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     

Drag reduction in polymer mixtures

Summary Drag reduction was studied in dilute toluene solutions of a mixture of two polymers: polyisobutylene (of three different molecular weights) and 1,4-cis-isoprene rubber in the turbulent region at low (up to 5000) Reynolds numbers. Experiments were carried out with mixed solutions at a concentration equal to optimum concentration of polyisobutylene or higher than it. Drag reduction of the polymer mixtures depending on the ratio of the two polymers showed a positive deviation from the additive straight line at all concentrations investigated. To evaluate the degree of deviation from additivity, the excess drag reduction, was introduced which represents the difference between the actually measured drag reduction and that read from the additive straight line. The excess drag reduction showed almost no dependence on the molecular weight of polyisobutylene in the investigated range of this magnitude. Deviation from additivity depending on the ratio of the two polymers in the mixture growed higher with increasing the flow rate at a given molecular weight of polyisobutylene. The highest excess drag reduction was observed in solutions containing a larger amount of the lower molecular isoprene rubber polymer. The effect of polymer coils on drag reduction in binary polymer solutions was studied. An assumption was made that higher drag reduction in the polymer mixtures as compared to the additive was due to the change of polymer coil dimensions caused by the copresence of the macromolecules of both polymers in the solution. It was further supposed that low shear stresses at which the experiments were carried out caused sufficient orientation and deformation of isoprene rubber enlarged molecules and the contribution of the latter in increasing drag reduction of the mixture was higher.

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Zusammenfassung Die Widerstandsverminderung in verdünnten toluolischen Lösungen einer Mischung von zwei verschiedenen Polymeren wird untersucht. Verwendet werden Polyisobutylene (mit drei verschiedenen Molekulargewichten) und 1,4-cis-Isopren-Kautschuk, und es wird im turbulenten Bereich bei Reynolds-Zahlen bis zu 5000 gemessen. Die Versuche werden bei Konzentrationen, die der Optimalkonzentration von Polyisobutylen entsprechen, oder höheren Konzentrationen durchgeführt. Die Widerstandsverminderung der Polymermischungen zeigt bei allen untersuchten Konzentrationen eine positive Abweichung von der additiven Geraden, deren Größe vom Mischungsverhältnis abhängt. Zur Beschreibung der Abweichung vom additiven Verhalten wird die überschüssige Widerstandsverminderung (excess drag reduction) eingeführt, welche die Differenz zwischen dem wirklich gemessenen Wert und dem zugeordneten Wert auf der additiven Geraden beschreibt. Diese Größe zeigt nur eine geringe Abhängigkeit vom Molekulargewicht der eingesetzten Polyisobutylene. Die Abweichung vom additiven Verhalten als Funktion des Mischungsverhältnisses beider Polymeren wächst mit zunehmendem Volumenstrom. Die größte überschüssige Widerstandsverminderung wird in Lösungen beobachtet, die einen größeren Anteil des weniger hochmolekularen Isopren-Kautschuks enthalten. Der Einfluß der Polymerverknäuelung auf die Widerstandsverminderung wird betrachtet. Es wird angenommen, daß die überschüssige Widerstandsverminderung auf eine Änderung der Knäuelgröße infolge der Anwesenheit des jeweils anderen Polymeren in der Lösung zurückzuführen ist. Weiter wird vermutet, daß die relativ niedrigen Schubspannungen, bei denen die Versuche ausgeführt wurden, doch schon eine hinreichend starke Orientierung und Deformation der aufgeweiteten Isopren-Kautschuk-Moleküle bewirken, so daß deren Beitrag zur Erhöhung der Widerstandsverminderung überwiegt.

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Notations D diameter of capillary - DR drag reduction - DR _add additive drag reduction - DR excess drag reduction,DR = DR – DR _add - DR _mixture theoretical drag reduction of the mixture - DR _mixture ^* actually measured drag reduction of the mixture - DR _1R drag reduction of an IR molecule in a separate IR solution - DR _1R ^* drag reduction of an IR molecule in the presence of molecules of another polymer in the solution - DR _PIB drag reduction of a PIB molecule in a separate PIB solution - DR _PIB ^* drag reduction of a PIB molecule in the presence of molecules of another polymer in the solution - L length of the capillary - flow rate - c concentration - n number of IR molecules - p number of PIB molecules - _w wall shear stress - CMC carboxymethylcellulose - IR isoprene rubber - PAA polyacrylic acid - PAM polyacrylamide - PEI polyethyleneimine - PEO polyethylene oxide - PIB polyisobutylene - PS polystyrene     

