An investigation of possible mechanisms of heterogeneous drag reduction in pipe and channel flows

When concentrated polymer solutions are injected into the core-region of a turbulent pipe or channel flow, the injected polymer solution forms a thread which preserves its identity far beyond the injection point. The resulting drag reduction is called heterogeneous drag reduction.This study presents experimental results on the mechanism of this type of drag reduction. The experiments were carried out to find out whether this drag reduction is caused by small amounts of polymer removed from the thread and dissolved in the near-wall region of the flow or by an interaction of the polymer thread with the turbulence. The friction behavior of this type of drag reduction was measured for different concentrations in pipes of different cross-sections, but of identical hydraulic diameter. The parameters of the injection, i.e. injector geometry as well as the ratio of the injection to the bulk velocity, were varied. In one set of experiments the polymer thread was sucked out through an orifice and the friction behavior in the pipe was determined downstream of the orifice. In another experiment, near-wall fluid was led into a bypass in order to measure its drag reducing properties. Furthermore, the influence of a water injection into the near-wall region on the drag reduction was studied.The results provide a strong evidence that heterogeneous drag reduction is in part caused by small amount of dissolved polymer in the near-wall region as well as by an interaction of the polymer thread with the turbulence.Nomenclature a channel height - b channel width - c _p concentration of the injected polymer solution - c _R effective polymer concentration averaged over the cross-section - d pipe or hydraulic diameter - d _i injector diameter - DR drag reduction - f friction factor - l downstream distance from injector - L length of a pipe segment - P polymer type - p differential pressure - Re Reynolds number - U bulk velocity - u ^* ratio of injection to bulk velocity - y ⁺ dimensionless wall distance - v kinematic viscosity - density of the fluid - _w wall shear stress     

High Reynolds number thick axisymmetric turbulent boundary layer measurements

Experimental measurements of the wall shear stress and momentum thickness for thick axisymmetric turbulent boundary layers are presented. The use of a full-scale towing tank allowed zero pressure gradient turbulent boundary layers to be developed on cylinders with diameters of 0.61, 0.89, and 2.5 mm and lengths ranging from 30 m to 150 m. Moderate to high Reynolds numbers (10⁴<Re <10⁵, 10⁸<Re _L<10⁹) are considered. The relationship between the mean wall shear stress, cylinder diameter, cylinder length, and speed was investigated, and the spatial growth of the momentum thickness was determined. The wall shear stress is significantly higher, and the spatial growth of the boundary layers is shown to be lower than for a comparable flat-plate case. The mean wall shear stress exhibits variations with length that are not seen in zero pressure gradient flat plate turbulent boundary layers. The ratio of outer to inner boundary layer length scales is found to vary linearly with Re , which is qualitatively similar to a flat plate turbulent boundary layer. The quantitative effect of a riblet cylindrical cross-sectional geometry scaled for drag reduction based on flat plate criteria was also measured. The flat plate criteria do not lead to drag reduction for this class of boundary layer shear flows.List of symbols a cylinder radius, mm - A _s total cylindrical surface area, m² - C _d tangential drag coefficient - D drag force, Newtons - boundary layer thickness, mm - * displacement thickness, mm - h riblet height, mm - L cylinder length, m - kinematic viscosity, m²/s - momentum thickness, mm - fluid density, kg/m³ - r radial coordinate, mm - Re _L Reynolds number based on length= - Re Reynolds number based on momentum thickness= - s riblet spacing, mm - _w mean wall shear stress, N/m² - u(r) mean streamwise velocity, m/s - u friction velocity= - U _o tow speed, m/s - x streamwise coordinate, m     

Tests of a fiber-optic velocity transducer

A novel transducer is developed and tested. The transducer utilizes optical fiber to measure mean and instantaneous flow rates in turbulent flows, and is capable of detecting flow reversal. Calibration of the transducer is conducted in both air and water. The dynamic response of the transducer is tested against hot-wire anemometery in the wake flow of a circular cylinder over a wide range of Reynolds number.List of symbols C _D drag coefficient - D diameter of cylinder - d diameter of fiber - E modulus of elasticity of the fiber - e output voltage - F drag force per unit length of a cylinder - f frequency (Hz) - L length of the fiber cantilever - M magnification factor - m mass per unit length of the fiber - Re Reynolds number - q dynamic pressure (= 1/2 U ²) - U free stream velocity - density - v kinematic viscosity     

