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.

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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.

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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).     

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.

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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.

     

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.

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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.

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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.     

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.

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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.

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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     

PIV measurements of velocities and concentrations of wood fibres in pneumatic transport

A method for simultaneous measurement of the concentration and velocity of wood fibres suspended in air was developed. The velocity of the wood fibres was measured by the use of particle image velocimetry (PIV). The concentration of wood fibres was measured using the raw images from the PIV equipment as input data. An image processing procedure was used to determine the volume fraction of the fibre particles in the images. The method gave good qualitative and quantitative results for low volume fractions of fibres; for higher volume fractions the quantitative results were unsatisfactory.Latin letters C Concentration of fibres [g/m³] - d Diameter of fibre [m] - M_w Water mass [kg] - M_f Fibre mass [kg] - m Calibration mass flow [kg/s] - m₂₅ Calibration mass flow at C=25 g/m³ [kg/s] - n Fan rpm [-] - t Thickness of light sheet [m] - t Time between laser pulses [s] - U_i Velocity component in i-direction [m/s] - v Streamwise velocity [m/s] - v_average Average streamwise velocity [m/s] - x_i Particle displacement in i-direction [m] Greek letters _f Volume fraction of fibres [-] - _average Average volume fraction of fibres [-] - Area fraction of fibres in image [-] - Density of fibre particle [kg/m³]     

Similarity Solutions for Thawing Processes with a Heat Flux Condition at the Fixed Boundary

Similarity solutions for a mathematical model for thawing in a saturated semi-infinite porous medium is considered when change of phase induces a density jump and a heat flux condition of the type is imposed on the fixed face x=0. Different cases depending on physical parameters are analysed and the explicit solution is obtained if and only if an inequality for the thermal coefficient q ₀ is verified. An improvement for the existence of a similarity solution for the same free boundary problem with a constant temperature on the fixed face x=0 is also obtained. Sommario. Vengono considerate soluzioni di similarità per un modello matematico di disgelo di un mezzo poroso saturo semi-infinito allorquando il cambiamento di fase induce un salto di densità ed una condizione di flusso di calore del tipo viene imposta sulla faccia fissa x=0. Si analizzano differenti casi dipendenti da parametri fisici e la soluzione esplicita viene ottenuta se e solo se risulta verificata una diseguaglianzo per il coefficiente termico q ₀. Si ottiene altresi un miglioramento della condizione di esistenza di una soluzione di similarità per lo stesso problema al contorno libero con temperatura costante sulla faccia fissa x=0.     

Predictions on momentum,heat and mass transfer in turbulent channel flow with the aid of a boundary layer growth-breakdown model

This paper deals with theoretical aspects of momentum, heat and mass transfer in turbulent channel flow and in particular with phenomena occurring close to the wall. The analysis presented involves the use of a boundary-layer growth-breakdown model. Theoretical expressions have been derived predicting heat and mass transfer at smooth surfaces in the fully developed and entrance region and at surfaces provided with ideal two-dimensional roughness elements. The analysis is restricted to fluids having Prandtl and Schmidt numbers larger than one. Good agreement appears to exist between theoretical predictions and experimental observations.

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Zusammenfassung Diese Arbeit behandelt die Theorie der Übertragungsvorgänge von Impuls, Wärme und Stoff in turbulenter Kanalströmung unter besonderer Berücksichtigung der Vorgänge in Wandnähe. Das verwendete Modell beruht auf dem Zusammenbruch der anwachsenden Grenzschicht. Für die ausgebildete Strömung und für den Einlaufbereich bei glatter Wand und bei Oberflächen mit idealen zweidimensionalen Rauhigkeitselementen werden theoretische Ausdrücke abgeleitet bei Beschränkung auf Prandtl- und Schmidt-Zahlen über Eins. Zwischen den theoretischen Voraussagen und den Versuchsergebnissen scheint gute Übereinstimmung zu herrschen.

