3.1. Asymmetrical Velocity Distribution
shows the distribution of dimensionless streamwise mean velocity (U
) 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
= 0 represents the position of the internal porous wall in the wall-normal direction.
In Figure 4
, it can be seen that the mean velocity profile for the water channel flow is typical as the conventional wall turbulence. The velocity has a large gradient in the near-wall region while it tends to be close in the core region of the channel. By contrast, the mean velocity profile shown in red symbols for the drag-reducing flow without the water layer becomes the typical characteristic of the laminar flow. The velocity gradients from the wall to the channel center are approximate. These results are consistent with those reported in all the studies of turbulent drag-reducing flow with viscoelastic additives [4
]. Furthermore, for the drag-reducing flow case, the mean velocity profile, as shown in the black symbols with an injected water layer, is changed in the near-wall region when compared with the drag-reducing flow without the water layer. The mean velocity distribution becomes asymmetric relative to the channel center under the effect of the injected ultrathin water layer.
The distribution of the mean velocity in the logarithmic layer normalized by the frictional velocity (U+
) 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
From Figure 5
, it can be found that the mean velocity profile in the logarithmic layer for the drag-reducing flow with the water layer is declined relative to that for the drag-reducing flow without the water layer. This may cause a varying velocity gradient in the near-wall region, which has also been shown in Figure 4
for the mean velocity normalized by Ub
. 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.
To further illustrate the symmetry of mean velocity and its break caused by the viscoelastic additives, the mean velocity for another drag-reducing flow case formed by injecting viscoelastic polymer solution in a preliminary study [13
] is presented in Figure 6
along with the water flow cases under the same conditions. The polymer solution at 50 ppm was injected into the water flow from the porous channel wall. It is also seen that the mean velocity for the drag-reducing flow with the injection of polymer solution shown as the red symbols is modified in the near-wall region. The symmetry about the channel center shown for the N and N2N flow cases is broken by the injected viscoelastic polymer solution.
The fields of velocity gradient in the half longitudinal section of the channel near the porous wall for the non-Newtonian flow without and with the injection of the ultrathin water layer are shown in Figure 7
to observe the whole velocity field further.
It was found that, for the viscoelastic drag-reducing flow, as shown in Figure 7
a, the uniform speed-gradient area in the near-wall region extends to the center of the channel (y
= 1), which was similar to the velocity distribution characteristic of the laminar flow. After the water layer is injected from the porous wall, as shown in Figure 7
b, there is an obvious high speed-gradient value near the wall. But the distribution of the velocity gradient is relatively uniform in the range of y
< 0.8 outside this high-gradient region. Meanwhile, the high-gradient area is in the inclined upward streaky structure, which is similar to that shown in Figure 7
a. These results show that under the influence of the injected water layer, the near-wall shear layer indicating by the high-velocity gradient is changed, and the symmetry is broken.
In the following section, more modifications in the near-wall region accompanied by the changes of the velocity distribution will be examined on the basis of the statistical analyses of velocity fluctuations.
3.2. Turbulence Intensity and Anisotropy
shows the distribution of dimensionless turbulence intensity (vrms
) 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 (urms
) for the four flow cases to characterize the anisotropy of the flow in the near-wall region.
It can be seen from Figure 8
that for the drag-reducing flow with the water layer, the wall-normal turbulence intensity has an obvious peak value in the range of y
< 0.2. Meanwhile, the near-wall turbulence intensity increases as the injection rate increases in general. With the increase of the wall-normal distance from y
= 0.3, the turbulence intensity tends to be consistent. This also shows that the drag-reducing flow with the ultrathin water layer presents obvious zonal characteristics and structures, as described in reference [18
]. The turbulence intensity changes obviously in the near-wall region while it appears the same as the NN flow without water layer outside this region. In addition, under the condition of an injection rate of 3 L/min, the peak value of the turbulence intensity appears at y
= 0.040, which moves to the center of the channel with the increase of the injection rate, i.e., the thickened water layer.
The ratio of intensity shown in Figure 9
can reflect the degree of anisotropy in the flow case. It can be seen from the figure that the anisotropy of drag-reducing flow is strengthened when compared with the pure water channel flow and N2N flow case. In particular, for the N2NN flow case, the anisotropy increases obviously in the near-wall region from y/h
= 0 to y/h
= 0.1. This indicates that the near-wall turbulence in viscoelastic drag-reducing flow tends to be dominated by the fluctuation in the streamwise direction. This result is consistent with the possible mechanism of drag reduction effect caused by anisotropic characteristics advocated in reference [21
]. Thus, this anisotropy caused by the injected water layer may bring certain enlightenment to the near-wall modulation of turbulence.
