Experiments in Fluids 36 (2004) 685–700 DOI 10.1007/s00348-003-0745-3
Influence of wavy structured surfaces and large scale polymer structures on drag reduction M. Vlachogiannis, T. J. Hanratty
685 molecular weights. Only a few (Lindgren and Hoot 1968; Virk 1971; Bewersdorff and Thiel 1993) have considered a roughened wall. This paper presents measurements of the influence of a copolymer of polyacrylamide and sodium acrylate on flow over a surface that consists of a train of six hundred sinusoidal waves with an amplitude of 0.25 mm and a wavelength of 5 mm. The structured surface constituted the bottom wall of a rectangular flow channel. The top wall was flat, so comparisons of the behaviors with smooth and wavy surfaces could be made in the same experiment. The study differs from previous researches in that turbulence measurements are obtained and a welldefined and well-documented roughness is used. The experiments involved the injection of a polymer solution with a high concentration, ci, through slots in the wall. An important aspect of this study is the use of a fluorescence imaging technique (Vlachogiannis and Bontozoglou 2001, 2002) to find out whether the injected polymer solution contains sheets or large filaments of entangled or gelatinous polymer molecules and how the presence of these structures affects drag reduction. To the knowledge of the authors, this type of study did not accompany previous research that used wall slots. There is a similarity between our experiments in which large polymer structures were present and ‘heterogeneous’ drag reduction, first studied by Vleggaar and Tels (1973). In this and later works (Hoyt and Sellin 1991; Bewersdorff 1982; Smith and Tiederman 1991) a highly concentrated polymer solution (ci‡4000 ppm) was found to move through the fluid as a thread when it was injected from a small tube. Discussions of the effect of these threads on drag reduction are presented in papers by Bewersdorff 1 et al. (1979, 1982, 1984), Usui et al. (1988) and Smith and Introduction Tiederman (1991). The experiments described in this paA large number of studies have been carried out which show that the drag of a turbulent fluid on a smooth surface per focus on structures formed by injection through a wall slot. They also differ from studies of heterogeneous drag can be reduced by the addition of polymers with high reduction, cited above, in that the concentration of the injected solution was smaller and polymer structures already existed in the injected solution. However, they Received: 22 October 2002/Accepted: 20 September 2003 support the claims in several previous works that the Published online: 13 February 2004 presence of large polymer structures can be beneficial. Springer-Verlag 2004 Studies of turbulence in drag reducing solutions were advanced by the works of Pinho and Whitelaw (1990), Wei M. Vlachogiannis, T. J. Hanratty (&) and Willmarth (1992), Harder and Tiederman (1991), and Department of Chemical Engineering, University of Illinois, Walker and Tiederman (1989, 1990). Their use of laser Urbana, IL 61801, USA Doppler velocimetry avoided problems that arise when E-mail:
[email protected] Sponsored by the Defense Advanced Research Projects Agency, polymer solutions flowed past a probe. In the experiments Advanced Technology Office, Friction Drag Reduction Program, of Wei and Willmarth (1992) and Harder and Tiederman ARPA order No: MDA972-01-C-0029. (1991) polymers were injected into the flow and Abstract Drag reduction was studied for turbulent flow over a structured wall that contained 600 sinusoidal waves with a wavelength of 5 mm and an amplitude of 0.25 mm. A concentrated solution of a co-polymer of polyacrylamide and sodium acrylate was injected into the flow through wall slots. Laser Doppler velocimetry was used to measure turbulence. A fluorescence technique was developed that enabled us to demonstrate the existence, under certain circumstances, of large gelatinous structures in the injected polymer solution and in the flow channel. At maximum drag reduction, the Reynolds shear stress was zero and the velocity field was the same as found for a smooth surface. Larger drag reductions could be realized for a wavy wall because the initial drag was larger. The influences of polymers on the turbulent fields are similar for smooth and wavy boundaries. These results are of interest since the interaction with the wall can be quite different for water flow over smooth and wavy boundaries (which are characterized as being completely rough). An important effect of polymers is a decreasing relative importance of high frequency fluctuations with increasing drag reduction that is characterized by a cut-off frequency. This cut-off is the same for smooth and wavy walls at maximum drag reduction. The sensitivity of drag reduction to the method of preparing and delivering the polymer solution suggests that aggregation of polymers could be playing an important role for the system that was studied. For example, drag reduction was enhanced when large polymer structures are present.
