Elastohydrodynamic lubrication in a plane slider bearing

Summary The present work deals with the case of a two-dimensional slider bearing with a rigid pad and an elastic bearing. Fluid viscosity is assumed to be only a pressure function. We determined the bearing deformation, the pressure distribution and the load capacity at different values of the inclination angle of the slider, with a numerical integration of the system consisting of the elasticity and Reynolds equations. The results show that, with an iso-viscous fluid, bearing elasticity causes a load capacity decrease. Instead bearing elasticity together with the variation of fluid viscosity due to pressure causes a load capacity greater than that of the iso-viscous case (=0).

---

Sommario Il presente lavoro studia il problema della coppia prismatica lubrificata con pattino rigido di allungamento infinito e cuscinetto deformabile; si suppone che la viscosità del fluido sia funzione della sola pressione. Il sistema di equazioni, costituito dall'equazione di Reynolds e dall'equazione dell'elasticità, è stato risolto numericamente, determinando la deformazione del cuscinetto, andamento della pressione e la capacità di carico per diversi valori dell'inclinazione del pattino. I risultati dimostrano che, con fluido isoviscoso, la deformabilità del cuscinetto determina una riduzione della capacità di carico. Se si considera, invece, effetto combinato dell'elasticità del cuscinetto e della variazione della viscosità del fluido, la capacità di carico risulta maggiore di quella che si ottiene con fluido isoviscoso (=0).

---

Nomenclature /L - /L - x/L - x/L - - ¯C CZ/h ₁ - E elasticity modulus - h film thickness - H elastic deformation of the bearing - h ₁ minimum film thickness - h ₂ inlet thickness - inclination of the pad - h Z/h ₁ - HZ/h ₁ - L pad length - viscosity - ₀ viscosity with no over-pressure - p over pressure - p P _ec-P _rc where:ec=elastic caserc=rigid case - P h ₁ ² /6₀VL - h ₂/h ₁=1+L/h ₁ - FV bearing velocity - W load capacity per unit width - Wh ₂ ¹ /6₀ VL ² - Z E h ₃ ¹ /12 ₀ VL ² A first version of this paper was presented at the 7th National AIMETA congress, held at Trieste, October 2–5, 1984. This work was supported by C.N.R.     

Convective fluid inertia forces in steady laminar lubrication: Solution of the inverse problem

Summary The present work deals with the problem of determining the influence of the inertial terms solving the inverse problem in the case of a plane slider bearing. The method determines the film geometry under a given pressure distribution when inertial terms are taken into account. The Volterra integral equation — which gives the velocity distribution in every section — was solved in a strictly numerical way. Our results showed that inertial terms determine an increase of the load capacity and a decrease of the flow rate. The friction coefficient proved scarcely influenced by inertial effects. The present method enables us to obtain the results of the linear theory as an asymptotic solution.

---

Sommario Il presente lavoro tratta il problema della determinazione dell'influenza dei termini inerziali, mediante la risoluzione del problema inverso, nel caso di un accoppiamento prismatico. Il metodo determina lo spessore del meato per una assegnata distribuzione di pressione. L'equazione integrale di Volterra, che permette di determinare la distribuzione di velocitá in ogni sezione, é stata risolta numericamente. I risultati dimostrano che i termini inerziali determinano un incremento della capacitá di carico ed un decrementa della portata. Il coefficiente di attrito ne é invece debolmente influenzato. I risultati della teoria lineare sono ottenuti come soluzione asintotica.

---

Nomenclature a ·L/h _e - h ₀ Q/V ₀=typical film thickness - h _e film thickness at trailing edge - L length of bearing pad - p overpressure - P ^* P/(·V ₀ ² )=u ₁ ^*2 /2 - Q flow rate - u, v velocity components - u ^* u/V ₀ - x, y axial and vertical coordinate - Re (V ₀·L/v)=Reynolds number - V ₀ pad velocity - z _* u ^*2(x ^*, ^*)+u ₁ ^*2 (x ^*) - Z ^* z ^*(x ^*+x ^*, ^*) - inclination angle of the pad - (L/h ₀)²·x ^*/Re - v kinematic viscosity - density - stream function - ^* /(h ₀ V ₀)=non dimensional stream function This research was funded both by the Italian Ministry for Education and the National Research Council (C.N.R.) of Italy     

Effect of nonlinear constitutive equation on vibrations of beams

Summary This paper is concerned with the analytical investigation of static and dynamic nonlinear behaviors of beams with different boundary conditions. While geometric type of nonlinearities on beams have been investigated extensively, material type nonlinearities have received very little attention. Therefore, material nonlinearities of the Ramberg-Osgood type are considered in this analysis. The use of Self-Generating functions for nonlinear beam problems is demonstrated for this type of nonlinearity. Transverse shear and rotatory inertia effects have been included in the formulation to study moderately thick beams. For all the cases investigated here nonlinear frequency ratios are calculated at various amplitudes of vibration and geometric parameters of beams. Numerical results indicate that the Ramberg-Osgood type nonlinearity produces softening-type responses. The study is limited to materials which are nonlinearly elastic and the effect of geometric nonlinearity is not considered in this paper.

