Experimental investigation of polymer diffusion in the drag-reduced turbulent channel flow of inhomogeneous solution

International Journal of Heat and Mass Transfer 77 (2014) 860–873

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Experimental investigation of polymer diffusion in the drag-reduced turbulent channel flow of inhomogeneous solution Zaiguo Fu a, Tomohiro Otsuki a, Masaaki Motozawa b, Taiki Kurosawa a, Bo Yu c, Yasuo Kawaguchi a,⇑ a

Department of Mechanical Engineering, Tokyo University of Science, Noda 278-8510, Japan Department of Mechanical Engineering, Shizuoka University, Hamamatsu 432-8561, Japan c National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, People’s Republic of China b

a r t i c l e

i n f o

Article history: Received 20 February 2014 Received in revised form 7 May 2014 Accepted 5 June 2014

Keywords: Inhomogeneous solution Planar laser-induced fluorescence Polymer diffusion Turbulent Schmidt number

a b s t r a c t Spatial polymer diffusion in the drag-reduced turbulent channel flow of an inhomogeneous polymer solution was investigated by simultaneously measuring velocity and concentration fields using particle imaging velocimetry and planar laser-induced fluorescence techniques. The polymer solution was dosed into the turbulent channel flow from the surface of one-side of the channel wall. The Reynolds number (based on channel height, bulk velocity and solvent viscosity) was set as 4.0  104 and the weight concentrations of dosed polymer solution were set to 25, 50 and 100 ppm. The measurements were obtained in the streamwise wall-normal (x–y) plane at three streamwise positions along the dosing wall. The detailed statistical analyses consisting of concentration distribution, turbulence modification, turbulent mass flux, and eddy diffusivities of momentum and of mass are presented. The results show that the polymer diffusion, which has a close relationship with the local polymer concentration and drag reduction in the drag-reduced turbulent channel flow, is suppressed due to the inhibited turbulence other than the diffusion of passive scalar in ordinary turbulence. Two characteristic regions exist in the near-wall region according to the diffusion characteristics and altered motions in the wall-normal direction. The wall-normal turbulent fluxes that control the transport of mass are reduced significantly in the near-wall region for the drag-reduced flow when compared with the case of dosing water. With the increase of local polymer concentration in the ‘‘effective position’’, the corresponding drag reduction rate (DR) increases. The turbulent Schmidt number (ScT), which represents the relative intensities of the eddy diffusivities of momentum and of mass, is also found to increase with increasing DR. The mean value of ScT for the drag-reduced flow can rise to 2.9, while it is 1.2 for the case of dosing water in the present measurements. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction It has been known for more than 60 years that the addition of some kinds of high-molecular-weight polymer solution to turbulent water flow can reduce the skin frictional drag as the Toms effect [1]. This drag-reducing effect can be applied to crude oil pipelines, district heating and cooling (DHC) systems and ship hulls to produce a major benefit in terms of energy conservation [2–4]. To ascertain the mechanism of drag reduction in wall turbulence, many researchers have concentrated on the complicated effects of polymer on turbulence in the drag-reduced flow of homogeneous polymer solutions in which the polymer was considered to be uniformly mixed. Meanwhile, the drag reduction effect ⇑ Corresponding author. Tel.: +81 04 7122 9589; fax: +81 04 7123 9814. E-mail address: [email protected] (Y. Kawaguchi). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.06.016 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

caused by inhomogeneous polymer solutions has gradually attracted more attention based on the increasing requirements of drag reduction for some external flows [5,6]. In this case, the polymer solution was added into the turbulent flow by injection as needed. Hence, the drag reduction effect was generated during the process of diffusion of the polymer additives [7,8]. At present, the most common and well-known method of polymer addition to achieve drag reduction for an external flow is to inject polymer solution into the bulk flow via a slot on the wall where the boundary layer forms [9–11]. On the basis of this slot-injection method, many researchers have investigated the turbulence characteristics modified by inhomogeneous polymer solutions using the particle imaging velocimetry (PIV) technique [9–14]. Some consensuses have been reached on some of the mechanisms including: the turbulent motion (e.g. Reynolds shear stress, bursting rate) is inhibited in the near-wall region, the mean

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Nomenclature C C0 Cf Cw c0 H h L Nk Nx PrT Q ScT Sh t U u u0 us v v0 x

concentration initial concentration frictional coefficient concentration at the internal surface of wall concentration fluctuation height of channel half height of channel distance between fixed pressure taps number of the frames number of the points in one streamwise line turbulent Prandtl number flow rate turbulent Schmidt number Sherwood number time mean streamwise velocity streamwise velocity component streamwise velocity fluctuation frictional velocity wall-normal velocity component wall-normal velocity fluctuation streamwise coordinate

velocity profile of logarithmic layer is shifted upward compared with that of Newtonian fluid turbulent flow and the near-wall turbulence structures including low-speed streak and vortex are modified. Beyond that, several researchers have also investigated polymer diffusion and concentration distribution in drag-reduced flow by sampling or the planar laser-induced fluorescence (PLIF) technique. Fruman et al. [15] investigated the concentration distribution of injected polymers along the plate and found it can be represented by two regions: one corresponding to a constant concentration and the other in which the concentration decreases as the inverse of the distance from the injection slot. Poreh et al. [16] measured the diffusion process in a turbulent boundary layer (TBL) and concluded that the diffusion rate of diluted polymers was reduced together with the skin frictional drag. Vdovin et al. [17] observed the differences between active admixtures of polymer solution in a drag-reduced TBL and passive admixtures of potassium chloride solution in a TBL. They also linked drag reduction to the polymer concentration. Walker et al. [18] measured the time-resolved polymer concentration in a drag-reduced channel flow with polymer injection from the spanwise slot on the wall and showed that high-concentration fluid moved from the nearwall region to the outer region in the form of long filaments lifted from the wall layer. Dimitropoulos et al. [19] investigated the dragreduced flow of inhomogeneous polymer solutions by using direct numerical simulation (DNS). The results showed that the transport of polymer decreased drag reduction downstream compared with that in a homogeneous case and the polymer concentration fluctuations were anti-correlated with streamwise velocity fluctuations. Winkel et al. [7] investigated the relationship between concentration of polymer injected from a spanwise slot and the local drag reduction in a TBL at a high Reynolds number. Elbing et al. [20] obtained the divisional characteristics of polymer diffusion in the streamwise direction based on the measurements in a roughwalled TBL with polymer degradation. These studies have clearly provided us with a basic and meaningful understanding of polymer diffusion in the near-wall region of drag-reduced flow with polymer injection. In more recent work, to link the diffusion process with the turbulent motion in

y z d dc Dp

s et em

wall-normal coordinate spanwise coordinate thickness of polymer-affecting region for mass eddy diffusivity thickness of concentration layer static pressure difference wall shear stress eddy diffusivity of momentum eddy diffusivity of mass

