Coupling effect of wall slip and spanwise oscillation on drag reduction in turbulent channel flow

The coupling effect of the isotropic wall slip and the spanwise oscillation boundary conditions on the drag reduction and turbulence properties are studied by direct numerical simulations. As the slip length increases, the drag reduction gradually changes from the oscillation dominated to the slip dominated. The increase of slip length will decrease the maximum spanwise velocity of the fluid on the wall, which is responsible for the decreased ability of the oscillatory wall motion to reduce the skin-friction drag. The drag reduction decomposition shows that the contribution from the modifications of turbulent dynamics will undergo a shift from drag increase to drag reduction as the isotropic slip length increases. Compared with the respective no-slip reference flow, the drag reduction of wall slip in laminar and turbulent channel flows can be expressed in a unified form versus the outer scale slip length. Furthermore, many aspects of the turbulence properties are influenced by the coupling effect. First, the wall slip condenses the envelope range of the phase fluctuations caused by the oscillatory wall motion, as well as the magnitude of the periodic fluctuation of the phase-averaged friction coefficient. Second, an unexpected property of the coupled boundary condition is found that the existence of the Stoke layer delays the relaminarization process caused by the large slip length. Third, the wall slip would narrow the periodic inclination of the streaks and then inhibit the energy transfer process in the horizontal direction. Fourth, in terms of phase, the Stokes strain and the shear angle have the same lag phase with the spanwise velocity, while the hysteresis of turbulent dynamics leads to the larger lag phase of the streaks and phase-averaged friction coefficient. These new features are valuable for increasing knowledge on this topic.

Figure 1

Schematic diagram of the physical problem. The coupled boundary condition of wall slip and spanwise oscillation. $W\left(t\right)$ is the oscillating wall velocity in the spanwise direction. $u_s$ and $w_s$ is the streamwise and spanwise slip velocities of the fluid on the wall, respectively. $l_b$ is the isotropic slip length.

Figure 2

Validation of the numerical method. (a) The RMS velocity fluctuations scaled with $u_{\tau 0}$ in global coordinates. (b) Streamwise one-dimensional energy spectra of $y^{+0}=5.4$ (scaled with $u_{\tau 0}^2/2$) versus the streamwise wave number $k_x$. “Present” represents the result of the current work. “KMM 87” represents the result of Kim et al. [47]. Case ${\mathrm{Re}}_{\tau 0}=180$.

Figure 3

(a) DR and (b) ${\mathrm{DR}}_{\mathrm{loss}}/{\mathrm{DR}}_{\mathrm{osci}}$ versus the slip length $l_b^{+0}$. The oscillation parameters corresponding to ${\mathrm{DR}}_{\mathrm{osci}}$ and ${\mathrm{DR}}_{\mathrm{coup}}$ are $I_m^{+0}=12$ and $T^{+0}=92$. The delta and gradient symbols represent the ${\mathrm{Re}}_{\tau 0}=180$ cases and the ${\mathrm{Re}}_{\tau 0}=396$ cases, respectively.

Figure 4

Contour lines of ${\mathrm{DR}}_{\mathrm{coup}}$ versus the slip length $l_b^{+0}$ and (a) $I_m^{+0}$, (b) $T^{+0}$. The dots represent the DNS cases, and their parameters are listed in Table 2. Case ${\mathrm{Re}}_{\tau 0}=180$.

Figure 5

Contour lines of ${\mathrm{DR}}_{\mathrm{loss}}/{\mathrm{DR}}_{\mathrm{osci}}$ versus the slip length $l_b^{+0}$ and (a) $I_m^{+0}$, (b) $T^{+0}$. The dots represent the DNS cases, and their parameters are listed in Table 2. Case ${\mathrm{Re}}_{\tau 0}=180$.