The influence of polyvinylacetate additive in water on turbulent velocity field and drag reduction

The effect of polymer concentration on drag reduction was studied experimentally with diluted water solutions of polyvinylacetate in a 2.4 cm I. D. pipe. The instantaneous local velocities of the velocity fields were measured by a one-channel differential laser-Doppler anemometer DISA Mark II, with forward scattering. Concentrations of water-polyvinylacetate over the range from 10 to 2,000 ppm were used. The drag reduction coefficient is proportional to the concentration and hydrolysis degree of the saponificated polyvinylacetate (PVAC) employed. A mechanical degradation in the turbulent shear flow was not observed.List of Symbols a ₁ coefficient in Eq. (3) - a ₂ coefficient in Eq. (3) - D pipe diameter - k coefficient in modified Blasius equation for friction factor - K consistency parameter given by (1 b) - K _i coefficients in Eq. (5) - m coefficient in Eq. (3) - n flow index Eq. (1a), coefficient in Eq. (3) - n ⁺ dimensionless position parameter defined by Eq. (4) - N ⁺ position parameter defined by Eq. (7) - r radial distance from the pipe center - R pipe radius - Re Reynolds number - Re _g generalized Reynolds number, Eq. (9) - t temperature - u ⁺ dimensionless local velocity, /u ^* - u ^* dynamic friction velocity, w(/8) ^0,5 - U ⁺ dimensionless local mean velocity defined by Eq. (6) - time-averaged local velocity - _m time-averaged local velocity at the pipe center - w average velocity over the cross-section of the pipe - X concentration of polymer in water, w · ppm - y distance from the pipe wall - y ⁺ dimensionless distance from the pipe wall, y u ^* / or as in Eq. (8) - friction factor in drag reduction flow - ₀ friction factor of pure water - degree of drag reduction - viscosity - standard deviation A version of this paper was presented at the 9th National Symposium on the measurement of turbulence with laser Doppler and other anemometers, Bratislava, CSSR, 1986     

Reynolds number dependence of the drag coefficient for laminar flow through fine-scale photoetched screens

The laminar steady flow downstream of fine-mesh screens is studied. Instead of woven-wire screens, high-uniformity screens are fabricated by photoetching holes into 50.8 m thick Inconel sheets. The resulting screens have minimum wire widths of 50.8 m and inter-wire separations of 254 m and 318 m for the two screens examined. A flow facility has been constructed for experiments with these screens. Air is passed through the screens at upstream velocities yielding wire width Reynolds numbers from 2 to 35. To determine the drag coefficient, pressure drops across the screens are measured using pressure transducers and manometers. Threedimensional flow simulations are also performed. The computational drag coefficients consistently overpredict the experimental values. However, the computational results exhibit sensitivity to the assumed wire cross section, indicating that detailed knowledge of the wire cross section is essential for unambiguous interpretation of experiments using photoetched screens. Standard semi-empirical drag correlations for woven-wire screens do not predict the present experimental results with consistent accuracy.List of symbols A ₁, A ₂ screen aspect ratios - c _d screen drag coefficient - d woven-wire diameter - D photoetched minimum wire width (spanwise) - f woven-wire screen drag function - M distance between adjacent wires - N spectral-element order - o woven-wire open area fraction - O photoetched open area fraction - _p pressure drop across screen - Re _d woven-wire diameter Reynolds number - Re _D photoetched wire width Reynolds number - U fluid velocity upstream of screen - W photoetched sheet thickness (streamwise) - x, y, z spatial coordinates - fluid density - fluid viscosity     

Polymer Induced Drag Reduction in a Turbulent Pipe Flow Subjected to a Coriolis Force

