Similarity solutions for creeping flow and heat transfer in second grade fluids

The two-dimensional equations of motions for the slowly flowing and heat transfer in second grade fluid are written in cartesian coordinates neglecting the inertial terms. When the inertia terms are simply omitted from the equations of motions the resulting solutions are valid approximately for Re?1. This fact can also be deduced from the dimensionless form of the momentum and energy equations. By employing Lie group analysis, the symmetries of the equations are calculated. The Lie algebra consist of four finite parameter and one infinite parameter Lie group transformations, one being the scaling symmetry and the others being translations. Two different types of solutions are found using the symmetries. Using translations in x and y coordinates, an exponential type of exact solution is presented. For the scaling symmetry, the outcoming ordinary differential equations are more involved and only a series type of approximate solution is presented. Finally, some boundary value problems are discussed.     

Rotational instability in second order fluids

Summary Using Rivlin and Erickson constitutive equation, the rotational instability of a second order fluid contained between two concentric rotating cylinders has been examined. It is found that the marginal instability is governed by a sixth order eigen value problem. The critical Taylor's number which determines the onset of instability has been determined as a function of m (=₂/₁), a (wave length) and S (=2S ₁+S ₂), with the help of a modified Galerkin technique. Figures 1 to 4 show the effect of second order terms on Taylor's number, radial velocity and streamlines.On study leave at the Department of Applied Mathematics, Indian Institute of Technology, Kharagpur, India.     

High‐order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations

In this paper, we investigate the accuracy and efficiency of discontinuous Galerkin spectral method simulations of under‐resolved transitional and turbulent flows at moderate Reynolds numbers, where the accurate prediction of closely coupled laminar regions, transition and developed turbulence presents a great challenge to large eddy simulation modelling. We take full advantage of the low numerical errors and associated superior scale resolving capabilities of high‐order spectral methods by using high‐order ansatz functions up to 12th order. We employ polynomial de‐aliasing techniques to prevent instabilities arising from inexact quadrature of nonlinearities. Without the need for any additional filtering, explicit or implicit modelling, or artificial dissipation, our high‐order schemes capture the turbulent flow at the considered Reynolds number range very well. Three classical large eddy simulation benchmark problems are considered: a circular cylinder flow at Re_D=3900, a confined periodic hill flow at Re_h=2800 and the transitional flow over a SD7003 airfoil at Re_c=60,000. For all computations, the total number of degrees of freedom used for the discontinuous Galerkin spectral method simulations is chosen to be equal or considerably less than the reported data in literature. In all three cases, we achieve an equal or better match to direct numerical simulation results, compared with other schemes of lower order with explicitly or implicitly added subgrid scale models. Copyright © 2014 John Wiley & Sons, Ltd.     

Heat transfer and pressure drop for purely viscous non-Newtonian fluids in turbulent flow through rectangular passages

Experimental measurements of friction factor and heat transfer for the turbulent flow of purely viscous non-Newtonian fluids in a 21 rectangular channel are compared with results previously reported for the circular tube geometry. Comparisons are also made with available analytical and empirical predictions.It is found that the rectangular duct fully established friction factor measurements are within ± 5% of the Dodge-Metzner prediction if the Kozicki generalized Reynolds number is used. A modified form of the simpler explicit equation proposed by Yoo, [i.e.f=0.079n ^0.675(Re ^*)^–0.25], is found to yield predictions for both the rectangular duct and the circular tube geometries with approximately the same accuracy as the Dodge-Metzner equation.Fully developed Stanton numbers for the rectangular duct are in good agreement with the circular tube data over a range ofn from 0.37 to 0.88 for a given Prandtl number,Pr _a , when compared at a fixed value of the Reynolds number based on the apparent viscosity evaluated at the wall shear stress. In general, the experimental data are within ± 20% of Yoo's equation,St=0.0152Re _a ^–0.155 Pr _a ^–2/3 . A new equation is proposed to bring the prediction for circular pipes as well as rectangular channels into better agreement with generally accepted Newtonian heat transfer results.