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Nomenclature a thermal diffusivity [m²/s] - c concentration [kg/m³] - c _p specific heat [J/kg °C] - D molecular diffusivity [m²/s] - G relative increase in friction factor due to surface roughening - d pipe diameter [m] - e height (depth) of roughness element [m] - e ^p+ dimensionless roughness height (depth) - F parameter denoting the ratio - f friction factor for smooth surface and isothermal conditions - f _h friction factor for heating conditions - f _r friction factor for artificially roughened surface - n _av average frequency of fluctuations at the wall [s^–1] - q heat flux [W/m₂] - q _w heat flux at the wall [W/m²] - q _wr heat flux at roughened wall [W/m²] - q _wx wall heat flux to growing laminar boundary layer at positionx [W/m²] - R _ma longitudinal correlation coefficient for mass transfer - R _mo longitudinal correlation coefficient for momentum transfer - T temperature [°C] - T _b bulk temperature of fluid [°C] - T ₀ fluid temperature at edge of viscous boundary layer (edge of viscous region) [°C] - T _w wall temperature [°C] - T _wx wall temperature at positionx for growing laminar boundary layer [°C] - t time [s] - t ₀ characteristic time period associated with boundary layer growth [s] - u local axial fluid velocity, at wall distancey, for turbulent flow also denoting the mean velocity at that distance [m/s] - u _b bulk fluid velocity [m/s] - u ₀ fluid velocity at edge of viscous boundary layer (edge of viscous region) [m/s] - u _0r fluid velocity at edge of viscous region for the case of an artificially roughened wall [m/s] - u axial fluid velocity fluctuation [m/s] - u ⁺ dimengionless fluid velocity,u/(_w/)^1/2 - u _i ⁺ instantaneous value ofu ⁺ - u _min ⁺ minimum value ofu _i ⁺ - u _r ⁺ root mean square value of dimensionless axial velocity - u ₀ ⁺ value ofu ⁺ at edge of viscous region - v fluid velocity normal to flow direction and normal to wall [m/s] - v fluctuation of the velocityv [m/s] - x coordinate in flow direction [m] - x axial distance interval [m] - x ⁺ dimensionless distance interval - x ₀ viscous boundary layer growth length [m] - x ₀ ⁺ dimensionless boundary growth length - x _r axial dixtance between roughness elements [m] - x _r ⁺ dimensionless distance between roughness elements - x _h value of viscous boundary growth length for heating conditions [m] - y distance from wall [m] - y ⁺ dimensionless wall distance - y _v thickness of viscous region [m] - y _v ⁺ dimensionless form ofy _v - z _u unheated (zero mass transfer) part of elementary viscous boundary layer in entrance region [m] - z _h heated (mass transfer) part of elementary viscous boundary layer [m] - z _v lateral extent of elementary viscous boundary layer [m] Greek symbols heat transfer coefficient defined with respect to bulk fluid temperature [W/m² °C] - ₀ viscous region heat transfer coefficient [W/m² °C] - _0h viscous boundary layer heat transfer coefficient averaged over lengthx ₀ for conditions of heating [W/m² °C] - _0hh viscous region heat transfer coefficient averaged over lengthx _h for conditions of heating [W/m² °C] - entrance region heat transfer coefficient at position [W/m² °C - _,t viscous boundary layer heat transfer coefficient at position and timet [W/m² °C] - mass transfer coefficient [m/s] - _av average value of mass transfer coefficient [m/s] - _x mass transfer coefficient for viscous boundary layer at positionx [m/s] - entrance region mass transfer coefficient at position [m/s] - thickness of laminar (viscous) boundary layer evaluated atu=1/2u ₀ [m] - _max maximum value of boundary layer thickness [m] - _i turbulent diffusivity for momentum transfer [m²/s] - _h turbulent diffusivity for heat transfer [m²/s] - _m turbulent diffusivity for mass transfer [m²/s] - turbulent intensity - thermal conductivity [W/m °C] - kinematic viscosity [m²/s] - ₀ value ofv at edge of viscous region [m²/s] - _w value ofv at the wall [m²/s] - density [kg/m³] - shear stress [N/m²] - _tx local value of wall shear stress associated with viscous boundary layer growth [N/m²] - ₀ value of wall shear stress averaged over lengthx ₀ [N/m²] - _0r value of ₀ for the case of an artificially roughened wall [N/m²] - _0h value of ₀ for heating conditions [N/m²] - _h value of wall shear stress for heating conditions, averaged over lengthx _h [N/m²] - _w wall shear stress for conditions of turbulent flow [N/m²] - _wh value of _w for heating conditions [N/m²] - dimensionless axial distancex/x ₀ in extrance region Dimensionless numbers Nu Nusselt number (d/) - Nu _x Entrance region Nusselt number at axial positionx - Nu _h Nusselt number for heating conditions - Nu _r Nusselt number for the case of artificially roughened surface - Pr Prandtl number (v/a) - Re Reynolds number (d u _b/v) - Re _b Boundary layer Reynolds number (1/2 u ₀/v) - Re _ber Critical value ofRe _b - Sh Sherwood number (d/D) - Sh _x entrance region Sherwood number at axial positionx - Sc Schmidt number (v/D)     

Buckling of inelastic arches: A simple model

Summary In analogy with the well-known Shanley's column model, a model has been devised to illustrate the buckling behaviour of three-hinged arches in the inelastic range. A bilinear stress-strain relationship has been assumed in the deformable elements. As in the case of the compressed strut, a wholesegment of bifurcation is found, and load values analogous to thetangent modulus (Shanley) andreduced modulus (Kàrmàn) loads can be defined. An example of load-displacement curves is fully developed.