The skin friction factor f in a turbulent wall-bounded flow can be greatly reduced by using polymer solutions. In this paper we discuss experimental results on the effect of the Coriolis force on turbulent drag reduction. To study this, a horizontal smooth-walled pipe with internal diameter 25?mm is placed on a horizontal table rotating about its vertical axis. The rotation is made non-dimensional with friction velocity and pipe diameter, to form the Rotation number Ro. For a range of bulk Rotation number (Ro _b ) between 0 and 0.6 for two different Reynolds numbers (Re _b = 15 & 30 × 10³), the pressure drop is measured, from which the average friction factor f is obtained. Additionally the effect of four different polymer concentrations has been investigated. The single-phase results show that the friction factor increases monotonic but gradual with Rotation. With polymer additives a drag reduction is found that increases with concentration, but which is not affected by the rotation.     

Blade manipulators in turbulent channel flow

We report here the results of a series of careful experiments in turbulent channel flow, using various configurations of blade manipulators suggested as optimal in earlier boundary layer studies. The mass flow in the channel could be held constant to better than 0.1%, and the uncertainties in pressure loss measurements were less than 0.1 mm of water; it was therefore possible to make accurate estimates of the global effects of blade manipulation of a kind that are difficult in boundary layer flows. The flow was fully developed at the station where the blades were mounted, and always relaxed to the same state sufficiently far downstream. It is found that, for a given mass flow, the pressure drop to any station downstream is always higher in the manipulated than in the unmanipulated flow, demonstrating that none of the blade manipulators tried reduces net duct losses. However the net increase in duct losses is less than the drag of the blade even in laminar flow, showing that there is a net reduction in the total skin friction drag experienced by the duct, but this relief is only about 20% of the manipulator drag at most.List of symbols A, A log law constants - c chord length of manipulator - D drag of the manipulator - dp/dx pressure gradient in the channel - h half height of the channel - H height of the channel (2h) - K log law constant - L length of the channel - L.E. leading edge of the manipulator - P static pressure - P _x static pressure at a location x on the channel - P _xm static pressure at the location x in the presence of manipulator - p _ref static pressure at any reference location x upstream of the manipulator - Re Reynolds number - t thickness of the manipulator - T.E. trailing edge of the manipulator - u velocity in the channel - U friction velocity - U ^* average velocity in the channel - u _c centre-line velocity in the channel - U ₊ U/U _* - u _m velocities in the channel downstream of the manipulators - u _ref velocities in the channel at reference location upstream of the manipulators - w Coles's wake function - W width of channel Also National Aeronautical Laboratory, Bangalore 560 017, India     

Experimental study of sedimentation characteristics of spheroidal particles

The drag of non-spherical particles is a basic, important parameter for multi-phase flow. As the first step in research in this area, the terminal velocities, Ut, of hemispherical and spherical segment particles with maximal diameters of 6-21 mm were measured in static fluids by using a high-speed video camera. The drag coefficient, CD, measured for Reynolds number, Re of 10^1-10^5, has been obtained and compared with those for a sphere. The Re based on the terminal velocity has a logarithmic linear relationship with Ar number for both the facet facing upwards or downwards for the two experimental spheroidal particles, and their Co values are greater than those of spheres. A shape function that depends on the initial orientation of the particle facet is presented to correct for the shape effects.     

Shell-side distribution and the influence of inlet conditions in a model of a disc-and-doughnut heat exchanger

Measurements of wall pressure and of three orthogonal velocity components with their corresponding fluctuations are reported for two systems of alternating and equi-spaced doughnut and disc baffles axisymmetrically located in a water turbulent pipe flow, simulating the isothermal shell-side flow in shell and tube heat exchangers. The influence of inlet Reynolds number and of asymmetric inlet flow conditions was studied for two geometries. The velocity field was dominated by the pressure gradient and the flow around each individual baffle was influenced by the relative position of its neighbouring baffles.List of symbols C _p wall static-pressure coefficient - D internal diameter of upstream and downstream pipes (mm) - D _s internal diameter of test section (mm) - d _d disc diameter (mm) - d _c doughnut-hole diameter (mm) - l baffle-pitch (mm) - l _i entrance length in the model before first baffle (mm) - l ₀ exit length in the model after last baffle (mm) - mass flow rate (kg/s) - p local wall-static pressure (mm H₂O) - p density of water (1.006 kg/dm³ at 17°C) - Re _b Reynolds number based on bulk velocity - U _b bulk velocity - U _max maximum centre-line axial velocity (ms^–1) - x, y, z Cartesian coordinates - mean and turbulent velocity components along x, y, z respectively     