On two distinct types of drag-reducing fluids,diameter scaling,and turbulent profiles

Two distinct scaling procedures were found to predict the diameter effect for different types of drag-reducing fluids. The first one, which correlates the relative drag reduction (DR) with flow bulk velocity (V), appears applicable to fluids that comply with the 3-layers velocity profile model. This model has been applied to many polymer solutions; but the drag reduction versus V scaling procedure was successfully tested here for some surfactant solutions as well. This feature, together with our temperature profile measurements, suggest that these surfactant solutions may also show this type of 3-layers velocity profiles (3L-type fluids).The second scaling procedure is based on a correlation of τ_w versus V, which is found to be applicable to some surfactant solutions but appears to be applicable to some polymer solutions as well. The distinction between the two procedures is therefore not simply one between polymer and surfactants. It was also seen that the τ_w versus V correlation applies to fluids which show a stronger diameter effect than those scaling with the other procedure. Moreover, for fluids that scale according to the τ_w versus V procedure, the drag-reducing effects extend throughout the whole pipe cross section even at conditions close to the onset of drag reduction, in contrast to the behavior of 3L fluids. This was shown by our measurements of temperature profiles which exhibit a fan-type pattern for the τ_w versus V fluids (F-type), unlike the 3-layers profile for the fluids well correlated by drag reduction versus V. Finally, mechanically-degraded polymer solutions appeared to behave in a manner intermediate between the 3L and F fluids.Furthermore, we also showed that a given fluid in a given pipe may transition from a Type A drag reduction at low Reynolds number to a Type B at high Reynolds number, the two types apparently being more representative of different levels of fluid/flow interactions than of fundamentally different phenomena of drag reduction. After transition to the non-asymptotic Type B regime, our results suggest that, without degradation, the friction becomes independent of pipe diameter and that the drag reduction level becomes also approximately independent of the Reynolds number, in a strong analogy to Newtonian flow.     

Turbulent flows of solutions of surface-active substances

Not only polymers, but also some surface-active substances have the capacity to reduce the hydrodynamic resistance in the turbulent regime of a fluid. From the point of view of stability against mechanical destruction, the addition of surface-active substances has certain advantages over polymers; in particular, they are capable of recovering their hydrodynamic effectiveness having passed through pumps and local points of resistance [1]. The basic possibility of reducing drag by the addition of surface-active substances was demonstrated in [2–6]. It was later shown that a reduction in the drag of liquids could be achieved only by micelle-forming surface-active substances [5, 7], in solutions of which spherical and platelike micelles, respectively, are formed at the critical concentrationsC _m1 andC _m2 for micelle formation. The values ofC _m1 andC _m2 depend both on the molecular structure of the surface-active substances as well as on external factors such as the temperature, the presence in the solution of electrolytes, polar organic substances, and so forth. The connection established in [5, 7] between the reduction in the drag and the value ofC _m2 suggests that the reduction in the turbulent friction will also depend on the above factors. It is therefore possible to control processes of turbulent transfer in a fluid as well as the drag. However, this possibility has not been sufficiently studied. In [1–7], the influence of surface-active substances on the drag was investigated. First data on the velocity profiles in solutions of surface-active substances were given in [9, 10]; they are not sufficiently complete. In the present work, we made measurements of the velocity profiles and the turbulence intensity in solutions of surfaceactive substances, and we have calculated the generation of turbulence energy and the dissipation of the energy of the averaged motion. We have studied the influence of electrolytes and the temperature on the reduction in the drag and give the results of full-scale experiments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–43, January–February, 1980.     