Figure 6

The phase-averaged spanwise velocity of the fluid on the wall when $l_b^{+0}=0,3.2,10,32$, respectively. $w^{+0}$ is the spanwise velocity of the fluid scaled with $u_{\tau 0}$. The oscillation parameters of the cases are $I_m^{+0}=12$ and $T^{+0}=184$. “PH1” and “PH16” stand for phase $\pi /8$ and $2\pi$, respectively. The interval between adjacent phases is $\pi /8$. Case ${\mathrm{Re}}_{\tau 0}=180$.

Figure 7

(a) The maximum spanwise velocity of the fluid on the wall $w_{max}^{+0}$ versus the slip length $l_b^{+0}$ ($I_m^{+0}=12$, $T^{+0}=184$). (b) The DNS results coincide with the Stokes theoretical solution after superimposing the lag phases $\phi$ ($l_b^{+0}=10$). Case ${\mathrm{Re}}_{\tau 0}=180$.

Figure 8

(a) The DNS results of ${\mathrm{DR}}_{\mathrm{osci}}$ in the present study and their fitting curve versus $I_m^{+0}$ when $T^{+0}=92$. The fitting function is ${\mathrm{DR}}_{\mathrm{osci}}\times 100=0.0015{\left(I_m^{+0}\right)}^3-0.1577{\left(I_m^{+0}\right)}^2+4.6666\left(I_m^{+0}\right)$. “QR 04” represent the DNS result of Quadrio et al. [42] at $T^{+0}=100$, and ${\mathrm{Re}}_{\tau 0}=200$. (b) Drag reduction loss ${\mathrm{DR}}_{\mathrm{loss}}$ and ${\mathrm{DR}}_{\mathrm{loss}}^{\mathrm{osci}}$ versus $l_b^{+0}$. Case ${\mathrm{Re}}_{\tau 0}=180$.

Figure 9

Decomposition of the mean velocity profile in turbulent channel flow. $①$ represents the mean slip velocity on the wall, i.e., $u_s$. $②$ is the remaining part after deducting $u_s$ from the mean velocity profile, and $②$ is zero on the wall.

Figure 10

The DNS results of [(a) and (c)] $C_1$, $C_2$, and [(b) and (d)] $C_2/C_1$ versus the inner scale slip length $l_b^{+0}$. “slip” represents the flow under the isotropic slip boundary condition. “couple” represents the flow under the coupled boundary condition with $I_m^{+0}=12$ and $T^{+0}=92$. Panels (a) and (b) are the results of the cases of ${\mathrm{Re}}_{\tau 0}=180$. Panels (c) and (d) are the results of the cases of ${\mathrm{Re}}_{\tau 0}=396$. In (b), “MK 04” and “BS 12” represent the DNS results of Min et al. [23] and Busse et al. [25] (isotropic slip boundary condition, ${\mathrm{Re}}_{\tau 0}=180$), respectively. In (d), “BS 12” represents the DNS result of Busse et al. [25] (isotropic slip boundary condition, ${\mathrm{Re}}_{\tau 0}=360$). The way to calculate $C_1$ and $C_2$ of Refs. [23, 25] is $C_1=h〈{\tau }_w^{\mathrm{ref}}〉/\mu U_b$ and $C_2\left(l_b^{*}\right)={\mathrm{DR}}_r/l_b^{*}\zeta -C_1$.

Figure 11

Mean velocity profiles (a) $〈u^{+}〉$ and (b) $〈u^{+}〉-〈u_s^{+}〉$ in wall units scaled with the actual friction velocity. “MK 04” and “TL 12” represent the DNS results of Min et al. [23] (the isotropic slip boundary condition) and Touber et al. [43] (the spanwise oscillation boundary condition), respectively.

Figure 12

The Reynolds shear stress $-〈u^{\prime }{v}^{\prime }〉$ normalized by $u_{\tau 0}^2$ in global coordinates. $y/h=-1$ represents the lower wall of the channel. The red lines represent the results under the isotropic slip boundary condition, and the green lines represent the results under the coupled boundary condition with $I_m^{+0}=12$ and $T^{+0}=184$. The arrows in the figure indicate the change of the peak position of the Reynolds shear stress as the slip length increases.

