3.1. Bifurcation diagram of EQ states

The bifurcation diagram of the various equilibrium solutions is shown in figure 1. Each of the solutions emerges through a saddle–node bifurcation. The solution that appears at the smallest Reynolds number is EQ0 from Nagata (Reference Nagata1990) at critical Reynolds number ${Re_c} \approx 127$, followed by EQ7 at ${Re_c} \approx 167$ from Gibson et al. (Reference Gibson, Halcrow and Cvitanovic2009). The numerical continuation over the computational domain size indicates that EQ7 does not exist in the domain $\varOmega _N$, and EQ0 does not exist in the domain $\varOmega _{HKW}$. A large number of solutions, including EQ1, EQ4 and EQ7, were found previously in $\varOmega _{HKW}$. The bifurcation curves for EQ1 and EQ4 are more complicated. EQ1 appears through a saddle–node bifurcation at ${Re_c}=386$, then second and third bifurcations occur on the upper branch at ${Re_c}=408$ and $435$, respectively. EQ4 first arises from a saddle–node bifurcation at ${Re_c}=275$, then a second saddle–node bifurcation appears at ${Re_c}=494$ from the lower branch, and at ${Re_c}=438$ from the upper branch, respectively, giving rise to a new branch. For further details about the invariant solutions in $\varOmega _{HKW}$, the reader may refer to Gibson et al. (Reference Gibson, Halcrow and Cvitanovic2009). The bifurcation curve for a periodic orbit (PO) identified in $\varOmega _{HKW}$ is also presented in the figure by a green dotted line. The saddle–node bifurcation behaviour of PO is very similar to that of EQ0 and EQ7, with a critical Reynolds number located at ${Re_c}=236.5$.

Figure 1.

Bifurcation diagram of the equilibrium solutions in the ${C_f}$–$Re$ plane: EQ0, black thick solid line; EQ1, red thin solid line; EQ4, pink dash-dotted line; EQ7, blue dashed line; PO, green dotted line. The vertical dashed line indicates $Re=400$.

It is known that there is an optimal oscillation period at ${T^{+}} \approx 100$ for spanwise wall oscillation control. Focusing on EQ0, we show that such an optimal oscillation period appears in the bifurcation diagram. Figure 2(a) presents the bifurcation curves for EQ0 with six different oscillation periods in the range $T {U_w}/h\in [1,150]$ for a given wall oscillation amplitude ${A_w}/{U_w}=0.1$. As the focus is on the saddle–node point, here the viscous wall units of the uncontrolled saddle–node point are used for the scaling of the oscillation period, which is more meaningful to identify the inner scaled optimal wall oscillation period. This gives ${T^{+}}=53$, $70.6$, $106$, $158.9$, $211.9$ and $264.9$ for the six studied cases. Here, the amplitude ${A_w}$ is selected deliberately to be small such that the solutions can exist at large oscillation periods, as will be discussed at the end of this section. As can be seen, the lower branch is much less affected by the control compared to the upper branch. The saddle–node point continues to move towards higher values as the oscillation period ${T^{+}}$ is increased, from ${Re_c}=127.3$ of the uncontrolled case to ${Re_c}=140.4$ at ${T^{+}}=106$ and ${Re_c}=142.5$ at ${T^{+}}=211.9$, then moves back to ${Re_c}=142.2$ at ${T^{+}}=264.9$. This indicates that the emergence of the unstable solutions is delayed to higher Reynolds numbers, implying the suppression of these solutions, a primitive form of turbulence. We note that the decreasing rate of ${Re_c}$ becomes slower eventually as ${T^{+}}$ becomes larger, and ${Re_c}$ increases only slightly for ${T^{+}}>106$. The skin friction of the upper branch does not decrease after ${T^{+}}>70.6$ at $Re\approx 170$. For larger oscillation periods, it was difficult to obtain converged solutions, and we were not able to continue numerically for ${T^{+}}>264.9$ with EQ0. The critical Reynolds number variation at the saddle–node point with the wall oscillation period is presented in figure 2(b). It is clear that the saddle–node Reynolds number saturates for ${T^{+}}>106$, consistent with the optimal oscillation period (i.e. ${T^{+}}\approx 100$) as observed in the DNS data as well as our previous bifurcation study of the wall-attached ECS under spanwise wall oscillation in Poiseuille flow (Yang & Hwang Reference Yang and Hwang2019), although the present solution shows a broader optimal ${T^{+}}$ regime for drag reduction compared to that of fully developed turbulence at much higher Reynolds numbers.