---

Sommario Questo lavoro riguarda lo studio dei comportamenti non lineari statici e dinamici di travi con diverse condizioni a contorno. Mentre le non linearità di tipo geometrico sono state studiate estesamente, quelle di tipo non lineare hanno ricevuto un'attenzione molto ridotta. Perciò in questa ultima analisi si considerano non linearità materiali del tipo di Ramberg-Osgood. Si dimostra uso delle funzioni autogeneratrici nei problemi non lineari per le travi con questo tipo di non linearità. Nella formulazione dello studio di travi moderatamente spesse si sono inclusi effetti di taglio e inerzia rotatoria. Per tutti i casi qui studiati si calcolano i rapporti di frequenza non lineare per varie ampiezze di vibrazione e parametri geometrici delle travi. I risultati numerici indicano che la non linearità del tipo di Ramberg-Osgood produce risposte del tipo ammorbidimento. Lo studio si limita a materiali con nonlinearità elastica e non si considera nel lavoro l'effetto della non linearitá geometrica.

---

List of symbols T _s Transverse shear - k ₁ Shear connection factor - A, B, m Material constants - I Moment of inertia - G Shear modulus - Mass per unit length - w Lateral displacement - h Beam thickness - b Bredth of beam - t Time - R _i Rotatory inertia - a Area of cross-section of beam - q(x) Lateral load on beam - _x Stress - _x Strain - Length of beam - x Beam coordinate - - r Radius of gyration - w Nondimensional maximum deflectionw _max/r - q ₀ ^* Nondimensional load, (q ₀ ³/Al - Thickness parameter,h/ - (T _sK₁/Ga) - ₀ Linear frequency - Nonlinear frequency - w _max w measured at the point of maximum deflection.     

The effect of non-steady wall shear on pressure surges in conduits carrying viscous liquids

Summary A study is made of the attenuation of pressure surges in a two-dimension a channel carrying a viscous liquid when a valve at the downstream end is suddenly closed. The analysis differs from previous work in this area by taking into account the transient nature of the wall shear, which in the past has been assumed as equivalent to that existing in steady flow. For large values of the frictional resistance parameter the transient wall shear analysis results in less attenuation than given by the steady wall shear assumption.Nomenclature c /, ft/sec - e base of natural logarithms - F(x, y) integration function, equation (38) - (x) mean value of F(x, y) - g local acceleration of gravity, ft/sec² - h width of conduit, ft - k (2m–1)^{2 2} L/h ² c, equation (35) - k* 12L/h ² c, frictional resistance parameter, equation (46) - L length of conduit, ft - m positive integer - n positive integer - p pressure, lb/ft² - p ₀ constant pressure at inlet of conduit, lb/ft² - P pressure plus elevation head, p+gz, equation (4) - mean value of P over the conduit width h - P ₀ p ₀+gz ₀, lbs/ft² - R frictional resistance coefficient for steady state wall shear, lb sec/ft⁴ - s positive integer; also, condensation, equation (6) - t time, sec - t ct/L, dimensionless time - u x component of fluid velocity, ft/sec - u _m mean velocity in conduit, equation (12), ft/sec - u ₀(y) velocity profile in Poiseuille flow, equation (19), ft/sec - transformed velocity - U initial mean velocity in conduit - x distance along conduit, measured from valve (fig. 1), ft - x x/L, dimensionless distance - y distance normal to conduit wall (fig. 1), ft - y y/h, equation (25) - z elevation, measured from arbitrary datum, ft - z ₀ elevation of constant pressure source, ft - isothermal bulk compression modulus, lbs/ft² - _n , equation (37) - _n (2n–1)/2, equation (36) - viscosity, slugs/ft sec - / = kinematic viscosity, ft²/sec - density of fluid, slugs/ft³ - ₀ density of undisturbed fluid, slugs/ft³ - ø angle between conduit and vertical (fig. 1) The research upon which this paper is based was supported by a grant from the National Science Foundation.     

