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Advances in Mechanical Engineering

Volume 2011 (2011), Article ID 367042, 9 pages

http://dx.doi.org/10.1155/2011/367042

## Analysis of Zero Reynolds Shear Stress Appearing in Dilute Surfactant Drag-Reducing Flow

^{1}School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China^{2}Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, 278-8510, Japan

Received 3 June 2011; Accepted 18 July 2011

Academic Editor: Jinjia Wei

Copyright © 2011 Weiguo Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Dilute surfactant solution of 25 ppm in the two-dimensional channel is investigated experimentally compared with water flow. Particle image velocimetry (PIV) system is used to take 2D velocity frames in the streamwise and wall-normal plane. Based on the frames of instantaneous vectors and statistical results, the phenomenon of zero Reynolds shear stress appearing in the drag-reducing flow is discussed. It is found that 25 ppm CTAC solution exhibits the highest drag reduction at and loses drag reduction completely at . When drag reduction lies in the highest, Reynolds shear stress disappears and reaches zero although the RMS of the velocity fluctuations is not zero. By the categorization in four quadrants, the fluctuations of 25 ppm CTAC solution are distributed in all four quadrants equally at , which indicates that turnaround transportation happens in drag-reducing flow besides Reynolds shear stress transportation. Moreover, the contour distribution of streamwise velocity and the fluctuations suggests that turbulence transportation is depressed in drag-reducing flow. The viscoelasticity is possible to decrease the turbulence transportation and cause the turnaround transportation.

#### 1. Introduction

If small amounts of polymer or surfactant are added into the water, the friction will decrease in a large degree in turbulent flow. This phenomenon of drag reduction is named “Toms effect” because it was firstly reported by Toms in 1948 [1]. Li et al. reported that ultimate 80% frictions were reduced in the very dilute aqueous solution of cetyltrimethylammoniumchloride (CTAC) while the concentration was only 30 ppm [2, 3]. For the energy conservation purpose, surfactant solution has increasingly received the significant attention during the past some decades studying by experiments and simulations in order to clarify the mechanism of drag reduction [4, 5]. Gyr and Tsinober summarized the dynamic characteristics of drag-reducing flow, and concluded that the small scale part of the flow (both in time and space) was suppressed, and the streamwise velocity fluctuations remained approximately the same whereas the transverse one was strongly reduced [6].

Reynolds shear stress is important to the turbulence transportation in wall-bounded flow [7], but it was strongly reduced in drag-reducing flow and some researchers ascribed the phenomenon to the less correlation between the two velocity fluctuations. Moreover, surfactant solution flow exhibited that the cyclic turbulent bursts were inhibited and the vortex structures were modified [8]. Kawaguchi et al. found that the penetration of the fluid from the low-speed fluid region into the high-speed region almost disappeared, and the strong fluctuations of spanwise vorticity near the wall also disappeared in surfactant solution channel flow [9, 10]. Li et al. found that the inclination angle of the low-momentum region below the hairpin vortices decreased and the frequency of bursts was reduced, which indicated the inhibition of bursting events by surfactant additives [11, 12]. However, all these studies have not given the reasonable explanation to the appearance of zero Reynolds shear stress.

Rheology of the solution, such as the viscoelasticity, was always considered to modify the turbulent structures and lead to the phenomenon of drag reduction [13], because surfactant additives added into the water would form the cross-linked micellar networks under a proper shear stress [14, 15]. The investigation by Cryo-TEM proved that the solution with wormlike micella displayed rich rheological behaviors such as drag-reducing agents and viscosity enhancers [16]. But the micella have not been tested dynamically because of the shortage of present measure method.

This paper will aim at the modification of turbulence transportation in the drag-reducing channel flow and analyze the disappearance of Reynolds shear stress. PIV is used to measure the velocity field in streamwise-wall-normal plane. The dynamic characteristics of the surfactant solution flow are studied based on the statistic results and instantaneous velocity distribution.