Figure 2.

(a) Plots of ${C_f}$–$Re$ bifurcation curves for EQ0 with different oscillation periods at ${A_w}/{U_w}=0.1$. (b) The saddle–node Reynolds number variation with the oscillation period. Here, the wall units of the uncontrolled case at the saddle–node point are used for the scaling of the wall oscillation periods.

In figure 3, we show the bifurcation curves of the equilibrium solutions EQ0, EQ1, EQ4 and EQ7 in the ${C_f}$–$Re$ plane for different spanwise wall oscillation amplitudes, each of which has a lower and an upper branch subject to the spanwise wall oscillation. The thick solid lines represent the uncontrolled reference cases of each equilibrium solution. For each solution, we fix the wall oscillation period ${T^{+}}$ and choose different values of amplitude ${A_w^{+}}$. As mentioned in table 2, here the viscous inner units of the corresponding uncontrolled DNS of each computational domain at $Re=400$ are used for the scaling. The upper branch of each equilibrium solution pair shows a significant skin-friction reduction as ${A_w^{+}}$ is increased. This strong effect on the upper branch of each solution leads to the emergence of sophisticated bifurcations along the upper branch on increasing ${A_w^{+}}$ (figures 3(a,b), in particular), although there is no common trend in those bifurcations. It is noted that the monotonic decrease of the ${C_f}$ value of the upper branch solution with the increase of ${A_w^{+}}$ is analogous to the effect of ${A_w^{+}}$ on turbulent drag reduction in fully developed turbulence at much higher Reynolds numbers (Quadrio & Ricco Reference Quadrio and Ricco2004; Quadrio et al. Reference Quadrio, Ricco and Viotti2009). On the contrary, the effect on the lower branch is nearly negligible over almost the entire range of Reynolds numbers. The effect of control on the lower branch is, however, not monotonic. We observe a slight decrease of ${C_f}$ for sufficiently high Reynolds numbers, while the skin-friction coefficient near the saddle–node point increases until the lower and upper branches collapse and disappear. As a consequence, the saddle–node bifurcation points move towards higher Reynolds numbers, indicating stabilisation of the equilibrium solutions.

Figure 3.

Bifurcation curves at different values of wall oscillation amplitudes ${A_w^{+}}$ in the ${C_f}$–$Re$ plane: (a) EQ0 with ${T^{+}}=79.8$; (b) EQ1 with ${T^{+}}=14.3$; (c) EQ4 with ${T^{+}}=14.3$; (d) EQ7 with ${T^{+}}=28.7$. Thick solid curves correspond to the uncontrolled reference cases. The arrows indicate increasing ${A_w^{+}}$.

Figure 4 shows further the bifurcation curves of the skin-friction coefficient with respect to the wall oscillation amplitude ${A_w^{+}}$ for various oscillation periods ${T^{+}}$ at $Re=400$. In general, commonly the upper branches of the solutions exhibit a significant reduction in $C_f$ on increasing ${A_w^{+}}$, consistent with figure 3. On the contrary, $C_f$ values of the lower branches depend on the individual solutions and the time period considered (see figure 4). However, the significant reduction of $C_f$ in the upper branch leads the two branches to meet at a certain value of ${A_w^{+}}$, forming a saddle–node bifurcation in terms of ${A_w^{+}}$. An increase of the wall oscillation period ${T^{+}}$ yields the solutions to form the saddle–node point at lower values of ${A_w^{+}}$. However, it should be mentioned that this behaviour may well be true only for the values of ${T^{+}}$ considered here, as the numerical continuation of the solutions (upper branches, in particular) is found to be difficult for very large ${T^{+}}$.