Figure 13

(a) The weighted Reynolds shear stress $-y〈u^{\prime }{v}^{\prime }〉$ of the upper half channel in wall units scaled with $u_{\tau 0}^{2}h$. Panels (b), (c), and (d) are the RMS velocity fluctuations (scaled with $u_{\tau 0}$) of the streamwise, wall-normal, and spanwise directions, respectively. The blue and yellow thin solid lines represent different phases of the “Osci” case and the “Coup” case, respectively. The thick lines represent the time-averaged results.

Figure 14

(a) The temporal variations of the wall-averaged friction coefficients of the four cases (solid lines). The large-scale components of the “Osci” case and the “Coup” case are represented by $r_{\mathrm{osci}}$ (dashed line) and $r_{\mathrm{coup}}$ (dash-dot line), respectively. (b) The small-scale fluctuations of the friction coefficients of the “Osci” case and the “Coup” case. For the “Osci” case, $n=3$. For the “Coup” case, $n=2$. Panels (c) and (d) are the periodic fluctuations of the phase-averaged friction coefficients of the “Osci” case and the “Coup” case, respectively. The vertical bars represent the standard deviation calculated from the data without large-scale components.

Figure 15

$P_k$ from each quadrant scaled with $u_{\tau 0}^3/{\delta }_{\nu }$ of the turbulent baseline flow. Black long dashed lines are the total $P_k$ of the four quadrants.

Figure 16

The joint PDF of $u^{\prime }$ and $v^{\prime }$ at $y^{+0}=10.5$, scaled with $u_{\tau 0}$. (a) The “Base” case (black dashed lines) compared with the “Slip” case (solid red lines). The probability density in this figure ranges from 100 to 10. (b) The “Osci” case (blue lines) compared with the “Coup” case (yellow lines), solid lines represent the time-averaged results, and the dotted lines represent the phase-averaged results. The probability density in this figure ranges from 800 to 100.

Figure 17

Contours of the phase-averaged Stokes strain scaled with $u_{\tau 0}/{\delta }_{\nu }$. (a) The “Osci” case. (b) The “Coup” case. The phases corresponding to the maximum and minimum of the periodic fluctuations of the friction coefficient are also marked with vertical dashed lines in the two figures. The maximum and minimum are obtained by Fourier fitting of the curves shown in Figs. 14 and 14. 0 and T/2 correspond to the zero-velocity states of the wall.

Figure 18

The distributions of the anisotropy ratio. (a) $〈u_i^{\prime }{u}_j^{\prime }〉/〈u_i^{\prime }{u}_i^{\prime }〉$. (b) $〈{\omega }_i^{\prime }{\omega }_j^{\prime }〉/〈{\omega }_i^{\prime }{\omega }_i^{\prime }〉$. In both figures, the line style is used to distinguish different physical quantities. The “Base,” “Slip,” “Osci,” and “Coup” cases are distinguished by the black, red, blue, and yellow lines, respectively. In (a), the delta and gradient symbols represent the $〈u^{\prime }{u}^{\prime }〉/〈u_i^{\prime }{u}_i^{\prime }〉$ and $〈w^{\prime }{w}^{\prime }〉/〈u_i^{\prime }{u}_i^{\prime }〉$ of the DNS results from Touber et al. [43].

Figure 19

Contours of the phase-averaged (a) $\widetilde{u^{\prime }{u}^{\prime }}$ and (b) $\widetilde{w^{\prime }{w}^{\prime }}$, normalized by the inner scale of the turbulent baseline flow. The contour maps represent the results of the “Osci” case, and the colored contour lines represent the results of the “Coup” case, using the same color legend. The vertical dashed lines have the same meaning as in Fig. 17.

