Drag Reduction Effect of Streamwise Traveling Wave-Like Wall Deformation with Spanwise Displacement Variation in Turbulent Channel Flow

Abstract

Direct numerical simulation of a fully developed turbulent channel flow controlled using a streamwise traveling wave having a periodicity not only in the streamwise direction but also in the spanwise direction, referred to as wave-machine-like traveling wave, is performed to investigate the impact of the spanwise variation of the streamwise traveling wave on the drag reduction effect. The maximum drag reduction rate attained in the present study is smaller than that in the case of the spanwise-uniform traveling wave. The drag reduction rate increases as the spanwise wavelength increases, and the drag reduction effect can be obtained in the range of $\lambda _z^+ > 400$. According to the analysis of the phased-averaged Reynolds shear stress (RSS), the wave-machine-like traveling wave makes the flow field more uniform in the streamwise direction and non-uniform in the spanwise direction, as compared to the case of spanwise-uniform traveling wave. While the turbulent component of RSS in the antinode plane is suppressed near the wall, that in the node plane is significantly suppressed far from the wall. An analysis using the Fukagata–Iwamoto–Kasagi identity shows that the drag reduction effect, which was primary due to the significant decrease in the contribution from the turbulent component of the RSS in the case of spanwise-uniform traveling wave, is partly deteriorated by the contribution from the periodic (i.e., dispersive) component of RSS in the case of wave-machine-like traveling wave.

Appendix 1: Parametric Study on \(\left( a,~k_x,~c \right)\)

As a preliminary study, we have performed a parametric study of the control parameters of the velocity amplitude, a, streamwise wavenumber, k_x, and phasespeed, c. The velocity amplitude, streamwise wavenumber, and phasespeed are determined based on the results for the spanwise-uniform cases (Nabae et al. 2020), i.e., $a^+ = 5,~10$, $k_x^+ = 0.006,~0.011,~0.022$, and $c^+ = 20,~25,~30,~35,~60,~90,~120$, while the spanwise wavelength, $\lambda _z^+$, is set to $\lambda _z^+ = 1131$. The grid resolution in each direction is $\left( \Delta \xi _1^+,~\Delta \xi _2^+,~\Delta \xi _3^+ \right) = \left( 8.8,~0.9-6.0,~4.4 \right)$, which is a little coarser than that in the present study, i.e., $\left( \Delta \xi _1^+,~\Delta \xi _2^+,~\Delta \xi _3^+ \right) = \left( 8.8,~0.9-6.0,~3.9 \right)$.

Figure 13 shows the contour of the drag reduction rate in $c^+$–$k_x^+$ plane at $\mathrm{Re}_\tau = 360$ and $a^+ = 5,~10$. For $c^+ \ge 30$, regardless of the velocity amplitude, lower phasespeed results in higher drag reduction rate. In the cases with lower phasespeed, i.e., $c^+ \lesssim 60$, the drag reduction rate increases as the wavenumber increases, while in the cases with higher phasespeed, i.e., $c^+ \gtrsim 60$, the opposite trend is observed. These trends are similar to those in the case of the spanwise-uniform traveling wave (Nabae et al. 2020). Therefore, the relationship between the control parameters ($a,~k_x,~c$) and the drag reduction rate in the case of wave-machine-like traveling wave is basically similar to that in the case of spanwise-uniform traveling wave.

Fig. 13

Contour of the drag reduction rate, $R_{D}$, in $c^+$–$k_x^+$ plane at $\mathrm{Re}_\tau = 360$ and $\lambda _z^+=1131$; a $a^+ = 5$; b $a^+ = 10$. Circle, square, cross, and triangle represent the maximum drag reduction, drag reduction, drag increase, and little change cases, respectively