Figure 4.

Bifurcation curves ${C_f}$–${A_w^{+}}$ for different oscillation periods ${T^{+}}$ at $Re=400$: (a) EQ0, (b) EQ1, (c) EQ4, (d) EQ7. The black filled circles and squares indicate the lower and upper branch solutions, respectively, at ${A_w^{+}}=0$. The open circle and triangle in (a) indicate, respectively, case A ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=2.6$) and case B ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=4.4$) presented in table 2.

3.2. Statistics

The spanwise wall oscillation generates an unsteady near-wall spanwise shear layer referred to as the Stokes layer (according to the Stokes’ second problem). For the laminar case, the flow introduced by the spanwise wall motion is decoupled from the streamwise flow, and the unsteady Stokes layer can be expressed analytically as (Schlichting Reference Schlichting1979)

(3.1)\begin{equation} W(y,t) = {A_w} \,{\rm e}^{-\eta}\sin(2{\rm \pi} t/T - \eta), \end{equation}

where $\eta = -y\sqrt {{\rm \pi} \,Re/T}$ is related to the Stokes layer thickness. A larger spanwise wall oscillation period leads to a thicker Stokes layer, and for a given oscillation period, the wall oscillation amplitude only affects the spanwise displacement of the wall. Because of the presence of two walls, (3.1) is, however, not valid in this context, especially when the Stokes layer thickness is comparable to the half channel width. Therefore, we have considered further the analytical solution for the uniform spanwise wall oscillation problem in channel flow, derived by Jozsa (Reference Jozsa2019). This solution is given by

(3.2)\begin{equation} W(y,t) = {{\rm Re}}{\left({A_w} \exp({{\rm i}2{\rm \pi} t/T})\, \frac{\cos(y\sqrt{-{\rm i}2{\rm \pi}\,Re/T})} {\cos(\sqrt{-{\rm i}2{\rm \pi}\,Re/T})}\right)}, \end{equation}

where ${{\rm Re}}(\cdot )$ indicates the real part.

In figure 5, we present the Stokes layer at eight equally separated phases for cases A and B of the EQ0 state (see also figure 4a). With the given wall oscillation period ${T^{+}}=166.4$, the analytical profiles of the Stokes layer decay to zero only near the channel centre. As the spanwise Stokes layer is affected by the equilibrium state, the Stokes layer profiles of EQ0 under control start to deviate from the analytical ones. We note that such a deviation is small in the region close to the wall, but becomes large away from the wall (see figure 5a). In general, it is found that at a small oscillation period, the Stokes layer is thin and the agreement with the analytical solution is improved (not shown here). This is because the Stokes layer of the equilibrium state is obtained from the spanwise mean momentum balance and is determined by the contribution from the wall-normal gradient of the spanwise Reynolds shear stress term $\partial \overline {v'w'} / \partial y$, which has been found to increase with the oscillation period and the Reynolds number (Ricco & Quadrio Reference Ricco and Quadrio2008; Yang Reference Yang2016). At a higher drag state, such as case A shown in figure 5(a), the Stokes layer of the equilibrium state differs largely from that of the analytical solution in (3.2), and some oscillating flow even appears in the channel centre. On the contrary, the agreement of the Stokes layer between the equilibrium state and the analytical solution is greatly improved for a lower drag state of case B (figure 5b). It is noted that under the control parameters for cases A and B, the difference between the analytical solutions from (3.1) and (3.2) is small.

Figure 5.

Unsteady spanwise Stokes layer at $8$ equally separated phases for: (a) case A ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=2.6$), and (b) case B ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=4.4$). The solid and dashed lines indicate, respectively, the equilibrium state of EQ0 and the analytical solution in (3.2).