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measurements were stopped when the polymer concentration in the holding tank built to an unacceptable level. The flow was recycled through a large pump in order to degrade the polymers to a point where the pressure drop was the same as would be measured for water flow. The experiments were then resumed. Wei and Willmarth introduced solutions of Polyox with concentrations of 500 to 2000 ppm. The mixed polymer concentration in the test section was 10 ppm and the amount of drag reduction was about 30 percent. They found that the energy of the normal velocity fluctuations is greatly reduced and that there is a redistribution of energy from high frequencies to low frequencies. Warholic et al. (1999b) used this approach and a solution of a co-polymer of polyacrylamide and sodium acrylate to explore a wider range of conditions for flow over a flat surface. Of particular interest is their finding that the Reynolds shear stress was zero, at all locations in the channel when maximum drag reduction was realized. The work described in this paper used the same technique and the same flow loop that was employed by Warholic et al. (1999a). When a fluid flows turbulently over a smooth wall, it is sustained by elongated flow-oriented vortices located near the wall. (See, for example, the collection of papers edited by Panton [1997]). Laboratory studies of flow over a wavy boundary, with large enough amplitude, a, that separation occurs, reveal that turbulence is generated in shear layers that form behind the crest. (Buckles et al. 1984; Hudson et al. 1996). The study of Hudson was carried out with a train of waves with a wave height, 2a, to wavelength ratio 0.1 that was the bottom wall of a rectangular channel. The Reynolds number based on the half-height of the channel, h, and the average velocity was Re=3380. The dimensionless height of the waves based on the friction velocity and the kinematic viscosity of the liquid was 2a+=73.5. Hudson et al., concluded that flow near the boundary is ‘‘fundamentally different from what is found for flow over a flat wall in that production is not associated with the floworiented vortices described by Brooke and Hanratty (1993) and by Kline and Robinson (1989).’’ Detailed descriptions of these separated flows have been obtained in direct numerical simulations by Cherukat et al. (1998) and by Na (Nakagawa et al. 2003a) for a wavy wall with a wave height to wavelength ratio of 0.1, Re=3400, and a+=69.6. Not surprisingly, vortical structures close to the wall were different from what is observed for a flat wall. Na showed contours of the streamwise fluctuating velocity in an x-z plane located just above the wave crest. No evidence of a streaky structure was observed. Cherukat et al. (1981) also observed that ‘‘coherent streak-like structures..are not found anywhere above the wave.’’ Na showed that floworiented vortical structures, identified by the scheme of Yong et al. (1990), form on the upstream side of the wave and disappear as they progress over the crest. They, therefore, appear to be the result of a centrifugal instability. Studies of water flow over the wavy surface used in this research (2a/k=0.1, k=5 mm) were carried out by Nakagawa and Hanratty (2003a, b) with LDV techniques for Re=46,000, Re=11,000 and Re=6000. Comparison PIV
measurements were also made by Nakagawa and Hanratty (2001) at Re=46,000. The rationale in these experiments was that the dimensionless wave amplitude, a+, at Re=46,000 was approximately the same as for the DNS studies at Re=3400. Therefore, the flow pattern observed very close to wall with the DNS might be similar to what exists for laboratory measurements at Re=46,000, since 2a/k and a+ are the same. Nakagawa and Hanratty (2003b) carried out visual studies for which dye was injected at the crest. The dye streamer separated from the crest both for Re=46,000 and Re=11,000 but the mixing was much more vigorous for Re=46,000. The streamer followed the wave contour for Re=6000. The surfaces studied by Nakagawa were characterized by equivalent sand roughnesses of kþ s ¼ 104; 22:4; 6:9. They were described as being completely rough, as possessing an intermediate roughness and as being hydraulically smooth. The experiments described in this paper were for Re=48,000, Re=20,700, Re=11,000 and Re=6000 for which kþ s ¼ 104; 46:8; 22:4; 3:4 lower flow. The results are presented in separate sections for high and low Reynolds numbers. The experiments at the two largest