#### 2. Experimental Facility

Experiments are performed on a closed-circuit water loop as shown in Figure 1. The tested fluid is through a two-dimensional channel made of transparent acrylic resin with the length () of 10 m, height () of 0.04 m, and width () of 0.5 m. A honeycomb rectifier with length of 0.15 m is set at the channel entrance for removing large eddies. The test segment lies 6 m downstream from the entrance of the channel. An electromagnetic flow meter with uncertainty of ±0.01 m^{3}/min is installed upstream of the channel for flow measurement. The water tank in the flow loop contains an electrical heater for heating the circulating solution with uncertainty of ±0.1°C when the fluid temperature is below 25°C. The wall shear stress was estimated from the static pressure gradient which is tested between two pressure tabs which are located on the vertical side wall of the channel with 1.5 m distance and the uncertainty of ±0.1 Pa.

The surfactant added to the water in this experiment is one kind of cationic surfactant, cetyltrimethylammoniumchloride (CTAC). Sodium salicylate (NaSal) is added to the solution with the same weight concentration with CTAC for providing counterions. The CTAC concentration of 25 ppm is selected in this experiment.

PIV system consists of a double-pulsed laser, laser sheet optics, charge-coupled device (CCD) camera, timing circuit, image-sampling computer, and image-processing software. The double-pulsed laser is a combination of a pair of Nd-YAG lasers. The timing circuit communicates with the CCD camera and computer and generates pulses to control the double-pulse laser. The CCD camera used in the experiment has a resolution of pixels. Figure 2 shows a schematic of the optical configuration of PIV to measure the flow on the streamwise and wall-normal (-) plane. Cartesian coordinates are also shown in the diagram. The laser sheets are aligned at mm, where represents the plane at the bottom of the channel. The tracers seeded in the flow are white water-based synthetic resins coatings with a diameter of 0.1–1 *μ*m. When adding tracers to the solution, particle density is adjusted and on average at least 10 particle pairs will be observed in an interrogation window.

In the experiments, five hundred of PIV image pairs are taken for one condition case. The photograph acquisition rate is 4 Hz. The picture frames cover the full width of the channel with the size of about mm^{2}. The interrogation area is set as pixels with 75% overlap in each direction. As a result, the vector numbers are about , respectively, in and direction. The spacing between adjacent vectors in both directions is about 0.45 mm.

#### 3. Results and Discussion

This section will analyze the instantaneous velocity images and the statistic results based on five hundred of 2D frames of velocity vectors in streamwise-wall-normal (-) plane. Water and CTAC solution with the weight concentration of 25 ppm are compared under the Reynolds number from 10000 to 40000. Reynolds number is defined as , where is the bulk velocity of the flow, is the width of the channel in direction, and is the dynamic viscosity of the solution. However, the CTAC solution of 25 ppm is very dilute, so the dynamic viscosity of the solution is selected as same as the water.

##### 3.1. Frictional Drag Factor and Drag Reduction Rate

Fanning friction factor and drag reduction (DR) are defined as following equations where the subscripts of “” and “” represent water and CTAC solution, respectively,

The profiles of friction factor and DR are presented in Figure 3. At the same time, Dean’s correlation of friction factor for a Newtonian fluid in a two-dimensional channel [17] and Virk’s ultimate correlation of friction factor for polymers solution flow [18] are also included in Figure 3 for comparison.

The figure exhibits that the friction factors of water obey Dean’s equation well especially during the high Reynolds numbers. The friction factors of CTAC solution are markedly smaller than the water and reach the profile of Virk’s ultimate friction factor before . According to the division of DR state [19], the Reynolds number at which DR reaches the maximum is called the 1st critical Reynolds number and at which DR disappears completely is called the 2nd critical Reynolds number. As a result, where DR reaches the maximum nearly 80% is the 1st critical Reynolds number for 25 ppm CTAC solution at the temperature of 25°C in this experiment. When the Reynolds number is larger than the 1st critical Reynolds number, the friction becomes large and DR decreases quickly while Re increases which indicates the degradation of drag reduction. DR almost disappears at .

##### 3.2. Mean Velocity and Fluctuations

At the same Reynolds number, five hundred of velocity frames are used to do statistical average. Then ensemble average is performed along streamwise direction based on about fifty thousand of vectors at one identical coordinate.