The time variations of the instantaneous skin-friction coefficient $C_{f,t}$ for cases A and B are shown in figure 6. For the spanwise wall oscillation cases, the statistics is sampled and averaged over one single wall oscillation period for the converged solution, i.e. ${C_f}=({1}/{T})\int _0^{T} C_{f,t}\,\mathrm {d}t$. Each solution is computed and compared with two different values of $n$ (i.e. $n=1, 2$) used for the calculation (see § 2.2). As expected, the converged solution does not depend on $n$, except that the solution with $n=2$ simply repeats that of $n=1$ twice. As already mentioned for figure 4(a), case A is a higher drag state with ${C_f}=0.0107$, while case B is a lower drag state with ${C_f}=0.0079$. Based on the derivation of the wave pattern of $C_{f,t}$ in time compared to the sinusoidal wall oscillation wave, the equilibrium solution of case A is modulated more strongly by the spanwise wall oscillation compared to that of case B, and each solution has the $C_{f,t}$ variation of $3.2\,\%$ and $1.6\,\%$ during one wall oscillation period, respectively. It is noted that the skin-friction coefficient for the upper branch of the uncontrolled EQ0 state at $Re=400$ is ${C_f}=0.0167$, thus the two controlled cases have drag reduction $\mathcal {DR}=35.9\,\%$ and $52.7\,\%$, respectively.

Figure 6.

Variation of instantaneous skin-friction coefficient $C_{f,t}$ over time for (a) case A ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=2.6$), and (b) case B ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=4.4$). The red dashed (black solid) lines are for equilibrium solutions computed with $n=1$ ($n=2$) in the Newton solver. The horizontal lines indicate the mean value of the skin-friction coefficient, i.e. ${C_f}=0.0107$ and $0.0079$ for cases A and B, respectively. The vertical dotted lines mark one wall oscillation period, ${T^{+}}=166.4$. The blue circles mark the $8$ equally separated phases investigated for each case, i.e. $\phi =0$, ${\rm \pi} /4$, ${\rm \pi} /2$, $3{\rm \pi} /4$, ${\rm \pi}$, $5{\rm \pi} /4$, $3{\rm \pi} /2$, $7{\rm \pi} /4$.

Figure 7 shows the root-mean-square (r.m.s.) velocity components, i.e. ${u_{rms}^{+}}$, ${v_{rms}^{+}}$, ${w_{rms}^{+}}$, and the streamwise Reynolds shear stress $-\overline {u'v'}^{+}$ profiles for case A at $8$ equally separated phases (dotted lines) over one wall oscillation period (see symbols in figure 6a). The phase variation in the statistics becomes clear with a modest value, which corresponds to $3.2\,\%$ variation in the skin friction (figure 6a). It is interesting to note that the r.m.s. profiles of the controlled state A appear to sit between those of the upper- and lower-branch solutions. For example, the upper-branch solution shows a strong near-wall peak in ${u_{rms}^{+}}$, while such a peak is located in the channel centre for the lower-branch solution. The controlled state A exhibits the two features in a mixed manner – the strength of ${u_{rms}^{+}}$ in the channel centre is increased compared to the upper branch, while the ${u_{rms}^{+}}$ peak in the near-wall region is weakened (figure 7a). The rest of the profiles behave in a similar manner (figures 7b–d). Furthermore, the reduction of $-\overline {u'v'}^{+}$ over the entire wall-normal locations compared to the upper-branch solution indicates a drag reduction according to the FIK identity (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002) – indeed, $35.9\,\%$ of $\mathcal {DR}$ from the upper-branch solution is observed. Similar statistics features are also observed from the other exact coherent states in the domain $\varOmega _{HKW}$, which are presented in Appendix B.

Figure 7.