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Wärmeübergang und Druckverlust für viskose nicht-Newtonsche Fluide in turbulenter Strömung durch rechteckige Kanäle

Zusammenfassung Es werden Messungen des Reibungsfaktors und des Wärmeübergangs bei turbulenter Strömung viskoser nicht-Newtonscher Fluide in einem rechteckigen Kanal mit dem Seitenverhältnis 21 verglichen mit früheren Ergebnissen, die an runden Rohren gewonnen wurden. Weiterhin werden Vergleiche mit aus der Literatur verfügbaren analytischen und empirischen Beziehungen gemacht.Es zeigte sich, daß die Messungen des Reibungsfaktors im rechteckigen Kanal bei vollausgebildeter Strömung auf ± 5% mit der Vorhersage von Dodge-Metzner übereinstimmen, wenn die von Kozicki verallgemeinerte Reynolds-Zahl verwendet wird. Eine modifizierte Form der einfachen von Yoo vorgeschlagenen einfachen Gleichung in explizierter Form (f=0,079n ^0,675(Re ^*)^–0,25) bewies, daß sie sowohl für den rechteckigen Kanal als auch das runde Rohr die Werte mit fast der gleichen Genauigkeit wie die Methode von Dodge-Metzner vorhersagen kann.Die Stanton-Zahlen für den rechteckigen Kanal bei vollausgebildeter Strömung sind in guter Übereinstimmung mit den Werten für das runde Rohr in einem Bereich vonn= 0,37 – 0,88 für eine gegebene Prandtl-Zahl, wenn man den Vergleich bei einem vorgegebenen Wert der Reynolds-Zahl anstellt, die auf die scheinbare Viskosität — abgeleitet aus der Wandschubspannungbezogen ist. Generell läßt sich sagen, daß die Werte auf ± 20% mit der Gleichung von Yoo (St=0,0152Re _a ^–0,155 )Pr _a ^–2/3 ) übereinstimmen. Es wird eine neue Gleichung vorgeschlagen, welche sowohl die Werte für runde Rohre als auch die für rechteckige Kanäle in bessere Übereinstimmung bringt mit den in der Literatur üblichen Ergebnissen für den Wärmeübergang an Newtonsche Fluide.

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Nomenclature a constant in Eq. (8) - A area of cross-section of channel [m²] - b constant in Eq. (8) - c _p specific heat of test fluid [J kg^–1 K^–1] - d capillary tube diameter [m] - D _h hydraulic diameter, 4A/P[m] - f Fanning friction factor, _w/(g9 V²/2) - h axially local (spanwise averaged) heat transfer coefficient,q _w /(T_wi-T_b) [Wm^–2 K^–1] - k _f thermal conductivity of test fluid [Wm^–1K^–1] - K consistency index of power law fluid - n power law index - Nu fully established, local (spanwise averaged) Nusselt numberh D _h /k _f - P perimeter of channel [m] - Pr _a Prandtl number based on apparent viscosjity, c _p /k _f - Pr ^* defined as (Re _a Pr _a )/Re ^* - q _w wall heat flux [Wm^–2] - Re _a Reynolds number based on apparent viscosity, VD _h/ - Re Metzner's generalized Reynolds number in Eq. (2) - Re ^* Reynolds number defined in Eq. (8) - St Stanton number,h/( V c_p) - T _b local bulk temperature of the fluid [K] - T _wi local inside wall temperature [K] - T _wo local outside wall temperature [K] - V bulk flow velocity [m s^–1] - x distance from the inlet of channel along flow direction [m] Greek symbols shear rate [s^–1] - apparent viscosity [Pa s] - density of test fluid [kg m^–3] - shear stress [Pa] - _w shear stress at the wall [Pa] Dedicated to Prof. Dr.-Ing. U. Grigull's 75th birthday     

Effects of buoyancy on heat transfer during turbulent flow of drag reducing fluids in vertical pipes

The effect of buoyancy on the heat transfer during upward turbulent flow of drag reducing fluids in vertical tubes has been theoretically analyzed. A criteria has been established for limiting the decrease in heat transfer to less than 5% for fluids of varying drag reducing ability. The final expression for quantitative evaluation of natural convection effect on forced convection could be applied to upward as well as downward turbulent flow of drag-reducing fluids merely by a change in sign of the controlling term.