Temperature variations across the lubricant film in hydrodynamic lubrication

Summary Temperature variations across the lubricant film in hydrodynamic lubrication have been taken into account. The consequent variations of viscosity cannot be neglected for high Prandtl number lubricants.Nomenclature A constant defined by eq. (13a) - b h ₀/h _L - c heat capacity - h film thickness - h ₀ h at X=0 - h _L h at X=L - H ₀ local heat transfer coefficient defined by (23a) - K thermal conductivity - L length of bearing - m* defined by (9) - Nu Nusselt number defined by (23b) - P pressure - Pe Peclet number (Re) (Pr) - q ₀ slider surface heat flux - q ₀ ^* dimensionless heat flux defined by (20) - Q ₀ slider surface total heat transfer - Q ₀ ^* dimensionless total heat transfer defined by (21) - Re Reynolds number Vh ₀/ - Pr Prandtl number c/ - T temperature - T ₀ slider surface temperature - T _B film bulk temperature - u longitudinal velocity - v transverse velocity - V slider velocity - x longitudinal coordinate - y transverse coordinate - x/h ₀ - u/V - h/h ₀ - y/h - density - viscosity - v/V - - - _S dimensionless stationary surface temperature - dimensionless average stationary surface temperature - _B dimensionless film bulk temperature     

Quasi- determinism of sea wave groups

Summary Provided we know that at a point x ₀,y ₀ within an irregular gravity wave field, like those generated by wind on sea, at a time instant t ₀ there is a wave with a height H great with respect to the mean, we can predict that that wave, with high probability, has been formed because of the transit of a well defined (deterministic) group, like that in Fig. 1. In mathematical terms: if the ratio between the known wave height H and the mean wave height tends to infinity, the probability that the true wave group is equal to the deterministic wave group plus a lower order random noise approaches 1 [1]. The effect of the random noise is the object of this paper. In particular, the effect on the mean heights and periods of the waves forming the group is estimated within errors of an order smaller than (H/ m ₀)^–1 (m ₀ being the variance of the free surface elevation of the irregular wave field), The knowledge of the infinitesimal differences between the true wave group and the deterministic wave group, for H/ m ₀,proves to be useful for assessing the differences in the case that H is realistically great for a sea state. To that end data from numerical simulations of irregular gravity wave fields are used too. The conclusion is that, for a realistically great H, the deterministic wave group closely reflects the essential features of the true wave group.

---

Sommario Ammesso di sapere che in un punto x ₀,y ₀ di un campo di onde irregolari di gravità, come quelle generate dal vento sul mare, ad un certo istante t ₀ c' è un'onda con un'altezza H grande rispetto alla media, si può prevedere che, con grande probabilità, tale onda si sia formata per il passaggio di un ben definito (deterministico) gruppo come quello di Fig. 1. In forma matematica: se il rapporto tra l'altezza H nota e l'altezza media tende ad infinito, la probabilità tende ad 1 che il gruppo di onde vero sia uguale al gruppo deterministico più un disturbo aleatorio di ordine inferiore [1].L'effetto del disturbo aleatorio è l'oggetto di questa memoria. In particolare l'effetto sull'altezza e sul periodo medio delle onde nel gruppo viene stimato a meno di errori di ordine inferiore a (H/ m ₀)^–1 (m ₀ essendo la varianza della quota del pelo libero del campo di onde irregolari). La conoscenza delle differenze inflnitesime tra il gruppo di onde vero e il gruppo di onde deterministico, per H/ m ₀,si dimostra utile per prevedere quali possano essere le differenze nel caso che H sia realisticamente grande per uno stato di mare. Allo scopo si utilizzano anche i dati di simulazioni numeriche dei campi di onde di gravita irregolari. La conclusione è che, per H realisticamente grande, il gruppo di onde deterministico rispecchia da vicino i caratteri essenziali del gruppo di onde vero.

     

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.

     

Heat transfer for laminar flow in internally finned pipes with different fin heights and uniform wall temperature

