Mathematical model of the entry length in the flow of suspensions of solid particles in a gas

A mathematical model consisting of equations of mass and momentum and for the velocity field has been used for computing the entry length of the flow of non-Newtonian fluids in laminar, transition and turbulent regions. Experimental data measured in a vertical flow of a suspension of solid particles in air have been used for verifying the predictions. n flow index for laminar flow - Re Reynolds number defined for the flow of the carrier medium - q exponent for turbulent flow - ratio of core radius with a flat velocity profile to pipe radius - _c ratio of the axial component of local velocity in the core to mean velocity - w mean flow velocity - ratio of axial distance from the pipe entrance to the pipe radius - ratio of the entrance length to the pipe radius - relative mass fraction of particles - ratio of the distance from the pipe wall to the pipe radius - coefficient of pressure loss due to friction     

Some measurements in a binary gas jet

Simultaneous measurements of species volume concentration and velocities in a helium/air binary gas jet with a jet Reynolds number of 4,300 and a jet-to-ambient fluid density ratio of 0.64 were carried out using a laser/hot-wire technique. From the measurements, the turbulent axial and radial mass fluxes were evaluated together with the means, variances and spatial gradients of the mixture density and velocity. In the jet near field (up to ten diameters downstream of the jet exit), detailed measurements of u/ ₀ U ₀, v/ ₀ U₀, u v/ ₀ U ₀ ² , u ² / ₀ U ₀ ² and v ² / ₀ U ₀ ² reveal that the first three terms are of the same order of magnitude, while the last two are at least one order of magnitude smaller than the first three. Therefore, the binary gas jet in the near field cannot be approximated by a set of Reynolds-averaged boundary-layer equations. Both the mean and turbulent velocity and density fields achieve self-preservation around 24 diameters. Jet growth and centerline decay measurements are consistent with existing data on binary gas jets and the growth rate of the velocity field is slightly slower than that of the scalar field. Finally, the turbulent axial mass flux is found to follow gradient diffusion relation near the center of the jet, but the relation is not valid in other regions where the flow is intermittent.     

A Turbulence Model Sensitivity Study for CH4/H2 Bluff-Body Stabilized Flames

The objective of this work is to assess the performances of different turbulence models in predicting turbulent diffusion flames in conjunction with the flamelet model.The k– model, the Explicit Algebraic Stress Model (EASM) and the k– model withvaried anisotropy parameter C (LEA k– model)are first applied to the inert turbulent flow over a backward-facing step, demonstrating the quality of the turbulence models. Following this, theyare used to simulate the CH₄/H₂ bluff-body flame studied by the University of Sydney/Sandia.The numerical results are compared to experimental values of the mixture fraction, velocity field, temperature and constituent mass fractions.The comparisons show that the overall result depends on the turbulence model used, and indicate that theEASM and the LEA k– models perform better than the k– model and mimic most of the significant flow features.     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

Combined effect of external turbulence and strong acceleration on boundary layer transition in a nozzle

The effect of external turbulence on the boundary layer flow in a convergent-divergent nozzle with a high expansion ratio has been studied numerically. The external turbulence was simulated by the turbulent viscosity _e, for which we used the partial differential equation that serves to close the system of boundary layer equations [1–4]. It was found that there exists a critical value _cr such that for all _e< _cr the flow regime in the nozzle remains perfectly laminar, whereas for _ecr a laminar-turbulent transition takes place and the boundary layer in the supersonic part of the nozzle becomes turbulent. For postcritical values of _e the heat fluxes and friction losses are approximately an order greater than for precritical values. With increase in the Reynolds number, determined from the parameters in the nozzle throat, the value of _cr decreases; as the coordinate of the onset of boundary layer formation is displaced in the direction of flow the value of _cr increases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 34–37, January–February, 1906.The authors are grateful to L. V. Gogish for participating in the discussion of the results.     

