Aspect ratio effect on flow-induced forces on circular cylinders in a cross-flow

Finite-span circular cylinders with two different aspect ratios, placed in a cross-flow, are investigated experimentally at a cylinder Reynolds number of 46,000. Simultaneous measurements of the flow-induced unsteady forces on the cylinders and the stream velocity in the wake are carried out. These results together with mean drag measurements along the span and available literature data are used to evaluate the flow mechanisms responsible for the induced unsteady forces and the effect of aspect ratio on these forces. The coherence of vortex shedding along the span of the cylinder is partially destroyed by the separated flow emanating from the top and by the recirculating flow behind the cylinder. As a result, the fluctuating lift decreases drastically. Based on the data collected, it is conjectured that the fluctuating recirculating flow behind the cylinder is the flow mechanism responsible for the unsteady drag and causes it to increase beyond the fluctuating lift. The fluctuating recirculating flow is a direct consequence of the unsteady separated flow. The unsteady forces vary along the span, with lift increasing and drag decreasing towards the cylinder base. When the cylinder span is large compared to the wall boundary layer thickness, a submerged two-dimensional region exists near the base. As the span decreases, the submerged two-dimensional region becomes smaller and eventually vanishes. Altogether, these results show that fluctuating drag is the dominant unsteady force in finite-span cylinders placed in a cross-flow. Its characteristic frequency is larger than that of the vortex shedding frequency.List of symbols a span of active element on cylinder, = 2.5 cm - C _D local rms drag coefficient, 2D/ U ² da - C _L local rms lift coefficient, 2l/ U ² da - C _D local mean drag coefficient, 2D/ U ² da - C _D spanwise-averaged C _D for finite-span cylinder - (C _D ) _2D spanwise-averaged mean drag coefficient for two-dimensional cylinder - C _p pressured coefficient - -(C _p ) _b pressure coefficient at = - d diameter of cylinder, = 10.2 cm - D fluctuating component of instantaneous drag - D local rms of fluctuating drag - D local mean drag - E _D power spectrum of fluctuating drag, defined as - E _L power spectra of fluctuating lift, defined as - f _D dominant frequency of drag spectrum - f _L dominant frequency of lift spectrum - f _u dominant frequency of velocity spectrum - h span of cylinder - H height of test section, = 30.5 cm - L fluctuating component of instantaneous lift - L local rms of fluctuating lift - R _Du () cross-correlation function of streamwise velocity and local drag, - R _Lu () cross-correlation function of stream wise velocity and local lift, - Re Reynolds number, U d/y - S _L Strouhal number based on f _L ,f _L d/U - S _D Strouhal number based on f _D ,f _D d/U - S _u Strouhal number based on f _u , f _u d/U - t time - u fluctuating component of instantaneous streamwise velocity - U mean streamwise velocity - mean stream velocity upstream of cylinder - x streamwise distance measured from axis of cylinder - y transverse distance measured from axis of test section - z spanwise distance measured from cylinder base - angular position on cylinder circumference measured from forward stagnation - kinematic viscosity of air - density of air - time lag in cross-correlation function - _D normalized spectrum of fluctuating drag - _L normalized spectrum of fluctuating lift     

Non-Linear response of a long,discontinuous fiber/melt system in elongational flows