Subscripts bulk bulk flow dosing with dosing of polymer solution k kth frame max maximum p dosing polymer solution rms root mean square w dosing water water water flow Superscript + normalized quantity by inner variables

drag-reduced flow by polymer additives, Somandepalli et al. [8] measured the concentration and velocity fields simultaneously and studied the spatial distribution and spread of the injected polymer solution in a drag-reduced flat-plate TBL. The results indicated that the action of polymer reduced the streamwise and wall-normal concentration fluxes in the boundary layer before the polymer lost its effectiveness. However, the intrinsic relationship between the concentration flux and turbulent momentum transfer has not been elucidated comprehensively for drag-reduced flow with inhomogeneous polymer solutions. On the other hand, these previous investigations of polymer diffusion have been conducted primarily in TBL, with polymer injection from a spanwise slot on the wall. The drag reduction rate (DR) obtained by this method cannot be sustained for long distances from the slot. Recently, against the background of applying the Toms effect to ship hulls, a new method of developing novel antifouling paint, which can release a small amount of polymer while a ship is sailing, was proposed for reducing drag [21]. To simulate the release process and investigate the interaction of polymer with external flow during this process in lab experiments, a channel with a permeable wall was used to carry out the related research. The polymer solution was dosed into the turbulent channel flow from the surface of one-side of the channel wall due to the attached dosing wall having micro pores. Conspicuous drag reduction via this wall-blowing method was obtained [22–23]. Nevertheless, owing to the injection process differing from the case of slotinjection, clear differences of concentration distribution and spatial development of polymer solution, especially in the streamwise direction, were found to emerge in this case. To acquire more valuable information from this complicated drag-reduced flow with inhomogeneous polymer solutions via the wall-blowing method, research on the characteristics and influences of polymer diffusion is necessary. In addition, owing to the connection between drag reduction and mass transfer in this drag-reduced flow, investigation of the intrinsic relationship between momentum transfer and mass transfer in the dragreduced flow seems to be important for furthering our understanding of the drag-reducing process and mechanism. A conceptual model of this intrinsic relationship is shown in Fig. 1.

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Suppression of momentum transfer (eddy diffusivity of momentum, t )

Reduction of drag (frictional coefficient, Cf)

Local viscoelasticity by polymer Mean concentration of polymer, C

Turbulent Schmidt number, ScT

Suppression of mass transfer (eddy diffusivity of mass, m)

Velocity field Concentration field

Reduction of diffusion (Sherwood number, Sh)

Fig. 1. Conceptual model of the relationship between momentum transfer and mass transfer in the drag-reduced flow of polymer solutions.

As described in the above model, the existence of the polymer solution with different mean concentrations induces local viscoelasticity. This viscoelasticity cause inhibition of turbulent eddies, which results in the suppression of momentum transfer (i.e. decrease of eddy diffusivity of momentum, et). Finally, it causes drag reduction. It is also interesting to note that the inhibition of turbulent eddies also suppresses mass transfer (i.e. decrease of eddy diffusivity of mass, em). This may cause a reduction of mass diffusion (i.e. decrease of Sherwood number, Sh). However, the decreases of et and of em are not the same because of the dissimilarities in the two phenomena of momentum and mass transfer, even though they have some links indeed. The magnitude of decreases may differ at different locations due to the varying local viscoelasticity caused by the local concentration of polymer. The dissimilarity and locality effect should be discussed in detail by examining the local distribution of et and em and checking their relationship (i.e. turbulent Schmidt number, ScT) with various concentration distributions. From this perspective, relevant research should be carried out to further our understanding of drag reduction by additives and its practical application. The objective of this paper is to investigate the detailed diffusion characteristics of dosed polymer in drag-reduced turbulent channel flow. The investigations include measurements of the spatial evolution of polymer solution along the dosing wall, the corresponding effects on turbulent motions and momentum transfer, the streamwise and wall-normal turbulent mass fluxes, the quantitative evaluation of et, em and their relationship, and the relationship among some characteristic parameters (e.g. DR,

Sh, C and ScT). Finally, on the basis of these investigations, the concept of ‘‘effective position’’ of polymer additives is discussed. 2. Experiment 2.1. Experimental facility The experiment was conducted in a closed-circuit water channel, as schematically shown in Fig. 2. It consisted of a water tank, a circulating pump, a two-dimensional channel, pipelines and valves. The channel was made of transparent acrylic resin and the test section was straight with a length of 6000 mm, a width of 500 mm and a height of 40 mm (2h). An electromagnetic flow meter with uncertainty of ±0.01 m3/min was installed in the upstream part of the water channel to measure the flow rate. The temperature of flowing fluid was kept stable at 298.15 ± 0.1 K by a temperature control system in the tank during the experiment. The dosing wall was attached to one-side wall of the channel via a chamber embedded in the channel wall. The surface of the dosing wall was flush with the rest of the channel surface. The dosing wall was made of seven laminated layers of sintered stainless wire mesh. The pore of the main layer of mesh was square (also shown in Fig. 2) and its nominal size was 150 lm. The size of the dosing wall was 450  1450 mm. The total thickness of the dosing wall was 1.7 mm. The distance between neighboring pores on the surface of the dosing wall was 0.5 mm. Polymer solution was dosed into the channel flow from the whole surface of this dosing wall. The three-dimensional Cartesian

Fig. 2. Schematics of the channel system and dosing system with various measurement locations.

Z. Fu et al. / International Journal of Heat and Mass Transfer 77 (2014) 860–873

coordinates are also shown in Fig. 2. The x-axis is parallel to the flow direction. The x, y and z directions denote the streamwise, wall-normal and spanwise directions, respectively. The direction of gravity is downward along z-axis, which is also marked in Fig. 2. A honeycomb rectifier with a length of 150 mm was set at the entrance of the water channel in order to remove large eddies. The leading edge of the dosing wall was located 2300 mm downstream from the entrance of the channel. We defined the position of the leading edge as x = 0. Three glass windows A were set on the opposite side of the dosing wall at x = 250 mm, x = 800 mm and x = 1350 mm downstream from the leading edge, which are defined as positions 1, 2 and 3, respectively. They were used to make the laser light pass through and illuminate the flow field. Three glass windows B for observation (acquiring images) were set at the bottom of the channel accordingly. Four pressure taps were located on the opposite side of the dosing wall at x = 0 mm (port 1), x = 550 mm (port 2), x = 1100 mm (port 3) and x = 1650 mm (port 4) in the streamwise direction. The relative locations of the windows and ports in the channel are shown in Fig. 2. The static pressure difference (Dp) between two fixed taps was measured using a precise differential pressure gauge with uncertainty of ±0.1 Pa. In addition, due to the high aspect ratio (W/H = 12.5) of the channel, this channel was assumed to be twodimensional and the wall shear stresses of the two side walls were ignored accordingly [24]. The wall shear stress of the opposite wall of the dosing wall was confirmed to be unaffected by dosing and be same as the wall shear stress in water flow case (detailed introduction is presented in Section 2.3). Thus, the wall shear stress of the dosing wall in the case of dosing polymer solution can be calculated from the pressure difference and the force balance as follows:

h L

sdosing ¼ 2  Dp  swater ;

ð1Þ

where sdosing and swater are the wall shear stress with and without dosing of polymer solution at the same Reynolds number, respectively. The DRs of three streamwise sections (port 1–port 2, port 2–port 3 and port 3–port 4) can be calculated using the four pressure taps by the following equation:

