Reynolds stresses transport in a turbulent channel flow subjected to streamwise traveling waves

Reynolds stresses transport in a turbulent channel flow under streamwise traveling waves is analyzed in detail using direct numerical simulations to gain physical insights into the mechanism of drag reduction. Streamwise traveling waves are known to produce larger drag reduction margins compared to simple homogeneous wall oscillations. The aim of the current investigation is to identify and analyze the direct effects arising from streamwise traveling waves that leads to larger drag reduction margins compared to simple homogeneous wall oscillations. Several cases were considered, with amplitudes ranging from 0.15 to 1.25 (in outer units) at fixed angular frequency and wave number of 0.16 and 1.66 (in outer units), respectively, to yield drag reduction margins ranging from 26% to 58%, respectively. Streamwise traveling waves of large amplitudes were found to block the intercomponent energy transfer, resulting in shut off of the near-wall buffer layer dynamics. The analyses here suggest that the combined effect of loss of communication between low and high buffer layers with damping in the wall-normal Reynolds stress component is associated to the traveling wave effect and results in larger drag reduction margins.

Figure 1

Phasewise variations of phase-averaged spanwise velocity $\langle W_0\rangle$ at different wall-normal locations $y_0$ for $A0.50$ case. The red contours in the left figure correspond to positive values, while the blue contours correspond to negative values.

Figure 2

(a) Initial response of the normalized skin friction coefficient $(C_f/C_{f0})$ after the actuation of control for different forcing amplitudes. (b) Mean velocity profiles for the uncontrolled and controlled cases.

Figure 3

Reynolds stresses profiles for uncontrolled and controlled cases. (a) streamwise component $\overline{uu}$, (b) spanwise component $\overline{ww}$ (scaled with local friction velocity), (c) wall-normal component $\overline{vv}$ and Reynolds shear stress $\overline{uv}$, and (d) spanwise component ${\overline{ww}}_0$ (scaled with the friction velocity of the uncontrolled flow). The profiles in panels (a), (b), and (c) were scaled with the local friction velocity, while the profiles in panel (d) were scaled with the friction velocity of the uncontrolled flow. Refer to Fig. 2 for markers corresponding to different cases.

Figure 4

Extra production terms (a) $\overline{P_{ww}^1}=-2\overline{\langle wu\rangle \partial \tilde{W}/\partial x}$ and (b) $\overline{P_{ww}^2}=-2\overline{\langle wv\rangle \partial \tilde{W}/\partial y}$ appearing in the transport equation of spanwise Reynolds shear stress component $\overline{ww}$. Refer to Fig. 2 for markers corresponding to different cases.

Figure 5

Wall-normal distributions of terms appearing in the transport equations of Reynolds stresses components. (a) Production $\overline{P_{uu}}$, dissipation $-\overline{{\epsilon }_{uu}}$; (b) velocity-pressure gradient $\overline{{\mathrm{\Pi }}_{uu}}$ term for streamwise component $\overline{uu}$; (c) velocity-pressure gradient $\overline{{\mathrm{\Pi }}_{vv}}$, dissipation $-\overline{{\epsilon }_{vv}}$ term for wall-normal component $\overline{vv}$; (d) velocity-pressure gradient $\overline{{\mathrm{\Pi }}_{uv}}$, production $\overline{P_{uv}}$ term for shear stress component $\overline{uv}$; (e) production $\overline{P_{ww}}$, dissipation $-\overline{{\epsilon }_{ww}}$; and (f) velocity-pressure gradient $\overline{{\mathrm{\Pi }}_{ww}}$ term for spanwise component $\overline{ww}$. Refer to Fig. 2 for markers corresponding to different cases.

Figure 6

Comparsion of transport terms in the $\overline{uu}$ budget for HWO case (solid lines) with Touber and Leschziner [21] data (broken lines) at ${\text{Re}}_{\tau }=500$ at same control parameters.

Figure 7

Wall-normal distribution of (a) the streamwise component of turbulent enstrophy $\overline{{\omega }_x{\omega }_x}$ and (b) its production term. Note in the above figure that the local maxima and minima emanating from different cases are relatively well regrouped for $A^{*}<0.75$ including HWO. The profiles gradually deviate once $A^{*}>0.75$, pointing at severe alterations of the near-wall coherent eddies regeneration process. Refer to Fig. 2 for markers corresponding to different cases.

Figure 8

(a) Phasewise variations of $-\langle uw\rangle$ and $-\langle vw\rangle$ at $y=15$ for $A1.25$ case. Phasewise variations of $\langle P_{ww}^1\rangle$, $\langle P_{ww}^2\rangle$, $\langle P_{ww}\rangle$, $-\langle {\epsilon }_{ww}\rangle$, and $\langle {\mathrm{\Pi }}_{ww}\rangle$ at $y=5$ for (b) $A1.25$ and (c) $A0.30$ cases, respectively. Note how $\langle {\mathrm{\Pi }}_{ww}\rangle$ for $A1.25$ case is entirely frozen compared to $A0.30$ case where there are large modulations.

Figure 9

Streamwise velocity fluctuations $\left(u\right)$ at $y=10$ for (a) uncontrolled, (d) HWO, (g) $A0.50$, and (j) $A1.25$ cases, respectively. The blue color represents the low-speed $u$-streaks $(u<0)$, while the red color represents the high-speed $u$-streaks $(u>0)$. The contours are in the range $-3$ to $+3$. Note that $u$ was scaled by the local friction velocity; had it been scaled by the friction velocity of the uncontrolled flow, the streaky structures would have disappeared, especially for the large DR cases. Spanwise velocity fluctuations $\left(w\right)$ at $y=10$ for (b) uncontrolled, (e) HWO, (h) $A0.50$, and (k) $A1.25$ cases, respectively. The blue color represents the low-speed $w$-streaks $(w<0)$, while the red color represents the high-speed $w$-streaks $(w>0)$. The contours are in the range $-2$ to $+2$. Note that $w$ was scaled by the local friction velocity. The instantaneous visualizations of the $\partial w/\partial x$ shear layers at $y=10$ for (c) uncontrolled, (f) HWO, (i) $A0.50$, and (l) $A1.25$ cases, respectively. The blue color represents the negative, while the red color represents the positive values. The contours are in the range $-0.1$ to $+0.1$. Here also $\partial w/\partial x$ was scaled by the local friction velocity. Note that the heavily modulated $\mathrm{\Lambda }$-shaped structures of $\partial w/\partial x$ start to appear for cases with large imposed amplitudes of STW. These structures are absent for both the uncontrolled and HWO cases.

Figure 10

(Top View) Instantaneous near-wall vortical structures (${\lambda }_2=-0.02$) scaled by the local inner variables for the $A1.25$ case. The regions where ${\omega }_x>0$ are colored in red and where ${\omega }_x<0$ are colored in blue.

Figure 11

(Side View) Instantaneous near-wall vortical structures (${\lambda }_2=-0.02$) scaled by the local inner variables for the $A1.25$ case. The regions where ${\omega }_x>0$ are colored in red and where ${\omega }_x<0$ are colored in blue.

Figure 12

Comparison of (a) root-mean-square velocity and (b) root-mean-square vorticity components in the streamwise, spanwise, and wall-normal directions with the data of Moser and Kim [47] (in markers).

Figure 13

Reynolds stresses profiles for the $A1.25$ case scaled with the local friction velocity with the corresponding error margins. (a) Streamwise component $\overline{uu}$, (b) spanwise component $\overline{ww}$, (c) wall-normal component $\overline{vv}$, and (d) shear stress $-\overline{uv}$.