In the intermixing region behind circular cylinders with stepwise change of the diameter

The flow within the intermixing region behind circular cylinders with stepwise change of the diameter of diameter ratio d/D of 0.5 has been examined. Based on the statistical analysis and conditional sampling of the velocity fluctuations and of flow visualization, the vortex wakes associated with the big and small cylinders have been established. Both wakes are found under the dominant primary mode, which corresponds to the vortex shedding Strouhal number of two dimensional cylinder, and the less dominant secondary mode. The Strouhal number of the secondary mode of the big vortex wake is higher than that of the primary mode and the opposite is found for the small vortex wake. Both vortex wakes and their modes are found convecting downstream and into region behind the other cylinder. Both wakes are observed to be different from that of two dimensional cylinder.List of symbols D, d diameter of big and small cylinder - f frequency - R ₁₂ (f) cross-power spectral function - R ₁₁, R ₂₂ auto-power functions - Re _D, Re_d Reynolds numbers U ₀ D/v, U ₀ d/v - t time relative to triggering instant - U ₀ freestream mean velocity - U, V, W streamwise, lateral and spanwise mean velocity, respectively - u, v, w streamwise, lateral and spanwise velocity fluctuations, respectively - U _f phase velocity - U _T convection velocity - u _R, v_r recovered u and v velocity fluctuations - uv Reynolds stress - x, y, z streamwise, lateral, and spanwise coordinates, respectively - separation - ₁₂ ² (f) coherence function - _R recovered coherent vorticity fluctuation - phase - ₁₂ (f) phase spectral function     

Near-wall velocity distributions within a straight conical diffuser

The measured mean velocity profiles at the various stations along a conical diffuser (8° total divergence angle) were found to consist of log regions, half-power law regions and linear regions. The describing coefficients for the inner half-power law region (which followed a rather narrow log region) differed from the standard values due to the axi-symmetric geometry and lack of moving equilibrium of the flow as it attempted to adjust to a varying adverse pressure gradient. However, these coefficients (like those for the linear region) correlated with the local wall shear stress and the kinematic pressure gradient.List of symbols A, B coefficients in logarithmic law velocity distribution (Eq. (1)) - C, D coefficients in half-power law velocity distribution (Eq. (5)) - D_i inside diameter of feed pipe (10.16 cm) - d _p outer diameter of Preston tube - E, F coefficients in linear law velocity distribution (Eq. (10)) - P _s local static pressure - R local radius of diffuser, (D _i /2) + x _w sin 4° - Re Reynolds number, D _i U _b /v - U local mean velocity in the x _w direction - U _b cross-sectional average mean velocity (x-direction) in feed pipe - U _c mean velocity at the diffuser centerline - u ^* local friction velocity - u ⁺ dimensionless local mean velocity, U/u ^* - axial distance along diffuser centerline (measured from inlet to diffuser) Fig. (2) - _w distance along diffuser wall (measured from inlet to difusser (Fig. 2) - y _w distance from wall in direction orthogonal to wall (Fig. 2) - y ⁺ dimensionless position, y _w u ^*/v - kinematic (axial static) pressure gradient, (1/g9) dP _s/dx - ^* displacement thickness (Eq. (4)) - dimensionless pressure gradient parameter, x v/(u*) ³ - Von Karman constant (0.41) - density - kinematic viscosity - shear stress     

Displacement effect of square-ended pitot tubes in shear flows

Measurements in uniformly sheared flows indicate that the displacement of total-pressure tube readings due to shear is roughly constant, even for values of the shear parameter smaller than previously believed.List of symbols D tube outer diameter - d tube inner diameter - h wind-tunnel height - K tube shear parameter - K _r reference tube shear parameter - L characteristic scale of turbulence - P pressure - P _D total pressure indicated by tube of diameter D - P ₀ free stream total pressure - P _r total pressure indicated by reference tube - P _s pressure indicated by static pressure tube - U mean velocity - U _c centerline mean velocity - U _D mean velocity indicated by tube of diameter D - U _r mean velocity indicated by reference tube - u r.m.s. velocity - y transverse coordinate - empirical coefficient - wind-tunnel shear parameter - displacement - fluid density     