DR ¼

swater  sdosing  100ð%Þ: swater

ð2Þ

2.2. Simultaneous PIV and PLIF arrangement The simultaneous PIV and PLIF measurements were arranged at three streamwise positions along the dosing wall. A double-pulse laser (New Wave Research Co., Ltd., MiniLase 30 Hz) composed of a pair of Nd: YAG lasers, each having 30 mJ per pulse of output and 532 nm of wavelength, was used. The laser sheet thickness and spread angle were set to 1 mm and 20°, respectively. Two charge-coupled device (CCD) cameras with a resolution of 2048  2048 were used for separately acquiring PIV images and PLIF images in the same streamwise wall-normal (x–y) plane. They were controlled by a timing circuit to take a picture at the same time. In addition, a dichroic beam splitter, with a 499–555 nm reflection band and a 569–730 nm transmission band, was installed in front of the two cameras to separate scatter from tracer particles and fluorescence from the dye. Fig. 3 shows the arrangement of the PIV and PLIF system in the measurement. The velocity and concentration in the x–y plane were measured by PIV and PLIF simultaneously. The PIV images were analyzed using the cross-correlation technique with an interrogation area of 64  64 pixels in size. Turbulent statistics were calculated using 500 velocity fields that contained 85  500 vectors at each wallnormal (y) position. The channel flow was seeded with polyethylene powder with a nominal mean diameter of 20 lm and a specific

863

gravity to water of 0.92 as the tracer particles. Meanwhile, to acquire the PLIF images, the dosed fluids of water and polymer solution were dyed with Rhodamine-WT (a non-toxic and noncarcinogenic dye) at 5 ppm. It can be fluoresced upon being subjected to 532 nm light and emit light with a wavelength of 580 nm. The size and number of PLIF images correspond to the PIV settings. To ensure that the acquired concentration corresponds to the velocity vector at the same position, the PLIF image was processed to obtain the concentrations by averaging the values of the 5  5 pixels in a square region centered at the position corresponding to the position of every velocity vector in the PIV image. 2.3. Polymer solution and dosing conditions The polymer used in the present experiment was polyethylene oxide, brand name PEO-18Z manufactured by Sumitomo Seika Chemicals Co., Ltd. It is a water-soluble polymer powder with a mean molecular weight of 4.3 million. The aqueous polymer solutions used for dosing were prepared by mixing polymer powder with water from which chlorine had been removed. Polymer solutions of 25, 50 and 100 ppm uniform weight concentrations (C0) were prepared for dosing in the simultaneous PIV and PLIF measurements. The effectiveness and repeatability of the mixing process were checked to ensure that the solutions were consistently prepared. The Reynolds number (based on channel height, bulk velocity and solvent viscosity) was set to 4.0  104. Thus, the bulk mean velocity of the channel flow was 0.88 m/s. The dosing rate (Qdosing) of polymer solution from the whole surface of the dosing wall was fixed at 10.5 L/min, which was controlled using a dosing system consisting of tube pumps, a reserve tank and flow meters. Under these conditions, the dosing velocity of polymer solution from the dosing wall was 2.9  104 m/s, and the ratio of the dosing velocity to the bulk mean velocity was 3.3  104. Special measurements of the water flow with and without dosing of water were performed under the same conditions as for the dosing of polymer solution to quantify the effect of the dosing process on channel flow. The mean velocity profiles and the turbulent intensity profiles of the wall-normal velocity component on the dosing wall measured at position 1, position 2 and position 3 are shown along with the same statistics of water flow in Fig. 4. From Fig. 4a, it can be seen that the dosing at different positions has no significant effect on the mean velocity profiles. The profiles almost overlap with the slope of the theoretical log-law profile in wall turbulence of Newtonian fluid. The profiles are slightly below the theoretical log-law profile due to the effect of the rough and permeable dosing wall [25]. As can be seen from Fig. 4b, the dosing process also has no discernible effect on the wall-normal velocity fluctuation intensity. The negligible changes in the statistics indicate that the dosing process itself does not result in turbulence modification of the channel flow. To ascertain the effects of the dosed water and polymer solution on the opposite wall of the dosing wall, the velocity statistics near the opposite wall were also conducted. Fig. 5 shows the mean streamwise velocity of the cases of water flow, dosing water and dosing 50 ppm polymer solution in the scope of the whole channel height. Fig. 6 shows the distribution of the sum of viscous stress and Reynolds shear stress near the opposite wall of the dosing wall for the same three cases. It can be seen in Fig. 5 that near the dosing wall (left side) the mean velocities of the three cases have larger difference than those near the opposite wall (right side). The dosed polymer solution caused increase of mean velocity near the dosing wall. In Fig. 6, the solid line is the fitting line for the stress distribution. The broken line is the extension of the solid line. The intercept indicates the wall shear stress of the opposite wall. It can be seen that the wall shear stress of the opposite wall

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Fig. 3. Schematic of PIV and PLIF arrangement.

Fig. 4. Comparison of velocity statistics showing effect of the dosing process including (a) mean velocity profiles and (b) turbulent intensity profiles for wall-normal velocity component. The solid line is the theoretical log-law profile in wall turbulence of Newtonian fluid.

1.6

Water flow Dosing water Dosing polymer, 50ppm

1.6

Water flow Dosing water Dosing polymer, 50ppm

1.2

1.2 0.8

0.8 0.4

0.4 0

0.5

1

1.5

2

Fig. 5. The mean velocities of the cases of water flow, dosing water and dosing 50 ppm polymer solution in the scope of the whole channel height.

of the dosing wall is uniform among the different cases. The dosed water and polymer solution have no effect on the opposite wall of the dosing wall. To ascertain the change of pressure during the process of dosing polymer solution, one typical plot of Dp vs. Time (t) of the case of dosing 50 ppm polymer solution measured between Port 3 and Port 4 is presented in Fig. 7. The original data were acquired for 15 min with a frequency of 25 Hz. To facilitate the presentation of the plot, we calculated one average value per 15 s (375 points). This figure shows the change of pressure difference around the typical dosing process clearly. In the figure, the first 1 min

0 0

0.2

0.4

0.6

0.8

1

Fig. 6. The distribution of the sum of viscous stress and Reynolds shear stress near the opposite wall of the dosing wall for the cases of water flow, dosing water and dosing 50 ppm polymer solution.

corresponded to the process of water flow. The process of dosing polymer solution was followed lasting 7 min (marked in the figure). Then, dosing was stopped until the end. From the figure, it can be seen that the dosed polymer solution results in decrease of Dp gradually. After dosing for 2 min, the magnitude of Dp tends to be uniform. Then the PIV and PLIF acquirement at position 3 can be carried out (marked in the figure). After stopping of dosing process for 4 min, the magnitude of Dp recovers to a stable value as the water flow (see the solid line). In addition, the initial dosing causes a little increase of Dp due to the effect of dosing itself,

Z. Fu et al. / International Journal of Heat and Mass Transfer 77 (2014) 860–873

Fig. 7. Change of pressure difference of the case of dosing 50 ppm polymer solution measured between Port 3 and Port 4 around the dosing process.