Influence of molecular weight and molecular conformation of polymers on turbulent drag reduction

Summary An investigation was carried out on drag reduction of diluted solutions of four polyisobutylenes of different molecular weight in diversely good and poor solvents in the turbulent region at small (up to 5000) and average (from 19 000 to 42 000) Reynolds numbers, as well as of mixtures of polyisobutylene and polystyrene and of two polyisobutylenes of different molecular weight. Concentration at maximum drag reduction grows, while drag reduction itself decreases with molecular weight going down. The universal curve established byVirk et al. for aqueous solutions of a polyethylene oxide family is also confirmed for a family of polyisobutylenes in an organic solvent. The effect of polymer coils of various dimensions on drag reduction is assessed. PIB coils of various dimensions are produced in two ways — dissolving polyisobutylenes of different molecular weight in a given solvent and dissolving polyisobutylene of a definite molecular weight in diversely good and poor solvent. Coil dimensions in the solution are increasing with the growth of intrinsic viscosity of a polymer by dissolving it in still better solvents, but probably due to impeded orientation and deformation of the larger polymer coils their drag reduction is smaller at low rather than at high shear stresses. Drag reduction of diluted solutions of two PIB differing in molecular weight shows almost no deviation from the additive straight line both when the overall concentration of solutions is equal and when it exceeds the one at maximum drag reduction of PIB of higher molecular weight. Drag reduction of diluted solutions of PIB and PS mixtures at an overall concentration higher than optimum concentration shows a positive deviation from the additive straight line.

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Zusammenfassung Untersucht wurde die Verringerung des Reibungswiderstandes (VRW) fließender, verdünnter Lösungen von vier Polyisobutylenen mit verschiedenem Molekulargewicht in unterschiedlich guten und schlechten Lösungsmitteln im turbulenten Bereich bei kleinen (bis 5000) und bei mittleren (von 19 000 bis 42 000) Reynoldsschen Zahlen, sowie von Mischungen aus Polyisobutylen und Polystyrol und aus zwei Polyisobutylenen mit verschiedenem Molekulargewicht. Die Konzentration bei maximalem VRW-Effekt nimmt zu, und der maximale VRW-Effekt selbst vermindert sich mit abnehmendem Molekulargewicht. Die vonVirk und Mitarbeitern für wäßrige Lösungen einer Reihe von Polyäthylenen festgestellte universale Kurve wird auch für die Polyisobutylenfamilie in einem organischen Lösungsmittel bestätigt. Der Einfluß der nach ihren Dimensionen unterschiedlichen Polymerknäuel auf die Verringerung des Reibungswiderstandes wurde ausgewertet. Polyisobutylen-Knäuel mit verschiedenen Dimensionen werden auf zwei Weisen realisiert, durch Auflösung von Polyisobutylenen mit verschiedenem Molekulargewicht in einem gegebenen Lösungsmittel und durch Auflösung von Polyisobutylen mit einem gegebenen Molekulargewicht in unterschiedlich guten und schlechten Lösungsmitteln. Mit der durch seine Auflösung in immer besseren Lösungsmitteln steigenden Grenzviskosität eines Polymeren nehmen die Knäueldimensionen in den Lösungen zu. Jedoch ist, wahrscheinlich wegen der damit verbundenen Behinderung der Orientierung und Deformierung der größeren Polymerknäuel, ihr VRW-Effekt bei den niedrigen Scherspannungen kleiner als bei den höheren. Der VRW-Effekt der verdünnten Lösungen aus zwei Polyisobutylenen mit verschiedenem Molekulargewicht weist fast keine Abweichung von der additiven Geraden auf, sowohl wenn die Gesamtkonzentration der Lösungen gleich als auch wenn sie größer als die Konzentration beim maximalen VRW-Effekt des Polyisobutylens mit höherem Molekulargewicht ist. Der VRW-Effekt der verdünnten Lösungen von Mischungen aus Polyisobutylen und Polystyrol mit einer über der optimalen liegenden Gesamtkonzentration zeigt eine positive Abweichung von der additiven Geraden.

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D diameter of capillary - [DR] intrinsic drag reduction - DR _F fractional drag reduction - DR _{F, max} maximum fractional drag reduction - DR_sp specific drag reduction - K constant - L length of capillary - M _v molecular weight of polymer, determined by the viscosimetric method - flow rate - c concentration - [c] intrinsic concentration - _equiv. equivalent sphere density - _{w, p} wall shear stress of polymer solution - _{w, s} wall shear stress of solvent - CMC carboxymethylcellulose - PAA polyacrylamide - PEO polyethylene oxide - PIB polyisobutylene - PS polystyrene - DR drag reduction With 8 figures and 2 tables     