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     

Steady flow of a liquid metal in a rectangular MHD channel

The known experimental studies of steady flows of a liquid metal in magnetohydrodynamic (MHD) channels of rectangular section [1–4] were performed only for a few values of the Reynolds number, which does not permit a clear delineation of the fundamental governing laws of the flow in the zone of transition from laminar to turbulent flow. In addition, the study of turbulent MHD flows has been limited to two-dimensional channels.Below we present some results of experimental studies of the effect of a transverse magnetic field on the resistance coefficient for mercury flow in an MHD channel with side ratio 1 to 2.5. The choice of a channel with this side ratio was dictated by the need for studies of the intermediate case between flows in two-dimensional and square channels, which differ significantly from one another because of the different effect of the walls parallel to the magnetic field. In our studies, for each value of the Hartmann number the investigations were made for 30–50 values of the Reynolds number.Notation B₀ flux density of the applied magnetic field - M Hartmann number - R Reynolds number - _tm resistance factor of turbulent MHD flow - _* critical value of the resistance factor - geometric parameter of channel - the component of resistance factor in ordinary hydrodynamics due to pulsations - normed function - electric conductivity of metal - viscosity of metal - R₀ shydraulic radius - N smagnetic field parameter     

Swirling turbulent flow through a curved pipe

An experimental study of a swirling turbulent flow through a curved pipe with a pipe-to-mean-bend radius ratio of 0.077 and a flow Reynolds number based on pipe diameter and mean bulk velocity of 50,000 has been carried out. A rotating section, six pipe diameters long, is set up at six diameters upstream of the curved bend entrance. The rotating section is designed to provide a solid-body rotation to the flow. At the entrance of the rotating section, a fully-developed turbulent pipe flow is established. This study reports on the flow characteristics for the case where the swirl number, defined as the ratio of the pipe circumferential velocity to mean bulk velocity, is one. Wall static pressures, mean velocities, Reynolds stresses and wall shear distribution around the pipe are measured using pressure transducers, rotating-wires and surface hot-film gauges. The measurements are used to analyze the competing effects of swirl and bend curvature on curved-pipe flows, particularly their influence on the secondary flow pattern in the crossstream plane of the curved pipe. At this swirl number, all measured data indicate that, besides the decaying combined free and forced vortex, there are no secondary cells present in the cross-stream plane of the curved pipe. Consequently, the flow displays characteristics of axial symmetry and the turbulent normal stress distributions are more uniform across the pipe compared to fully-developed pipe flows.List of symbols B calibration constant - e bridge voltage - e ₀ bridge voltage at zero flow - C _f total skin friction coefficient, = 2 _w/ W ₀ ² - D pipe diameter, = 7.62 cm - De Dean number, = ^1/2 Re - M angular momentum - n calibration constant - N _s swirl number, = D/2 W ₀ - r radial coordinate - R mean bend radius of curvature, = 49.5 cm - Re pipe Reynolds number, = DW ₀/ - S axial coordinate along the upstream (measured negative) and downstream (measured positive) tangent - U, V, W mean velocities along the radial, tangential and axial directions, respectively - u, v, w mean fluctuating velocities along the radial, tangential and axial directions, respectively - u, v, w root mean square normal stress along the radial, tangential and axial directions, respectively - v ^{ov2}, u^{ov2} normal stress along the tangential and radial direction, respectively - W ₀ mean bulk velocity, 10 m/s - W _c W measured at pipe axis - W total wall friction velocity, - total wall friction velocity measured at S/D = -18 - ,v vw, w7#x016B; turbulent shear stresses - pipe-to-mean-bend radius ratio, = D/2 R = 0.077 - axial coordinate measured from bend entrance - fluid kinematic viscosity - fluid density - _w mean total wall shear stress - instantaneous total wall shear - azimuthal coordinate measured zero from pipe hori zontal diameter near outer bend - angular speed of the rotating section     

Experimental investigation of the flow characteristics within a shallow wall cavity for both laminar and turbulent upstream boundary layers