which is shown as the extreme point. Moreover, a short decrease of Dp appears after the abrupt stop of dosing due to the effect of remaining polymer near the dosing wall. 3. Results and discussion 3.1. Spatial evolution of polymer solution To observe the polymer diffusion fields near the dosing wall along the streamwise direction visually, instantaneous PLIF images at the aforementioned three streamwise positions were taken after the polymer solution had been dosed stably for a sufficient time till the acquired pressure difference was stable. Fig. 8 presents typical PLIF images obtained for the cases of the dosing of water and of polymer solutions at 25, 50 and 100 ppm. From Fig. 8, it can be seen that a polymer concentration layer corresponding to bright fluorescence for the case of dosing polymer is formed in the near-wall region and it thickens downstream along the dosing wall. A concentration gradient of polymer solution in the near-wall region can be clearly observed in terms of the strong brightness in the region adjacent to the wall, but weak brightness at positions distant from the wall. The observed boundaries between the dyed solution layer and the outer region are wavy at position 2 and position 3, which indicates that the turbulence tends to mix the filament-like polymer solution into the outer region during the diffusion process. In addition, with the increase of polymer concentration, it can be seen that the brightness of fluorescence strengthens progressively at the same measurement position, although the dye is used at the same concentration. In contrast, the brightness for the case of dosing water is relatively weak in the near-wall region as shown in Fig. 8a, because the dyed water can disperse into the core region easily. This indicates that the polymer solutions of higher concentrations diffuse more slowly after being dosed from the wall. To analyze the evolution of polymer solutions quantitatively, the polymer concentrations from the wall to the centerline of the channel in the wall-normal direction normalized by the initial concentration of the dosed solution, which was derived from the brightness acquired in PLIF images, were calculated. The concentration distributions for the case of dosing dyed water at the three positions are shown in Fig. 9a. By contrast, the results of the case of dosing 100 ppm polymer solution under the same flow conditions are shown in Fig. 9b. It can be seen that the dimensionless concentration of dyed water at the downstream position is higher than that at the upstream position in Fig. 9a. The dimensionless concentration also decreases with increasing distance from the wall along the y direction. The same characteristics are observed in Fig. 9b for dosing of polymer solution, but with higher magnitudes near the wall. The maximum value of dimensionless polymer concentration is about

865

0.55 in the near-wall region. This confirms that the diffusion of polymer is slower than that of water. Beyond that, the dosed solution seems to accumulate in the downstream regions due to the consecutive release from the dosing wall. Hence, the concentration boundary layer thickens along the dosing wall. To investigate the effect of the initial concentration of the dosed polymer solution on concentration distribution, the normalized mean concentrations for the cases of dosing of water and polymer solutions measured at position 3 are presented in Fig. 10a. The absolute mean concentrations for the cases of dosing polymer are presented in logarithmic form in Fig. 10b. From Fig. 10, it can be seen that the normalized concentrations for cases of dosing polymer are higher than those for dosing water. The concentration of the 100 ppm polymer case in the entire halfchannel region is highest at the same position. Meanwhile, the concentration gradient of the 100 ppm case in the near-wall region is the lowest among the dosing polymer cases, which indicates that the polymer solution with a high concentration diffuses badly due to the inhibited turbulence. Nevertheless, the concentrations for the cases of dosing polymer decrease almost linearly in the outer region from y/h = 0.40, as shown in Fig. 10b, which implies that the polymer in this region is easily mixed to become uniform. In Fig. 10b, the absolute mean concentrations in logarithmic form are used for convenience to compare the development of the concentration layer among the three cases. At the same position, different concentrations appear for each case. The solid line representing the concentration of 4 ppm is easily used to ascertain the boundaries of this concentration for the three cases. The reason for choosing 4 ppm and the values of dc1, dc2 and dc3 will be discussed with other characteristic parameters in Section 3.3.2. 3.2. Effects of polymer diffusion on turbulence and momentum transfer The interaction of the viscoelastic polymer solution with turbulence generates drag reduction during polymer diffusion in the channel flow. Thus, the diffusion process of polymer solution can cause the variation of DR. Table 1 lists the obtained DRs of the cases of dosing of 25, 50 and 100 ppm polymer solutions in the present experiment. In Table 1, the DR is observed to increase with increasing downstream distance under the same conditions, which can be explained by the well-mixed and accumulated polymer solution in the near-wall region at positions further downstream along the dosing wall. For the case of dosing 100 ppm polymer solution, the DR increases to the largest value of 44% at position 3 among all the cases. In terms of the DR at position 1, the appearance of a negative value for the 100 ppm case is interesting. This phenomenon has also been observed by Tiederman et al. [9] via the slot-injection method. The abrupt injection of polymer at a high concentration at the initial position results in high viscosity and a large velocity gradient in the near-wall region. In addition, the injected polymers staying in the viscous sublayer cannot act on turbulence effectively before they spread to a more distant region. The negative DR corresponding to the increasing frictional stress should be attributable to the above two factors. In contrast, positive DRs are obtained in cases with lower polymer concentrations due to the rapid dispersion in the near-wall region. It can be speculated that the dosed polymer solution with high concentration may produce large DR after it diffuses well in the channel flow, similar to the homogeneous drag-reduced flow. Nevertheless, the DRs tend to be saturated from a certain dosing rate according to the maximum drag reduction (MDR) phenomenon in the drag-reduced flow by additives. This means that the dosed polymer solution with a certain concentration has an extreme effect in the near-wall region, even with a greater dose. In other words, polymer will act as a drag reducer to suppress the

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Z. Fu et al. / International Journal of Heat and Mass Transfer 77 (2014) 860–873

Fig. 8. Typical PLIF images obtained for various cases: rows a, b, c and d represent cases of dosing of polymer solution at 0 (water), 25, 50 and 100 ppm; numbers 1, 2 and 3 represent observation at positions 1, 2 and 3, respectively.

0.5 0.4 0.3

0.3

0.2

0.2

0.1

0.1

0 0

0.2

0.4

0.6

0.8

Dosing polymer, position 1 Dosing polymer, position 2 Dosing polymer, position 3

0.4 C/C0

C/C0

0.5

Dosing water, position 1 Dosing water, position 2 Dosing water, position 3

1

0 0

0.2

0.4

0.6

y/h

y/h

(a)

(b)

0.8

1

Fig. 9. Mean concentration for the case of dyed solution dosed from the wall of (a) water and (b) 100 ppm polymer solution. For the case of dosing water, C/C0 denotes the ratio of local brightness to the original brightness of the dyed water.

Z. Fu et al. / International Journal of Heat and Mass Transfer 77 (2014) 860–873

0.5

867

Dosing water Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm

0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

(a)

(b)

Fig. 10. Mean concentration for the cases of dosing water and polymer solutions measured at position 3 (a) normalized by initial concentration and (b) in logarithmic value. The additional lines are used to demonstrate the boundary position of 4 ppm for each case of dosing polymer. dc denotes the distance from the wall to the position.

Table 1 DRs of the dosing polymer cases at three streamwise positions along the dosing wall.

25 ppm 50 ppm 100 ppm

Position 1

Position 2

Position 3

7.0% 2.0% 6.0%

18% 20% 21%

30% 40% 44%

turbulence, with its ultimate effect being due to the limited retention of polymer in a special region near the wall under fixed flow conditions. This special region is defined as the ‘‘effective position’’ of polymer, similar to the effective region of polymer discussed elsewhere [9,26]. Because the boundary and mechanism of generation of this special region are not easy to determine, this concept is only proposed here and will be discussed in more detail later. Fig. 11 shows the mean velocity profiles of the cases of water flow and dosing polymer solution with various concentrations measured at position 3 to quantify the effect of dosed polymer solution on velocity profile near the dosing wall. From the figure, it can be seen that the profile of the log-law layer for the case of dosing polymer solution is displaced upward with the increase of DR when compared with that of the case of water flow. The slope of the log-law region of the mean velocity profile is also increased. These tendencies agree well with the common phenomenon in drag-reduced flow by additives [6]. To compare the modified burst events by dosing of polymer solution at near-wall and distant-wall locations, the joint probability density functions (JPDF) of the streamwise and wall-normal fluctuation velocities (u0 , v0 ) in the planes at y/h = 0.10 and y/h = 0.50 for the dosing cases measured at position 3 are presented in Fig. 12. This analysis of JPDF is useful to judge the probability of ejection and sweep motions (corresponding to the second and fourth quadrant statistics, respectively) in burst events [10,12].