Reynolds numbers are in or close to the ‘‘fully rough’’ regime. Those for the two smallest Reynolds numbers are in the intermediate and hydraulically smooth regimes. Another reason for this method of presentation is that drag reduction was not realized (for the range of polymer concentrations used in this research) at the two lowest Reynolds numbers unless large scale polymer structures were present in the solution. The contributions of this paper are (1) a study of the influence of a drag reducing polymer on flow over a welldefined and well-documented roughened wall, (2) the demonstration that the delivery of polymers through wall slots could result in the formation of large polymer structures, and (3) the further documentation that the presence of these structures can enhance drag reduction. However, some of the results could impact theoretical understanding of drag reduction. The interaction of a turbulent fluid with a wavy wall can be quite different from the interaction with a smooth wall; yet, there is a similarity in the turbulence measurements in drag reducing flows. The effect of polymers on spectral functions is of particular interest since attention has been given to the notion that the changes in turbulence with the addition of polymers is due to the elastic behavior of stretched or partiallystretched polymers (Tabor and de Gennes 1986; de Gennes 1990). This is manifested by a decrease in the relative importance of high frequency (or low wavenumber) fluctuations and by the existence of a cut-off frequency. The documentation of these effects in this paper is, therefore, useful.
2 Description of the experiments 2.1 The flow facility The flow facility, which is depicted in Fig. 1, has been described in several previous papers (Warholic et al. 1999; Niederschulte et al. 1990; Nakagawa et al. 2001). The
polymers after this treatment was determined to be completely destroyed, by comparing the pressure drop of the degraded polymer solution and the pressure drop obtained with a water flow. Also, agreement was realized between measurements made at the beginning and the end of an injection period.
Fig. 1. Diagram of the flow facility
closed loop water channel has an aspect ratio of 12:1 and a height, 2 h, of 50 mm. The temperature was maintained at 25.0±0.8C, by using the reservoir’s cooling coils. The rectangular channel had a length of 11 m. The test section consisted of the final 3 m. As depicted in Fig. 1, three sets of optical grade windows were located on both sides of the channel in order to allow the insertion of laser light. Measurements were carried out at the third set of windows. Figure 2 is a sketch of the test section. The top wall (referenced as 1) was flat and the removable bottom wall (referenced as 2), contained six hundred sinusoidal waves that extended over the whole width of the channel.
2.2 Polymer solution A master polymer solution of Percol 727, a copolymer of polyacrylamide and sodium acrylate, was prepared over a period of 12–16 hours before each experiment. Details of the mixing procedure and the design of the system are given in the paper by Warholic et al. (1999). The concentrated polymer solution could be injected without operating a pump, by using a mixing tank that was located 10 m above the flow loop. The polymer was admitted to the system 8 m upstream of the test section through two inclined slots in the bottom of the channel. (Tests which used two slots in the top wall or all four slots gave the same results.) The concentration of the injected solution, ci, varied from 100 to 2500 ppm. The concentration of the polymer in the test section, cm, was adjusted by changing the flow rate of the injected solution. The operational procedure was similar to that developed in the laboratory of Tiederman. The liquid was circulated with a 5 hp centrifugal pump, which did not completely degrade the polymers in a single pass. The injected polymers were diluted to such an extent in the piping and holding tank that the concentration of undegraded polymers at the inlet did not change appreciably during the injection period. In order to avoid the buildup of undegraded polymers, the injection was stopped after 7–10 min and the liquid was circulated through a 60 hp pump for more than 15 min. The effectiveness of the