Figure 4 presents the profiles of mean streamwise velocities handled dimensionlessly with friction velocity () which is marked as the superscript “+”. Because the CTAC solution of 25 ppm exhibits the maximal DR at and drag reduction degradation at , these two Reynolds numbers are selected for comparison with Newtonian.

The log law equation of mean velocity for Newtonian turbulent flow and the ultimate velocity profile of polymeric solution flow suggested by Virk are included in Figure 4, where .

Figure 4 shows that mean streamwise velocities of water are distributed in close agreement with the profile of (4) for Newtonian turbulent channel flow. This phenomenon also appears in the case of 25 ppm CTAC solution at because drag reduction disappears completely. At , the mean streamwise velocities of 25 ppm CTAC solution are distributed upward and close to the line of (5).

Root mean square (RMS) is defined as (6), where represents the number of the vectors. RMS is usually regarded as the representation of turbulence intensity.

Figures 5(a) and 5(b) exhibit the RMS-distributed profiles, respectively, of streamwise velocity fluctuations () and wall-normal velocity fluctuations (). The abscissa of the figure is the wall-normal coordinate () in the channel where the superscript “*” represents the dimensionless of geometry by the half of channel width ().

Small difference appears in the distribution of streamwise velocity fluctuations of the two fluids at two Reynolds numbers in Figure 5(a), whereas the wall-normal velocity fluctuations decrease in a large degree for 25 ppm CTAC solution at . This phenomenon indicates that the wall-normal fluctuations are depressed in drag reducing flow. As mentioned above, some references suppose that the viscoelasticity of the micella in the solution suppresses the turbulent fluctuations toward the bulk flow when drag reduction appears. The decrease of wall-normal velocity fluctuations in 25 ppm CTAC solution at will confirm this assumption. Although 25 ppm CTAC solution at almost loses DR, the wall-normal velocity fluctuations are also smaller than the water in the centre of the channel, which indicates that the suppression also happens in CTAC solution and the micella degrade only near the wall because of the high shear rate.

Figure 6 shows the statistical correlation coefficients between streamwise and wall-normal velocity fluctuations which represents the Reynolds shear stress in turbulence. As we know, the sum of Reynolds shear stress and viscous shear stress obeys the equation of , and Reynolds shear stress dominates in the main flow because the shear rate and viscous shear stress decrease quickly. The water agrees well with the law, but the Reynolds shear stress of 25 ppm CTAC solution at is diminished and surprisingly approaches zero. This phenomenon is called “zero Reynolds shear stress.” In Figure 5, the auto-correlations of both streamwise and wall-normal velocity fluctuations are not zero, but the cross-correlation of the two fluctuations almost becomes zero. Some researches explained this phenomenon with the reason of correlation degradation. In fact, this phenomenon relates to special fluid motions of the solution.

##### 3.3. Distribution of Velocity Fluctuations

In turbulent flow, the distribution of velocity fluctuations is categorized in four quadrants according to the plus-minus of fluctuations as shown in Figure 7. The abscissa of the figure is the streamwise velocity fluctuations (), and the ordinate is the wall-normal velocity fluctuations (). The plus or minus of velocity fluctuations in two direction is dependent on the fluid motions in the flow.

Figures 8 and 9 show the statistical categorization of velocity fluctuations based on one group of 2D velocity frames, about ten thousand of vectors in one identical position. Three positions in direction within the channel flow are selected for comparison. In water flow, the fluctuations are distributed mainly in the second and fourth quadrants with elliptical shape. And with the increase of position, the horizontal angle of ellipse also increases because the streamwise velocity fluctuations () become weak. However, the fluctuations of 25 ppm CTAC solution at are equally distributed in all quadrants. Moreover, the characteristics are almost not changed versus different positions.

It should be pointed out that the products of the two fluctuations are both minus in second quadrant ( and ) and fourth quadrant ( and ). As a result, the Reynolds shear stress of water channel flow will be larger than zero based on this definition. At the same time because the fluctuations of drag-reducing flow are equally distributed in all quadrants, the products of the tow fluctuations will be minus in second and fourth quadrant, plus in first and third quadrant, then the sum will probably be zero.