An analysis is presented for fully developed laminar convective heat transfer in a pipe provided with internal longitudinal fins, and with uniform outside wall temperature. The fins are arranged in two groups of different heights. The governing equations have been solved numerically to obtain the velocity and temperature distributions. The results obtained for different pipe-fins geometries show that the fin heights affect greatly flow and heat transfer characteristics. Reducing the height of one fin group decreases the friction coefficient significantly. At the same time Nusselt number decreases inappreciably so that such reduction is justified. Thus, the use of different fin heights in internally finned pipes enables the enhancement of heat transfer at reasonably low friction coefficient.Nomenclature A_f dimensionless flow area of the finned pipe, Eq. (8) - a_f flow area of the finned pipe - C_p specific heat at constant pressure - f coefficient of friction, Eq. (12) - H₁, H₂ dimensionless fin height h₁/r_o h₂/r_o - h₁, h₂ fin heights - average heat transfer coefficient at solid-fluid interface - KR fin conductance parameter, k_s/k_f - k_f thermal conductivity of fluid - k_s thermal conductivity of fin - l pipe length - mass flow rate - N number of fins - Nu Nusselt number, Eqs. (15) and (16) - P pressure - Q total heat transfer rate at solid fluid interface - Q_f1, Q_f2 heat transfer rate at fin surface - q_w average heat flux at pipe-wall, Q/(2 r_ol) - R dimensionless radial coordinate r/r_o - Re Reynolds Number, Eq. (13) - r radial coordinate - r_o radius of pipe - r₁, r₂ radii of fin tips - T temperature - T_b bulk temperature - U dimensionless velocity, Eq. (2) - U_b dimensionless bulk velocity - u_z axial velocity - z axial coordinate - angle between the flanks of two adjacent fins - half the angle subtended by a fin - angle between the center-lines of two adjacent fins - angular coordinate - dynamic viscosity - density - dimensionless temperature, Eq. (6) - _b dimensionless bulk temperature     

Determination of rough-surface skin friction coefficients from wake profile measurements

A technique for determining the skin friction coefficients from wake profile measurements is described, and is applied to symmetric turbine airfoils with rough surfaces, which operate in a compressible, high-speed flow environment. The procedure involves the measurement of profiles of streamwise momentum in the wakes which form downstream of different airfoils with different levels of surface roughness. Different physical phenomena which affect such wake profiles are discussed and related to different effects, such as surface roughness, form drag, flow separation zones, and laminar-to-turbulent transition. With the same inlet experimental condition for each case, overall skin friction coefficients for the rough airfoils are determined. Resulting values increase considerably as the magnitude of equivalent sandgrain roughness size increases.List of symbols A test airfoil surface area - A _i test section inlet area - A _e test section exit area - c chord length of airfoil - C _f/2 skin friction coefficient - (C _f/2)_smooth skin friction coefficient for smooth airfoil - (C _f/2)_rough skin friction coefficient for rough airfoil - F _s force from surface shear stress - F _p force from form drag due to airfoil blockage and separated flow - F _w force imposed by the top and bottom test section walls - F _s,smooth force from surface shear stress for smooth airfoil - F _s,rough force from surface shear stress for rough airfoil - h height of test section - k roughness height - k _s equivalent sand grain roughness - L total length of airfoil surface from leading edge to trailing edge - p airfoil passage effective pitch - P _o stagnation pressure - P _oe exit local stagnation pressure - P _oe, exit freestream stagnation pressure - P _oi inlet stagnation pressure - P _s static pressure - P _se exit static pressure - s distance along airfoil surface from leading edge - u local streamwise velocity - u _i local streamwise velocity at test section inlet - u _e local streamwise velocity at test section exit - u local freestream streamwise velocity at test section exit - w width of test section - x linear distance along airfoil centerline from airfoil leading edge - y normal coordinate measured from airfoil centerline Greek symbols ratio of specific heats - _s roughness parameter - _i local static air density at test section inlet - _e local static air density at test section exit - local static air density in freestream at test section exit     

The Kepler problem spinor regularization and quantization on the manifolds S2

Summary In a previous research we have shown that the KS-transformation, developed by Kustaanheimo and Stiefel for the regularization of the Kepler problem, may be interpreted as the correspondence which associates to each null 4-vector of the space of Minkowski a one-index spinor, defined up to a phase factor, and we have obtained a new form of the KS-transformation. In the present research we show that this formulation allows a straight derivation of the Hopf fibering of the sphere S³ (characterized by unit spinors) having the base space given by the section (sphere S²) of the light cone, and we show that the KS-transformation allows the quantization of the symplectic manifold S² in the sense of Souriau. The sphere S³ turns out to be a contact quantized manifold. The bilinear relation characteristic of the KS-theory and the column vectors of the KS-matrix are intimately related to the contact structure.

---

Sommario In un precedente lavoro si è mostrato che la trasformazione KS, introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero, è riconducibile alla ben nota corrispondenza fra vettori del cono isotropo dello spazio di Minkowski e spinori semplici, definiti a meno della fase, e si è pervenuti ad una nuova formulazione della KS. Nel presente lavoro si mostra come da tale formulazione scaturisca in modo naturale la fibrazione di Hopf della sfera S³ (caratterizzata dagli spinori unitari) avente quale base una sezione (sfera S²) del cono isotropo e si mette in luce come la trasformazione KS consenta di effettuare la quantizzazione della varietà simplettica S² nel senso di Souriau e di ottenere la sfera S³ quale varietà quantica di contatto. La relazione bilineare caratteristica della teoria KS ed i vettori colonna della matrice KS risultano intimamente legati alla struttura di contatto.