The authors investigated the transient elongational behavior of a highly-aligned 600% volume fraction long, discontinuous fiber filled poly-ether-ketone-ketone melt with a computer-controlled extensional rheometer at 370°C. Prior experiments at controlled strain rate and stress produced _E ⁺ (t, ) and ^– (t, _E) similar to a shear dominated flow of a non-linear viscoelastic fluid. Stress relaxation following steady extension showed nonlinear effects in the change in stress decay rate with increasing strain rate. Continuous relaxation spectra showed a shift in the spectral peak to smaller values of with increasing strain rate. The Giesekus nonlinear constitutive relation modeled the elongation and stress relaxation with shearing rate at the fiber surface set by a strain rate magnification factor. Suitable for elongation, the model produced insufficient shift in the stress relaxation spectrum to account for the large change in stress decay rate exhibited in the experiments.English alphabet a _r aspect ratio of the fibers or l/d - A ₀ initial uniform cross-section area of the specimen - d fiber diameter - f fiber volume fraction - H() relaxation spectrum found by the method of Ferry and William l length of the fiber - L(t) time function specimen length - L ₀ initial specimen length - r radial coordinate across the shear cell - R _i fiber radius and inner cell dimension - R _o outer cell radius - t time in s - t _max duration of the extension - T _g glass transition temperature of the polymer - v velocity of the moving end of the test specimen - x axial position where is calculated Greek alphabet nonlinearity parameter in the Giesekus relation - axial mass distribution along the specimen major axis - shear strain rate - strain tensor - ₍₁₎ first convected derivative of the strain tensor - ₍₂₎ second convected derivative of the strain tensor - average strain at the end of extension as determined from - extension strain rate - average extension strain rate determined from - ^– transient strain rate under controlled stress, creep, test - _E elongational viscosity - _Eapp apparent elongational viscosity determined from - _E ⁺ transient elongational viscosity - ₀ zero shear rate viscosity - relaxation parameter - ₁ relaxation parameter in either Jeffrey's or Giesekus fluid - ₂ retardation parameter in either Jeffrey's or Giesekus fluid - _max relaxation value at which 99.9% of the H spectrum had occurred - _p relaxation value at which H reaches a maximum - volumetric composite density - _E elongational stress - _E ⁺ transient elongational stress - _E ^– controlled elongational stress, creep stress - _E ^y peak elongational stress in controlled experiment - shear stress at surface of the fiber in a shear cell - _yx simple shear component of the strain rate tensor - stress tensor - ₁ first convected derivative of the stress tensor     

Evaluation of the performance characteristics of a thermal transient anemometer

The Goertler instability of a hypersonic boundary layer and its influence on the wall heat transfer are experimentally analyzed. Measurements, made in a wind tunnel by means of a computerized infrared (IR) imaging system, refer to the flow over two-dimensional concave walls. Wall temperature maps (that are interpreted as surface flow visualizations) and spanwise heat transfer fluctuations are presented. Measured vortices wavelengths are correlated to non-dimensional parameters and compared with numerical predictions from the literature.List of symbols c _p Specific heat coefficient at constant pressure of the free stream - F Input (true) image - F ₀ Fourier number - Restored image - G Recorded (degraded) image - G Goertler number based on the boundary layer thickness, as defined by Eq. (3) - H System transfer function - M Mach number - Pr Prandtl number - p ₀ Stagnation pressure - Exchanged net heat flux - Convective heat flux - Radiative heat flux - r Recovery factor - Re _m Unit Reynolds number - Re _x Local Reynolds number based on the distance from the leading edge - Re Local Reynolds number based on the boundary layer thickness - Curvature radius - St Stanton number, as defined by Eq. (7) - T _aw Adiabatic wall temperature - T _w Wall temperature - T ₀ Stagnation temperature - t Time - V Free stream velocity - x Streamwise spatial coordinate - y Normal-to-wall spatial coordinate - z Spanwise spatial coordinate - Thermal diffusivity coefficient - Disturbance wavenumber - Non dimensional wavenumber - Boundary layer thickness - Goertler number based on the vortices wavelength - Vortices wavelength - Free stream density - Disturbance total amplification, as defined by Eq. (3) - Disturbance (spatial) growth rate - Non-dimensional growth rate - Perturbation amplitude of a generic quantity - Perturbation amount     

