# Asymmetrical Velocity Distribution in the Drag-Reducing Channel Flow of Surfactant Solution Caused by an Injected Ultrathin Water Layer

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## Abstract

**:**

^{−4}m/s) into the drag-reducing flow of surfactant solution. The instantaneous concentration and flow fields were measured by using planar laser-induced fluorescence (PLIF) and particle imaging velocimetry (PIV) techniques, respectively. Moreover, analyses on turbulent statistical characteristics and spatial distribution of viscoelastic structures were carried out on the basis of comparison among various flow cases. The results showed that the injected ultrathin water layer under present experimental conditions affected the anisotropy of the drag-reducing flow. The characteristics, such as turbulence intensity, showed the zonal feature in the wall-normal direction. The Reynolds shear stress was enhanced in the near-wall region, and the viscoelastic structure was modified severely due to the redistributed stress. These results may provide experimental supports for the near-wall modulation of turbulence and the exploration of the drag-reducing mechanism by viscoelastic additives.

## 1. Introduction

## 2. Experiments

#### 2.1. Experimental Equipment

#### 2.2. PIV/PLIF Arrangements

#### 2.3. Experimental Conditions

^{−6}m

^{2}/s. This viscoelastic CTAC solution flowed in the loop, which formed a drag-reducing flow in the channel. The average bulk velocity (U

_{b}) of the channel flow was set as 0.44 m/s. The experimental Reynolds number is 20,000, and the adopted characteristic length was the full height of the channel. The temperature of the CTAC solution was maintained at 25 °C, and the accuracy is ±0.50 °C. This condition was defined as the viscoelastic drag-reducing flow without water injection. For the convenience of comparison, it is also called the condition of the injection rate as 0 L/min.

^{−5}, 1.80 × 10

^{−4}and 2.70 × 10

^{−4}m/s, respectively. The injection rate of 3 L/min was 0.56% of the average flow rate. The injected water formed a water layer near the porous wall. We use an equivalent thickness to quantify the amount of the injected water in terms of the dosing rate and the dosing time, which was 120 s for each measuring case determined by the stable period after injection. The near-wall velocity of the supposed water layer was assumed to be 0.4 U

_{b,}according to the general mean velocity profile. Thus, the equivalent thickness was estimated to be 0.03 h when the dosing rate is 3 L/min. The equivalent thickness was 0.09 h when the dosing rate is 9 L/min. This thickness of the water film at the rate of 3 L/min was 16.5 in y

^{+}, which is the non-dimensional wall-normal distance from the porous wall and equals yu

_{τ}/ν (ν is the kinematic viscosity of solvent). Actually, because the water layer remains flowing downstream and spreads on a long channel wall rather than only the porous plate, the thickness of the water layer will be smaller than the estimated value. The equivalent thickness of the water layer is finite and thought as ultrathin.

## 3. Results and Discussion

#### 3.1. Asymmetrical Velocity Distribution

_{b}) in the longitudinal section of the channel under the same Reynolds number for the viscoelastic drag-reducing flow with and without water layer along with the water flow. For the shown drag-reducing channel flow with the water layer, the relative large injection flow rate of 9 L/min was chosen to reflect the velocity difference more clearly. In the figure, y/h = 0 represents the position of the internal porous wall in the wall-normal direction.

^{+}= U/u

_{τ}) for the water flow case and the drag-reducing flow case are shown in Figure 5 to examine the modification of velocity in the near-wall region carefully. The phenomenon of the profiles slightly below the theoretical log-law profile that emerged in the figure, has been validated to be caused by the effect of the permeable and rough wall [19].

_{b}. In addition, it is observed that the slope of the profile in the logarithmic region is increased relative to the water flow case. It is still lower than the slope of Virk’s asymptote, which has also been universally recognized and reported by White et al. [20]. However, the water flow case with and without the water layer has the uniform mean velocity profile, which indicates the water layer itself does not affect the velocity profile. The existing results also showed that the low-speed water injection did not affect the mean velocity profile in [13]. Thus, by comparison, it is speculated that the changed viscoelasticity in the near-wall region of the NN flow case caused by the injected water results in the modification of velocity distribution. The symmetrical mean velocity turns into asymmetrical distribution.