50

U+

40 30

Water flow Dosing polymer, 25ppm (DR = 30 %) Dosing polymer, 50ppm (DR = 40 %) Dosing polymer, 100ppm (DR = 44 %) U+=2.5lny ++5.5 + U++=11.7lny -17.0 + U =y

20 10 0 0 10

101

102

103

y+ Fig. 11. Mean velocity profiles of the cases of water flow and dosing polymer solution with various concentrations measured at position 3.

As expected, the motions in quadrants II and IV are more probable than those in quadrants I and III for the case of dosing water as reflected in Fig. 12a. From the left side of Fig. 12b–d, it can be seen that the magnitudes of u0 increase while values of v0 decrease, and the probability of occurrence for quadrants II and IV statistics also decreases relative to the case of dosing water in the near-wall plane at y/h = 0.10. In contrast, the magnitudes of u0 , v0 , and the probability of quadrants II and IV events exhibit no obvious changes at y/h = 0.50, as shown on the right side of Fig. 12b–d. In addition, the principal axes of JPDF for the cases of dosing polymer at y/h = 0.10 tend to be horizontal, while the others are inclined. With the increase of polymer concentration, the JPDFs become more symmetric about the u-axis. These findings indicate that the ejection and sweep motions in the near-wall plane at y/h = 0.10 are inhibited greatly, with a low probability of occurrence. However, the effects of polymer are limited in the distant-wall plane at y/h = 0.50. It can be speculated that this plane may be beyond the ‘‘effective position’’ of polymer. In order to investigate the effect of polymer diffusion on turbulence further, the distributions of the eddy diffusivity of momentum (et, equal to u0 v 0 ðdu=dyÞ) for the dosing cases measured at position 3, which is normalized by inner variables, are shown in Fig. 13. From Fig. 13, it can be seen that the eddy diffusivities of momentum of the cases of dosing polymer are lower than that of the case of dosing water in the near-wall region of y/h < 0.40, with no significant differences, although the local polymer concentrations are different (shown in Fig. 10). The data points in the distant-wall region are noisy due to the unstable Reynolds shear stress and low velocity gradient there. However, the mean eddy diffusivities of momentum for the dosing polymer cases are also lower than that of the case of dosing water. The eddy diffusivity of momentum decreased dramatically in the near-wall region, which implies that the polymer solution in that region acts as a powerful inhibitor of turbulence. Therefore, it seems that the local retention of polymer in the near-wall region has an important effect on momentum transport during the diffusion process. To examine in detail the effect of the polymer with a local concentration on et in the near-wall region, the decreases of et caused by dosed polymer solutions relative to that of the case of dosing water in the near-wall region of y/h < 0.40 are presented in Fig. 14. The solid line in the figure was fitted from average values in the region from the wall to the position of y/h = 0.25. It can be seen that all data points are distributed around the line, which indicates that the magnitudes of the decrease of et are at the same level for the cases of three concentrations in this region. From y/h = 0.25, the data points fluctuate markedly and broken lines are presented to show an idealized trend for the decrease of et in the rest of the region.

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160 Dosing water Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm

εt

120 80 40 0 0

0.2

0.4

0.6

0.8

1

y/h Fig. 13. Eddy diffusivities of momentum for the dosing cases measured at position 3.

60 Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm fit

εtw -εtp

40 20 0 0

0.1

0.2

0.3

0.4

y/h Fig. 14. Decreases of et for the cases of dosing polymer in the near-wall region of y/h < 0.40.

Fig. 12. JPDF for u0 and v0 measured at position 3 for the dosing cases of (a) water, (b) 25 ppm polymer solution, (c) 50 ppm polymer solution and (d) 100 ppm polymer solution. The left side shows results in the plane at y/h = 0.10, while the right side shows results in the plane at y/h = 0.50.

3.3. Turbulent mass transfer characteristics 3.3.1. Turbulent mass fluxes Turbulent mass fluxes were obtained by multiplying the concentration fluctuations by the velocity fluctuations at the same locations. Then the ensemble average value over all frames and points in one streamwise line was obtained for each y position. The streamwise turbulent mass flux can be defined as follows: N

c0 u0 ðyÞ ¼

Nx X k 1 X c0 u0 ðx; yÞ; Nx  Nk x¼1 k¼1 k k

ð3Þ

where c0 is the concentration fluctuation, Nx is the number of points in one streamwise line of each frame, Nk is the number of frames

and k denotes the kth frame. The wall-normal turbulent mass flux can be obtained by changing u0 to v0 in Eq. (3). These turbulent mass fluxes and their variations can be used to understand the polymer diffusion as a result of the interaction between the dosed polymer solution and turbulence in the channel flow. Fig. 15 shows the dimensionless streamwise turbulent mass fluxes normalized by C0 and frictional velocity (us) at the three streamwise positions along the dosing wall for the cases of dosing of water and 100 ppm polymer solution. It can be seen in Fig. 15 that the streamwise turbulent mass fluxes for the cases of dosing polymer are larger than that of the case of dosing water in the near-wall region at the same position. With increasing distance downstream, the streamwise turbulent mass flux increases for each case. This phenomenon should be attributable to the increasing c0 at downstream positions. The polymer concentration fluctuations are also observed to correlate inversely with streamwise velocity fluctuations. Fig. 16 shows the dimensionless streamwise turbulent mass fluxes normalized by C0us and Cmaxus for the dosing cases measured at position 3, which is presented to investigate the effect of polymer concentration on streamwise turbulent mass flux. Here, Cmax denotes the local maximum polymer concentration measured near the dosing wall. It can be seen in Fig. 16a that the streamwise turbulent mass flux of the cases of dosing polymer increases significantly in the near-wall region of y/h < 0.40 relative to that of the case of dosing water at position 3. The distribution of streamwise flux does not change markedly with the increase of polymer concentration. To ensure that the magnitudes of stream fluxes are not artificially reduced by using the initial dosing concentration, the local maximum concentration near the wall was used. It can be seen in Fig. 16b that the streamwise fluxes are also close for the different cases of dosing polymer. They are a little larger than the flux of the case of dosing water in a very small near-wall region of y/h < 0.10,

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0.08

0.24 Dosing water, position 1 Dosing water, position 2 Dosing water, position 3

-c'u'/C0uτ

0.04

0.12

0.02 0 0

Dosing polymer, position 1 Dosing polymer, position 2 Dosing polymer, position 3

0.18

-c'u'/C0uτ

0.06

0.06

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

0.6

y/h

y/h

(a)

(b)

0.8

1

Fig. 15. Streamwise turbulent mass flux along the dosing wall at three positions for the cases of (a) dosing water and (b) dosing 100 ppm polymer solution.