Non-isothermal flow of polymer melts in a curved tube

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions. Br — Brinkman number - c specific heat, J/kg K - De — Dean number - E dimensionless apparent viscosity, eq. (14d) - G dimensionless shear rate, eq. (19) - k parameter of the power-law model, °C^–1, eq. (7) - mass flow rate, kg/s - m ₀ parameter of the power-law model, Pa · s ⁿ , eq. (7) - n parameter of the power-law model, eq. (7) - Nu 2r _p/ — Nusselt number, eqs. (28,31) - p pressure, Pa - Pe — Péclet number - P —(p/)/r _c — pressure gradient, Pa/m - dissipated energy, W, eq. (29) - total energy, W, eq. (30) - r radial coordinate, m - r _c radius of tube-curvature, m, fig. 1 - r _p radius of tube, m, fig. 1 - r _t variable, m, eq. (6) - R dimensionless radial coordinate, eq. (14a) - R _c dimensionlessr _c, eq. (14a) - R _t dimensionlessr _t, eq. (14a) - t temperature, °C - bulk temperature, °C, eq. (27) - t ₀ inlet temperature of the melt, °C - t _w tube wall temperature, °C - T dimensionless temperature, eq. (14c) - T _w dimensionless tube wall temperature - T dimensionless bulk temperature - u ₁ variable, s^–1, eq. (4) - u ₂ variable, s^–1, eq. (5) - U ₁ dimensionlessu ₁, eq. (18) - U ₂ dimensionlessu ₂, eq. (18) - v velocity in-direction, m/s - average velocity of the melt, m/s - V dimensionlessv, eq. (14b) - dimensionless , eq. (15) - z r _c — centre length of the tube, m - Z dimensionlessz, eq. (14e) - heat transfer coefficient, W/m² K - shear rate, s^–1, eq. (8) - — shear rate, s^–1 - apparent viscosity, Pa · s, eq. (7) - ₀ — apparent viscosity, Pa · s - angular coordinate, rad, fig. 1 - thermal conductivity, W/m K - melt density, kg/m³ - axial coordinate, rad, fig. 1 - rate of strain tensor, s^–1, eq. (8) - (—p) pressure drop, Pa     

Flow about yawed,stranded cables

The development of a steady lift force on a stranded cable, which is yawed with respect to a flow, is a unique characteristic of a cable when compared to a circular cylinder. Comparisons of lift and normal drag coefficients and wake characteristics were made between stranded cable models and the cylinder. These were based upon surface pressure and hot-wire measurements and flow visualization studies conducted in a low speed wind tunnel on rigid cables and cylinders. The models were yawed to four different yaw angles and tested within the Reynolds number range of 5,000 and 50,000. Pressure profiles for the yawed cables indicated that the lift force is directed towards the side where the primary strands are more nearly aligned with the flow. The pressure profiles also indicated that the lift force is generated by asymmetric separation. The small scale irregularities associated with wires within individual strands also appeared to have an effect on the cable's lift and drag characteristics. Results show that cables have significantly different shedding characteristics and near-wake shear layer structure when compared to the circular cylinder. For the flow regime tested, the Strouhal number showed no dependence on Reynolds number nor spanwise position along the cable.List of symbols C _dn normal drag coefficient - C _l lift coefficient - C _p pressure coefficient - D actual diameter, based on circumscribing circle for the cable - f _v shedding frequency - L/D length to actual diameter ratio - ppd peak-to-peak distance, unit span - Re Reynolds number based on actual diameter - S Strouhal number, - V free stream velocity - cable angle - azimuthal angle     

Response of a viscous annular liquid layer in zero-gravity to different axial excitations of the rigid boundary plates

A cylindrical annular liquid layer between two plates and around a rigid center-core consisting of incompressible and viscous liquid is subjected to different axial excitations, such as one-sided, counter-directional and double-sided unequal excitations. The response of the free liquid surface, the velocity- and pressure-distribution has been determined.

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Zusammenfassung Eine zylindrische Flüssigkeitsschicht bestehend aus inkompressibler und viskoser Flüssigkeit wurde verschiedenen harmonischen Anregungsformen ausgesetzt. Dabei wurden die Fälle einseitiger, doppelseitiger entgegengesetzter und ungleicher doppelseitiger Anregung mit Phase behandelt. Die Vergrößerungsfunktionen für die freie Flüssigkeitsoberfläche, für die Geschwindigkeits- und Druckverteilung wurden bestimmt.

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List of symbols a radius of liquid layer - b radius of inner cylindrical core - (a–b) thickness of layer - e _r , e , k unit vectors in the radial, angular and axial direction resp. - h length of layer - I _m , K _m modified Bessel functions of first and second kind and order m - diameter ratio - p pressure - q 2na/h - q* na/h - r, , z cylindrical coordinates - complex frequency - S sa ²/ - t time - u, w velocity components in the radial- and axial direction - ₀ excitation amplitude - abbreviation - surface tension parameter - surface tension - dynamic viscosity - kinematic viscosity - density of liquid - free liquid surface elevation - dimensionless time - _rz shear stress - reduced forcing frequency - forcing frequency - stream function - _mn natural frequency of non-viscous liquid     

Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Velocity field in isothermal turbulent bubbly gas-liquid flow through a pipe