Figure 20

Budgets of the phase-averaged [(a) and (b)] $\widetilde{u^{\prime }{u}^{\prime }}$ and [(c) and (d)] $\widetilde{w^{\prime }{w}^{\prime }}$, normalized by the inner scale of the turbulent baseline flow. The left-hand and the right-hand columns represent the results of the “Osci” case and the “Coup” case, respectively.

Figure 21

Contours of the phase-averaged (a) $\widetilde{{\omega }_x^{\prime }{\omega }_x^{\prime }}$ and (b) $\widetilde{{\omega }_z^{\prime }{\omega }_z^{\prime }}$, normalized by the inner scale of the turbulent baseline flow. The contour maps represent the results of the “Osci” case, and the colored contour lines represent the results of the “Coup” case, using the same color legend. The vertical dashed lines have the same meaning as in Fig. 17.

Figure 22

Phase variations of [(a) and (b)] the $\widetilde{{\omega }_x^{\prime }{\omega }_x^{\prime }}$ budget and [(c) and (d)] the $\widetilde{{\omega }_z^{\prime }{\omega }_z^{\prime }}$ budget, as well as [(e) and (f)] the productions of $\widetilde{{\omega }_x^{\prime }{\omega }_x^{\prime }}$ and [(g) and (h)] the productions of $\widetilde{{\omega }_z^{\prime }{\omega }_z^{\prime }}$, normalized by the inner scale of the turbulent baseline flow. The left-hand and the right-hand columns represent the results of the “Osci” case and the “Coup” case, respectively.

Figure 23

Imbalances in the budgets [sum of the right-hand sides of Eq. (34)] for (a) the “Osci” case and (b) the “Coup” case. The imbalances of $\widetilde{{\omega }_y^{\prime }{\omega }_y^{\prime }}$ are also shown in this figure.

Figure 24

Vortices identified by the $Q$ criterion. $Q=1$, normalized by the outer scale of the turbulent baseline flow, i.e., $U_b^2/h^2$. The color on the vortices represents the value of $u^{\prime }/U_b$, i.e., the streamwise velocity fluctuations scaled with the bulk mean velocity. The background is $u^{\prime }/U_b$ at $y^{+0}=5.4$, and the regions where $u^{\prime }/U_b<-0.05$ are colored by dark grey (the low-speed streaks). The “Osci” case and the “Coup” case are in their respective phase of maximum inclination angle.

Figure 25

Panels (a) and (b) are the contours of $\widetilde{R_{u^{\prime }{u}^{\prime }}}$ at $y^{+0}=5.4$ of the “Osci” case and the “Coup” case, respectively. For panels (a) and (b), both cases are in their respective phase of maximum inclination angle. Panels (c) and (d) are the phase variations of the isoline with $\widetilde{R_{u^{\prime }{u}^{\prime }}}=0.4$, including the maximum inclination angle ${\gamma }_{max}$ measured from the angle between the tangent line and the streamwise direction. Panels (c) and (d) are the results of the “Osci” case and the “Coup” case, respectively. “PH1” and “PH16” stand for phase $\pi /8$ and $2\pi$, respectively. The interval between adjacent phases is $\pi /8$.

Figure 26

(a) The phase variations of the shear angle ${\gamma }_s$ at the $y^{+0}=5.4$, 10, and 15 wall-normal planes and (b) the periodic change of the inclination angle $\gamma$ at the $y^{+0}=5.4$ plane. In (a), the blue and yellow lines represent the results of the “Osci” case and the “Coup” case, respectively. The solid, dashed, and dash-dot lines represent the results of the $y^{+0}=5.4$, 10, and 15 planes, respectively. The lag phases of $\widetilde{\partial w/\partial y}$ and $\widetilde{C_f}$ are also indicated in the legend for comparison (the width of the gray rectangle). In both figures, the horizontal axis coordinates of the blue and yellow bars represent the phases corresponding to the maximum or minimum shear angles. The width of the gray area between the blue and yellow bars at the same wall-normal plane represents the lag phase.