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Sommario Si studia un modello teorico di arco inelastico a tre cerniere, analogo al ben noto modello di asta compressa proposto da Shanley, assumendo una relazione bilineare tra sforzi e deformazioni negli elementi del concio deformabile. Si ritrova anche in questo caso l'esistenza di unsegmento di biforcazione, si definiscono gli analoghi delcarico di Shanley e delcarico di Von Kàrmàn, e si traccia un esempio di curve carico-spostamento.

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The results presented in this paper were obtained in the course of researches sponsored by the Plasticity Group of the National Research Council of Italy (C.N.R.). An Italian version of this paper has been published in litographed form in July, 1967 by the Istituto di Scienza delle Costruzioni of Naples University.

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(presently, on leave at the Division of Engineering, Brown University, Providence, R. I.).     

Mixed convection in vertical channels with a discrete heat source

Two dimensional laminar mixed convection flow in vertical channels with a discrete heat source was numerically analyzed. An isoflux discrete heating element was located on the left wall, while the isothermal conditions were imposed on the other wall. The governing equations were solved using a finite difference method based on the control volume approach. The mean Nusselt number was calculated and the maximum component temperature was determined. The computations were carried out for different Grashof number, Reynolds number, heater locations and the channel width. It was observed that the location of the heating element does not play a considerable role on the flow. At low Reynolds numbers (Re<50), the mean Nusselt number and the maximum temperature are mainly controlled by the Grashof number. However, at higher Reynolds numbers, the Reynold number plays an important role on the flow. It was also found that at low Reynolds numbers, cooling is more effective when the channel width is large (W/H>1). However, at high Reynolds numbers more effective cooling is obtained in narrow channels.

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Mischkonvektion in vertikalen Kanälen mit einer lokalen Wärmequelle

Zusammenfassung Die zweidimensionale laminare Mischkonvektion in vertikalen Kanälen mit einer lokalen Wärmequelle wird numerisch untersucht. Ein Heizelement konstanter Wärmeleistung befindet sich auf der linken Kanalwand, die rechte hat konstante Temperatur. Die Lösung der Grundgleichung erfolgte mit Hilfe der auf dem Kontrollvolumenprinzip basierenden Finitdifferenzenmethode. Die mittlere Nusselt-Zahl sowie die Maximaltemperatur des Heizelementes wurden berechnet, und zwar unter Variation der Grashof-Zahl, der Reynolds-Zahl, der Lage des Heizelements und der Kanalbreite. Letztere hatte nur geringen Einfluß auf den Strömungsverlauf. Bei kleinen Reynolds-Zahlen (Re<50) werden Nusselt-Zahl und Maximaltemperatur vorrangig durch die Grashof-Zahl bestimmt, während bei hohen Reynolds-Zahlen letztere den Strömungsvorgang beherrscht. Ferner zeigte sich, daß bei niedrigen Reynolds-Zahlen die Kühlung für große Kanalbreite (W/H>1) effektiver wird und bei hohen Reynolds-Zahlen die Verhältnisse gerade umgekehrt liegen.

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Nomenclature g gravitational acceleration - Gr Grashof number (Gr=gqH ⁴/v²k) - H heater hight - k thermal conductivity of fluid - L height of the channel - Nu Nusselt number - P pressure - Pr Prandtl number - Re Reynolds number (Re=V ₀H/v) - S position of heater center - T temperature - T _c cold wall temperature - T ₀ inlet temperature - u velocity component inx-direction - U dimensionless velocity component inx-direction (U=u/V ₀) - x horizontal axis - X dimensionless horizontal axis (x/H) - v velocity component iny-direction - V dimensionless velocity component iny-direction (V=v/V ₀) - V ₀ inlet velocity - W width of the channel - y vertical axis - Y dimensionless vertical axis (y/H) Greek symbols a thermal diffusivity - thermal expansion coefficient - density of fluid - kinematic viscosity - dimensionless temperature (=(T–T _c)/[qH/k])     