2.3 Velocity measurements The velocity field was measured with the three-beam, two color LDV system used by Gu¨nther et al. (1998) and Warholic et al. (1999). The accuracy of the measurements is discussed in these papers. A beam-expanding module (TSI model 9832) was used with a fiber optic probe (TSI model 9253) to produce a measuring volume, which had a diameter of 45 lm and a length in the spanwise direction of 0.44 mm. The water and the injected polymer solution were seeded with polystyrene spheres with an average diameter of 500 nm and a specific gravity of 1.005. These were manufactured by a process described by Goodwin et al. (1973). The receiving optics (TSI model 9176), which were located at the opposite side of the channel, gathered the scattered light from the particles. Both the transmitting and receiving optics were located on an I-beam, which was mounted on a transversing mechanism. The latter was used in order to change the position of the measuring volume and to assure that all of the optics were moved together. The output signals from the photo-multipliers (TSI model 9162) were analyzed with a two-channel correlation processor (TSI model IFA655). We acquired four samples (files) of 5000 points, with a maximum data rate of 1000 to 3000 Hz, for each y-location. Very small differences were noted between a 5000 point file and the sum of the four files so this assured that 20000 points provided converged statistics. The position in the spanwise direction was kept constant at 210 mm away from the sidewall, where the transmitting optics were located. Previous measurements of the mean streamwise velocity and the root-mean-square of the fluctuating velocity at different locations indicate that the results were not affected by the proximity of the sidewall. The signals were corrected for the presence of white noise, which was estimated from measurements of the frequency spectra. The procedure used is described by Gu¨nther et al. (1998) and by Warholic et al. (1999). The noise plateau was clearly seen in the spectra measured with the polymer solutions, so that the method of correction was easily implemented. 2.4 Measurements of the pressure drop Measurements of the pressure gradient were obtained with a Validyne reluctance pressure transducer (model DP103). The pressure range can be varied by changing the diaphragm, so that measurements of high-accuracy were obtained for all the Reynolds numbers. The pressure taps were located in the top flat wall; they had a separation distance of 1.01 m. As depicted in Fig. 2, the location of zero Reynolds shear stress may not be at the center of the channel
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very long length of channel. This was not possible in the system used in this study. However, wall shear stresses determined from pressure drop measurements and by extrapolating Reynolds shear stress measured in the central regions of the channel were the same for most of the runs. This suggests that the flow in the test section is approximated by a fully-developed condition at the locations where measurements were made. The possibility has been mentioned that the turbulence observed at maximum drag reduction are remnants of upstream turbulence events. This would mean that a fully developed flow at Fig. 2. Sketch of the test section and schematic representation of maximum drag reduction is laminar. But this has never the measurement of sw,1 and sw,2 been mentioned in the literature. because of the difference in the roughnesses of the top and bottom walls. The distortion of the profile, for the same wavy wall and for a range of Reynolds numbers, is described in detail in a paper by Nakagawa et al. (2003a). A force balance across the test section was used in order to determine the average wall shear stresses: DP TopwallðsmoothÞ: and sw1 ¼ h1 ð1Þ Dx measured DP ð2Þ BottomwallðwavyÞ: and sw2 ¼ h2 Dx measured where hi is the distance between the position of zero stress and the wall (i=1 for the top wall and i=2 for the bottom wall). The wall shear stress was also obtained from the extrapolation of measurements of the Reynolds shear stress in the central regions of the channel to the wall. A very good agreement with measurements obtained from pressure drops was found in studies with water and with polymer solutions for all runs except those close to maximum drag reduction. The amount of drag reduction (DR) is defined in terms of the wall shear stresses for the polymer solution and for the Newtonian fluid, sw2;water sw2;polymer %DR2 ¼ 100 ð3Þ sw2;water The use of a channel with different walls has the advantage that drag reductions with flat and structured surfaces could be determined simultaneously. The results for the flat surface were obtained