylindrical waves - E _o maximum strength of the electric field at the source of a cylindrical wave - E _s electric field of a scattered wave - E _y time-dependent electric field in the case of electric polarization - f characteristic length for the phase of the scattering amplitude - f _a anisotropic frequency - f _D Doppler frequency - F _DCW transfer function of DCW system for particle sizing - F _pDA Phase Doppler transfer function - g wall velocity gradient - g _m measured wall velocity gradient - I ₀, I₂ integrals in the asymptotic expansion of the scattering amplitude - I _s intensity of the scattered light - k wave number of laser light in the fluid medium - m refractive index of the particle relative to the surrounding medium - N ₀ nominal number of fringes resulting from interference of two cylindrical waves - P phase of a plane wave - P ₁, P₂ phases of plane waves from downstream and upstream cylindrical waves respectively - P _s scattered light power at a receiving aperture - r unit vector in the direction of light scattering - r _D distance of the signal detector from the particle center - S scattering amplitude of a cylindrical wave - S ₁, S₂ Scattering amplitudes of the cylindrical waves emanating from S₁ and S₂ respectively - magnitude of the scattering amplitude for a plane wave - S _c combined scattering amplitude of two cylindrical waves - S₁, S₂ downstream and upstream sources of cylindrical waves, respectively - S scattering amplitude of a plane wave - s half-spacing between sources of the cylindrical waves - t time - u velocity along x-axis - w ₀ 1/e half-width of the field distribution at the waist of a laser sheet - X ₀ nominal width of the fringe volume along the particle path - X particle position in the measuring volume - x, y, z Cartesian coordinates Greek symbols angle of the direction of wave propagation from x-axis - coefficient of the second-order term in the phase function of a cylindrical wave - angular size of the signal receiving aperture - incremental used for numerical differentiation the ratio of to _p - — ₀ integration parameter for I₀ and I₂ - half-angle between the directions of propagation of two waves - wavelength of laser in the fluid medium - ₀ wavelength of laser in vacuum - parameter defining the direction of propagation of a plane wave - 1/e 1/e half-width of the function A - ₀ direction of propagation of the dominant plane wave in the spectrum - _0s the direction of propagation of the plane wave that contributes predominantly to scattering in a particular direction - _p the value of . corresponding to one cycle of P - _s change in corresponding to a lobe of the scattering amplitude - a dimensional form of that determines lobes in the scattered field - signal phase, ₀ + _a - _a anisotropic phase shift - ₀ phase difference between two indicent waves - off-axis angle - elevation angle - circular frequency of laser light     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

Bifurcations and complex instability in a 4-dimensional symplectic mapping

Summary We study the main periodic solutions of a 4-dimensional symplectic mapping composed of two coupled 2-dimensional mappings. Their bifurcations were calculated numerically and also theoretically for small values of the coupling parameter . Most bifurcating families of period 2ⁿ (n0) have complex unstable regions that extend from =0 to the maximum allowed value of for each family. These complex unstable regions do not allow the transmisssion of the stability of the corresponding families to families of higher order. We found only one family with a complex unstable region not extending to the maximum , but in this case also the transmission of the stability is stopped at an inverse bifurcation. Thus although there are infinite sequences of bifurcations (of the Feigenbaum type) in the limiting 2-dimensional case =0, all such sequences are interrupted when the system is 4-dimensional (i.e. for 0). The appearance of complex instability for =0 can be predicted by studying the cases =0 and applying the Krein-Moser theorem.