---

---

Presented at the VI Congresso Nazionale dell'Associazione Italiana di Meccanica Teorica ed Applicata (AIMETA), Genova, October 1982. Work performed under the auspices of G.N.F.M. of the C.N.R. (Consiglio Nazionale delle Ricerche).     

Perturbation analysis for periodic heat transfer in radiating fins

A perturbation analysis is presented for periodic heat transfer in radiating fins of uniform thickness. The base temperature is assumed to oscillate around a mean value. The perturbation expansion is carried out in terms of dimensionless amplitude of the base temperature oscillation. The zero-order problem which is nonlinear, and corresponds to the steady state fin behaviour, is solved by quasilinearization. A method of complex combination is used to reduce both the first and the second order problems to two, coupled linear boundary value problems which are subsequently solved by a noniterative numerical scheme. The second-order term is composed of an oscillatory component with twice the frequency of base temperature oscillation and a time-independent term which causes a net change in the steady state values of temperature and heat transfer rate. Within the range of parameters used, the net effect is to decrease the mean temperature and increase the mean heat transfer rate. This is in constrast to the linear case of convecting fins where the mean values are unaffected by base temperature oscillations. Detailed numerical results are presented illustrating the effects of fin parameter N and dimensionless frequency B on temperature distribution, heat transfer rate, and time-average fin efficiency. The time-average fin efficiency is found to reduce significantly at low N and high B.

---

Störungsanalyse für periodische Wärmeübertragung an Strahlungsrippen

Zusammenfassung Eine Störungsanalyse wird für periodische Wärmeübertragung in Strahlungsrippen gleicher Dicke vorgelegt. Die Fußtemperatur wird als um einen Mittelwert schwingend angenommen. Die Störungsentwicklung wird in Termen einer dimensionslosen Amplitude e dieser Schwingung angesetzt. Das Problem nullter Ordnung, das nichtlinear ist und dem stationären Verhalten der Rippe entspricht, wird durch Quasilinearisierung gelöst. Eine Methode der komplexen Kombination wird angewandt, um die Probleme erster und zweiter Ordnung auf zwei gekoppelte Grenzwertprobleme zu reduzieren, die nacheinander nach einem nichtiterativen Schema gelöst werden. Der Term zweiter Ordnung besteht aus einer Schwingungskomponente mit der doppelten Frequenz der Schwingung der Fußtemperatur und einem zeitunabhängigen Term, der eine Nettoänderung der stationären Werte der Temperatur und der Wärmeübertragung verursacht. Im verwendeten Bereich der Parameter tritt eine Abnahme der mittleren Temperatur und eine Zunahme der mittleren Wärmeübertragung auf. Das steht im Gegensatz zum linearen Fall der Konvektionsrippe, bei dem die Mittelwerte durch Schwingungen der Fußtemperatur nicht beeinflußt werden. Detaillierte numerische Ergebnisse zeigen die Einflüsse des Rippenparameters N und der dimensionslosen Frequenz B auf Temperatur Verteilung, Wärmeübertragung und zeitliches Mittel des Rippengütegrades. Dieses zeitliche Mittel nimmt merklich ab bei kleinem N und hohem B.

---

Nomenclature b fin thickness - B dimensionless frequency, L²/ - E emissivity - f₀, f₁ functions of X - g₀, g₁, g₂ functions of X - h₀, h₁, h₂ functions of X - k thermal conductivity - L fin Length - N fin parameter, 2EL²T_bm/bk - q heat transfer rate - Q dimensionless heat transfer rate, qL/kbT_bm - t time - T temperature - T_b fin base temperature - T_S effective sink temperature - T_bm mean fin base temperature - x axial distance - X dimensionless axial distance, x/L - dimensionless amplitude of base temperature (s. Eq.2) - thermal diffusivity - instantaneous fin efficiency - time-average fin efficiency - _ss steady state fin efficiency - dimensionless temperature, T/T_bm - ₀ zero-order approximation - ₁ first-order approximation - ₂ second-order approximation - _2s steady component of ₂ - , ₁, ₂ constants - complex function of X - ₁ real part of - ₂ imaginary part of - complex function of X - ₁ real part of Y - ₂ imaginary part of - dimensionless time, t/L² - frequency of base temperature oscillation     

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

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

Normal and oblique collisions of a vortex ring with a wall

The oblique collision of a vortex ring with a solid wall, atRe=/=1389, has been analysed by the direct simulation of the Navier-Stokes equations in Cartesian coordinates. In accordance with a previous experimental study [1], the secondary vorticity produced at the wall is organized into a loop-like vortex in the region of the ring furthest away from the wall. As the ring approaches the wall, the region closest is subjected to a high rate of stretching which increases the vorticity in the core. The vorticity gradients along the core generate bi-helical vortex lines continually displaced towards the region of the ring furthest away from the wall. The analysis of the vorticity and straining fields revealed that the pressure gradient along the core is responsible for the convective motion that displaces these vortex lines and accumulates secondary vorticity in the region far from the wall. This vorticity rolls up and forms a secondary structure which by self-induction moves away from the wall.The fundamental role of the differential stretching has been demonstrated by comparing the case of oblique collision with that of normal collision and with the collision of a two-dimensional vortex pair with an oblique wall.