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Einfluß des Auftriebes auf den Wärmeübergang bei turbulenter Strömung mit widerstandsreduzierenden Fluiden in vertikalen Rohren

Zusammenfassung Es wird der Einfluß des Auftriebes auf den Wärmeübergang bei turbulenter Aufwärtsströmung von widerstandsreduzierenden Fluiden in vertikalen Rohren theoretisch analysiert. Dabei wird ein Kriterium für die Begrenzung der Abnahme des Wärmeübergangs auf weniger als 5% für Fluide unterschiedlicher Widerstandsverringerung aufgestellt. Die endgültige mathematische Formulierung für die quantitative Beschreibung des Einflusses der natürlichen Konvektion auf die erzwungene Konvektion konnte sowohl auf Aufwärts-als auch auf Abwärtsströmung eines Fluids mit vermindertem Strömungswiderstand einfach dadurch angewandt werden, daß man ein Vorzeichen in dem die Strömung bestimmenden Term ändert.

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Nomenclature C _p specific heat at constant pressure (KJ/kg) - D pipe diameter (m) - De Deborah number - f friction factor - f reduced friction factor - g acceleration due to gravity (m/sec²) - Gr Grashof number - k thermal conductivity (KW/m °K) - Nu Nusselt number - Nu reduced Nusselt number - Pr Prandtl number - R pipe radius (m) - Re Reynolds number - Re reduced Reynolds number - T temperature (°C or °K) - T _b temperature of bulk of the fluid (°C or °K) - T _i initial temperature (°C or °K) - T _w wall temperature (°C or °K) - u ^* friction velocity (m/sec) - u ⁺ dimensionless axial velocity - V _m average velocity (m/sec) - y ⁺ dimensionless distance from the wall Greek symbols , functions ofDe (Table 1) - wall shear rate (sec^–1) - _fl fluid relaxation time (sec) - kinematic viscosity (m²/sec) - _B buoyant boundary layer thickness (m) - _m thickness of boundary sub-layer plus buffer layer (m) - _m ⁺ dimensionless thickness of boundary sub-layer plus buffer layer - _t thermal boundary layer (m) - density (kg/m³) - integrated density (kg/m³) - _b density of bulk of the fluid (kg/m³) - _w density of fluid at the wall (kg/m³) - _b viscosity of bulk of the fluid (Pa · sec) - _w viscosity of fluid at the wall (Pa · sec) - _w wall shear stress (N/m²) - _w reduced wall shear stress (N/m²) - change of shear stress across buoyant layer (N/m²)     

Variational second order flow estimation for PIV sequences

We present in this paper a variational approach to accurately estimate simultaneously the velocity field and its derivatives directly from PIV image sequences. Our method differs from other techniques that have been presented in the literature in the fact that the energy minimization used to estimate the particles motion depends on a second order Taylor development of the flow. In this way, we are not only able to compute the motion vector field, but we also obtain an accurate estimation of their derivatives. Hence, we avoid the use of numerical schemes to compute the derivatives from the estimated flow that usually yield to numerical amplification of the inherent uncertainty on the estimated flow. The performance of our approach is illustrated with the estimation of the motion vector field and the vorticity on both synthetic and real PIV datasets.     

Consequences of the second law for a turbulent fluid flow

Second Law statements in thermomechanics applicable to turbulent fluid flow, in which the internal energy in a macroscopic field theory includes contributions both from molecular vibrations and from turbulent fluctuations, are discussed. In the absence of turbulence, these statements naturally reduce to the known and accepted Second Law statements for a nonturbulent medium. The usual version of the Second Law statements — which deny the existence of perpetual motion and place restrictions on the constitutive equations —is extended here in the presence of turbulence; and an additional statement is introduced associated with the tendency of turbulent fluctuations to decay in the absence of external work or the addition of thermal heat. The mathematical representations of various Second Law statements are then used to derive several restrictions on the response variables of the macroscopic turbulence theory. Examples of such variables include the rates of production and dissipation of turbulent fluctuations, the rate of thermal entropy production, internal energy (involving constitutive coefficients which may be taken to be the thermal and turbulent specific heats), turbulent viscosity coefficients and other response functions which control the degree of flow anisotropy in the medium. These Second Law restrictions are then applied to a recent theory of macroscopic turbulent flow by the present authors in which fairly general constitutive equations are presented for the dependent variables of the theory. It is found that not only is the range of values of several constitutive coefficients limited by these Second Law restrictions, but the presence of a number of terms in the constitutive equations is entirely denied.     