0.2

0.5 Dosing water Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm

-c'u'/Cmaxuτ

0.12 0.08 0.04 0 0

Dosing water Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm

0.4

-c'u'/C0uτ

0.16

0.3 0.2 0.1

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

0.6

y/h

y/h

(a)

(b)

0.8

1

Fig. 16. Streamwise turbulent mass flux along the dosing wall measured at position 3 for dosing cases normalized by (a) initial concentration and (b) local maximum concentration.

0.008

0.016 Dosing water, position 1 Dosing water, position 2 Dosing water, position 3

0.004 0.002 0 0

Dosing polymer, position 1 Dosing polymer, position 2 Dosing polymer, position 3

0.012 c'v'/C0uτ

c'v'/C0uτ

0.006

0.008 0.004

0.2

0.4

0.6

0.8

1

0 0

0.2

0.4

0.6

y/h

y/h

(a)

(b)

0.8

1

Fig. 17. Wall-normal turbulent mass flux along the blowing wall at three positions for the case of (a) dosing water and (b) dosing 100 ppm polymer solution.

but they decrease progressively in the rest of the region due to the action of polymer. From the position of y/h = 0.15, the streamwise fluxes of the cases of dosing polymer are lower than that of the case of dosing water. This modification of streamwise turbulent mass flux is similar to the trend obtained via the slot-injection method in a previous study [8]. Fig. 17 presents the dimensionless wall-normal turbulent mass fluxes normalized by C0 and us at the three streamwise positions along the dosing wall for the cases of dosing water and 100 ppm polymer solution. Fig. 18 shows the dimensionless wall-normal turbulent mass fluxes normalized by C0us and Cmaxus for the dosing cases at position 3. From Fig. 17, it can be seen that the wall-normal turbulent mass flux increases with increasing downstream distance for each case. This phenomenon should also be attributable to the increasing c0 at

downstream positions. In Fig. 18a, it is observed that the wall-normal turbulent mass flux decreases with increasing concentration. In Fig. 18b, it can be seen that the wall-normal flux profiles for the cases of dosing polymer are lower than that for the case of dosing water in almost the entire half-height region of the channel, except the initial layer located at less than y/h = 0.15. The decrease of the wall-normal turbulent mass flux for the 100 ppm case is largest among the three cases when compared with the case of dosing water. This can be explained by the intensive suppression caused by the polymer solution of high concentration at the downstream position, where polymer solution can accumulate in the near-wall region. In addition, it can be seen that the peak values of wall-normal turbulent mass flux for the three cases appear at almost the same position near y/h = 0.10, and the suppression is more obvious after the peak location. It also indicates that the

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0.025

0.1 Dosing water Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm

0.015 0.01 0.005 0 0

Dosing water Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm

0.08 c'v'/Cmaxuτ

c'v'/C0uτ

0.02

0.06 0.04 0.02

0.2

0.4

0.6

0.8

1

0 0

0.2

0.4

0.6

y/h

y/h

(a)

(b)

0.8

1

Fig. 18. Wall-normal turbulent mass flux along the blowing wall measured at position 3 for dosing cases normalized by (a) initial concentration and (b) local maximum concentration.

modification of wall-normal turbulent mass flux presents a special region where polymer can interact with turbulence effectively. It is interesting to note that the wall-normal turbulent mass fluxes are also very close to the level of DR for the 50 ppm and 100 ppm cases, which indicates that the suppression of wall-normal turbulent mass flux in drag-reduced turbulent flow has a strong relationship with the drag reduction. In fact, the wall-normal turbulent flux can control the mass transportation from the near-wall region to the bulk flow. It can be suppressed significantly in the dragreduced channel flow with inhomogeneous polymer solutions via the wall-blowing method. 3.3.2. Eddy diffusivity of mass and turbulent Schmidt number The eddy diffusivity of mass was calculated and adopted to evaluate the polymer diffusion in the drag-reduced flow of inhomogeneous solution. Moreover, the turbulent Schmidt number (ScT) which represents the relative intensities of the eddy diffusivities of momentum and of mass, was also obtained to evaluate the polymer diffusion quantitatively [27]. It is defined as follows:

ScT ¼

et u0 v 0 =ðdu=dyÞ u0 v 0 dc=dy ¼ ¼ : ; em c0 v 0 =ðdc=dyÞ c0 v 0 du=dy

ð4Þ

where em equals c0 v 0 =ðdc=dyÞ, dc/dy is the gradient of the mean concentration and du/dy is the gradient of the mean streamwise velocity. Fig. 19 presents the eddy diffusivities of mass normalized by inner variables for the dosing cases measured at position 3. The solid lines shown in the figure were used to fit the trend of the varying eddy diffusivities of mass. It can be seen that the eddy diffusivities of mass of the cases of dosing polymer are smaller than that of the case of dosing water in a wide region near the wall due to the suppression of turbulence caused by dosed polymer. In addition, the eddy diffusivities of

mass are observed to decrease with the increase of polymer concentrations in the near-wall region. They vary noisily in the distant-wall region from y/h = 0.40 due to the unstable and small concentration gradients there. In the figure, d is used to mark the distance from the wall to the position where the slope of the trend line changes abruptly. d1, d2 and d3, which correspond to the cases of dosing water and dosing 25 and 100 ppm polymer solutions, equal 0.12, 0.27 and 0.41, respectively. The location of the slope change for the 50 ppm case in the profile is almost same as that for the 100 ppm case. These feature thicknesses of the polymeraffecting region for em seem to be related to the local concentration of polymer. They will also be discussed later along with other parameters in this section. Fig. 20 presents the decreases of em caused by dosed polymer solution in the near-wall region of y/h < 0.40 based on the above results. The solid line in the figure was fitted from average values. For the 25 ppm case, it covers the region with a significant decrease of em from the wall to the position of y/h = 0.23, which is thought to be reasonably affected by the polymer. The broken line is also presented as a trend line for the rest of the region. It can be seen that the decreases of em for the cases of 50 and 100 ppm are larger than that of the 25 ppm case due to the more powerful suppression. The magnitudes of the decrease for the cases of 50 and 100 ppm are also at the same level in the near-wall region of y/h < 0.40. From the position of y/h = 0.25, the em of the 25 ppm case does not change sharply, which indicates that the effect of the polymer with a local concentration on mass transport is limited around this position. When compared with the decreases of et shown in Fig. 14, it can be found that the decreases of em and et for the cases of dosing polymer present different trends. They are not analogous phenomena in the drag-reduced flow. However, ScT can be used to reveal the quantitative relationship between em and

60 Dosing polymer, 25ppm Dosing polymer, 50ppm Dosing polymer, 100ppm fit

εmw -εmp

50 40 30 20 10 0 0

Fig. 19. Eddy diffusivities of mass for the dosing cases measured at position 3.

0.1

0.2 y/h

0.3

0.4

Fig. 20. Decreases of em for the cases of dosing polymer in the near-wall region of y/ h < 0.40.