Root-mean-square (r.m.s.) velocity profiles scaled with viscous inner units of the uncontrolled DNS for case A ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=2.6$): (a) $u_{rms}^{+}$, (b) $v_{rms}^{+}$, (c) $w_{rms}^{+}$, and (d) $-\overline {u^{\prime }v^{\prime }}^{+}$. Dash-dotted (dashed) lines represent the upper (lower) branch of the uncontrolled state, and dotted lines indicate the controlled state at $8$ equally separated phases, with the mean profile indicated by the solid line.

3.3. Streak variation

Now, the time and space variations of the streaks are studied using cases A and B of EQ0. We note that the choice of EQ0 is made because it appears to be more useful than the other solutions for the following reasons. First, currently EQ0 is known as the invariant solution emerging at the lowest Reynolds number in plane Couette flow. Importantly, turbulence emerges directly from the upper branch of EQ0 through a bifurcation cascade (Kreilos & Eckhardt Reference Kreilos and Eckhardt2012) – note that all the other equilibrium solutions considered here do not have this feature. Furthermore, the skin friction of the upper branch of EQ0, often believed to be more relevant to the turbulent state, is $C_f=0.0168$ at $Re=400$, not very far from the time-averaged turbulent state with $C_f=0.0138$ at the same Reynolds number. Finally, at $Re=400$, the lower branch of EQ0 is the edge state in the domain $\varOmega _N$ (Schneider et al. Reference Schneider, Gibson, Lagha, De Lillo and Eckhardt2008). A particular feature for spanwise wall oscillation control of turbulence is the modulation of the near-wall streaks by the Stokes layer, which has been studied extensively using instantaneous flow fields or conditionally averaged flow structures obtained from DNS. As a result, some key dynamical features of the controlled flow may not be well observed in the complex three-dimensional flow field or are averaged out in the conditional flow fields. Such a drawback may be bypassed by inspecting some equilibrium solutions modulated by the Stokes layer, as they are the exact solutions of the Navier–Stokes equations.

Figure 8 presents the streamwise-averaged streaks visualised by $\langle u' \rangle _x$ contour and the vortical motions visualised by $( \langle v' \rangle _x, \langle w' \rangle _x - \langle w' \rangle _{x,z} )$ vectors for cases A and B. The features of the upper- and lower-branch solutions of EQ0 at $Re=400$ without control are shown in the top row of figure 8. It is clear that the shapes of the low- and high-speed streaks of the lower-branch solution are almost symmetrical to each other, while this is not the case for the upper-branch solution, with the high-speed streak spreading a much wider area in the spanwise direction of the near-wall region and the low-speed streak more localised. Therefore, in the upper-branch solution, the high-speed streak is expected to make a strong contribution to the elevation of local skin friction, and this should be significantly greater than the local reduction created by the low-speed streak.

Figure 8.

Streamwise-averaged low- and high-speed streaks (contour by $\langle u' \rangle _x$) and in-plan vortex (arrows plotted by ($\langle v' \rangle _x, \langle w' \rangle _x - \langle w' \rangle _{x,z}$)) variation during one oscillation period for (a–h) case A ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=2.6$), and (i–p) case B ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=4.4$), at different phases: (a,i) $\phi =0$; (b,j) $\phi ={\rm \pi} /4$; (c,k) $\phi ={\rm \pi} /2$; (d,l) $\phi =3{\rm \pi} /4$; (e,m) $\phi ={\rm \pi}$; ( f,n) $\phi =5{\rm \pi} /4$; (g,o) $\phi =3{\rm \pi} /2$; (h,p) $\phi =7{\rm \pi} /4$. The Stokes profiles at the corresponding phases are shown on the right-hand side of each contour plot. The top row presents the flow fields of the corresponding upper- and lower-branch solutions of uncontrolled EQ0 at $Re=400$. Viscous inner units of the uncontrolled DNS case are used for the scaling.