---

Sommario Si svolge uuno studio dettagliato delle soluzioni periodiche principali di due mappe simplettiche bidimensionali accoppiate, calcolandone sia analiticamente che numericamente le biforcazioni per piccoli valori del parametro di accoppiamento . Quasi tutte le famiglie di periodo 2ⁿ (n0) prodotte dalle biforcazioni presentano regioni di instabilità complessa che si estendono da =0 fino al massimo valore di considerato. Queste regioni di instabilità complessa impediscono il trasferimento della stabilità di una famiglia a famiglie di ordine più elevato. In un solo caso si osserva una famiglia la cui regione di instabilità complessa non arriva ad estendersi fino al valore massimo di ; in questo caso però il trasferimento della stabilità viene interrotto da una biforcazione inversa. Se ne conclude che, nonostante I'esistenza di una famiglia di infinite biforcazioni di tipo Feigenbaum nel caso limite bidimensionale (=0), tutte le sequenze si interrompono se il sistema è a quattro dimensioni. Il formarsi di regioni di instabilità complessa per 0 può essere previsto studiando il caso =0 ed appplicando il teorema di Krein-Moser.

     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

An extended numerical method of calculating twodimensional or axisymmetric heated flows allowing for dissipation

Summary The analysis of [1] is extended to cover viscous flow for both plane twodimensional and axisymmetric flow. The equations used are the Navier-Stokes equations in streamwise coordinates, except that for simplicity terms of order are neglected, where is a coefficient of viscosity and the flow direction. The supersonic combustion chamber of [1] is used as an example, but the technique of specifying the streamlines in advance has to be modified for a boundary layer with pressure gradient. Thermal conductivity is included but does not have a very important effect. Flow separation is indicated in some circumstances.

---

Übersicht Das Rechenverfahren von [1] wird erweitert auf Strömungen mit Reibung und zwar für zweidimensionale ebene und rotationssymmetrische Strömungen. Benutzt werden die Navier-Stokesschen Gleichungen in Stromlinienkoordinaten; zur Vereinfachung werden Glieder der Größenordnung vernachlässigt, wobei der Reibungskoeffizient und die Richtung der Strömung sind. Als Beispiel dient die Überschallbrennkammer von [1]. Das Verfahren, im voraus Stromlinien festzulegen, muß bei Grenzschichten mit Druckgradient abgeändert werden. Wärmeleitfähigkeit wird berücksichtigt, aber ihr Einfluß ist gering. In einigen Fällen wird Strömungsablösung vorausgesagt.

     

Zum Impuls-, Wärme- und Stoffaustausch in turbulenten Strömungen reagierender Binärgemische

Zusammenfassung Für ein reagierendes Binärgemisch konstanter Dichte werden die Reynolds'schen Gleichungen angegeben. Die Transportkoeffizienten sowie die Wärmekapazitäten der beiden Komponenten werden als konstant angenommen. Für die unbekannten Reynolds'schen Terme werden Transportgleichungen angegeben. Weiter wird der Einfluß der Turbulenz auf die chemische Produktionsdichte diskutiert.

---

About the transfer of momentum, heat and mass in turbulent flows of binary mixturesPart I: The reynolds equations and the transport equations

The Reynolds equations for a reacting binary mixture of constant density are given. The transport coefficients as well as the specific heats of the components are assumed to be constant. Transport equations for the unknown Reynolds-terms are given. The influence of turbulence on the chemical production of species in discussed.

---

Formelzeichen c Massenkonzentration - c_p soezifische Wärme bei konstantern Druck - D binärer Diffussionskoeffizient - h spezifische Enthalpie - h_o=h + v _k ² /2 totale spezifische Enthalpie - h^o Reaktionsenthalpie - j_k Massendiffusionsstromvektor - k Reaktionsgeschwindigkeitskonstante - p Druck - Pr= c_p/ Prandtl-Zahl - q²/2 kinetische Energie der Schwankungsbewegung - q_k Energiestromvektor - R universelle Gaskonstante - Sc=/D Schmidt-Zahl - t Zeit - T absolute Temperatur - v_k Geschwindigkeitsvektor - x_k Ortsvektor Griechische Symbole Dissipationsfunktion - Wärmeleitfähigkeit - dynamische Viskosität - =/ kinematische Viskosität - Dichte - Produktionsdichte - _jk viskoser Spannungstensor Indizes auf die Komponente bezogen - 1 auf die Komponente 1 bezogen - 2 auf die Komponente 2 bezogen - mol molekularer Anteil - tur turbulenter Anteil - res resultierender Anteil     