Viscoelasticity of a surfactant and its drag-reducing ability

There is considerable interest in the use of viscoelastic cationic surfactant-counterion mixtures in district heating and cooling systems to reduce pressure losses. A recent field test in a secondary system near Prague showed a 30+% reduction in pumping energy requirements.We have studied a number of commercial surfactants and we report here results of rheological, drag reduction and turbulence measurements on Arquad 18–50 (octadecyl trimethyl ammonium chloride (AR 18)) with an excess of sodium salicylate (NA). The concentration studied was 1.6 mM AR 18 and 4.0 mM NA which is about one third the concentration for excellent drag reduction in this surfactant's effective temperature range 30–90°C.Viscosity, , vs. shear rate,D, first normal stress difference,N ₁, vs. shear rate, drag reduction (as pressure drop,i=P/1) vs. average velocity,U _ave, in a 39.4 mm tube for AR 18, and turbulence intensity data for three drag reducing surfactants are reported.Of particular interest are the generally low turbulence intensities in all three directions which correspond to reduced heat, mass and momentum transfer rates compared to water, and the existence of large normal stress differences at 20°C for AR 18, a temperature at which no drag reduction occurs with this surfactant, indicating that normal stress effects do not correlate directly with drag reduction.The effect of time of pumping on increasing drag reduction demonstrates that this factor overwhelms the expected increase in drag reduction as temperature is raised from 18–19°C to 40.5°C.     

Numerical simulation of sedimentation of rectangular particle in Newtonian fluid

The sedimentation of a rectangular particle falling in a two-dimensional channel filled with Newtonian fluid was simulated with finite element arbitrary Lagrangian–Eulerian domain method. The numerical procedure was validated by comparison of the simulation results with existing numerical work. Moreover, good agreement was obtained between the simulation results and experimental measurements performed in the current study. The equilibrium position, stable orientation and drag coefficient of a rectangular particle for different particle Reynolds numbers (Re_p) were studied. The results show that there is a critical particle Reynolds number for the preferred orientation of a rectangular particle falling in a Newtonian fluid. When Re_p is smaller than the critical value, the particle falls with its long side parallel to gravity; otherwise the particle falls with its long side perpendicular to gravity. The critical particle Reynolds number is a decreasing function of the blockage ratio and aspect ratio. The distributions of pressure and shear stress on rectangular particle surface were analyzed. Moreover, the drag coefficient of the rectangular particle decreases as Re_p or the blockage ratio increases; however, it appears to be independent of aspect ratio.     

Turbulent pipe flow of a drag-reducing rigid “rod-like” polymer solution

Fully developed turbulent pipe flow of an aqueous solution of a rigid “rod-like” polymer, scleroglucan, at concentrations of 0.005% (w/w) and 0.01% (w/w) has been investigated experimentally. Fanning friction factors were determined from pressure-drop measurements for the Newtonian solvent (water) and the polymer solutions and so levels of drag reduction for the latter. Mean axial velocity u and complete Reynolds normal stress data, i.e. u′, v′ and w′, were measured by means of a laser Doppler anemometer at three different Reynolds numbers for each fluid. The measurements indicate that the effectiveness of scleroglucan as a drag-reducing agent is only mildly dependent on Reynolds number. The turbulence structure essentially resembles that of flexible polymer solutions which also lead to low levels of drag reduction.     

Effects of riblet bends on pipe flows

Four riblet bends were tested to investigate the effects of riblets on pipe flows including the secondary flow on the Reynolds numbers; Re _D =6×10³–4×10⁴. The pressure gradients on the smooth pipe downstream from the riblet bends were measured, and also the pressure losses of the bends only were measured. All riblet bends reduced the pressure gradient on the smooth pipe downstream from them, which means a drag reduction. Two of the riblet bends showed the maximum drag reduction of about 4 percent at Re _D = 6500; this reduction rate was significant considering the uncertainty of the present experiments. Since the pressure losses of these two riblet bends were almost identical to that of the smooth bend at Re _D = 6500, they could cause a net drag reduction of about 4 percent on the piping system including these bends at that Reynolds number. Furthermore, the velocity profiles measured by LDV indicated that the secondary flow becomes weaker downstream from the riblet bends when a drag reduction is recognized there.Nomenclature D pipe diameter - D ₀ the distance from the valley to the valley passing through the pipe center - H height of groove - P nondimensional static pressure (p/it/(U ₀ ² ):p is gauge pressure) - dP/dX nondimensional pressure gradient - Rc curvature of bend - Re _D Reynolds number based on bulk velocity and pipe diameter - s spacing of groove - U mean streamwise velocity along the horizontal diameter - U ₀ bulk velocity - V mean vertical velocity along the horizontal diameter - x streamwise direction along the pipe axis (see Fig. 1) - X nondimensionalx (=x/D) - y radial direction in the horizontal plane which is perpendicular to the plane including the bend (see Fig. 1) - yUV swirl intensity (nondimensional swirl intensity:yUV/(DU ₀ ² ))     