Once spanwise wall oscillation is applied, the streaks are modulated significantly by the Stokes layer. For case A, (figures 8a–h), similarly to the upper-branch solution of the uncontrolled case, the high-speed streak is stronger than the low-speed one. However, under the control, the high-speed streak in the near-wall region is tilted strongly in the spanwise direction. In particular, the near-wall part of the high-speed streak penetrates deeply underneath the low-speed streak, and vice versa. Such a spanwise motion appears to enhance fluid mixing between the low- and high-speed streaks. Therefore, the locally high-friction region from the high-speed streaks is balanced out with the locally low-friction region from the low-speed streaks, resulting in a drag reduction – indeed, the asymmetry between the low- and high-speed streaks appears to be weaker than that of the upper-branch solution of the uncontrolled case (see figures 8d,h). For case B (figures 8i–p), a lower drag state, the low- and high-speed streaks are almost symmetric to each other, and their movements in the spanwise direction are nearly in phase. Moreover, the low- and high-speed streaks have the largest fluctuation in the channel centre. We also note that the streak variation in the spanwise direction is very similar to the conditionally averaged results in the DNS study by Touber & Leschziner (Reference Touber and Leschziner2012). This indicates that the equilibrium solutions in this study do capture some important dynamics of the near-wall flow under the spanwise wall oscillation, although the Reynolds number considered here is much lower than that of Touber & Leschziner (Reference Touber and Leschziner2012).

To explore further the streak angle variation during one wall oscillation period, the high-speed streak and vortical structure are visualised in figures 9 and 10 for cases A and B, respectively. At first glance, the structures for case A are more active than those for case B. During one oscillation period, two dominant streak-tilting angles are seen with opposite sign, as also observed in previous DNS studies (Touber & Leschziner Reference Touber and Leschziner2012; Chung et al. Reference Chung, Hurst and Yang2013). Each of these angles is seen dominantly for half an oscillation period. For cases A and B, these tilting angles of the streaks are found to be $\alpha \approx \pm 18^{\circ }$ and $\alpha \approx \pm 14^{\circ }$, respectively. By monitoring the evolution of the streaks in figure 9, the high-speed streak is first tilted in the negative spanwise direction until $\phi = {\rm \pi}/2$, then breaks down suddenly into two streaks, a stronger one that grows fast, and a weaker one that dissipates quickly. By checking the temporal skin-friction variation of case A (see figure 6a), it is found that $C_{f,t}$ reaches its minimum value at the streak breakdown stage, while it reaches the peak value at the phase with the strongest streak. For case B (figure 6b), the switch from a negatively tilted streak to a positively tilted one ($\phi ={\rm \pi} /2\unicode{x2013}3{\rm \pi} /4$) is also very quick, but the streak breakdown process is not observed clearly as in case A.

Figure 9.

Streak variation viewed from the bottom of the channel during one oscillation period for case A ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=2.6$) at different phases: (a) $\phi =0$, (b) $\phi ={\rm \pi} /4$, (c) $\phi ={\rm \pi} /2$, (d) $\phi =3{\rm \pi} /4$, (e) $\phi ={\rm \pi}$, ( f) $\phi =5{\rm \pi} /4$, (g) $\phi =3{\rm \pi} /2$, and (h) $\phi =7{\rm \pi} /4$. A yellow isosurface ($u'^{+}=2.0$) indicates the high-speed streak, and a green isosurface ($v'^{+}=-0.2$) represents the vortical structure. The arrows represent the streak angle. Viscous inner units of the uncontrolled DNS are used for the scaling.

Figure 10.

Streak variation viewed from the bottom of the channel during one oscillation period for case B ($Re=400$, ${T^{+}}=166.4$, ${A_w^{+}}=4.4$) at different phases: (a) $\phi =0$, (b) $\phi ={\rm \pi} /4$, (c) $\phi ={\rm \pi} /2$, (d) $\phi =3{\rm \pi} /4$, (e) $\phi ={\rm \pi}$, ( f) $\phi =5{\rm \pi} /4$, (g) $\phi =3{\rm \pi} /2$, and (h) $\phi =7{\rm \pi} /4$. A yellow isosurface ($u'^{+}=2.0$) indicates the high-speed streak, and a green isosurface ($v'^{+}=-0.1$) represents the vortical structure. The arrows represent the streak angle. Viscous inner units of the uncontrolled DNS are used for the scaling.