Zur Berechnung der Filmverdunstung von Kohlenwasserstoffen in einem Heißluftstrom

The results of an analytical approximation method to predict the film vaporization are compared with the predictions of a finite difference method of Hermitian type. The analytically estimated rate of vaporization of different hydrocarbons, which is the most important value for practical applications, deviates only a few percents from the numerically estimated value.

---

Zur Berechnung der Filmverdunstung von Kohlenwasserstoffen in einem Heißluftstrom

Zusammenfassung Es wird ein Näherungsverfahren zur Berechnung der Filmverdunstung dargestellt, bei dem eine vollständige Lösung der miteinander gekoppelten Grenzschichtgleichungen entfallt. Die nach dieser analytischen Methode ermittelte Verdunstung verschiedener Kohlenwasserstoffe wird mit Werten verglichen, die nach einem Differenzenverfahren vom Hermiteschen Typ berechnet wurden. Es zeigt sich, daß die analytisch berechnete Verdunstungsrate, die für praktische Anwendungen wichtigste Größe, nur wenige Prozent von dem numerisch ermittelten Wert abweicht.

---

Formelzeichen c _ew Konzentrationsdifferenz c_1e -c _1w - c _i Massenkonzentration der Komponentei - c_p, c_pi spezifische Wärmekapazität bei konstantem Druck des Gemisches — der Komponentei - D ₁₂ binärer Diffusionskoeffizient - f dimensionslose Stromfunktion - f dimensionslose Geschwindigkeit - g () allgemeine Funktion - m ₁ Massenstromdichte der Komponente 1 - m ^* dimensionslose Massenstromdichte, G1. (4.8) - M, M_i Molgewicht, — der Komponentei - P, P _i Druck, Partialdruck der Komponentei - Pr Prandtlzahl,C _p/ - q Wärmestromdichte - r ₁ Verdampfungswärme - R allgemeine Gaskonstante - Sc Schmidtzahl/D ₁₂ - T absolute Temperatur - u Geschwindigkeitskomponente inx-Richtung - v Geschwindigkeitskomponente iny-Richtung - x Längskoordinate - y Querkoordinate - z dimensionslose Konzentration - dimensionslose Funktion/ _{e e} - transformierte Koordinatey - dimensionslose Temperatur (T-T _w)/(T_e-T_w) - Wärmeleitfähigkeit des Gemisches - Zähigkeit des Gemisches - transformierte Koordinate - Dichte des Gemisches - Stromfunktion Indizes e am Außenrand der Grenzschicht - i Stoffi - w an der Filmoberfläche - 1, 2 Komponente 1, 2 - () Ableitung ()/ _n      

Laser velocimetry measurements in a horizontal gas-solid pipe flow

This paper presents laser measurements of particle velocities in a horizontal turbulent two-phase pipe flow. A phase Doppler particle analyzer, (PDPA), was used to obtain particle size, velocity, and rms values of velocity fluctuations. The particulate phase consisted of glass spheres 50 m in diameter with the volume fraction of the suspension in the range _p=10^-4 to _p=10^-3. The results show that the turbulence increases with particle loading.List of symbols a particle diameter - C _va velocity diameter cross-correlation - d pipe diameter - Fr ² Froude number - g gravitational constant - p(a) Probability density of the particle diameter - Re pipe Reynolds number based on the friction velocity - T characteristic time scale of the energy containing eddies - T _L integral scale of the turbulence sampled along the particle path - u, U, u characteristic fluid velocities: fluctuating, mean and friction - v characteristic velocity of the paricle fluctuations - f expected value of any random variable f - f¦g expected value of f given a value of the random variable g - _p particle volume fraction - _p particle response time - absolute fluid viscosity - v kinematic fluid viscosity - _p, _f densities, particle and fluid - _a ² particle diameter variance - _va ² velocity variance due to the particle diameter variance - _vT ² total particle velocity variance - _vt ² particle velocity variance due to the response to the turbulent field     