Instability of Buckley-Leverett flow in a heterogeneous medium

We study the simultaneous one-dimensional flow of water and oil in a heterogeneous medium modelled by the Buckley-Leverett equation. It is shown both by analytical solutions and by numerical experiments that this hyperbolic model is unstable in the following sense: Perturbations in physical parameters in a tiny region of the reservoir may lead to a totally different picture of the flow. This means that simulation results obtained by solving the hyperbolic Buckley-Leverett equation may be unreliable.Symbols and Notation f fractional flow function varying withs andx - value off outsideI - value off insideI - local approximation off around¯x - f ^–,f ⁺ values of - f _j ⁿ value off atS _j ⁿ andx _j - g acceleration due to gravity [ms^–2] - I interval containing a low permeable rock - k dimensionless absolute permeability - k ^* absolute permeability [m²] - k _c ^* characteristic absolute permeability [m²] - k _ro relative oil permeability - k _rw relative water permeability - L ^* characteristic length [m] - L ₁ the space of absolutely integrable functions - L the space of bounded functions - P _c dimensionless capillary pressure function - P _c ^* capillary pressure function [Pa] - P _c ^* characteristic pressure [Pa] - S similarity solution - S _j ⁿ numerical approximation tos(x_j, t_n) - S ₁, S₂,S ₃ constant values ofs - s water saturation - value ofs at - s ^L left state ofs (wrt. ) - s ^R right state ofs (wrt. ) - s s for a fixed value of in Section 3 - T value oft - t dimensionless time coordinate - t ^* time coordinate [s] - t _c ^* characteristic time [s] - t _n temporal grid point,t _n=n t - v ^* total filtration (Darcy) velocity [ms^–1] - W, , v dimensionless numbers defined by Equations (4), (5) and (6) - x dimensionless spatial coordinate [m] - x ^* spatial coordinate [m] - x _j spatial grid piont,x _j=j x - discontinuity curve in (x, t) space - right limiting value of¯x - left limiting value of¯x - angle between flow direction and horizontal direction - t temporal grid spacing - x spatial grid spacing - length ofI - parameter measuring the capillary effects - argument ofS - _o dimensionless dynamic oil viscosity - _w dimensionless dynamic water viscosity - _c ^* characteristic viscosity [kg m^–1s^–1] - _o ^* dynamic oil viscosity [kg m^–1s^–1] - _w ^* dynamic water viscosity [k gm^–1s^–1] - _o dimensionless density of oil - _w dimensionless density of water - _c ^* characteristic density [kgm^–3] - _o ^* density of oil [kgm^–3] - _w ^* density of water [kgm^–3] - porosity - dimensionless diffusion function varying withs andx - ^* dimensionless function varying with s andx ^* [kg^–1m³s] - _j ⁿ value of atS _j ⁿ andx _j This research has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).     

Effect of entrained air on dynamic characteristics of hydraulic servosystem with asymmetric linear motor

Summary A theoretical investigation into the effect of entrained air on dynamic behaviour of hydraulic servosystem is made. The nonlinear system equation developed in dimensionless form is linearised to obtain stiffness, damping ratio and natural frequency in generalised dimensionless form that includes the effects of underlap, leakage, area ratio, per cent air content and the process of change of state of air. A nomogram is developed that helps in quick determination of the dynamic properties directly without any computation. The analysis shows that a decrease in area ratio or an increase in the amount of air entrained into the system reduces both damping ratio and natural frequency. An excessive air entrainment may even lead to instability. The original nonlinear system equation is also solved numerically by fourth order RK-method which shows qualitative agreement with the linear solution. The numerical solution shows that the process of change of state of air also has significant influence on the said dynamic behaviour. A stability chart is prepared relating two most important variables, viz. area ratio and per cent air content, with underlap as a parameter, that facilitates the estimation of allowable air content for a system with given area ratio, or alternatively where the air content in oil is likely to be high; with the help of this graph suitable of underlap and/or area ratio can be selected.