Effect of longitudinal riblets on the aerodynamic characteristics of a straight wing

The results of balance aerodynamic tests on model straight wings with smooth and ribbed surfaces at an angle of attack =–4°–12°, Mach number M=0.15–0.63, and Reynolds number Re=2.4·10⁶–3.5·10⁶ are discussed. The nondimensional riblet spacings ⁺, which determines the effect of the riblets on the turbulent friction drag, and the effect of riblets on the upper and/or lower surface of a straight wing on its drag, lift, and moment characteristics are estimated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 33–38, March–April, 1995.     

Vortex shedding in cylinder flow of shear-thinning fluids: I. Identification and demarcation of flow regimes

An experimental study on the flow of non-Newtonian fluids around a cylinder was undertaken to identify and delimit the various shedding flow regimes as a function of adequate non-dimensional numbers. The measurements of vortex shedding frequency and formation length (l_f) were carried out by laser-Doppler anemometry in Newtonian fluids and in aqueous polymer solutions of CMC and tylose. These were shear thinning and elastic at weight concentrations ranging from 0.1 to 0.6%. The 10 and 20 mm diameter cylinders (D) used in the experiments had aspect ratios of 12 and 6 and blockage ratios of 5 and 10%, respectively. The Reynolds number (Re^*) was based on a characteristic shear rate of U_∞/(2D) and ranged from 50 to 9×10³ thus encompassing the laminar shedding, the transition and shear-layer transition regimes. Increasing fluid elasticity reduced the various critical Reynolds numbers (Re_etr^*, Re_lf^*, Re_bbp^*) and narrowed the extent of the transition regime. For the 0.6% tylose solution the transition regime was even suppressed. On the other end, pseudoplasticity was found to be indirectly responsible for the observed reduction in Re_otr^*: it increases the Strouhal number which in turn increases the vortex filaments, precursors of the transition regime. Elasticity was better quantified by the elasticity number Re′/We than by the Weissenberg number. This elasticity number involves the calculation of the viscosity at a high characteristic shear rate, typical of the boundary layer, rather than at the average value (U_∞/(2D)) used for the Reynolds number, Re^*.     

Near Field Round Jet Flow Downstream from an Abrupt Contraction Nozzle with Tube Extension

Round air jet development downstream from an abrupt contraction coupled to a uniform circular tube extension with length to diameter ratio L/D?=?1.2 and L/D?=?53.2 is studied experimentally. Smoke visualisation and systematic hot film velocity measurements are performed for low to moderate Reynolds numbers 1130?<?Re _b ?<?11320. Mean and turbulent velocity profiles are quantified at the tube exit and along the centerline from the tube exit down to 20 times the diameter D. Flow development is seen to be determined by the underlying jet structure at the tube exit which depends on Reynolds number, initial velocity statistics at the tube exit and the presence/absence of coherent structures. It is shown that the tube extension ratio L/D as well as the sharp edged abrupt contraction influence the initial jet structure at the tube exit. For both L/D ratios, the presence of the abrupt contraction results in transitional jet flow in the range 2000?<?Re _b ?<?4000 and in flow features associated with forced jets and high Reynolds numbers Re _b ?>?10⁴. The tube extension ratio L/D downstream from the abrupt contraction determines the shear layer roll up so that for L/D?=?1.2 flow visualisation suggests the occurrence of toroidal vortices for Re _b ?<?4000 whereas helical vortices are associated with the transitional regime for L/D?=?53.2. Found flow features are compared to features reported in literature for smooth contraction nozzles and long pipe flow.     