Z. Fu et al. / International Journal of Heat and Mass Transfer 77 (2014) 860–873

10 Dosing water Dosing polymer, 25ppm (DR = 30 %) Dosing polymer, 50ppm (DR = 40 %) Dosing polymer, 100ppm (DR = 44 %)

8 ScT

6 4 2 0 0

0.2

0.4

0.6

0.8

1

y/h Fig. 21. ScT for the dosing cases measured at position 3.

et. Fig. 21 presents ScT for the cases of dosing of water and polymer solutions measured at position 3. The corresponding DRs of the cases of dosing polymer solution are also shown in the figure. It can be seen that the ScT are large and noisy for the cases of dosing 50 ppm and 100 ppm polymer solutions. However, the ScT for the cases of dosing water and 25 ppm polymer solution are stable over the half-height region. For the case of dosing water, the ScT tends to be uniform and its mean value is 1.20 in the near-wall region of y/h < 0.40. With the increase of polymer concentration, the mean ScT increases to 1.60, 2.40 and 2.90 for the 25, 50 and 100 ppm cases, respectively. The high ScT of the case of dosing polymer signifies that the turbulent mass flux from the near-wall region is suppressed due to the action of polymer solution, which agrees well with the trend of ScT obtained elsewhere [8] via the slot-injection method. The ScT also has similar variation to the turbulent Prandtl number (PrT), which was defined as the ratio of the eddy diffusivity of momentum to that of heat. The PrT was observed to increase with increasing DR, as investigated by Gupta et al. [28] in a polymer drag-reduced channel flow and by Li et al. [29] in a surfactant drag-reduced channel flow. Gasljevic et al. [30] pointed out that constant PrT with a value of 5–8 occurred in a drag-reduced pipe flow with homogeneous polymer or surfactant solutions. The value of ScT exceeding one also indicates that the suppression of turbulent momentum transportation due to the polymer is smaller than that of mass transport. Thus, the dosed polymer diffuses as an active scalar rather than a passive scalar which can simply diffuse following the turbulent momentum transfer. The quantitative results for ScT for the drag-reduced flow were thought to be appropriate as benchmark numerical simulations [8]. Furthermore, in order to relate the characteristics of mass transfer to the local concentration of the inhomogeneous polymer solution and to the DRs obtained during the diffusion process, the characteristic parameters including ScT, Sh (calculated by Sh = mHSc/Cwm = us HSc/Am, where Sc is the molecular Schmidt number and set to 55, A is a constant and set to 144; referenced from [31]), the thicknesses d of the polymer-affecting region for em, the absolute concentrations C at y/h = 0.25 for the 25 ppm case and at y/h = 0.40 for the 50 and 100 ppm cases, and the thicknesses (dc) of the concentration layers bounded by 4 ppm for the three dosing polymer cases measured at position 3 are listed in Table 2.

Table 2 Characteristic parameters of mass transfer for the cases of dosing polymer. Case (DR)

ScT

Sh

d (y/h)

C (ppm)

dc (y/h)

25 ppm (30%) 50 ppm (40%) 100 ppm (44%)

1.60 2.40 2.90

739.35 729.47 710.68

0.27 0.41 0.41

2.41 3.85 7.94

0.10 0.38 0.88

871

From Table 2, it can be seen that ScT increases while Sh, which represents the strength of convective mass transfer, decreases with the increase of DR. All of these findings indicate that mass transfer is suppressed in the drag-reduced flow and becomes weaker with higher DR. The thicknesses (d) of the polymer-affecting region for em increase with increasing DR for the 25 and 50 ppm cases. They are almost the same for the 50 and 100 ppm cases, which is similar to the change of DRs in the two cases. These phenomena regarding DR and d are analogous. At y/h = 0.25, the rapid decrease of em for the 25 ppm case almost stops. There, the absolute concentration (C) for the 25 ppm case is 2.41 ppm. For the 50 and 100 ppm cases, the decreases of em affected by polymer end at around the position of y/h = 0.40. The corresponding concentrations at the position are 3.85 and 7.94 ppm, respectively. According to these feature concentrations, a local concentration of 2–4 ppm is thought to be very important. Thus, the concentration of 4 ppm was chosen to be a reference value used to evaluate the thicknesses (dc) of the concentration layers on the basis of Fig. 7b. It can be seen that, for the 25 ppm case, dc is 0.10, which is less than the thickness (d) of the polymer-affecting region, namely, 0.27. This implies that a concentration less than 4 ppm can still act on turbulence, resulting in a decrease of em. For the 50 ppm case, dc is 0.38, which is close to the thickness of the polymer-affecting region, namely, 0.41. This implies that the concentration of 4 ppm can decrease em effectively to the position of y/h = 0.40 and can be used as a threshold for suppressing mass transfer for the 50 ppm case. For the 100 ppm case, dc is 0.88, which is larger than the thickness of the polymer-affecting region, namely, 0.41, which indicates that a higher concentration of polymer has an ultimate effect on turbulence. This also agrees with the MDR phenomenon in drag-reduced flow. In addition, it is apparent that dc increases with the increase of initial concentration. The growth rate of this thickness is larger than that of DR. This also indicates that the ‘‘effective position’’ of polymer for reducing drag is in the near-wall region from another side. It seems that the local concentration of dosed polymer solution determines the level of drag reduction and the level of the decrease of em, but the relationship is not linear. These characteristic parameters listed here offer valuable clues for ascertaining the relationships among local concentration, drag reduction and mass transfer features, even the ‘‘effective position’’ of polymer. 4. Discussion The concept of the ‘‘effective position’’ of polymer mentioned above has also been discussed in the literature. Injected polymer solution was thought to act on turbulence effectively and cause drag reduction when it is located in the buffer region (10 < y+ < 100) [9,26]. The drag-reducing effect could not appear immediately if the polymer was injected from the core region of a turbulent pipe or channel flow [26,32]. The effectiveness of near-wall viscoelastic fluids was also identified in a drag-reduced channel flow by surfactant additives [33]. Therefore, it seems that the ‘‘effective position’’ of polymer solution exists accordingly in wall-bounded turbulent flow. However, it is not easy to determine the exact boundary before we ascertain the reason why the polymer solution is just effective in the special region. For further consideration, the difference between the special region and the core region should also be investigated from different perspectives, including the mass transfer characteristics and modified characteristic structures. As is well known, the viscoelasticity of polymer additives plays a key role in polymer drag reduction. It may modify the energy cascade and turbulence regeneration cycle, according to existing findings. We speculate that polymer solution with a certain local concentration in the special region, which can effectively interact