The mean and turbulent flow fields were measured upstream, within, and downstream of a non-resonating shallow wall cavity subject to low Mach number flows with both laminar and turbulent upstream boundary layers. The laminar case displayed a cavity vortex that was stronger and more localized towards the trailing edge compared to the turbulent case with the same freestream velocity. The location of the maximum Reynolds shear stress in the shear layer rises slightly above the cavity mouth near the cavity centerline for the laminar case in contrast to the turbulent case, where it remains near or slightly below the cavity mouth across the entire cavity. Downstream of the cavity, the laminar and turbulent cases converged towards a common turbulent boundary layer. The non-resonating condition of the cavity was explored through comparisons with resonance criteria from previous experimental investigations.List of symbols D Depth of cavity - L Length of cavity in streamwise direction - M Mach number - Re Reynolds number based on U and L - Re Reynolds number based on U and - St Strouhal number based on frequency, L, and U - U Velocity of freestream (streamwise) flow - W Width of cavity - x, y, z Coordinates in streamwise, cavity depth, and cavity width directions - u Velocity in streamwise direction - v Velocity normal to streamwise direction (in cavity depth direction) - Reynolds shear stress - Average of the quantity - Boundary layer thickness immediately upstream of the cavity opening - * Boundary layer displacement thickness immediately upstream of the cavity opening - _t Eddy viscosity - Boundary layer momentum thickness immediately upstream of the cavity opening - Vorticity     

The râle of microscopic cross flow in idealized porous media

In this paper, results from a combined network/averaging study are presented. The emphasis is placed on understanding the flow phenomena, rather than predicting results for real porous media. Idealized porous media, consisting of networks of tubes, are used to interpret two of the terms in the averaged momentum equation. In particular, it is demonstrated that the pressure term accounts for microscopic cross flow, and that the magnitude of this term is proportional to the variation of the cross-sectional areas of the tubes in the macroscopic flow direction. For one-dimensional macroscopic flow in these idealized porous media, the agreement of network theory and averaging theory permeabilities depends on areosity (a term related to the area open to flow in a direction) remaining constant in the macroscopic flow direction; it may vary in other directions.     

Flows in the initial sections of channels with permeable walls

Calculations of two types of flows in the initial sections of channels with permeable walls are carried out on the basis of semiempirical turbulence theories during fluid injection only through the walls and during interaction of the external flow with the injected fluid. Experimental studies of the first type [1–3] show that at least within the limits of the lengths L/h<30 and L/a< 50 (2h is the distance between permeable walls of a flat channel anda is the tube radius) the velocity distributions in the laminar and turbulent flow regimes differ little and are nearly self-similar for solutions obtained in [4]. For sufficiently large Reynolds numbers, Re₀>100 (Re₀=v₀h/ or Re₀=v₀ a/, where v₀ is the injection velocity), and small fluid compressibility, the axial velocity component is described by the relations for ideal eddying motion: u=(/2)x× cos (y/2) in a flat channel and u=x cos (y²/2) in atube (the characteristic values for the coordinates are, respectively, h anda). Measurements indicate the existence of a segment of laminar flow; its length depends on the Reynolds number of the injection [3]. In the turbulent regime the maximum generation of turbulent energy occurs significantly farther from the wall than in parallel flow. Flows of the second type in tubes were studied in [5–7]. These studies disclosed that for Reynolds numbers of the flow at the entrance to the porous part of the tube Re=u₀ a/<3.10³ fluid injection with v₀/u₀>0.01 leads to suppression of turbu lence in the initial section of the tube. An analogous phenomenon was observed in the boundary layer with v₀/u₀>0.023 [8, 9]. Laminar-turbulent transition in flows with injection was explained in [10, 11] on the basis of hydrodynamic instability theory, taking into account the non-parallel character of these flows. The mechanisms for the development of turbulence and reverse transition in channels with permeable walls are not theoretically explained. Simple semiempirical turbulence theories apparently are insufficient for this purpose. In the present work results are given of calculations with two-parameter turbulence models proposed in [12, 13] for describing complex flows. Due to the sharp changes of turbulent energy along the channel length, a numerical solution of the complete system of equations of motion was carried out by the finite-difference method [14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 43–48, September–October, 1976.     