#### 3.2. Turbulence Intensity and Anisotropy

_{rms}/u

_{τ}) in the wall-normal direction from the porous wall to the channel center for the water flow, drag-reducing flow without and with the injection of water layer at various injection rates. The N2N flow case is with the injection rate of 9 L/min. Figure 9 shows the ratio of wall-normal turbulence intensity (u

_{rms}/v

_{rms}) for the four flow cases to characterize the anisotropy of the flow in the near-wall region.

#### 3.3. Discussion of the Possible Cause

_{CTAC}/C

_{0}is a non-dimensional concentration deduced from 1 − C

_{water}/C

_{0}. The C

_{water}responds to the mean fluorescence intensity at the wall-normal location, and C

_{0}is the largest fluorescence intensity near the wall [22].

## 4. Conclusions

- Under the experimental conditions, the injected ultrathin water layer breaks the symmetry of the mean velocity distribution about the channel centerline. The normalized velocity in the logarithmic layer is descended, and the near-wall structure of the shear layer indicted by the velocity gradient is modified in the viscoelastic drag-reducing flow with the water layer when compared with the case without water layer.
- The injected water layer leads to the enhancement of turbulence fluctuation intensity. The turbulence anisotropy in the drag-reducing flow with the injected water layer is also promoted while it does not change in the water flow with the injection of the water layer. The change of the viscoelastic solution near the wall plays a key role.
- Under the influence of the ultrathin water layer, the viscoelastic drag-reducing flow presents the zonal characteristics of layers of water and viscoelastic solution in the wall-normal direction with a wavy boundary. The redistributed viscoelastic solution in the near-wall region causes the modification of stress distribution. The near-wall Reynolds shear stress is increased in the drag-reducing flow. This redistributed stress results in the break of the symmetric velocity and the related structure.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematics of turbulent flow cases of (

**a**) N, (

**b**) NN, (

**c**) N2N, and (

**d**) N2NN in the channel.

**Figure 4.**Distributions of mean velocities in the longitudinal section of the channel for different flow cases.

**Figure 5.**Distributions of mean streamwise velocities in the logarithmic layer for different flow cases.

**Figure 6.**The mean velocities of the water flow cases with injecting 50 ppm polymer solution, without and with the injection of water.

**Figure 7.**Instantaneous fields of streamwise velocity gradients (in s

^{−1}) for the drag-reducing flow (

**a**) without water layer and (

**b**) with water layer.

**Figure 9.**Ratios of streamwise turbulence intensity to wall-normal intensity for different flow cases.

**Figure 11.**PLIF measurements for the N2NN flow case about (

**a**) typical instantaneous PLIF images and (

**b**) concentration distribution.

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**MDPI and ACS Style**

Fu, Z.; Liang, X.; Zhang, K.
Asymmetrical Velocity Distribution in the Drag-Reducing Channel Flow of Surfactant Solution Caused by an Injected Ultrathin Water Layer. *Symmetry* **2020**, *12*, 846.
https://doi.org/10.3390/sym12050846

**AMA Style**

Fu Z, Liang X, Zhang K.
Asymmetrical Velocity Distribution in the Drag-Reducing Channel Flow of Surfactant Solution Caused by an Injected Ultrathin Water Layer. *Symmetry*. 2020; 12(5):846.
https://doi.org/10.3390/sym12050846

**Chicago/Turabian Style**

Fu, Zaiguo, Xiaotian Liang, and Kang Zhang.
2020. "Asymmetrical Velocity Distribution in the Drag-Reducing Channel Flow of Surfactant Solution Caused by an Injected Ultrathin Water Layer" *Symmetry* 12, no. 5: 846.
https://doi.org/10.3390/sym12050846