Drag reduction by polymer solutions in a riblet-lined pipe

The flow of 3 to 100 wppm aqueous solutions of a polyethyleneoxide polymer,M _w=6.2×;10⁶, was studied in a 10.2 mm i.d. pipe lined with 0.15 mm V-groove riblets, at diametral Reynolds numbers from 300 to 150000. Measurements in the riblet pipe were accompanied by simultaneous measurements in a smooth pipe of the same diameter placed in tandem. The chosen conditions provided turbulent drag reductions from zero to the asymptotic maximum possible. The onset of polymer-induced drag reduction in the riblet pipe occurred at the same wall shear stress, * _w =0.65 N/m², as that in the smooth pipe. After onset, the polymer solutions in the riblet pipe initially exhibited linear segments on Prandtl-Karman coordinates, akin to those seen in the smooth pipe, with specific slope increment . The maximum drag reduction observed in the riblet pipe was independent of polymer concentration and well below the asymptotic maximum drag reduction observed in the smooth pipe. Polymer solution flows in the riblet pipe exhibited three regimes: (i) Hydraulically smooth, in which riblets induced no drag reduction, amid varying, and considerable, polymer-induced drag reduction; this regime extended to non-dimensional riblet heightsh ⁺<5 in solvent andh ⁺<10 in polymer solutions. (ii) Riblet drag reduction, in which riblet-induced flow enhancementR>0; this regime extended from 5<h ⁺<22 in solvent and from 10<h ⁺<30 in the 3 wppm polymer solution, with respective maximaR=0.6 ath ⁺=14 andR=1.6 ath ⁺=21. Riblet drag reduction decreased with increasing polymer concentration and increasing polymer-induced flow enhancement S. (iii) Riblet drag enhancement, whereinR<0; this regime extended for 22<h ⁺<110 in solvent, withR;–2 forh ⁺>70, and was observed in all polymer solutions at highh ⁺, the more so as polymer-induced drag reduction increased, withR<0 for allS>8. The greatest drag enhancement in polymer solutions,R=–7±1 ath ⁺=55 whereS=20, considerably exceeded that in solvent. Three-dimensional representations of riblet- and polymer-induced drag reductions versus turbulent flow parameters revealed a hitherto unknown dome region, 8<h ⁺<31, 0<S<10, 0<R<1.5, containing a broad maximum at (h ⁺,S,R) = (18, 5, 1.5). The existence of a dome was physically interpreted to suggest that riblets and polymers reduce drag by separate mechanisms.     

Theoretical and experimental investigations on the formation of air core in a swirl spray atomizing nozzle

Theoretical and experimental studies have been made to investigate the variations of air core diameter, the most important hydrodynamic picture inside a swirl nozzle, with the pertinent guiding parameters like injection condition expressed as the Reynolds number at inlet to the nozzle and the geometrical dimensions of the nozzle, namely, the length and diameter of the swirl chamber, angle of spin chamber and the orifice diameter. The theoretical relations have been established through an approximated analytical solution of the hydrodynamics of flow of a viscous incompressible fluid in a swirl nozzle. A series of experiments have been carried out to support and compare the theoretical results. Finally, it has been recognized that for any nozzle, the air core diameter becomes a direct function of Reynolds number Re _i at inlet to the nozzle only at its lower range and then remains constant. Amongst the nozzle geometrics, the ratio of orifice to swirl chamber diameter D ₂/D ₁ has got the most predominant effect on the air core diameter. An increase in the ratio of orifice to swirl chamber diameter D ₂/D ₁, and in the spin chamber angle 2 and a decrease in the swirl chamber length to diameter ratio L ₁/D ₁ increase the ratio of air core to orifice diameter and vice versa.Nomenclature A _E Area of tangential inlet ports of the nozzle - A ₂ Area of the orifice - a Air core radius - D ₁ Swirl chamber diameter - D ₂ Orifice diameter - d ₂ Air core diameter - E A nondimensional parameter defined by equation (14) - E _R A nondimensional parameter defined by equation (33) - L ₁ Length of the swirl chamber - P Static pressure - P _b Back pressure of the nozzle - Q Volume flow rate - R Radius vector or the longitudinal co-ordinate with respect to spherical co-ordinate system (figure 3) - R ₁ Radius of the swirl chamber - R ₂ Radius of the orifice - Re _i Reynolds number at inlet to the nozzle - R _z Radius of the nozzle at any section - r Radial distance from the nozzle axis - U Longitudinal component of velocity with respect to spherical co-ordinate system (figure 3) - V Component of velocity in the axial plane perpendicular to R as defined in (figure 3) - V _r Radial velocity component - V _z Axial velocity component - V _Ø Tangential velocity component - Average tangential velocity at inlet to the nozzle - w Component of velocity perpendicular to axial plane with respect to the spherical co-ordinate as defined in figure 3 - z Distance along the nozzle axis from its inlet plane - Half of the spin chamber angle - Boundary layer thickness - ₂ Boundary layer thickness at the orifice - Angle which a radius vector according to the system of spherical coordinates (figure 3) makes with the nozzle axis - Dynamic viscosity - Kinematic viscosity - Density - Ø Running co-ordinate in the azimuthal direction with respect to the cylindrical polar co-ordinate system as shown in figure 3 - Circulation constant     