Characterization of micellar structure dynamics for a drag-reducing surfactant solution under shear: normal stress studies and flow geometry effects

Some surfactant solutions have been observed to exhibit a strong drag reduction behavior in turbulent flow. This effect is generally believed to result from the formation of large cylindrical micelles or micellar structures. To characterize and understand better these fluids, we have studied the transient rheological properties of an efficient drag-reducing aqueous solution: tris (2-hydroxyethyl) tallowalkyl ammonium acetate (TTAA) with added sodium salicylate (NaSal) as counter ion. For a 5/5 mM equimolar TTAA/NaSal solution, there is no measurable first normal stress difference (N ₁) immediately after the inception of shear, but N ₁ begins to increase after a well-defined induction time — presumably as shear-induced structures (SIS) are formed — and it finally reaches a fluctuating plateau region where its average value is two orders of magnitude larger than that of the shear stress. The SIS buildup times obtained by first normal stress measurements were approximately inversely proportional to the shear rate, which is consistent with a kinetic process during which individual micelles are incorporated through shear into large micellar structures. The SIS buildup after a strong preshear and the relaxation processes after flow cessation were also studied and quantified with first normal stress difference measurements. The SIS buildup times and final state were also found to be highly dependent on flow geometry. With an increase in gap between parallel plates, for example, the SIS buildup times decreased, whereas the plateau viscosity increased.     

Surfactant systems for drag reduction: Physico-chemical properties and rheological behaviour

The first part of the work presents an overview of the physical chemistry of surfactants which in aqueous solutions reduce the frictional loss in turbulent pipe flow. It is shown that these surfactants form rodlike micelles above a characteristic concentraionc _t. The experimental evidence for rodlike micelles are reviewed and the prerequisites that the surfactant system must fulfill in order to form rodlike micelles are given. It is demonstrated by electrical conductivity measurements that the critical concentration for the formation of spherical micelles shows little temperature dependence, whereasc _t increases very rapidly with temperature. The length of the rodlike micelles, as determined by electric birefringence, decreases with rising temperature and increases with rising surfactant concentration. The dynamic processes in these micellar systems at rest and the influence of additives such as electrolytes and short chain alcohols are discussed.In the second part, the rheological behaviour of these surfactant solutions under laminar and turbulent flow conditions are investigated. Viscosity measurements in laminar pipe and Couette flow show the build-up of a shear induced viscoelastic state, SIS, from normal Newtonian fluid flow. A complete alignment of the rodlike micelles in the flow direction in the SIS was verified by flow birefringence. In turbulent pipe flow, drag reduction occurs in these surfactant systems as soon as rodlike micelles are present in the solution. The extent and type of drag reduction, i.e. the shape of the friction factor versus Reynolds number curve, depends directly on the size, number and surface charge of the rodlike micelles. The friction factor curve of each surfactant investigated changes in the same characteristic way as a function of temperature. For each surfactant, independent of concentration, an upper absolute temperature limit,T _L, for drag reduction exists which is caused by the micellar dynamics.T _L is influenced by the hydrophobic chain length and the counter-ion of the surfactant system. A first attempt is made to explain the drag reduction of surfactants by combining the results of these rheological measurements with the physico-chemical properties of the micellar systems.     