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Sommario Si effettua una ricerca teorica sugli effetti dell'aria catturata sul comportamento dinamico di un servo-sistema idraulico. L'equazione non lineare del sistema, messa in forma adimensionale, viene linearizzata per ottenere la rigidità, il rapporto di smorzamento e la frequenza naturale in una forma adimensionale generalizzata che include gli effetti di restringimento, perdita, rapporto d'area, contenuto percentuale d'aria e il processo di cambiamento di stato dell'aria. Si sviluppa un nomogramma che aiuta a determinare rapidamente le proprietà dinamiche in maniera diretta senza alcun calcolo. L'analisi mostra che una diminuzione del rapporto d'area o un aumento nella quantità d'aria trascinata dentro il sistema riduce sia il rapporto di smorzamento sia la frequenza naturale. Un'eccessiva cattura d'aria può portare a instabilità. L'equazione non lineare viene anche risolta numericamente con un metodo di Runge-Kutta del quart'ordine che mostra un accordo con la soluzione lineare. La soluzione numerica mostra che anche il processo di cambiamento di stato dell'aria ha un'influenza significativa sul suddetto comportamento dinamico. Si prepara un diagramma di stabilità che collega due variabili di grande importanza, cioè il rapporto d'area e il contenuto percentuale d'aria, col restringimento come parametro, che facilita la stima del contenuto d'aria ammissibile per sistemi con un dato rapporto d'area, o alternativamente quando è vero-simile che il contenuto d'aria nell'olio sia elevato; con l'aiuto di questo grafico si possono scegliere valori opportuni di restringimento e/o rapporto d'area.

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Notation V _i initial enclosed volume of fluid - A ₁ piston end cylinder area - A ₂ rod end cylinder area - x nett valve displacement - y output displacement - L cylinder length - d valve sleeve diameter - u _s supply underlap - u _t exhaust underlap - r reference step input - x _p reference port length - P ₁ piston end cylinder pressure - P _s constant supply pressure - p fluid density - c _d orifice discharge coefficient - _oil bulk modulus of oil - K _L laminar leakage coefficient - {ie51-01} reference flow {ie51-02} - t time - M mass of the load - f _b viscous load coefficient - P _at atmospheric pressure - T reference time ({ie51-03}) - percentage of air entrained (by volume) - n polytropic index - X dimensionless nett valve displacement (x/x _p ) - Y dimensionless output displacement (y/x _p ) - Z dimensionless reference step input (r/x _p ) - dimensionless time (t/T) - n dimensionless supply underlap (u _s /x _p ) - v lap ratio (u _t /u _s ) - k area ratio (A ₂/A ₁) - ø dimensionless leakage parameter ({ie51-04}) - P ₁ dimensionless piston end cylinder pressure (P ₁/P _s ) - p _at dimensionless atmospheric pressure (P _at /P _s ) - p dimensionless load pressure (P _L /P _s ) - ₀ dimensionless oil bulk modulus ( _oil/P _s ) - dimensionless cylinder length (L/x _p ) - m dimensionless inertia load parameter (Mx _p /A ₁ P _s T ²) - ₁ dimensionless viscous load parameter (f _b x _p /A ₁ P _s T) - _mix effective bulk modulus of air-oil mixture - compressibility parameter ( _oil/{}_mix)     

A parametric study of boiling in a porous bed

The boiling process in a granular porous medium creates several different layers of the bed — a capillary layer [fixed bed], a cracked layer with horizontal cracks, and a chimney layer with vertical channels. Underneath those layers the bed becomes dry. An analysis is presented to calculate the heights of these different layers.

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Eine parametrische Studie des Siedevorganges in einer körnigen Schichtung

Zusammenfassung Der Siedevorgang in einer körnigen Schichtung bewirkt oft, daß sich in ihr mehrere Schichten ausbildeneine unveränderte kapillare Schicht, darüber eine Zwischenschicht mit horizontalen Sprüngen und darüber eine Schicht durchsetzt von vertikalen Kanälen. Unter diesen Schichten trocknet eine genügend tiefe Schichtung aus, d. h. ist frei von Flüssigkeit und mit überhitztem Dampf erfüllt. Die vorliegende Arbeit berichtet über eine analytische Studie mit dem Ziel, die Höhen der verschiedenen Schichten zu ermitteln.