Steady rise of a small spherical gas bubble along the axis of a cylindrical pipe at high Reynolds number

Steady irrotational flow of inviscid liquid of density ρ_l around a spherical gas bubble which lies on the axis of a cylindrical pipe is investigated using the analysis of Smythe (Phys. Fluids 4 (1961) 756). The bubble radius b=qa is assumed small compared to the pipe radius a, and the interfacial tension between gas and liquid is γ. Far from the bubble, in the frame in which the bubble is at rest, the liquid velocity along the pipe is v₀, whereas the liquid velocity at points on the wall closest to the bubble is U_zw=v₀(1+1.776q³+⋯). The decrease in wall pressure as the bubble passes is therefore Δp=1.776ρ_lv₀²q³. When the Weber number W=2bv₀²ρ_l/γ is small, the bubble deforms into an oblate spheroid with aspect ratio χ=1+9W(1+1.59q³)/64. If the fluid viscosity μ is non-zero, and the Reynolds number Re=2v₀ρ_lb/μ is large, a viscous boundary layer develops on the walls of the pipe. This decays algebraically with distance downstream of the bubble, and an exponentially decaying similarity solution is found upstream. The drag D on the bubble is D=12πμv₀b(1−2.21Re^−1/2)(1+1.59q³)+7.66μv₀bRe^1/2q^9/2, larger than that given by Moore (J. Fluid Mech. 16 (1963) 161) for motion in unbounded fluid. At high Reynolds numbers the dissipation within the viscous boundary layers might dominate dissipation in the potential flow away from the pipe walls, but such high Reynolds numbers would not be achieved by a spherical air bubble rising in clean water under terrestrial gravity.     

Effects of the bifurcation angle on the steady flow structure in model saccular aneurysms

The effects of the bifurcation angle on the steady flow structure in a straight terminal aneurysm model with asymmetric outflow through the branches have been characterized quantitatively in terms of laser-Doppler velocimetry (LDV)-measured mean velocity and fluctuating intensity distributions. The bifurcation angles investigated were 60°, 90°, and 140° and the Reynolds number based on the bulk average velocity and diameter of the afferent vessel was 500. It is found that the size of the recirculating zones in the afferent vessel, the flow activity (both mean and fluctuating motions) inside the aneurysm, and the shear stresses acting on the aneurysmal wall increase with increasing bifurcation angle. More importantly, both LDV-measured and flow-visualized results of the present study suggest the presence of a critical bifurcation angle below which the aneurysm is susceptible to thrombosis, whereas above this the aneurysm is prone to progression or rupture.List of symbols a aneurysm height - b distance from orifice to fundus - c orifice diameter - D afferent conduit diameter - d fundus diameter - Hz frequency unit = cycle/second - L length of bifurcation zone - Re Reynolds number = U · D/v - U streamwise mean velocity - U _m streamwise bulk mean velocity - u streamwise fluctuating component - X ^* normalized streamwise coordinate: X ^* 0: X ^* = X/a; X ^* <0: X ^* = X/L - Y ^* normalized transverse coordinate: Y ^* = Y/D - Z ^* normalized spanwise coordinate: Z ^* = Z/D - kinematic viscosity - _b angle of bifurcation - _c critical bifurcation angle     

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