---

Sommario L'interazione di un vortice ad anello con una parete obliqua, aRe=1389, è stata analizzata mediante la simulazione diretta delle equazioni di Navier-Stokes in coordinate cartesiane. In accordo con un precedente esperimento [1] è stato evidenziato che la vorticità secondaria, prodotta alla parete, si organizza in una strutura vorticosa a loop nella regione dell'anello più lontana dalla parete. Quando il vortice si avvicina alla parete, la parte più vicina è soggetta ad un'elevata deformazione che aumenta il valore della vorticità nel core. La distribuzione non uniforme di vorticità lungo il core del vortice genera delle linee di vorticità elicoidali che vengono transportate verso la regione dell'anello più lontana dalla parete. L'analisi dei campi di vorticità e di deformazione ha rivelato che il gradiente di pressione, dovuto al campo di deformazione non uniforme lungo il core del vortice, è responsabile di un moto convettivo che trasporta le linee di vorticità ed accumula la vorticità secondaria nella regione del vortice più lontana dalla parete, dove la struttura secondaria viene generata.Il ruolo fondamentale della deformazione non uniforme è stato evidenziato mediante il confronto della collisione obliqua coni casi di collisione normale e di collisione di una coppia di vortici bidimensionali con una parete obliqua.

     

Mixed convection from a horizontal plate to fluids of any Prandtl number

Laminar mixed convection over a horizontal plate with uniform wall temperature or uniform wall heat flux is analyzed by introducing proper buoyancy parameters and transformation variables for fluids of any Prandtl number between 0.001 and 10,000. Both cases of buoyancy assisting and opposing flow conditions are investigated. For the buoyancy-assisting case, the obtained numerical results are very accurate over the entire range of mixed convection intensity from pure forced convection limit to pure free convection limit. For the buoyancy-opposing case, solutions are obtained from the forced convection limit to the point of breakdown.

---

Mischkonvektion an einer horizontalen Platte für Fluide mit beliebiger Prandtl-Zahl

Zusammenfassung Es wurde laminare Mischkonvektion an einer horizontalen Platte mit einheitlicher Wandtemperatur oder einheitlicher Wandwärmestromdichte bei Einführung zweckmäßiger Auftriebsparameter und Transformationsvariablen für Fluide mit beliebiger Prandtl-Zahl zwischen 0,001 und 10 000 untersucht. Es wurden die Fälle der Strömung entgegen und in Richtung der Auftriebskraft untersucht. Für den Fall der Strömung in Richtung der Auftriebskraft wurden sehr genaue numerische Ergebnisse für den gesamten Bereich der gemischten Konvektion von rein erzwungener Konvektion bis zu rein freier Konvektion erhalten. Für den Fall der Strömung entgegen der Auftriebsrichtung wurden Lösungen für erzwungene Konvektion bis zum Umkehrpunkt erhalten.

---

Nomenclature C _f local friction coefficient - f reduced stream function - g gravitational acceleration - Gr local Grashof number for UWT,g (T _w –T )x ³/ ² - Gr* local Grashof number for UHF,g q _w x ⁴/k ² - m =10 for UWT; and =6 for UHF - n =5 for UWT; and =3 for UHF - Nu local Nusselt number - p pressure - Pr Prandtl number,/ - q _w wall heat flux - Ra local Rayleigh number for UWT,Gr Pr - Ra* local Rayleigh number for UHF,Gr*Pr - Re local Reynolds number,u x/ - T fluid temperature - T _w wall temperature - T free-stream temperature - u velocity component inx-direction - u free-stream velocity - v velocity component iny-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - =0 for UWT; and =1 for UHF - buoyancy parameter, =( Ra)^1/5/( Re)^1/2 for UWT; and =( Ra*)^1/6/( Re)^1/2 for UHF - pseudo-similarity variable, (y/x) - dimensionless temperature, =(T–T )/(T _w –T ) for UWT; and =(T–T )/(q _w x/k) for UHF - =[( Re)^1/2+( Ra)^1/5] for UWT; and =[( Re)^1/2+( Ra*)^1/6] for UHF - dynamic viscosity - kinematic viscosity - /(1+) - dimensionless pressure - density - Pr/(1+Pr) - _w wall shear stress,(u/y) _y=0 - stream function - Pr/(1+Pr)^1/3     

Computational evaluation of ship manoeuvring performance based on scale model tests

Summary A brief review of the most important existing mathematical models for predicting the manoeuvring performance of a ship at the design stage is presented. A model based on the derivation of the hydrodynamic coefficients from force measurements on scale models is used to develop a computer program for the evaluation of the ship performance in some standard manoeuvres such as turning circle and zig-zag manoeuvres.