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Nomenclatsre b channel width, m - c specific heat, J/kg K - f porosity, m³/m³ - g gravitational acceleration, m/s² - h _lv heat of evaporation, J/kg - k _t thermal conductivity, W/K m - Kg gravitational moisture transport coefficient, s - K _p single phase permeability, m² - K_w moisture transport coefficient, m²/s - l height, m - m mass flow, kg/m²s - N number of channels per unit cross-sectional area, l/m² - p pressure, N/m² - q heat flux from walls per unit area, W/m² - q volume heat source, W/m³ - q₀ heat flux conducted upward atx=0 per unit area - r, R effective particle radius, m - R radius of channels in chimney region, m - t temperature, K - velocity, m/s - W moisture content, kg water/kg dry bed - x coordinate, m Greek Symbols contact angle - mass generation rate of vapor, kg/m³s - tortuosity + vapor injection effect - two-phase to the permeabilityK _p - viscosity, kg/s m - density, kg/m³ - surface tension, N/m - wall shear stress, N/m² Indices d dry substance - e exit - i inlet - 1 liquid - s saturation - v vapor - c capillary - cr cracked - ch chimney - 0 atx=0 - d dry - f fluidized - t total=dry+cracked+chimney We dedicate this paper to Fran Bonjakovi on his 80th birthday. His early work enhanced our understanding of boiling     

Response variables correlation in stochastic finite element analysis

Summary Stochastic finite element techniques are used for estimating the uncertain response of discretized structural systems which are defined by random quantitites or are subjected to random excitations. Such techniques are especially useful in view of the development of probabilistic models for the estimation of the residual lifetime of components undergoing damage-accumulation processes. A special procedure for calculating the probability distribution of the single response variable was discussed in Ref. [1].Attention is focused in this paper on the improvements to be included for estimating the correlation between the response variables of interest and their joint probability distribution.

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Sommario La dicitura elementi finiti stocastici contraddistingue quell'insieme di tecniche numeriche che consentono una stima dell'incertezza che affligge la risposta di sistemi strutturali discreti o discretizzabili, caratterizzati da parametri aleatori o/e soggetti ad eccitazioni aleatorie. L'impiego di queste tecniche si è mostrato particolarmente utile nella definizione delle variabili di ingresso di quei modelli probabilistici con cui si stima la vita residua di elementi strutturali soggetti ad accumulo di danno. Un procedimento efficiente per ricavare la legge di distribuzione di probabilità della singola variabile di risposta era stato messo a punto in [1].In questo lavoro si discutono gli sviluppi necessari per includere la stima della correlazione tra due variabili e la determinazione di distribuzioni di probabilità congiunta.

     

Convective heat transfer within fibrous insulation slabs

A numerical study of convective heat flow within a fibrous insulating slab is presented. The material is treated as an anisotropic porous medium and the variation of properties with temperature is taken into account. Good agreement is obtained with available experimental data for the same geometry.

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Zusammenfassung Für den konvektiven Wärmestrom in einem faserförmigen Isolierstoff wird eine numerische Berechnung angegeben. Der Stoff wird als anisotropes poröses Medium mit temperaturabhängigen Stoffwerten angesehen. Die Übereinstimmung mit verfügbaren Versuchswerten ist gut.

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Nomenclature C_p specific heat of the gas at the mean temperature - Da Darcy number=k_y/H² - Gr^* modified Grashof number=gTHk_y/²= (Grashof number) × (Darcy number) - H thickness of the specimen - P gas pressure - Pr^* modified Prandtl number= C_p/_x - Ra^* modified Rayleigh number=Gr^* Pr^* - R_p ratio of permeabilities=k_y/k_x - R_k ratio of conductivities= _y/_x - T absolute temperature of the gas - t₁ absolute temperature of the hot face - T₂ absolute temperature of the cold face - T_m mean temperature of the gas=(T₁+T₂)/2 - k_x specific permeability of the porous medium along the x-direction - k_y specific permeability of the porous medium along the y-direction - p T/T_m - q exponent - r exponent - u gas velocity along the x-direction - v gas velocity along the y-direction - X_* distance along the x-direction - y_* distance along the y-direction - T temperature difference=t₁–T₂ - thermal coefficient of expansion of the gas - _m thermal coefficient of expansion of the gas at the mean temperature - _* T–T_m - dimensionless temperature= ^*/T - _a apparent thermal conductivity of the porous medium along the x-direction - _al local apparent thermal conductivity of the porous medium along the x-direction - _x thermal conductivity of the porous medium along the x-direction in the absence of convection - _y thermal conductivity of the porous medium along the y-direction in the absence of convection - dynamic viscosity of the gas - _m dynamic viscosity of the gas at the mean temperature - kinematic viscosity of the gas - _m kinematic viscosity of the gas at the mean temperature - density of the gas - _m density of the gas at the mean temperature - ^* stream function at any point - dimensionless stream function= ^*/( _m/_m)     

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.

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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.

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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.     