from Eq. 3 with sw2 replaced by sw1. The total shear stress consists of the sum of the Reynolds shear stress, the viscous stress and the polymer stress (Warholic et al. 1999). Except for maximum drag reduction or conditions close to it, the viscous stress and the polymer stress are negligible in a region around the center of the channel. The determination of the location of zero shear stress was also made by using the observation of Nakagawa and Hanratty (2003) that it is the same as the location of the maximum in the velocity profile. For maximum drag reductions the location of zero shear stress is at the center of the channel. The usual method for determining whether a flow is fully-developed is to measure the pressure profile over a
3 Visual studies of aggregated polymers In order to visualize the flow of the injected polymer solutions, a fluorescence imaging method, described by Vlachogiannis et al. (2001), was used. The injected polymer solution was doped with 100–300 ppm, of a sodium salt of fluorescein, C20H10O5Na2. The resulting solution fluoresces under ultraviolet illumination and emits light in a specific wavelength range of 520–575 nm. The ultraviolet source consisted of four high-intensity lamps (Philips, TL20/05), which were located above the top wall of the channel. The polymer particles have an average diameter of 550 nm, which is at least ten times greater than the dye particles. The powder of the polymer was mechanically mixed with the dye before adding it to water. The color of the mixed powder changed from white to a yellow–green color. Since the dye adhered to the polymer particles, by using sieves of different sizes we assured that only dyed polymer particles, and not pure dye grains, were mixed with the water. The mixing of the dyed powder with the water produced a green polymer solution. The concentration of the dye in the final polymer solution varied between 100 to 2500 ppm. A spectrometer (Perkin-Elmer Lambda 40) was used to examine the resulting polymer solution. The intensity of the light was higher for a dyed polymer solution than for dyed water because of the adherence of dye particles to the polymer. This behavior provided a method that used measurements of spatial variations of light intensity to determine how the polymers distribute in the fluid. A CCD video camera (shutter speed, 1/2000 s) and a digital video camera were employed to acquire flow images in the test section. The light intensity in the images varies linearly with both the flow of the injected polymer solution and the concentration of dispersed dye in the flowing water. The use of tap water, doped with the same amount of dye as was contained in the injected solution, showed that the dye was uniformly mixed with the circulated liquid. This measurement was the baseline for the image processing, which used commercial (CorelDraw) as well as in-house software (built in MatLab). The digitization noise was eliminated by applying appropriate convolution filters to the incoming images. A representative example of the fluorescence imaging method is depicted in the image series of Fig. 3. The
distance from the side or top wall were kept constant in the different experiments.
4 Homogeneous and heterogeneous drag reduction Two methods were used to introduce the polymers into the mixing tank. One involved pre-wetting of the polymer particles with a venturi injector (Warholic et al. 1999). Injected concentrations of ci £ 500 ppm could be obtained in this way. Solutions for which the fluorescence imaging technique did not reveal any large structures are called ‘homogeneous’. This was the case for all experiments that used the venturi mixing device. The ability to observe structures with the fluorescence imaging method depends on the light that is emitted from the dyed polymer solution. Small-scale structures of the order of 200 lm or less, are not clearly seen since the emitted light intensity is not enough to distinguish these structures from the surrounding fluid. It is possible (or, perhaps, likely) that smaller aggregates exist in solutions that are called ‘homogeneous’. For situations in which maximum drag reduction existed, thin threadlike structures were visible to the eye under intense illumination. These were not picked up by imaging techniques. The other method of mixing involved the sprinkling of polymer powder that was sieved on the top of the liquid in the mixing tank. The particles had a size range of 400 lm to 800 lm. Large structures were observed if ci‡500 