Shear stability studies on polymer-polymer and polymer-fibre mixtures

The shear stability of drag reducing polymer-polymer and polymer-fibre mixtures has been studied at a Reynolds number of 14,000 using a turbulent flow rheometer. The ratio of the drag reduction at a particular pass number to the initial drag reduction has been determined for the mixtures at various pass numbers and compositions in order to determine the effect of composition on the shear stability of the mixtures.It has been found in both cases that when there is a drastic difference in the shear stabilities of the constituents of the mixtures, the incorporation of a small amount of the less shear stable drag reducing agent reduces the shear stability drastically. On the other hand, when the shear stability of the constituents are of the same order, there is only a proportional change in the shear stability of the mixtures on addition of one component to the other. A correlation between the decay coefficient of the mixture (R _M ), the decay coefficients of the constituents (R ₁ andR ₂ ) and the weight fractions of the mixture components (W ₁ andW ₂) is suggested. An efficacious method for preparing asbestos fibre stock suspensions is also described.     

Aerodynamic characteristics of a square cylinder with a rod in a staggered arrangement

The aerodynamic characteristics of a square cylinder with an upstream rod in a staggered arrangement were examined. The pressure measurement was conducted in a wind tunnel at a Reynolds number of Re_D=82,000 (based on the width of the square cylinder) and the flow visualization was carried out in a water tunnel with the hydrogen bubble technique at Re_D=5,200. When the rod and the square cylinder were in tandem, the reduction of drag was mainly caused by the increase of the rear suction pressure. When the staggered angle was introduced, the shield and disturbance effect of the rod on the square cylinder diminished, which results in the increase of the cylinder drag. The side force induced by the staggered angle is small (the maximum value is 20% of the drag of the isolate square cylinder). There were six different flow modes with various staggered angles and spacing ratios, and the corresponding flow patterns are presented in present paper.     

Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium

This investigation is concerned with the plane strain deformation of an infinite slab, containing a circular cavity, within the theory of finite elastostatics for a particular homogeneous isotropic compressible material, the so-called Blatz-Ko material. The body is subjected to uniform pressure, either internal or external. Exact closed-form solutions for the axisymmetric deformation and stress fields are obtained. In the case of internal pressure, it is found that the applied pressure may not exceed a certain maximum value p _max. At a value of pressure p _e (<p _max), the governing equations lose ellipticity at the cavity wall. For greater values of pressure this solution remains smooth, though involving both elliptic and non-elliptic regions. Non-existence of axisymmetric solutions with discontinuous strain fields is established. The possibility of bifurication into a surface mode is considered and it is shown that this occurs at a value of pressure slightly smaller than p _e. Such surface wrinkling leads to a periodic distribution of points of stress concentration, from which shear bands may initiate.This work was supported by the U.S. Army Research Office under Grant DAAG29-83-K-0145 (R.A. & C.O.H.) and by the U.S. National Science Foundation under Grant MEA 78-26071 (C.O.H.).     

Direct drag measurements in a turbulent flat-plate boundary layer with turbulence manipulators