A new transient method for analysis of solidification in the continuous casting of metals

Solidification processes involve complex heat and mass transfer phenomena, the modelling of which requires state-of-the art numerical techniques. An efficient and accurate transient numerical method is proposed for the analysis of phase change problems. This method combines both the enthalpy and the enhanced specific heat approaches in incorporating the effects of latent heat released due to phase change. The sensitivity and accuracy of the proposed method to both temporal and spatial discretization is shown together with closed-form solutions and the results from the enhanced specific heat approach. In order to explore the proposed method fully, a non-linear heat release, as is the case for binary alloys, is also examined. The number of operations required for the new transient approach is less than or equal to the enhanced heat capacity method depending on the averaging method adopted. To demonstrate the potential of this new finite-element technique, measurements obtained on operating machines for the casting of zinc, aluminum and steel are compared with the model predictions. The death/birth technique, together with the proper heat-transfer coefficients, were employed in order to model the casting process with minimal error due to the modelling itself.Nomenclature [A] conductance matrix - [B] matrix containing the derivative of the element shape functions - c, C _p specific heat (J kg^–1°C^–1) - effective specific heat (J kg^–1°C^–1) - f(T) local liquid fraction - f thermal load vector - H enthalpy (J kg^–1) - [H] capacitance matrix - h, h _r,h _c heat transfer coefficient (W m^–2°C^–1) - K thermal conductivity (W m^–1°C^–1) - L latent heat of solidification (J kg^–1) - l overall length (m) - N _i shape functions - Q rate of heat generation per unit volume (J m^–3) - q heat flux (W m^–2) - R residual temperature (°C) - T temperature (°C) - T _s solidus temperature (°C) - T _l liquidus temperature (°C) - T _pouring pouring temperature (°C) - T _top temperature at the top of the mould (°C) - T _w temperature of the water spray (°C) - approximated temperature (°C) - T surrounding temperature (°C) - cooling rate (°C/s) - t time (seconds) - x _i,x, y, z spatial variables (m) - t time step (s) - x element size (m) - diffusivity (m²s^–1) - density (kg m^–3) - time marching parameter - _n direction cosines of the unit outward normal to the boundary     

Influence of stacking sequence and interlaminar layer on stress singularities at free edge of composite laminates

Summary The topic concerning the singularity in the state of stress existing at the free edges of plane composite laminates with resin interlaminar layer, has been dealth with. The problem has been studied by means of a F.E. approach, founded on a method able to determine the free edge state of stress without aprioristical hypotheses about the presence of a singularity. Then the singularity itself has been studied through a linear-logarithmic formulation which enables, for the stacking sequences considered — all belonging to the [°/°–90°] _s family — and for all meaningful stress components, to compute the power of the singularity and the boundary layer stress intensity factor (or free edge stress intensity factor). Taking into account these parameters, various laminations and stacking sequences have been compared, in order to provide a practical measure of the severity of the singular stress field and its influence on failure modes.Then a comparative analysis has been worked out, in order to evaluate the effects due to the thickness of the resin interlaminar layer between two fiber layers.

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Sommario Si prende in considerazione il problema connesso con l'esistenza di singolarità nel campo di sforzo presente ai bordi liberi di laminati piani in materialxe composito, in presenza di uno strato di resina interlaminare.La questione viene affrontata tramite un approccio numerico agli elementi finiti, basato su di un metodo in grado di determinare il campo di sforzo al bordo libero senza assumere aprioristicamente la presenza di singolarità. La eventuale singolarità viene poi studiata mediante una trattazione logaritmico-lineare che consente di calcolare, per tutte le laminazioni prese in considerazione, (appartenenti alla famiglia [°/°–90°] _s )e per le componenti di sforzo più significative, il valore della potenza della singolarità e quello che viene definito fattore d'intensità degli sforzi al bordo libero. Sulla scorta di tali valori si confrontano le varie laminazioni e sequenze di impaccamento.Per tuttii questi aspetti viene inoltre condotta un'indagine comparativa, per valutare l'effetto dovuto allo spessore dello strato di resina interlaminare.

     

Oil feed influence on the behaviour of a journal bearing

Summary The result of a research, presented at a recent AIMETA conference [1],are reported once again and rediscussed on the basis of further investigations. The research is concerned with the oil feed influence on the behaviour of a lubricated journal, rotating in a cylindrical bearing. Experiments showed that the journal locus varies with the oil flow rate or the oil feed pressure, depending on the LID ratio with L/D=0.5 the actual journal locus appears to depend on the oil flow rate, while with L/D=1 the main factor appears to be the feed pressure. It is also affected by viscosity; only at the lowest viscosity and with L/D=0.5 are the experimental journal positions very close to the theoritical locus, no matter what the feed pressure or the oil flow rate are. With L/D=1, attitude angles greater than 90 degrees were observed.