---

Sommario Viene presentata una breve descrizione delle metodologie attuali più seguite per la identificazione di un modello matematico atto alla previsione delle caratteristiche di manovrabilità di una nave. Utilizzando coefficienti idrodinamici ricavati da prove su modelli in scala si è sviluppato un codice di calcolo che consente di ottenere la risposta della nave in alcune manovre standard quali quelle di evoluzione e zig-zag.

---

Symbols G Center of gravity - g Acceleration due to gravity - I _ZZ Moment of inertia aboutz-axis - i _EP Effective moment of inertia about propeller axis - L Length between perpendiculars - m Ship mass - N Hydrodynamic moment aboutz-axis - n Rate of revolutions of propeller - O Origin of shipbound coordinate system - Q Propeller torque - Q _E Engine torque - q _F Engine fuel rate - R _T Total hull resistance - r Rate of turn aboutz-axis (yaw rate) - U Along-track velocity of0 - u, v Components ofU alongx, y-axes - X, Y Hydrodynamic forces alongx, y-axes - x,y,z Shipbound coordinate axes - x _G ,y _G ,z _G Coordinate of center of gravity in the shipbound system - x _o,y ₀,z ₀ Coordinate of 0 in the earthbound system, Fig. 1 - Drift angle - Rudder angle - Characteristic time - Heading angle Presented at the II Convegno AIMETA di Meccanica Computazionale, Rome, June 2–5, 1987.     

On the numerical computation of Generalized Burmester Points

This note describes a procedure for plane higher-curvature path analysis and synthesis. All coefficients have been written in terms of elementary instantaneous invariants. This facilitates the numerical computation of Generalized Burmester Points for a moving link of a planar mechanism in a non-symmetric position. FORTRAN subroutines have been written and a numerical example is provided.

---

Sommario Si descrive una procedura di analisi e sintesi per meccanismi piani generatori di traiettoria con approssimazione del quarto ordine. Nella formulazione adottata, l'impiego degli invarianti istantanei elementari consente di valutare analiticamente i termini delle equazioni per la ricerca dei punti generalizzati di Burmester. Sono state implementate subroutines in linguaggio FORTRAN ed è stato sviluppato un esempio numerico.

---

Nomenclature P _o velocity pole - inflection circle diameter - angle of rotation of the moving body - r _f ,r _m radii of curvature of the fixed and moving polodes, respectively - dl infinitesimal arc length measured on the polode - a,b coordinates ofP _o , in the canonical reference system¹ - a _i ,b _i i-th derivatives ofa andb, respectively, computed at the initial position (i.e. =0). These are the elementary instantaneous invariants - h,* polar coordinates of the moving point in the canonical reference system (0 ) - radius of curvature of the point-path trajectory - _E radius of curvature of the evolute of the point-path trajectory - _E ^/(2) radius of curvature of the evolute of the evolute of the point-path trajectory A canonical reference system is a rectangular right-handed cartesian system having they-axis directed toward inflexion pole, origin in the velocity pole.     

Numerical modelling of unbounded domains in coupled fleld problems

Summary Two- and three-field problems are often defined in domains which may be assumed as unbounded. The traditional approach for their numerical simulation, within the framework of the finite element method, is by simple truncation of the mesh at a finite boundary. This fact both results in a large number of degrees of freedom and causes often errors in the analysis, due to the difficulty of setting correct conditions at the finite boundary.This paper shows the possible errors of the ensuing numerical solution and points out the usefulness of the infinite elements to simulate the far field response. Three examples from the field of isothermal and nonisothermal consolidation are presented where the improvements in the numerical simulation obtained by the use of infinite elements are evidenced. These examples may be considered as representative for a series of other coupled problems involving partial differential equations with first order time derivatives.