Three-dimensional boundary-layer transition on a cone at Mach 3.5

A boundary-layer transition study on a sharp, 5° half-angle cone at various angles of attack was conducted at Mach 3.5. Transition data were obtained with and without significantly reduced freestream acoustic disturbance levels. A progressive downstream and upstream motion of the transition front on the windward and leeward rays, respectively, of the cone with angle of attack was observed for the high noise level data in agreement with data trends obtained in conventional (noisy) wind tunnels. However, the downstream movement was not observed to the same degree for the low noise level data in the present study. Transition believed to be crossflow dominated was found to be less receptive to freestream acoustic disturbances than first-mode (Tollmien-Schlichting) dominated transition. The previously-developed crossflow transition Reynolds number criterion, _tr,max 200, was found to be inadequate for the current case. An improved criterion is offered, which includes compressibility and flow-geometry effects.     

On a generalized nonlinearK-ɛ model for turbulence that models relaxation effects

In this paper, based on a similarity that exists between the constitutive relations for turbulent mean flow of a Newtonian fluid and that for the laminar flow of a non-Newtonian fluid, and making use of extended thermodynamics, we develop a generalized nonlinearK- model, whose approximate form includes the standardK- model and the nonlinearK- model of Speziale (1987) as special cases. Our nonlinearK- model, which is frame indifferent, can predict relaxation of the Reynolds stress, unlike most standardK- models. Also, our model is in keeping with that of Yakhotet al. (1992). Most interestingly, the linearized form of our model bears a striking resemblance to the model due to Yoshizawa and Nisizima (1993); however, it has been obtained from a totally different perspective.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

A method for the analysis of recirculating turbulent flows in a cavity

The complex fluid-dynamic aspects of a turbulent recirculating flow in a cavity with axial throughflow, and a rotating wall, were investigated by adopting a simple procedure for evaluating the turbulent stresses. The flow field was divided into two regions, a core and a wall region respectively. A wall function was adopted in the zones near to the solid boundaries, while a constant eddy diffusivity was assumed, in the core, following the indications of computed heat transfer coefficients in comparison with existing experimental data. The distributions of the stream function and of the tangential velocity are presented for a range of the rotational Reynolds number of the rotating wall and of the Reynolds number of the throughflow.

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Turbulente Rezirkulationsströmung in einem Hohlraum

Zusammenfassung Die komplizierten fluiddynamischen Aspekte einer turbulenten Rezirkulationsströmung in einem Hohlraum mit axialem Durchfluß und einer rotierenden Wand werden unter Verwendung einer vereinfachten Methode zur Berechnung der turbulenten Spannungen betrachtet. Das Strömungsfeld wird in einen Kern und einen Wandbereich aufgeteilt. Für die wandnahen Zonen wird eine Wandfunktion angenommen, während im Kern mit konstanter Wirbeldiffusivität gerechnet wird, was durch den Vergleich berechneter mit gemessenen Wärmeübergangskoeffizienten gerechtfertigt erscheint. Verteilungen der Stromfunktion und der tangentialen Geschwindigkeit sind für einen bestimmten Bereich der Reynoldszahlen für die Wandrotation und der für den Durchfluß angegeben.