ppm. These solutions are called ‘heterogeneous’. It is to be noted that either homogeneous or heterogeneous solutions could be obtained at ci=500 ppm depending on whether the venturi device was used. For ci>500 ppm mixing was done only by sprinkling the polymer particles. Sections 3 and 5.3 present observations of polymer structures in the channel. An experiment was performed in which the concentrated solution was allowed to flow down an inclined plane rather than to flow through the injection slots in the flow channel. This produced a thin flowing film with a free surface. The polymer damped the waves that would be present for a water flow. Therefore, filaments were easily observed in the film flow under conditions that a heterogeneous solution would exist in the flow channel. This indicates that filaments were present in the solutions Fig. 3. The implementation of the fluorescence imaging method that were injected, when large polymer structures were in polymer drag reduction studies at Re=11,000. a Original image observed in the flow channel. acquired by the color digital camera. b The light reflection from the wall in the image plane. c The baseline image obtained from 5 dyed water experiments. d The resulting image after the Results implementation of the image processing analysis
original image is shown in Fig. 3a, with the wavy wall clearly seen at the bottom. Some of the dye in the injected polymer escaped from the filaments to produce a greenish tinge to all of the fluid. The light that is reflected because of the wavy wall, Fig. 3b, was subtracted from the final images. The baseline images, Fig. 3c, were used in order to subtract the light that is transmitted from the scattered dye in the circulated liquid. Thus, only the light that is associated with the dyed polymers remained in the final grayscale image, as illustrated in Fig. 3d. The concentration of dye, the intensity of the ultraviolet source and the
5.1 Measurement conditions Measurements with laser Doppler velocimetry were carried out for four Reynolds numbers, Re=6000, 11,000, 20,700 and 48,000, based on the bulk velocity, the half-height of the channel and the viscosity of the water. The parameters are the concentration of the injected polymer solution, ci, the concentration of the polymer in the test section (so-called mixed concentration), cm, and the Reynolds number. These are listed in Table 1 for the different experiments. More than sixty pressure drop measurements were obtained in order to establish the dependence of the
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Table 1. Summary of velocity
measurements. Homogeneous drag-reduction: Re=20,700, 48,000. Heterogeneous dragreduction: Re=6000, 11,000
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Re
Cm
Ci
6000 6000 11,000 11,000 11,000 11,000 11,000 20,700 20,700 20,700 20,700 20,700 20,700 48,000 48,000 48,000 48,000 48,000 48,000
– 35.4 – 5.4 8.6 15 20 – 0.28 1 0.62 1.5 7.7 – 1.05 5 2.9 10.2 14.5
– 500 – 1000 1000 1000 1000 – 100 100 500 500 500 – 500 500 500 500 500
drag reduction on the above-defined parameters. A range of drag reductions of 10 to 85% and a range of concentrations in the test section of 0.2 to 25 ppm were covered. Visual observations, which were performed for all the Reynolds numbers, indicated when sheets or filaments of polymer molecules were present. If these structures were not observed, drag reduction was not realized at Re=6000 and Re=11,000. Therefore, the work described in this paper is separated into three parts: Homogeneous drag reduction studies are described for high Reynolds numbers in section 5.2. The structures of heterogeneous polymer solutions are discussed for a range of Reynolds numbers in section 5.3. The turbulence properties of polymer solutions at low Reynolds numbers, when structures are present, is examined in section 5.4.
u*2 1.40 1.36 2.35 2.12 2.06 1.95 1.87 4.51 4.21 4.16 3.96 3.78 1.90 10.19 9.24 6.32 8.58 5.44 4.51
sw2
DP/Dx
ymax/h
%DR1
%DR2
1.95 1.84 5.52 4.47 4.25 3.81 3.48 20.36 17.72 17.29 15.67 14.30 3.61 103.98 85.26 39.93 73.5 29.50 20.32
0.74 0.70 1.96 1.63 1.58 1.46 1.35 6.92 6.09 5.95 5.45 4.99 1.42 32.74 29 15 25 11.5 8
1.04 1.04 1.11 1.08 1.06 1.03 1.01 1.16 1.15 1.14 1.13 1.13 1.00 1.25 1.157 1.048 1.15 1.01 1
– 5 – 15 16 20 25 – 11 13 19 25 76 – 0.45 41.82 14.19 53.62 67.41
– 6 – 19 23 31 37 – 13 15 23 30 82 – 18.00 61.60 29.31 71.63 80.46
2 k ¼ 8 u =Ub
ð4Þ
Here u* is the friction velocity and Ub is the bulk-averaged velocity. The maximum drag reduction was reached at cm=7.7 ppm. A comparison with data obtained by