Shear Instability at the “Explosion Product–Metal” Interface for Sliding Detonation of an Explosive Charge

Periodic perturbations at the explosion product–metal interface were studied experimentally. Experiments were performed for both spherical and plane geometry. Critical conditions of wave formation (detonation velocity of an explosive charge D 6.9 mm/sec) are determined, and an explanation of this effect is given. It is found experimentally that a dynamic pulse causes intense plastic strains at the explosion products–metal interface, leading to thermal softening of the steel boundary layer. In this layer, Kelvin–Helmholtz instability occurs. Calculationanalytical estimates of the critical boundary unstable wavelength agree satisfactorily with experimental results.     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

Flow of soft solids squeezed between planar and spherical surfaces

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

A fourfold generalization of Peaucellier's inversion cell

New inversors are proposed which are generalizations of the well-known inversor of Peaucellier. It appears that a kite in the Peaucellier cell is replaceable by an arbitrary 4-bar linkage (abhk) whereas the direction and length of the straight line, produced by the inversor, can be manipulated through the particular choice of the relative polar coordinates of a vertex A of the triangular input link. Formulas are derived for practical inversors with a revolving input link. The ones selected are basically governed by the choice of two transmission angles, ₁ and ₃, by the length L of the acquired line, as well as by its direction represented by the angle /2- comprised between the line L and the frame.

---

Sommario Vengono proposti nuovi inversori quali generalizzazioni del ben noto inversore di Peaucellier. Si mostra che un quadrilatero isoscele nella cella di Peaucellier è sostituibile da un'arbitraria connessione a quattro barre, mentre la direzione e la lunghezza della retta, prodotta dall'inversore, possono essere manipolate con la particolare scelta delle relative coordinate polari di un vertice A del collegamento triangolare di input. Vengono derivate formule per inversori funzionali con un collegamento di input rotante. Quelli solezionati sono principalmente governati dalla scelta di due angoli di trasmissione, ₁ e ₃, dalla lunghezza L della linea ottenuta, così come dalla sua direzione rappresentata dall' angolo /2- compreso tra la linea L ed il riferimento.

     

Unique Periodic Orbits for Delayed Positive Feedback and the Global Attractor

The delay differential equation, (t)=–x(t)+f(x(t–1)), with >0 and a real function f satisfying f(0)=0 and f>0 models a system governed by delayed positive feedback and instantaneous damping. Recently the geometric, topological, and dynamical properties of a three-dimensional compact invariant set were described in the phase space C=C([–1, 0], ) of initial data for solutions of the equation. In this paper, for a set of and f which include examples from neural network theory, we show that this three-dimensional set is the global attractor, i.e., the compact invariant set which attracts all bounded subsets of C. The proof involves, among others, results on uniqueness and absence of periodic orbits.     

Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

Laminar assisting mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties

The steady laminar boundary layer flow, with an external force, along a vertical isothermal plate is studied in this paper. The external force may be produced either by the motion of the plate or by a free stream. The fluid is water whose density-temperature relationship is non-linear at low temperatures and viscosity and thermal conductivity are functions of temperature. The results are obtained with the numerical solution of the boundary layer equations with , k and variable across the boundary layer. Both upward and downward flow is considered. It was found that the variation of , k and with temperature has a strong influence on mixed convection characteristics.Nomenclature c_p water specific heat - f dimensionless stream function - g gravitational acceleration - Gr_x local Grashof number - k thermal conductivity - Nu_x local Nusselt number - Pr Prandtl number - Pr_a ambient Prandtl number - Re_x local Reynolds number - s salinity - T water temperature - T_a ambient water temperature - T_o plate temperature - u vertical velocity - u_a free stream velocity - u_o plate velocity - v horizontal velocity - x vertical coordinate - y horizontal coordinate - pseudo-similarity variable - nondimensional temperature - dynamic viscosity - _f film dynamic viscosity - _o dynamic viscosity at plate surface - kinematic viscosity - buoyancy parameter - water density - _a ambient water density - _f film water density - _o water density at plate surface - physical stream function     