In Reynolds turbulence transportation as shown in Figure 10(a), if the fluid jumps from the location to () and mixes with the local fluid, the streamwise velocity fluctuation is minus (). This case will lie in the second quadrant. In contrast, if the fluid jumps from the location to () and mixes with the local fluid, the streamwise velocity fluctuation is plus (). This case will lie in the fourth quadrant. Consequently, the fluctuations of water are mainly distributed in the second and fourth quadrants as shown in Figures 8 and 9 because Reynolds transportation as shown in Figure 10(a) dominates the turbulence transportation in the water flow.

In 25 ppm CTAC solution flow at , the fluctuations are distributed not only in the second and fourth quadrants but also in the first and third quadrants, which indicates that there is another type of turbulence transportation besides Reynolds transportation. Figure 10(b) exhibits the assumption of the new turbulence transportation. When the fluid jumps from the location to and does not mix enough with the local fluid then turns back to the location at once, the fluctuations will show and in location which lies in the third quadrant. At the same time, the fluid jumps from the location to and does not mix enough with the local fluid then turns back to the location at once, consequently and which lies in the first quadrant. So two important events occur in drag-reducing flow that the mixture process of fluids is delayed and turnaround transportation happens by some stress driving. In other words, the fluids in drag-reducing flow are vibrating during its flowing toward. Because the wall-normal fluctuations are small, vibration is slight. It is supposed that turnaround transportation in drag-reducing flow possibly relates to the viscoelasticity of the solution.

##### 3.4. Instantaneous 2D Velocity Distribution

Figure 11 shows the instantaneous 2D contour of streamwise velocity (), where is the bulk velocity. Large difference appears between drag-reducing flow and water flow. The contour lines of the velocity in 25 ppm CTAC solution flow at are regular and almost parallel to the wall surface, whereas the contour lines in water flow are irregular. Because 25 ppm CTAC solution flow at loses drag reduction, the contour lines are also irregular as same as the water.

The regular and parallel contour lines indicate that turbulence transportation is depressed in drag-reducing flow. But it does not imply that there is no turbulence transportation which causes the zero Reynolds shear stress. On the other hand, the regular and parallel contour lines also indicate the possibility of the above-mentioned vibration.

Figure 12 shows the contour lines of streamwise velocity fluctuations. It clearly exhibits that the contours of streamwise velocity fluctuations are band-like in drag-reducing flow, but block-like in water flow. Moreover, the bands as shown in Figure 12(a) are developed toward streamwise direction with small declined angle. This phenomenon indicates that turbulence transportation is prevented from the wall toward the main flow by the viscoelasticity of the solution. During this progress, the micella will produce deformation, and the counter elastic force will drive the fluids turning back and then vibrating.

But when the Reynolds numbers is high enough, the micella cannot sustain the large deformation and then break. In this case, the contour of fluctuations exhibits the large declined angle, and high turbulence transportation occurs as shown in Figure 12(b), which is the degradation of DR at .

#### 4. Conclusions

The velocity fluctuations of water and dilute CTAC surfactant solution flow in a two-dimensional channel are studied experimentally by using a PIV system. The phenomenon of “zero Reynolds shear stress” is analyzed by the categorization in four quadrants. The following conclusions are drawn from the present study.(1)The dilute CTAC surfactant solution with the concentration of 25 ppm exhibits the high drag reduction when Reynolds number is below 25000, and then DR degrades quickly and disappears completely at . When DR reaches the highest at , the Reynolds shear stress of the solution decreases and reaches zero at the same time, but the RMS of both two velocity fluctuations is not zero.(2)The fluctuations are analyzed by the categorization in four quadrants. The fluctuations are distributed in all four quadrants equally for 25 ppm CTAC solution at , but distributed mainly in the second and fourth quadrants for water. It is indicated that Reynolds transportation dominates the water flow during the turbulence transportation, whereas turnaround transportation also happens besides Reynolds transportation in drag-reducing flow.(3)Turbulence transportation is depressed in drag-reducing flow, which leads to the regular and parallel contour lines of the streamwise velocity. Band-like contour lines with small declined angle of streamwise velocity fluctuations are found in drag-reducing flow. The phenomenon and turnaround transportation are supposed to depend on the viscoelasticity of the solution.

#### Acknowledgment

This research is supported by National Natural Science Foundation of China and China Scholarship Council.

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