5.2 Measurements for high Reynolds numbers—homogeneous solutions Results for Reynolds numbers Re=20,700 and Re=48,000, for which a venturi device was used in the mixing process, are presented in this section. The dimensionless sand roughness, kþ s was estimated by Nakagawa et al. (2003a) from the roughness function proposed by Nikuradse (Schlichting 1960) for a fully rough surface. It had values of 46.6 and 108, for water flows. The transition to a fully rough region occurs at kþ s ffi 70. The dependence of the drag reduction on the mixed concentration is depicted in Fig. 4a, for a Reynolds number of 20,700. The average wall shear stresses for the wavy and the flat wall were determined from pressure drop measurements by using Eqs. 1,2. The estimation of the location of zero Reynolds shear stress was established from LDV measurements. Increases in the mixed concentration are accompanied by increases in drag reduction for both flat and wavy walls. However, the percent drag reduction for the wavy wall is larger than for the flat wall. This is consistent with the observation that the friction factor for a water flow, k=0.0297, is larger for the Fig. 4. Percentage drag reduction as a function of the mixed wavy wall than for the flat wall, k=0.01917, where k is concentration for flat and wavy surfaces. a Re=20,700. defined as b Re=48,000
Warholic et al. (1999) shows that the maximum drag reduction occurs at a larger mixed concentration, cm=13 ppm, when both walls of the channel were flat. Noteworthy, is the observation of a drag reduction of 20% with a mixed concentration of only 0.25 ppm. Similar results are found for a Reynolds number of 48,000, as shown in Fig. 4b. However, a lower drag reduction of 12% was observed at cm=0.25 ppm. Also, the mixed concentration that is required for maximum drag reduction increased to cm=14.5 ppm when the Reynolds number was increased from 20,700 to 48,000. The variation of the average streamwise velocity with the distance from the averaged location of the wave surface, made dimensionless with the half-height of the channel, is depicted in Fig. 5 for maximum drag reduction. The profiles over the crest and the trough of the waves were averaged. Despite the difference in the roughnesses of the top and bottom wall, the velocity profile was symmetric for both Reynolds numbers, 20,700 and 48,000. Furthermore, if a scale of 160 to 200 cm/s, rather than 0 to 210 cm/s, were used in Fig. 5 one could see that the maximum is located at the center of the channel. Therefore, the mean flow is not dependent on the structure of the wall at maximum drag reduction. The profiles are more diffuse, close to the wall, at maximum drag reduction than is observed for water, so the shear rate at the wall, (dU/dy)wall, could be directly determined. For a Reynolds number of 20,700, the shear rate was 117.7 s)1; for Re=48,000, it was 582 s)1. The dimensionless mean velocities, U+, versus the distance from the wall, y+, normalized with the friction velocity and the viscosity of the polymer solution near the wall, are close to Virk’s profile for maximum drag reduction, only for y/h £ 0.5. Figure 6a presents selected measurements of the rootmean-square (RMS) of the streamwise velocity fluctuations, u¢, for Re=20,700 over a structured surface. The peaks in u¢ are displaced outward with increasing drag reduction. The peak values are roughly the same for drag reductions of 30% or less. However, they increase when normalized with the friction velocity, because of the
decrease in u* with increasing drag reduction. The magnitude of u¢ decreases in the outer flow with increasing drag reduction. A plateau region which starts at y/h>0.17 for water flow over a wavy wall and at y/h>0.2 for a flat wall becomes less prominent with increasing drag reduction. For a flow over a wavy wall at Re=48,000, the dimensional u¢ decreases, over the whole flow, and the peaks are displaced outward with increasing drag reduction, as depicted in Fig. 6b. For conditions close to maximum drag reduction, the magnitudes of u¢ are greatly decreased and broad maxima occur in the neighborhoods of y/h=0.1 for Re=48,000 and at 0.19 for Re=20,700. Measurements of the dimensional root-mean-square of the normal velocity fluctuations are plotted in Fig. 7. They decrease systematically with increasing drag reduction for Re=20,700 and for Re=48,000. The influence of wave-induced flows for Re=48,000 is clearly seen in Fig. 7b, by comparing the v¢ for y/h