The effect of turbulence manipulators on the turbulent boundary layer above a flat plate has been investigated. These turbulence manipulators are often referred to as Large Eddy Break Up (LEBU) devices. The basic idea is that thin blades or airfoils are inserted into the turbulent flow in order to reduce the fluctuating vertical velocity component v above the flat plate. In this way, the turbulent momentum transfer and with it the wall shear stress downstream of the manipulator should be decreased. In our experiments, for comparison, a merely drag-producing wire also was inserted into the boundary layer.In particular, the trade-off between the drag of the turbulence manipulator and the drag reduction due to the shear-stress reduction on the flat plate downstream of the manipulator has been considered. The measurements were carried out with very accurate force balances for both the manipulator drag and the shear stress on the flat plate. As it turns out, no net drag reduction is found for a fairly large set of configurations. A single thin blade as a manipulator performed best, i.e., it was closest to break-even. However, a further improvement is unlikely, because the device drag of the thin blade elements used here has already been reduced to only that due to laminar skin friction, and is thus the minimum possible drag. Airfoils performed slightly worse, because their device drag was higher. A purely drag-producing wire device performed disastrously. The wire device, which consisted of a wire with another thin wire wound around it to suppress coherent vortex shedding and vibration, was designed to have (and did have) the same drag as the airfoil manipulator with which it was compared. The comparison showed that airfoil and blade manipulators recovered 75–90% of their device drag through a shear-stress reduction downstream, whereas the wire device recovered only about 25–30% of its device drag.Conventional LEBU manipulators with airfoils or thin blades produce between 0.25% and 1% net drag increase, whereas the wire device (with equal device drag) produces as much as 4% net drag increase. These data are valid for the specific plate length of our experiments, which was long enough in downstream extent to realize the full effect of the LEBU manipulators. Turbulence manipulators do indeed decrease the turbulent momentum exchange in the boundary layer by rectifying the turbulent fluctuations. This generates a significant shear-stress reduction downstream, which is much more than just the effect of the wake of the manipulator. However, the device drag of the manipulator cannot be reduced without simultaneously reducing the skin friction reduction. Thus, the manipulator's device drag exceeds, or at best cancels, the drag reduction achieved by the shear-stress reduction downstream. A critical survey of previous investigations shows that the suggestion that turbulence manipulators may produce net drag reduction is also not supported by the available previous drag force measurements. The issue had been stirred up by less conclusive measurements based on local velocity data, i.e., data collected using the so-called momentum balance technique.List of symbols b lateral breadth of test plate - c chord length of turbulence manipulator - d diameter of wire manipulator - e distance of the elastic center from the leading edge of the manipulator airfoil - h height of manipulator above test plate - q dynamic pressure of the potential flow above the test plate - s spacing of turbulence manipulator elements - t thickness of turbulence manipulator elements - u,v,w fluctuating velocities in downstream, platenormal, and lateral directions - x distance from the leading edge of the test plate in the downstream direction - x ₀ location of the trailing edge of the first manipulator - z distance from test plate center in the lateral direction - C _D drag coefficient - C _L lift coefficient - D _m drag of manipulated plate including device drag and shear stress, calculated from manipulator location to downstream location - D ₀ drag of unmanipulated plate boundary layer, consisting of the shear stress calculated from manipulator location to downstream location - F drag force - F ₀ total skin friction force, measured over a distance from 0.4 m upstream of manipulator to 6.35 m downstream of manipulator, measured without turbulence manipulator - F _LEBU device drag force of the LEBU, i.e., the turbulence manipulator - F _m total drag force of manipulated plate, consisting of - F _LEBU and skin friction force, measured over a distance from 0.4 m upstream of manipulator to 6.35 m downstream - F _cf skin friction force as measured by the floating element balance, manipulated case - F _cfo skin friction force, as measured by the floating element balance, unmanipulated case - F _cf skin friction saving, defined as F _cf = F _cf – F _cfo - F _cf cumulative skin friction savings, i.e., the sum of the skin friction savings F _cf , added up from the location of the manipulator to the downstream location , as shown in Fig. 11. In Fig. 13 the cumulative skin friction savings are summarized up to their asymptotic value, reached at 200 - Re _c Reynolds number of the manipulator elements, calculated with the chord length c and the local velocity in the boundary layer - Re ₀ Reynolds number at the location x ₀ of the manipulator, calculated with the momentum thickness of the boundary layer and the mean flow velocity U - U mean flow velocity in the potential regime of the wind tunnel test section - angle of attack of the manipulator airfoils - ₀ boundary layer thickness at the location x ₀ of the manipulator - dimensionless distance from the manipulator in the downstream direction, defined as - density of the air - ₀ local skin friction shear stress, unmanipulated case - _{0 Average} skin friction shear stress, average value over the lateral span (b = 2 m) of the test plate, unmanipulated case - _m local skin friction shear stress, manipulated case - momentum thickness of the undisturbed turbulent boundary layer at the location x ₀ The authors would like to thank Prof. H. H. Fernholz for his scientific and administrative support. The hardware for the experiments was designed and built by C. Daase, W. Hage and R. Makris. Funding for the project was provided by the Deutsche Forschungsgemeinschaft and is gratefully acknowledged.     

One Equation Model for Turbulent Channel Flow with Second Order Viscoelastic Corrections

A modified second order viscoelastic constitutive equation is used to derive a k–l type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by P _w , for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori P _w could be either positive or negative. When P _w is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, P _w  < 0 leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the k–l formulation rather than from the closure for P _w . An alternative closure for P _w , mimicking the viscoelastic stress work predicted by DNS using the Finitely Extensible Nonlinear Elastic-Peterlin fluid model, which is mostly characterized by P _w  > 0 but has also a small region of negative P _w in the buffer layer, was also successfully tested. This second model for P _w leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.     