Asymmetric snap-buckling of a column restrained by a stiff wire

Summary A relatively simple example of asymmetric snap-through buckling in a continuous structure is the nonlinear problem of a cantilevered column restrained at its tip by a stiff wire, which is inclined at an acute angle to the column centerline, and loaded at its tip by a force perpendicular to the centerline. A parameter called , which is the nondimensional ratio of the flexural rigidity of the column to the combined extensional stiffness of the wire and the column centerline, determines the essential features of the buckling. If is zero, or is small compared to unity, the bending of the column is small enough to justify the use of linear bending theory for the column. Hence, even though the constraint is nonlinear, the solution to this problem is obtained in closed form. The critical point for the structure is found to be an asymmetric branching point for =0, while for positive, the critical point is a snap-through type. The effect of is similar to that induced by initial imperfections in more complex structures. For very small , the critical load is markedly decreased from the value for =0. Moreover, the graph of the load vs. tip deflection has the appearance of having an acute discontinuity in slope at the critical point for very small, although it is actually found that the graph has a horizontal tangent there.

---

Sommario Un esempio relativamente semplice di collasso asimmetrico per carico di punta in una struttura continua è il problema non lineare di una colonna incastrata ad una estremità e vincolata all'altra estremità da un filo rigido che forma un angolo acuto con l'asse della colonna e caricata a quella estremità con una forza perpendicolare allo stesso asse. Un parametro chiamato , che è il rapporto adimensionale fra la rigidità flessionale della colonna e la rigidezza longitudinale del filo e dell'asse della colonna, determina le caratteristiche essenziali del cedimento. Se è zero o è piccolo rispetto all'unità, l'inflessione della colonna è sufficientemente piccola per giustificare l'uso della teoria lineare di inflessione per la colonna. Di conseguenza anche se il vincolo non è lineare la soluzione del problema è ottenuta in forma compatta. Il punto critico della struttura si trova nel punto di biforcazione asimmetrica per =0, mentre per positivo, il punto critico rappresenta un punto di collasso. L'effetto di è simile a quello prodotto da imperfezioni iniziali in strutture più complesse. Per molto piccolo il carico critico è notevolmente ridotto rispetto al valore per =0. Inoltre il grafico del carico in funzione della curvatura all'estremità sembra avere una netta discontinuità nella pendenza dal punto di biforcazione per molto piccolo benchè, in realtà si trovi che il grafico ha lì una tangente orizzontale.

---

This research was supported by the Advanced Research Projects Agency (Project DEFENDER) and was monitored by the U.S. Army Research Office, Durham, under Contract DA-31-124-ARO-D-257.     

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     

A Wollaston prism interferometer implemented with a digitizer

Objective of this work was to implement a Wollaston prism interferometer with a digitizer so as to perform the evaluation procedure of interferograms automatically by means of a microcomputer. The adopted opto-electronic sensing device is a linear array of 1,024 bar-shaped (15 m × 26 m) silicon photodiodes with center-to-center spacing 25 m. A stepping motor can move the array board in order to investigate the whole interferometric image. Scanning time is about 10 ms/line. Application of the system to measurements of concentration profiles in some mass diffusion processes of a binary gas mixture proves the ability of the digitizer to accurately measure gas densities. The performance and the potentiality of the sensor are analyzed and discussed.     