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with the turbulence in a limited range, may correspond to a certain kind of state. This state will change and has no effect on turbulence in the core region due to the varied local concentration of polymer solution and turbulence intensity. In turn, the effective region of polymer may enlarge or shrink if the state changes. As such, the process and features of polymer diffusion in the special region with different concentrations and Reynolds numbers should be investigated in detail, combined with modified characteristic structures, which may provide valuable information for furthering our understanding of the phenomenon of ‘‘effective position’’, even the interactions between polymer and turbulence. Coincidentally, the simultaneous measurements by PIV and PLIF can not only provide data for calculating the turbulent flux and other derived quantities to estimate the diffusion of polymer quantitatively, but can also be used to analyze the turbulent characteristic structures affected by polymer additives during the diffusion process based on the instantaneous velocity and concentration fields. A previous investigation [34] by our team showed that some variations of structures appear in the streamwise direction during the diffusion process of polymer solution. For instance, a strong wall-ward interaction motion (quadrant III) emerged under the shear layer with a small inclination angle, and the hairpin vortex cores along the shear layer were observed to almost disappear, while groups of clockwise and counterclockwise vortex cores ranged alternately bordering the shear layer with the evolution of the polymer concentration boundary. However, further investigations on polymer diffusion and characteristic structures around the ‘‘effective position’’ are required to further our understanding of the mechanism of drag reduction by polymer additives. 5. Conclusions Characteristics of polymer diffusion in the drag-reduced turbulent channel flow with dosing of polymer solution were investigated experimentally using the PIV and PLIF techniques. The measurements of velocity and concentration fields were obtained in the streamwise wall-normal (x–y) plane along the dosing wall of the channel. The present analysis showed that the polymer diffusion is suppressed due to the inhibited turbulence other than the diffusion of passive scalar in ordinary turbulence. This trend is more evident for the dosing of polymer solution with a higher concentration. The DR was found to increase with the increase of local polymer concentration in the ‘‘effective position’’ downstream. Two characteristic regions exist in the near-wall region according to the diffusion characteristics and altered motions in the wallnormal direction. Turbulent mass fluxes were also obtained by multiplying the concentration fluctuations by the velocity fluctuations at the same locations. They were found to increase with increasing downstream distance along the dosing wall. The wall-normal turbulent fluxes that control the transport of mass were reduced significantly in the near-wall region when compared with the case of dosing. The decrease of wall-normal turbulent flux also increased with increasing concentration of polymer solution, while it is almost same for the streamwise turbulent mass flux. The eddy diffusivities of momentum and of mass and turbulent Schmidt number were calculated and adopted to evaluate the polymer diffusion quantitatively. The decreases of the eddy diffusivity of momentum were found to be at the same level for the cases of dosing polymer in the near-wall region of y/h < 0.40. The eddy diffusivity of mass also decreased with increasing polymer concentration in the near-wall region. However, the decreases of the eddy diffusivities of momentum and of mass presented different trends and they are not analogous phenomena in the drag-reduced flow. The turbulent Schmidt number obtained for the case of dosing polymer is larger than that of dosing water. It was also found to

increase with increasing polymer concentration in the downstream region. The mean value of turbulent Schmidt number for the drag-reduced flow with dosing of 100 ppm polymer solution can rise to 2.90 from 1.20 for the dosing of water. Conflict of interests None declared. Acknowledgments The first author would like to thank the support of the scholarship from Japanese Ministry of Education, Culture, Sports, Science and Technology. Dr. YU Bo also acknowledges the support from the National Natural Science Foundation of China (No. 51325603). We also thank Dr. Wang Yi, Mr. Yuichiro Iwaki and Mr. Ryoto Goto for valuable discussions. References [1] B.A. Toms, Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers, in: Proceedings of the First International Congress on Rheology, North-Holland, Amsterdam, vol. 2, pp. 135–141, 1948. [2] E.D. Burger, L.G. Chorn, T.K. Perkins, Studies of drag reduction conducted over a broad range of pipeline conditions when flowing Prudhoe bay crude oil, J. Rheol. 24 (5) (1980) 603–626. [3] F.C. Li, B. Yu, J.J. Wei, Y. Kawaguchi, Turbulent Drag Reduction by Surfactant Additives, Higher Education Press, China, 2012, pp. 242–253. [4] K. Gasljevic, E.F. Matthys, Ship reduction by microalgal biopolymers: a feasibility analysis, J. Ship Res. 51 (4) (2007) 326–337. [5] A. Gyr, H.W. Bewersdorff, Drag Reduction of Turbulent Flows by Additives, Kluwer Academic, The Netherlands, 1995, pp. 126–131. [6] C.M. White, M.G. Mungal, Mechanics and prediction of turbulent drag reduction with polymer additives, Annu. Rev. Fluid Mech. 40 (2008) 235–256. [7] E.S. Winkel, G.F. Oweis, S.A. Vanapalli, D.R. Dowling, M.J. Solomon, S.L. Ceccio, High Reynolds-number turbulent boundary layer friction drag reduction from wall-injected polymer solutions, J. Fluid Mech. 621 (2009) 259–288. [8] V.S.R. Somandepalli, Y.X. Hou, M.G. Mungal, Concentration flux measurements in a polymer drag-reduced turbulent boundary layer, J. Fluid Mech. 644 (2010) 281–319. [9] W.G. Tiederman, T.S. Luchik, D.G. Bogard, Wall-layer structure and drag reduction, J. Fluid Mech. 156 (1985) 419–437. [10] A.A. Fontaine, H.L. Petrie, T.A. Brungart, Velocity profile statistics in a turbulent boundary layer with slot-injected polymer, J. Fluid Mech. 238 (1992) 435–466. [11] Y.X. Hou, V.S.R. Somandepalli, M.G. Mungal, A technique to determine total shear stress and polymer stress profiles in drag reduced boundary layer flows, Exp. Fluids 40 (2006) 589–600. [12] D.T. Walker, W.G. Tiederman, Turbulent structure in a channel flow with polymer injection at the wall, J. Fluid Mech. 218 (1990) 377–403. [13] H.L. Petrie, A.A. Fontaine, Comparison of turbulent boundary layer modification with slot-injected and homogeneous drag-reducing polymer solutions, ASME Publ. Fluids Eng. Div. 237 (1996) 205–210. [14] C.M. White, V.S.R. Somandepalli, M.G. Mungal, The turbulence structure of drag reduced boundary layer flow, Exp. Fluids 36 (2004) 62–69. [15] D.H. Fruman, M.P. Tulin, Diffusion of a tangential drag reducing polymer injection of a flat plate at high Reynolds numbers, J. Ship Res. 20 (1976) 171– 180. [16] M. Poreh, K.S. Hsu, Diffusion of drag reducing polymers in a turbulent boundary layer, J. Hydronaut. 6 (1) (1972) 27–37. [17] A.V. Vdovin, A.V. Smol’yakov, Diffusion of polymer solutions in a turbulent boundary layer, J. Appl. Mech. Tech. Phys. 19 (1978) 196–201. [18] D.T. Walker, W.G. Tiederman, The concentration field in a turbulent channel flow with polymer injection at the wall, Exp. Fluids 8 (1989) 86–94. [19] C.D. Dimitropoulos, Y. Dubief, E.G. Shaqfeh, P. Moin, Direct numerical simulation of polymer-induced drag reduction in turbulent boundary layer flow of inhomogeneous polymer solutions, J. Fluid Mech. 566 (2006) 153–162. [20] B.R. Elbing, M.J. Solomon, M. Perlin, D.R. Dowling, S.L. Ceccio, Flow-induced degradation of drag-reducing polymer solutions within a high-Reynoldsnumber turbulent boundary layer, J. Fluid Mech. 670 (2011) 337–364. [21] M. Motozawa, Y. Onose, S. Sugita, K. Iwamoto, H. Ando, T. Senda, Y. Kawaguchi, Experimental investigation on turbulent drag reduction with blowing polymer solution from the wall, in: 7th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, Krakow, pp. 2327–2334, 2009. [22] M. Motozawa, S. Ishitsuka, K. Iwamoto, H. Ando, T. Senda, Y. Kawaguchi, Experimental investigation on turbulent structure of drag reducing channel flow with blowing polymer solution from the wall, Flow Turbul. Combust. 88 (2012) 121–141.

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