Interaction between the near wake and the cross-section variation of a circular cylinder in uniform flow

The flow around a circular cylinder with a cross-section variation is experimentally investigated. Particle Image Velocimetry (PIV) is used to scrutinize the interaction of the cylinder’s wall with its near wake. The Reynolds number based on the cylinder’s diameter and freestream velocity is 80 × 10³, corresponding to the upper subcritical flow regime. At a forcing Strouhal number of St _f = 0.02, the maximum vorticity level around the cylinder is reduced by more than 50% as compared to its uncontrolled value. The topology of the bulk flow confined between the primary vortical structure and the cylinder surface is modified resulting in substantial drag reduction.     

On the skin friction coefficient in viscoelastic wall-bounded flows

Analysis of the skin friction coefficient for wall bounded viscoelastic flows is performed by utilizing available direct numerical simulation (DNS) results for viscoelastic turbulent channel flow. The Oldroyd-B, FENE-P and Giesekus constitutive models are used. First, we analyze the friction coefficient in viscous, viscoelastic and inertial stress contributions, as these arise from suitable momentum balances, for the flow in channels and pipes. Following Fukagata et al. (Phys. Fluids, 14, p. L73, 2002) and Yu et al. (Int. J. Heat. Fluid Flow, 25, p. 961, 2004) these three contributions are evaluated averaging available numerical results, and presented for selected values of flow and rheological parameters. Second, based on DNS results, we develop a universal function for the relative drag reduction as a function of the friction Weissenberg number. This leads to a closed-form approximate expression for the inverse of the square root of the skin friction coefficient for viscoelastic turbulent pipe flow as a function of the friction Reynolds number involving two primary material parameters, and a secondary one which also depends on the flow. The primary parameters are the zero shear-rate elasticity number, El₀, and the limiting value for the drag reduction at high Weissenberg number, LDR, while the secondary one is the relative wall viscosity, μ_w. The predictions reproduce both types A and B of drag reduction, as first introduced by Virk (Nature, 253, p. 109, 1975), corresponding to partially and fully extended polymer molecules, respectively. Comparison of the results for the skin friction coefficient against experimental data shows good agreement for low and moderate drag reduction which is the region covered by the simulations.     

Turbulent channel flow of dilute polymeric solutions: Drag reduction scaling and an eddy viscosity model

Direct numerical simulation of viscoelastic turbulent channel flows up to the maximum drag reduction (MDR) limit has been performed. The simulation results in turn have been used to develop relationships between the flow and fluid rheological parameters, i.e. maximum chain extensibility, Reynolds number, Re_τ, and Weissenberg number, We_τ and percent drag reduction (%DR) as well as the slope increment of the mean velocity profile. Moreover, based on the trends observed in the mean velocity profile and the overall momentum balance three different regimes of drag reduction (DR), namely, low drag reduction (LDR; 0 ≤ %DR ≤ 20), high drag reduction (HDR; 20 ≤ %DR ≤ 52) and MDR (52 ≤ %DR ≤ 74) have been identified and mathematical expressions for the eddy viscosity in these regimes are presented. It is found that both in LDR and HDR regimes the eddy viscosity varies with the distance from the channel wall. However, in the MDR regime the ratio of the eddy viscosity to the Newtonian one tends to a very small value around 0.1 within the channel. Based on these expressions a procedure that relies on the DNS predictions of the budgets of momentum and viscoelastic shear stress is developed for evaluating the mean velocity profile.     

Fatigue gage utilizing slip-initiation phenomenon in electroplated copper foil

Fundamental experiments are carried out to examine the parameter that dominates the slip-band initiation in electroplated copper foil under the condition where the mean stress as well as the stress amplitude varies. In the case of constant-amplitude stressing, the relation between the critical stress for the slip-band initiation σ _p and the number of cyclesN is well represented by σ _p ^α N=C. In other words, the slip bands appear when the total hysteresis energy applied to the copper foil attains a critical value. In the case of variable stresses, the range-pair mainly dominates the occurrence of the slip bands, and Miner's linear cumulative damage rule holds for the accumulation of the fatigue damage for the slip-band initiation. Accordingly, the parameter (Σσ _i ^α n _i/Σn _i)^1/α is equivalent to the critical stress σ _p under constant amplitude stressing, where σ _i andn _i are the stress amplitude and the number of cycles counted by the range-pair method, respectively, and α is the exponent of the σ _p -N relation. Based on these results, the applicability of the copper foil to the fatigue gage that accumulates and indicates a load experience is discussed.