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Sommario Vengono riproposti e ridiscussi, integrati da ulteriori esperienze, i risultati di una ricerca presentata ad un recente congresso AIMETA [1],L'indagine riguarda l'influenza che ha l'alimentazione sul comportamento di un perno lubrificato in un cuscinetto cilindrico. Sono statt utilizzati rapporti L/D=0.5 ed 1 e si è operato con valori diversi della viscosità dell'olio. Le esperienze hanno mostrato che il luogo delle posizioni di equilibrio del perno varia con la portata o con la pressions di alimentazione del lubriflcante, in dipendenza del valore del rapporto L/Dper L/D=0.5 il luogo effettivo appare condizionato dal valore della portata, mentre per L/D=1 il parametro determinante appare essere la pressione di alimentazione. Il fenomeno è influenzato anche dalla viscosità. Per L/D=1 sono stati osservati angoli di attitudine maggiori di 90 gradi.

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Symbols C Radial clearance - e Eccentricity - D Bearing diameter - L Bearing axial width - N Shaft speed (r.p.m.) - N Shaft speed (r.p.s.) - Q Oil flow rate (m ³·s ^–1) - Qt Theoretic oil flow rate - p Oil feed pressure (Pa) - S Sommerfeld number=(R/C)² NLD/W - W Load - Eccentricity ratio=e/C - Absolute viscosity (Pa·s) - Attitude angle     

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.

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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.

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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.     

Momentum and heat transfer in the entrance region of a parallel plate channel: developing laminar flow with constant wall temperature

A new integral method of solution is presented for developing laminar flow and heat transfer in the entrance region of a parallel plate channel with uniform surface temperature. Unlike earlier Karman-Pohlhausen analyses, the new analysis provides solutions which are free from jump discontinuities in the gradients of the velocity and temperature distributions throughout and at the end of the entrance region. The hydrodynamic and thermal results from the present analysis therefore join smoothly and asymptotically to their fully-developed values. The heat transfer results obtained are further found to agree well with previously published numerical solutions.Nomenclature a half width of the channel, m - D _h hydraulic diameter (=4a), m - h local heat transfer coefficient,W/(m²·K) - h _m mean heat transfer coefficient defined by Eq- (9),W/(m²·K) - k thermal conductivity, W/(m·K) - L _H axial length of the hydrodynamic entrance region, m - L _T axial length of the thermal entrance region, m - L _in,H axial length of the hydrodynamic inlet region, m - L _in,T axial length of the thermal inlet region, m - Nu _x local Nusselt number,hD _h /k, dimensionless - Nu _m mean Nusselt number defined by Eq. (9),h _mD_h/k, dimensionless - P pressure, N/m² - P _O pressure at the entrance, N/m² - Pr Prandtl number,c _p /k, dimensionless - Re Reynolds number, 4aU _o /v, dimensionless - T absolute temperature, K - T _b fluid bulk temperature, K - T _c centerline temperature, K - T _w wall temperature, K - U _c centerline velocity, m/s - U ₀ velocity of the fluid at entrance, m/s - U core velocity, m/s - u velocity component inx direction, m/s - v velocity component iny direction, m/s - x spatial coordinate, axial distance, m - y spacial coordinate measured from channel wall, m Greek Letters molecular thermal diffusivity, m²/s - hydrodynamic shape factor, dimensionless - _T thermal shape factor, dimensionless - hydrodynamic boundary layer thickness, m - * /a, dimensionless - _T thermal boundary layer thickness, m - * _{T T} /a, dimensionless - dimensionless distance,y/ ory/a - Pohlhausen's shape factor, dimensionless - dynamic viscosity coefficient, kg/(m·s) - v kinematic viscosity,/, m²/s - dimensionless axial distance,x/(a·Re) - _H dimensionless axial length of the hydrodynamic entrance region (=L _H /(a·Re)) - _T dimensionless axial length of the thermal entrance region (=L _T /(a·Re)) - _in,H dimensionless axial length of the hydrodynamic inlet region (=L _in,H/(a·Re)) - _in,T dimensionless axial length of the thermal inlet region (L _in,T /(a·Re)) - fluid density, kg/m³