ON THE SECOND ORDER WAVE DIFFRACTION IN TWO LAYER FLUIDS

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Heat transfer in turbulent pipe flow based on a new mixing length model

A new model for the heat transfer in turbulent pipe flow is presented based on a modified form of the mixing length theory developed by Cebeci [1] for boundary layer flow problems. The model predicts the velocity and temperature distributions and the Nusselt number for fluids with low, medium and high Prandtl numbers (Pr=.02 to 15) and fits the available experimental data very accurately for values of Reynolds number exceeding 10⁴. Expressions for the eddy conductivity and for the turbulent Prandtl number are presented and shown to be dependent upon the Reynolds number, the Prandtl number, and the distance from the tube wall.     

GENERAL SECOND ORDER FLUID FLOW IN A PIPE

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Heat transfer and friction in turbulent vortex flow

Summary This paper presents experimentally measured heat transfer and friction coefficients for air and water flowing through a pipe with several types of inserts designed to induce a swirl in the flow. It was observed that inside-surface heat transfer coefficients in swirling flow can, under favourable conditions, be at least four times as large as heat transfer coefficients at the same mass flow rate in purely axial flow. At the same time the pumping power per unit rate of heat transfer can be reduced. The increase in heat transfer coefficients was found to depend on the degree of swirl and on the density or temperature gradient. However, at comparable Reynolds numbers and swirling motions the heat transfer coefficients for air were found to be smaller than the coefficients for water. The reason for this difference is not definitely known, but the phenomenon is qualitatively compatible with that causing the cooling effect in Ranque-Hilsch vortex tubes. The observed phenomena are analyzed qualitatively and it is shown that they are primarily the result of a centrifugal force which induces a radial inward motion of warmer fluid and a radial outward motion of cooler fluid. The application of vortex flow to boiling heat transfer and other high heat flux systems is discussed briefly.

Nomenclature

Symbols c _p Specific heat at constant pressure, BTU/(lb)(deg F) - D _H Hydraulic diameter, (ft) - D Tube diameter, (ft) - f ₀ Fanning friction factor for axial flow, - f Fanning friction factor for swirling flow, - g Acceleration due to gravity, ft/(sec)² - G Mass velocity, lb/(sec) (sq ft) - h _i Inside surface coefficient of heat transfer, BTU/(hr)(sq ft)(deg F) - k Thermal conductivity, BTU/(hr)(sq ft)(deg F/ft) - L Characteristic length used in Grashof numbers, ft - p Frictional pressure drop in a duct, lbs/sq ft - r Radius of tube, ft - t Temperature potential in Grashof number, deg F - U _i Over-all coefficient of heat transfer based on inside tube area, BTU/(hr)(sq ft)(deg F) - V Axial velocity, ft/sec - Coefficient of thermal expansion, (deg F)^–1 - Absolute viscosity, (lbs)/(ft)(hr) - Density, lbs/(ft)³ - Angular velocity of fluid, rad/sec Dimensionless Parameters Nu ₀ Nusselt Number in axial flow, h _i D _H /k - Nu Nusselt Number in swirling flow, h _i D _H /k - Re Reynolds Number, VD _Hp / - Pr Prandtl Number, c _p /k - j Colburn j-Factor, (Nu/RePr)Pr ^2/3 Member of Technical Staff, Bell Telephone Laboratories, Murray Hill, N. J. formerly Baldwin Research Fellow, Lehigh University.     

Heat transfer to non-Newtonian fluids in laminar flow through rectangular ducts

Heat transfer to non-newtonian fluids flowing laminarly through rectangular ducts is examined. The conservation equations of mass, momentum, and energy are solved numerically with the aid of a finite volume technique. The viscoelastic behavior of the fluid is represented by the Criminale-Ericksen-Filbey (CEF) constitutive equation. Secondary flows occur due to the elastic behavior of the fluid, and, consequently, heat transfer is strongly enhanced. It is observed that shear thinning yields negligible heat transfer enhancement effect, when compared with the secondary flow effect. Maximum heat transfer is shown to occur for some combinations of parameters. Thus, there are optimal combinations of aspect ratio and Reynolds numbers, which depend on the fluid's mechanical behavior. This result can be usefully explored in thermal designs of certain industrial processes.     

Heat transfer in turbulent flow with radiation for small optical depths

A study of the effects of radiation on the heat transfer in fully developed turbulent flow in a channel is carried out. The analysis is valid for both small optical depths and for the optically thin limit. Nongrey effects are included through use of the total band absorptance.