Velocity field was measured by laser Doppler velocimetry in isothermal, turbulent bubbly gas-liquid flow through a 26.6 mm inner diameter vertical pipe. The measurements were made about 33 diameters downstream from the pipe entrance, gas injection being just upstream of the entrance. The gas phase radial distribution at the measurement plane exhibited influence of the injection device in that higher gas fraction existed in the central region of the pipe. For comparison, velocity field was also measured in isothermal, turbulent single-phase liquid flow through the same pipe at the same axial plane. Measured were the radial distributions of liquid mean axial and radial velocities, axial and radial turbulent intensities, and axial Reynolds shear stress. The radial distributions of gas bubble mean axial velocity and axial velocity fluctuation intensity were also measured by LDV. A dualsensor fiberoptic probe was used at the same time to measure the radial distributions of gas fraction, bubble mean axial velocity and size slightly downstream of the LDV measurement plane.List of Symbols an average gas bubble diameter - f, f _TP friction factor, friction factor for gas-liquid flow - k _L liquid turbulent kinetic energy - , gas, liquid mass flow rate - R inner radius of pipe - r, {sitR}^* radial coordinate; nondimensional radial coordinate (=r/R) - Re _L liquid Reynolds number - U _G mean axial velocity of gas bubble - U _L mean axial velocity of liquid - U _LO mean axial velocity for flow at the total mass velocity with properties of the liquid phase - u _L ⁺ nondimensional mean axial velocity of liquid in wall coordinate - friction velocity - axial velocity fluctuation intensity of liquid - axial velocity fluctuation intensity of gas bubbles - V_L mean radial velocity of liquid - v _L radial velocity fluctuation intensity of liquid - (uv)_L single-point cross-correlation between axial and radial velocity fluctuations of liquid ( axial Reynolds shear stress) - T _in mean liquid temperature at test section inlet - x flow quality - y normal distance from wall - y ⁺ nondimensional normal distance from wall in wall coordinate (=yu/v_L) - _G gas phase residence time fraction - _L rate of dissipation in the liquid - _L Kolmogorov length scale in the liquid - _L liquid kinematic viscosity - _L characteristic turbulence length scale in the liquid - _G, _L density of gas, liquid - _m gas-liquid mixture density This work was partly supported by National Science Foundation, Thermal Transport and Thermal Processing Program, Chemical and Thermal Systems Division, under Grant No. CTS-9411898.     

The finite element solutions of laminar flow and combined convection of air in a staggered or an in-line tube-bank

A finite element method is used to solve the full Navier-Stokes and energy equations for the problems of laminar combined convection from three isothermal heat horizontal cylinders in staggered tube-bank and four isothermal heat horizontal cylinders in in-line tube-bank. The variations of surface shear stress, pressure and Nusselt number are obtained over the entire cylinder surface including the zone beyond the separation point. The predicted values of total, pressure and friction drag coefficients, average Nusselt number and the plots of velocity flow fields and isotherms are also presented.

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Die Finite-Elemente-Lösung von laminarer Strömung und kombinierter Konvektion von Luft in einer versetzten oder fluchtenden Rohranordnung

Zusammenfassung Eine Methode der finiten Elemente wird zur Lösung der vollständigen Navier-Stokes- und der Energiegleichung für die Probleme der laminaren kombinierten Konvektion an drei isothermen geheizten horizontalen Zylindern in versetzter Rohranordnung sowie für vier isotherme geheizte horizontale Zylinder in fluchtender Anordnung verwendet.Die Veränderung der Wandschubspannung, des Druckes und der Nusselt-Zahl werden für die gesamte Zylinderoberfläche, einschließlich des Bereiches nach dem Ablösepunkt, bestimmt. Die Werte des gesamten Widerstandsbeiwertes aufgrund von Druck und Reibung, die durchschnittliche Nusselt-Zahl und die Diagramme des Geschwindigkeitsfeldes und der Isothermen werden ebenfalls aufgezeigt.