Untersuchungsergebnisse über die kapillare Leitfähigkeit von Baustoffen

Zusammenfassung Krischer hat die kapillare Flüssigkeitsbewegung als Potentialströmung beschrieben, deren Ursache ein Feuchtegefälle ist und führte als Stoffeigenschaft die Flüssigkeitsleitzahl als Funktion des Feuchtegehaltes ein. Trennt man durch einen modifizierten Ansatz Kapillar- und Reibungskräfte, so erhält man Kapillarfunktionen, die für den Fall der stationären Strömung bei horizontaler Flüssigkeitsbewegung oder bei lotrechter Flüssigkeitsbewegung unter Vernachlässigung der Schwerkraft in der Krischerschen Flüssigkeitsleitzahl (Kapillarleitkoeffizient) zusammengefaßt werden können.Diese Kapillarfunktionen für Wasser wurden von Quarzsand, Ziegel, Kalksandstein, Gasbeton und Bimsbeton ermittelt und der Kapillarleitkoeffizient als Funktion des Feuchtegehaltes für den Befeuchtungsvorgang angegeben. Zur experimentellen Bestimmung des Feuchtegehaltes war das Durchstrahlungsverfahren mit Gammastrahlen gewählt worden, um den volumenbezogenen Feuchtegehalt während eines quasistationären Vorganges der kapillaren Flüssigkeitsbewegung in Abhängigkeit von Zeit und Ort ohne Störung des Vorganges ermitteln zu können.

---

Results of investigations on the capillary motion of moisture in building materials

Krischer described the capillary motion of moisture as a water transfer proportional to the gradient of water content by volume, and defined a coefficient of capillary conductivity as a function of moisture content. Equations of general validity, however, can be developed by separation in terms for capillary and gravity forces and capillary resistance. These capillary functions can be transferred in the coefficient for processes with horizontal motion and for those cases where gravity does not have any impact on the motion in small capillary pore spaces.The capillary functions and the coefficients of capillary conductivity for quasi-steady processes of humidification were determined of quartz sand, brick, sandlime brick, cellular concrete and pumice concrete. The temporally and locally changing moisture content during capillary rising tests was measured non-destructively by means of the attenuation effect of penetrating gamma rays.

---

Formelzeichen F Stoffquerschnitt - H() feuchtigkeitsabhängige maximale kapillare Steighöhe - H_max maximale kapillare Steighöhe beim maximalen Feuchtegehalt - I₀ Intensität der auffallenden Gammastrahlung - I Intensität der durchfallenden Gammastrahlung - R() feuchtigkeitsabhängiger kapillarer Reibungskoeffizient - R_max kapillarer Reibungskoeffizient beim maximalen Feuchtegehalt - V Volumstrom - h kapillare Steighöhe - q_S Volumanteil des Feststoffes - q_W Volumanteil des Wassers - q_L Volumanteil der Luft - s Weglänge - t Zeit - x Schichtdicke - y Impulszahl - Neigungswinkel gegen die Lotrechte - statistischer Fehler bei der Impulsmessung - Kapillarleitkoeffizient bzw. Flüssigkeitsleitzahl na ch Krischer - Schwächungskoeffizient für Gammastrahlen - Dichte - / Massenschwächungskoeffizient - volumenbezogener Feuchtegehalt - _max maximaler volumenbezogener Feuchtgehalt - _S Schwächungskoeffizient des Feststoffes - _W Schwächungskoeffizient des Wassers - _L Schwächungskoeffizient der Luft Herrn Professor Dr.-Ing. H. Glaser, Stuttgart, zum 70. Geburtstag gewidmet.Die Untersuchungen erfolgten mit Mitteln der AIF (Arbeitsgemeinschaft industrieller Forschungsvereinigungen e.V., Köln). Der Aufbau der Versuchs-anordnung und die Gammastrahlungsmessungen mit Auswertung wurden von H. Perk durchgeföhrt, der zugleich der för den Strahlenschutz Verantwortliche des Instituts im Sinne des § 20 der I. Strahlenschutzverordnung ist.