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Nomenclature L axial length of enclosure - P dimensionless pressure, p^*2 - p static pressure - R dimensionless radial coordinate, r/r^* - r radial coordinate - r^* reference length, equal to r_O for enclosure - r_i radii of inlet and exit apertures - Re Reynolds number, v^*r^*/ - Re_i pipe Reynolds number, ¯v_zi(2r_i)/ - Re_t turbulent Reynolds number, Re(/) - Re rotational Reynolds number, r ₀ ² / - t dimensionless time,t/(r^*/v^*) - t time - V_r, V, Vz dimensionless velocity components, V_r/v^*, v, v_z/v^* - v_i turbulent fluctuation of the i-component of velocity - v_r, v, v_z velocity components - v^* reference velocity, equal to ¯v_zi for enclosure - X coordinate along a wall, x/r^* - Y coordinate normal to a wall, y/r^* - Z dimensionless axial coordinate, z/r^* - z axial coordinate - eddy diffusivity for momentum - dynamic viscosity - kinematic viscosity - density - shear stress - dimensionless shear stress, /v^*2 - dimensionless stream function, /r*²v*² - stream function - angular velocity - tangential vorticity component - ()_eff effective - ()_l laminar - ()_t turbulent - mean over the time     

Laminar flow performance of a heated body in particle-laden water

The effects of small uniformly sized spherical particles seeded into the freestream flow of a water tunnel on the delayed transition of a heated laminar flow control body is examined experimentally. In separate trials, four different mean diameter particle seedings were added to the flow and the approach flow velocity was cycled from subcritical to supercritical conditions at three different body heating conditions. The transition Reynolds number based on the body arc length and the approach flow velocity decreases monotonically with increasing d/ ^*, where d is the particle diameter and ^* is the displacement thickness at a critical location. The location of initial turbulent spot formation defines the critical location, and, within the range of experimental conditions reported here, is independent of particle size, heating condition and the approach velocity. For the high unit Reynolds numbers considered (Re ^u 1.88 × 10⁷ per metre), there is no observed critical particle diameterbased Reynolds number threshold; all sizes of particles considered in the experiments (d = 37 to 218 m) have some effect on transition. In a second set of experiments, particles were injected into the laminar boundary layer from a small orifice located at the forward stagnation point. These injected particles have no observable effect on the laminar layer or transition, which suggests that the injected particles fail to produce wakes or vorticity within the laminar layer that may lead to turbulent spot production.Also with the Graduate Program in Acoustics, Penn State UniversityThis work has been supported by the Applied Research Laboratory of The Pennsylvania State University under contracts with the Office of Naval Research and the Naval Sea Systems Command. The authors are particularly indebted to Professor Ron Blackwelder and his colleagues for sharing their yet unpublished findings from particle-induced transition experiments being conducted at the University of Southern California.     

Zusammenhang zwischen Geschwindigkeitsprofilen und Wärmeübergang bei turbulent pulsierender Rohrströmung

Zusammenfassung Über die Strömungsverhältnisse und deren Einfluß auf den Wärmeübergang pulsierend durchströmter Rohre mit Turbulenz liegen bisher kaum Ergebnisse vor. Die Definition der den Wärmeübergang bei turbulenter Strömung wesentlich mitbestimmenden laminaren Unterschicht wird erneut aufgegriffen, die Verhältnisse bei laminarer und turbulenter Strömung werden verglichen, und es werden zum ersten Mal Geschwindigkeitsprofile turbulent oszillierender Rohrströmungen auf der Basis von LDA-Messungen dargestellt. Außerdem wird eine untere Grenzfrequenz abgeleitet, ab der die Geschwindigkeitsänderungen der pulsierenden Strömung die Laminarisierung der wandnahen Schicht verhindert und somit der Wärmeübergang wesentlich verbessert wird.

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Connection between velocity distribution and heat transfer at turbulent pulsating flow

About the conditions of turbulent pulsating pipe flow only a small number of results has been submitted. The definition of the turbulent heat transfer determining laminar sublayer will be taken up again, laminar and turbulent flow will be compared, and in this paper velocity distributions at turbulent oscillating pipe flow on the basis of LDA-measurements will be presented for the first time. A low-end frequency has been calculated from that onward the velocity distribution of the pulsating flow inhibits the laminarization of the boundary layer and leads to an improvement of the heat transfer.