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Nomenclature C specifie heat - C _D total drag coefficient - C _f friction drag coefficient - C _p pressure drag coefficient - D diameter of cylinder,L=2R ₀ - G, g gravitational acceleration - Gr Grashof number, g(T_w–T )D ³/v ² - h local heat transfer coefficient - K thermal conductivity - L spacing between the centers of cylinder - M _l shape function - N _i shape function - Nu, local and average Nusselt numbers - P dimensionless pressure, p^*/u ² - p ^*,p pressure, free stream pressure - Pe Peclet number,RePr - Pr Prandtl number, c/K - Ra Rayleigh number,Gr Pr - Re Reynolds number,Du /v - R ₀ radius of cylinder - T temperature - T _w temperature on cylinder surface with fixed value - T free stream temperature - v dimensionless x-direction component of velocity,v ^*/u - u ^* x-direction component of velocity - u free stream velocity - v dimensionless Y-direction component of velocity,v ^*/u - v ^* Y-direction component of velocity - X x-direction axis - x dimensionless x-direction coordinate,x ^*/D - x^* x-direction coordinate - Y Y-direction axis - y dimensionless Y-direction coordinate,y ^*/D - y ^* Y-direction coordinate Greek symbols coefficient of volumetric thermal expansion - plane angle - dynamic viscosity - kinematic viscosity, / - density of fluid - _w dimensionless surface shear stress, _* ^w /u ² - skw/* surface shear stress - dimensionless temperature,      

Wärmeübertragung in einem axial rotierenden,durchströmten Rohr im Bereich des thermischen Einlaufs

Zusammenfassung Der Einfluß der Rotation auf das Temperaturprofil und die Wärmeübergangszahl einer turbulenten Rohrströmung im Bereich des thermischen Einlaufs wird theoretisch untersucht und mit Meßwerten verglichen. Es wird angenommen, daß das Geschwindigkeitsprofil voll ausgebildet ist. Die Rotation hat aufgrund der radial ansteigenden Zentrifugalkräfte einen ausgeprägten Einfluß auf die Unterdrückung der turbulenten Bewegung. Dadurch verschlechtert sich die Wärmeübertragung mit steigender Rotations-Reynoldszahl und die thermische Einlauflänge nimmt beträchtlich zu.

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Heat transfer in an axially rotating pipe in the thermal entrance region. Part 1: Effect of rotation on turbulent pipe flow

The effects of rotation on the temperature distribution and the heat transfer to a fluid flowing inside a tube are examined by analysis in the thermal entrance region. The theoretical results are compared with experimental findings. The flow is assumed to have a fully developed velocity profile. Rotation was found to have a very marked influence on the suppression of the turbulent motion because of radially growing centrifugal forces. Therefore, a remarkable decrease in heat transfer with increasing rotational Reynolds number can be observed. The thermal entrance length increases remarkably with growing rotational Reynolds number.

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Formelzeichen a Temperaturleitzahl - C _n , ,C ₁,C ₃ Konstanten - c _p spezifische Wärme bei konstantem Druck - D Rohrdurchmesser - E Funktion nach Gl. (30) - H _n Eigenfunktionen - l hydrodynamischer Mischungsweg - l _q thermischer Mischungsweg - Massenstrom - N=Re /Re Reynoldszahlenverhältnis - Nu Nusseltzahl - Nu Nusseltzahl für die thermisch voll ausgebildete Strömung - Pr Prandtlzahl - Pr _t turbulente Prandtlzahl - Wärmestromdichte - Re _* Schubspannungsreynoldszahl - R _n Eigenfunktionen - Durchfluß-Reynoldszahl - Re v =D/ Rotations-Reynoldszahl - Ri Richardsonzahl - R Rohrradius - r Koordinate in radialer Richtung - dimensionslose Koordinate in radialer Richtung - T Temperatur - T Temperaturschwankung - T _b bulk temperature - mittlere Axialgeschwindigkeit - v Geschwindigkeit - v Geschwindigkeitsschwankung - turbulenter Wärmestrom - dimensionsloser Wandabstand - =1/6 Konstante - Integrationsvariable - Integrationsvariable - , ₁, ₂, dimensionslose Temperaturen - Wärmeleitzahl - _n Eigenwerte - kinematische Viskosität - Dichte - tangentiale Koordinate - , Hilfsfunktionen Indizes m in der Rohrmitte - r radial - w an der Rohrwand - z axial - 0 am Rohreintritt - 0 ohne Rotation - tangential     