However, in our previous study for the N2N flow case with injection from the porous channel wall, it is also found that the injected water layer affects the turbulence intensity, but it does not affect the distribution of turbulence fluctuation, i.e., the anisotropy, although the injection rate is large. Figure 10
shows the ratios of streamwise to wall-normal turbulence intensity for the water flow without and with the injection of water layer at 18 L/min and 27 L/min [16
]. It shows that the ratios in the near-wall region from y/h
= 0 to y/h
= 0.2 are relatively large and limited in 4 due to the influence of the wall boundary when compared with the ratio as 1 from y/h
= 0.2 in the core region. This indicates once again that the Newtonian water layer in the water flow may not affect the turbulence anisotropy.
3.3. Discussion of the Possible Cause
The typical visual injected water layer obtained by the PLIF technique is shown in Figure 11
a to illustrate the near-wall modification in the channel. This typical water layer is for the drag-reducing flow case with water injection at 9 L/min. The fluorescent region denotes the existing injected dyed water. Although the water diffuses to the location around y/h
= 0.2, as shown in Figure 11
a, the main water layer is found to locate in the region of y/h
= 0.1. This can also be validated from the quantitative concentration distribution, as shown in Figure 11
b. In the figure, CCTAC/C0
is a non-dimensional concentration deduced from 1 − Cwater
. The Cwater
responds to the mean fluorescence intensity at the wall-normal location, and C0
is the largest fluorescence intensity near the wall [22
It can be seen from Figure 11
a that there is a wavy interface between the injected water layer and the main CTAC solution along the flow direction. The injected water forms a striped structure, in which there are viscoelastic solutions with unexcited fluorescence. From Figure 11
b, it can be found that with the increase of injection rate, the concentration of CTAC solution at the same wall-normal position is lower, and the water layer is wider. The maximum value of the average concentration gradient exists near the wall at y/h
= 0.1, which indirectly reflects the presence of a boundary of the ultrathin water layer at this location.
Therefore, under the experimental conditions, the stable viscoelastic fluid region coexists with the injected ultrathin water layer. The injected water layer causes the redistribution of the viscoelastic solution near the wall, which changes the viscoelasticity and reduces the modified distribution of stress near the porous wall. These may be the main reason for the asymmetrical velocity distribution emerged in the N2NN flow case.
In our previous study, the modification of viscoelastic stress and its contribution to the enhanced drag reduction effect was reported [18
]. Here, the near-wall Reynolds shear stress distribution in the measured N and NN flow cases are reexamined. Figure 12
shows the distribution of Reynolds shear stress in the wall-normal direction for the N and NN flow with the water layer injected at various rates. The Reynolds shear stress (
) shown in the figure is normalized by the square of the frictional velocity.
It can be seen from Figure 12
that in the NN flow without water layer, the Reynolds shear stress at different positions is greatly reduced relative to the water flow and N2N flow case and its value is close to zero, which conforms to the general characteristics of the drag-reducing flow with viscoelastic additives [23
]. However, the Reynolds shear stress for the N2NN flow case increases slightly within the range of y/h
< 0.40 and increases with the increase of the injection rate. It can be verified that the injected water layer reduces the near-wall viscoelasticity of the NN flow from the other perspective. Consequently, the relaxation characteristic of a near-wall viscoelastic solution is changed, leading to the modification of velocity distribution. Accordingly, the redistributed stress plays a key role in the break of the symmetry for the mean velocity distribution in the drag-reducing channel flow. A physical hypothesis that the redistributed stress causes the modification of the hairpin eddy structure, leading to the redistribution of velocity, may contribute to explain the relationship between the redistributed Reynolds stress and the mean velocity distribution.
It is worth noting that the adopted Re = 20,000 is adopted in the present study because the obtained drag reduction effect is stable and moderate. It indicates the experimental condition in general. We conducted measurements at other Reynolds number (Re = 30,000, 40,000) for the N2NN flow case. The similar asymmetric velocity in the channel and the corresponding drag reduction enhancement were also obtained [18
]. The stress distribution for all the NN flow cases will be modified due to the injected solution.