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Formelzeichen E _L ^* Dimensionslose Reibungsenergie der laminaren Zähigkeit - E _T ^* Dimensionslose Turbulenzenergie - Nu Nusselt-Zahl - Pr Prandtl-Zahl - R Rohrradius, m - Re Reynolds-Zahl - W Welligkeit (definiert in Bild 7) - d Rohrinnendurchmesser, m - f Pulsationsfrequenz, Hz - r Abstand von der Rohrmitte, m - r* Reibungsradius - u, v Turbulente Schwankungsgeschwindigkeiten in axialer und radialer Richtung, m/s - v Axiale Strömungsgeschwindigkeit, m/s - Über den Rohrquerschnitt gemittelte axiale Strömungs-geschwindigkeit, m/s - Pulsierender Geschwindigkeitsanteil, m/s - v* Schubspannungsgeschwindigkeit, m/s - y Wandabstand, m - Grenzschichtdicke, m - Kinematische Zähigkeit, m²/s - Dichte, kg/m³ - Schubspannung, N/m² Indizes G Grenzwert - L Laminar - max Maximalwert - P Pulsation - S Stationär - T Turbulent     

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.     

Kritische Reynolds-Zahlen bei oszillierenden und pulsierenden Rohrströmungen

Zusammenfassung Mit dem elektrolytischen Meßverfahren wurde der Umschlag laminar — turbulent bei oszillierenden und pulsierenden Rohrströmungen untersucht. Es zeigt sich, daß die kritische Reynolds-Zahl Re_c bei oszillierenden Strömungen von der Schwingungsamplitude und Frequenz abhängig ist. Bei pulsierenden Strömungen kommt noch eine starke Abhängigkeit von der stationären Grundgeschwindigkeit hinzu. Es wurden Werte bis zu Re₀ 15300 beobachtet.

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Critical reynolds numbers for oscillating and pulsating tube flow

The transition from the laminar to the turbulent flow has been measured for oscillating and pulsating flow within a pipe by means of the electrolytic method. It has been shown, that the critical Reynolds-number Re_c of the oscillating flow is dependent on the amplitude and the frequency of the oscillation. At pulsating flows there has been observed an additional dependence from the superimposed stationary flow. There have been observed values of Re_c up to 15300.

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Formelzeichent C Konzentration der reagierenden kmol/m³Ionen im Flüssigkeitskern - D Diffusionskoeffizient m²/s - d Rohrdurchmesser m - F Faraday-Konstante=A · s/kg äq=9,65 · 10⁷ - i_G elektrische Grenzstromdichte, bezogen auf die Fläche der A/m² Meßkathode - L Abstand vom Beginn des Stoffaustausches bis zum Ende der m Meßkathode - 1 Länge der Meßkathode in der m Strömungsrichtung - t Zeit s - v Strömungsgeschwindigkeit, gemittelt über den Rohrquerm/s schnitt - z Wertigkeit der Elektrodenkg äq/kmol reaktion - Stoffübergangskoeffizient m/s - kinematische Viskosität m²/s - Dichte kg/m³ - _W Wandschubspannung N/m² Dimensionslose Größen - f=2·_W/·v² Widerstandsbeiwert - Re=d · v/ Reynolds-Zahl - Sc=/D Schmidt-Zahl - Sh= · d/D Sherwood-Zahl - Sk=d²/·t₀ Stokes-Zahl Indizes c Kritisch - O Oszillation - P Pulsation - S Stationär     

Numerical analysis of the transient conjugated heat transfer in a circular duct with a power-law fluid

We treat numerically in this paper, the transient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluids is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the transverse and longitudinal heat conduction effects are taken into account. The numerical results for the temporal evolution of the average temperature of the tube wall, _av, is plotted for different nondimensional parameters such as conduction parameter, , the aspect ratios of the tube, and ₀ and the index of power-law fluid, n.