Base pressure measurements on a cone at Mach numbers from M∞ = 5 to 7

Base pressure measurements were performed on a blunt cone in the Ludwieg-Tube facility at the DLR in Göttingen at Mach numbers from M = 4.99 to 6.83. The angle of incidence was varied between = 0° and 15°. The results show the influence of Mach number and angle of incidence on the base pressure.List of symbols D base diameter - R base half diameter; D = 2R - r _n nose radius - angle of incidence - cone apex angle - p free stream static pressure - p _B base pressure (one pressure tap only) - p _ref reference pressure - U free stream velocity - M free stream Mach number - Re _L free stream Reynolds number based on cone length - free stream density - v free stream kinematic viscosity - ratio of specific heats - q free stream dynamic pressure - c _pB base pressure coefficient      

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

A turbulence model for the heat transfer near stagnation point of a circular cylinder

A one-equation low-Reynolds number turbulence model has been applied successfully to the flow and heat transfer over a circular cylinder in turbulent cross flow. The turbulence length-scale was found to be equal 3.7y up to a distance 0.05 and then constant equal to 0.185 up to the edge of the boundary layer (wherey is the distance from the surface and is the boundary layer thickness).The model predictions for heat transfer coefficient, skin friction factor, velocity and kinetic energy profiles were in good agreement with the data. The model was applied for Re 250,000 and Tu0.07.Nomenclature µ,C _D Constants in the turbulence kinetic energy equation - C ₁,C ₂ Constants in the turbulence length-scale equation - Skin friction coefficient atx - D Cylinder diameter - F Dimensionless flow streamwise velocityu/u _e - k Turbulence kinetic energy =1/2 the sum of the squared three fluctuating velocities - K Dimensionless turbulence kinetic energyk/u _e ^/2 - I Dimensionless temperature (T–T _w )/(T –T _w ) - l Turbulence length-scale - l _e Turbulence length-scale at outer region - Nu _D Nusselt number - p Pressure - Pr Prandtl number - Pr _t Turbulent Prandtl number - Pr _k Constant in the turbulence kinetic energy equation - R Cylinder radius - Re _D Reynolds number u D/µ - Re _x Reynolds number u x/µ - R _K Reynolds number of turbulence - T Mean temperature - T Mean temperature at ambient - T _s Mean temperature at surface - Tu Cross flow turbulence intensity, - u Mean flow streamwise velocity - u Fluctuating streamwise velocity - u _e Mean flow velocity at far field distance - u Mean flow velocity at ambient - u* Friction velocity - v Mean velocity normal to surface - V Dimensionless mean velocity normal to surface - x,x ₁ Distance along the surface - y Distance normal to surface - Dimensionless pressure gradient parameter - Boundary layer thickness atu=0.9995u _e - Transformed coordinate iny direction - Fluid molecular viscosity - _t Turbulent viscosity - _eff + _t - µ Fluid molecular viscosity at ambient - Kinematic viscosity/ - Density - Density at ambient - _w Wall shear stress - _w,0 Wall shear stress at zero free stream turbulence     

On the universality of the velocity profiles of a turbulent flow in an axially rotating pipe

If a fluid enters an axially rotating pipe, it receives a tangential component of velocity from the moving wall, and the flow pattern change according to the rotational speed. A flow relaminarization is set up by an increase in the rotational speed of the pipe. It will be shown that the tangential- and the axial velocity distribution adopt a quite universal shape in the case of fully developed flow for a fixed value of a new defined rotation parameter. By taking into account the universal character of the velocity profiles, a formula is derived for describing the velocity distribution in an axially rotating pipe. The resulting velocity profiles are compared with measurements of Reich [10] and generally good agreement is found.Nomenclature b constant, equation (34) - D pipe diameter - l mixing length - l ₀ mixing length in a non-rotating pipe - N rotation rate,N=Re /Re _D - p pressure - R pipe radius - Re _D flow-rate Reynolds number, - Re rotational Reynolds number, Re =v _w D/ - Re* Reynolds number based on the friction velocity, Re*=v*R/ - (Re*)₀ Reynolds number based on the friction velocity in a non-rotating pipe - Ri Richardson number, equation (10) - r coordinate in radial direction - dimensionless coordinate in radial direction, - v _r ,v ,v _z time mean velocity components - v _r ,v ,v _z velocity fluctations - v _w tangential velocity of the pipe wall - v* friction velocity, - axial mean velocity - v _ZM maximum axial velocity - dimensionless radial distance from pipe wall, - y ⁺ dimensionless radial distance from pipe wall - y ₁ ⁺ constant - Z rotation parameter,Z =v _w/v _* =N Re _D /2Re* - _m eddy viscosity - ( _m )₀ eddy viscosity in a non-rotating pipe - coefficient of friction loss - von Karman constant - ₁ constant, equation (31) - density - dynamic viscosity - kinematic viscosity     