---

Sommario Problemi di interazione fra due e tre campi sono spesso deflniti in domini che possono essere assunti come illimitati. Il modo tradizionale per la modellazione numerica di questi casi, nell'ambito del metodo degli elementi finiti, è quello di assumere frontiere fittizie in corrispondenza alle quali si devono imporre condizioni al contorno spesso di non facile valutazione. Questo modo di procedere comporta un numero di gradi di libertà assai elevato e può essere fonte di errori derivanti dall'imposizione di condizioni al contorno non corrette in corrispondenza della frontiera fittizia.Nel presente lavoro vengono evidenziati possibili errori delle conseguenti soluzioni numeriche e viene rimarcata l'utililità di elementi infiniti nella trattazione di questi problemi. In tre esempi di consolidazione isoterma e non, vengono messi in luce i miglioramenti della soluzione dovuti all'uso di elementi infiniti. Questi esempi sono rappresentativi di una più vasta classe di problemi accoppiati govemati da equazioni differenziali alle derivate parziali con derivate del primo ordine rispetto al tempo.

---

Notation b body force vector - c coefficient of consolidation - c strain independent vector defining the creep strain rate - C _s specific heat of the solid phase - C _w specific heat of the fluid phase - D _T tangential stiffness matrix - g gravity acceleration - k absolute permeability matrix - k coefficient of thermal diffusivity - k _s bulk modulus of the solid phase - k _w bulk modulus of the fluid - L differential operator which relates displacements to strains - m (1 1 1 0 0 0) ^T - p pore pressure - Q _e volumetric outflow of the fluid per unit volume of the solid - Q _h outflow of heat per unit volume of solid - t time variable - dimensionless time parameter - T temperature increase over an equilibrium state - boundary traction vector - u displacement vector - V ^a apparent velocity of the fluid - z elevation above some datum - _s thermal expansion coefficient of the solid phase - _w thermal expansion coefficient of the fluid - total strain vector of the soil skeleton - ₀ represents all other strains not directly associated with stress changes - thermal conductivity matrix of the soil - dynamic viscosity - _s density of the solid - _w density of the fluid - effective stress in the soil skeleton - porosity Paper presented at the First Italian Meeting on Computational Mechanics (Milan, June 1986).     

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     

A simplified flood routing model: variable prameter muskingum (VPM)

Summary Flood routing methods are numerical methods for estimating the movement of a flood wave along a channel reach, on the basis of the knowledge of the discharge hydrograph at the upstream end and of the hydraulic characteristics of the reach and, usually, in the hypothesis that no perturbation is coming from downstream (free boundary condition). The flood routing method wich is proposed is similar to the Muskingum one, but with variable and hydraulic parameters; it is able to estimate water levels too; is effective even if kinetic terms are not completely negligible; take advantage of the insignificance of the downstream condition and make it possible to obtain results starting upstream and proceeding downstream; for simplicity's sake, take advantage of the fact that the discharge loop of normal flood waves is quite small. Obtained results are much better that those obtainable from constant parameters methods and indeed, if the flood loop is less that 10%, very similar to those obtainable from more complex and time consuming models.

---

Sommario I metodi di flood routing studiano la propagazione di un'onda di piena lungo un tratto di un corso d'acqua, assegnato l'andamento temporale della portata nella sezione di monte e le caratteristiche dell'alveo, e usualmente nell'ipotesi di assenza di perturbazioni provenienti da valle (condizione di valle passiva). Viene qui proposto un procedimento di flood routing, formalmente simile ad un Muskingum ma con i parametri variabili e calcolati per via idraulica; idoneo a stimare anche i livelli idrici; valido anche se i termini cinetici non sono del tutto trascurabili; che sfrutta l'irrilevanza della condizione di valle procedendo a cascata da monte a valle; che sfrutta, a vantaggio della semplicità, il fatto che per le normali onde di piena dei corsi d'acqua il cappio di portata è di dimensioni modeste. I risultati ottenuti sono molto migliori di quelli ottenibili con metodi a parametri costanti e, almeno per i casi in cui il cappio relativo è inferiore al 10%, paragonabili a quelli ottenuti con metodi molto più complessi ed onerosi.

---

List of symbols x, t channel distance, starting upstream; time - z water surface height above datum - Q volumetric rate of discharge - P(x, z) steady rating curve - q=Q–P flood loop - g acceleration of gravity - A, B cross section wetted area and free surface width - I, S water surface slope and friction slope - c kinematic wave velocity - F Froude number - L,L ₀,L ₁,L ₂,L ₃ characteristic lengths of the channel - T=L/c characteristic time of the channel - D diffusion - p, l time and space steps - K, X Muskingum parameters - C ₁,C ₂,C ₃,C ₄ Muskingum coefficients - f _x=f/x,f _t=f/t etc. for the partial derivatives Paper presented at the First Italian Meeting of Computational Mechanics held in Milan, June 24–26, 1986.     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

---

Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

---

---

Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.