Laminar source flow between parallel porous disks with equal suction and injection

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.Nomenclature a distance between disks - A, B, ..., J functions of R _w only - F static pressure - p dimensionless static pressure, p(a ²/ ²) - Q volumetric flow rate of the source - r radial coordinate - r dimensionless radial coordinate, r/a - R radial coordinate of a point in the flow region - R dimensionless radial coordinate of a point in the flow region, R - Re source Reynolds number, Q/2a - R _w wall Reynolds number, Va/ - reduced Reynolds number, Re/r ² - critical Reynolds number - velocity component in radial direction - u dimensionless velocity component in radial direction, a/ - average radial velocity, Q/2a - u dimensionless average radial velocity, Re/r - ratio of radial velocity to average radial velocity, u/u - velocity component in axial direction - v dimensionless velocity component in axial direction, v - V magnitude of suction or injection velocity - z axial coordinate - z dimensionless axial coordinate, z a - viscosity - density - kinematic viscosity, / - shear stress at lower disk - shear stress at upper disk - ₀ dimensionless shear stress at lower disk, - ₁ dimensionless shear stress at upper disk, - dimensionless stream function     

Experimental investigations of fluid flow and heat transfer characteristics of a slot jet impinging on a rectangular cylinder

Experimental investigations on flow characteristics and average heat transfer due to slot jet impinging on a rectangular cylinder have been carried out for different non-dimensional parameters. The minimum value of the pressure coefficient is found on the lower face of the rectangular cylinder at an angle of inclination of 15°. Drag coefficient calculated from the measured pressure distribution is found to be maximum within a range of breadth/width ratio of 0.67 to 1.5 of rectangular cylinders. The maximum value of heat transfer rate is obtained at the angle of inclination of 15° of the cylinder to the jet axis. An increasing trend of heat transfer rate is observed with higher Reynolds numbers. A correlation of average Nusselt number is presented for rectangular cylinders.

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Experimentelle Untersuchung der Strömungs- und Wärmeübergangs-charakteristik eines auf einen rechteckigen Zylinder auftreffenden Strahls aus einer Schlitzdüse

Zusammenfassung Es wurden experimentelle Untersuchungen des Strömungs- und Wärmeübergangsverhaltens an einem rechteckigen, durch einen Strahl aus einer Schlitzdüse beaufschlagten Zylinders für verschiedene dimensionslose Parameter durchgeführt. Der Kleinstwert des Druckbeiwertes tritt an der Unterfläche des rechteckigen Zylinders bei einem Neigungswinkel von 15° auf. Der ausder gemessenen Druckverteilung berechnete Widerstandsbeiwert erreicht bei einem Breiten-Dicken-Verhältnis des Zylinders zwischen 0,67 und 1,5 Maximalwerte. Den maximalen Wärmestrom erhält man bei einem Neigungswinkel zwischen Zylinder und Strahlachse von 15°. Mit steigenden Reynoldszahlen erhöht sich der abgeführte Wärmestrom. Eine Korrelation für die mittlere Nusseltzahl an rechteckigen Zylindern wird mitgeteilt.

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Nomenclature A surface area of the rectangular cylinder - a width of the rectangular cylinder - b breadth of the rectangular cylinder - C _D drag coefficient =D/ - C _p pressure coefficient = (p – p _a )/ - C _pb base pressure coefficient = (p _b –p _a )/ - D drag force - h _f free convection heat transfer coefficient - average heat transfer coefficient - k thermal conductivity of air - L distance of the axis of the square cylinder from the nozzle exit - l length of the rectangular cylinder - Pr Prandtl number - p static pressure - P _a atmospheric pressure - P _b base pressure on the rear face - Nu _f free convection Nusselt number - average Nusselt number - q heat loss - q _f heat loss due to free convection - Re Reynolds number =u _j W/ _a - T _a ambient air temperature - average surface temperature - u _j average jet velocity at the nozzle exit - W nozzle width - angle of inclination of the rectangular cylinder to the jet axis in degrees - _a kinematic viscosity of air - _a density of air     

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)