Finally, it is worth mentioning that the tilting dynamics of the near-wall streaks is found to depend strongly on the strength of the Stokes layer. Unfortunately, this feature does not allow for a direct comparison with that observed in the previous DNS at high Reynolds numbers (Touber & Leschziner Reference Touber and Leschziner2012). In fact, as can be seen from figure 4, the equilibrium solutions of EQ1, EQ4 and EQ7 under the spanwise wall oscillation disappear at large wall oscillation amplitudes and periods. In other words, the Stokes layer, which was able to be examined here, was not strong enough to generate a large variation in the streak angle.

3.4. State space dynamics

Now, to visualise the relevant state space using state portraits, suitable observables are defined. Similarly to previous studies (e.g. Kawahara & Kida Reference Kawahara and Kida2001; Ibrahim et al. Reference Ibrahim, Yang, Doohan and Hwang2019), the volume-averaged equation for the mean kinetic energy (MKE) is first introduced:

(3.3a)\begin{align} \frac{{{\rm d}}E_{\boldsymbol{U}}}{{{\rm d}}t} &= \underbrace{\frac{U_{w}\tau_{uw}}{h}}_{\mathcal{I}_U} - \underbrace{\frac{1}{V}\int_\varOmega -\langle { u^{\prime}v^{\prime}} \rangle_{x,z} \frac{\partial U}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{uv}} - \underbrace{\frac{\nu}{V} \int_{\varOmega} \left(\frac{\partial U}{\partial y}\right)^{2} {{\rm d}}V}_{\mathcal{D}_U} \nonumber\\ &\quad + \underbrace{\frac{W_w\tau_{ww}}{h}}_{\mathcal{I}_W} - \underbrace{\frac{1}{V}\int_{\varOmega} -\langle { v^{\prime}w^{\prime} }\rangle_{x,z}\, \frac{\partial W}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{vw}} - \underbrace{\frac{\nu}{V} \int_{\varOmega} \left(\frac{\partial W}{\partial y}\right)^{2} {{\rm d}}V}_{\mathcal{D}_W}, \end{align}

with the MKE given by

(3.3b)\begin{equation} E_{\boldsymbol{U}}=\frac{1}{V}\int_\varOmega \boldsymbol{U}\boldsymbol{\cdot} \boldsymbol{U}\,{{\rm d}}V, \end{equation}

where $\varOmega$ ($=\varOmega _N$ or $\varOmega _{HKW}$) is the given computational domain, ${U_w}$ and ${W_w}$ are the streamwise and spanwise velocities at the wall, and $\tau _{uw}=\mu \langle \partial u/\partial y|_{y=0,2h} \rangle _{x,z}$ and $\tau _{ww}=\mu \langle \partial w/ \partial y|_{y=0,2h} \rangle _{x,z}$ are the spatial averaged streamwise and spanwise wall shear stresses, respectively. The terms indicated by $\mathcal {I}_U$, $\mathcal {P}_{uv}$ and $\mathcal {D}_U$ in the first line of (3.3a) are the rate of energy input, the rate of energy transport by streamwise mean shear to turbulent kinetic energy (TKE), and the dissipation rate by the streamwise mean velocity, respectively. Similarly, the three terms in the second line, i.e. $\mathcal {I}_W$, $\mathcal {P}_{vw}$ and $\mathcal {D}_W$, correspond to the rate of energy input by the spanwise wall oscillation, the rate of energy transport by the spanwise mean shear to TKE, and the dissipation rate by the spanwise mean velocity.

The volume-averaged equation for TKE is also written as

(3.4)\begin{equation} \frac{{{\rm d}}E_{\boldsymbol{u}}}{{{\rm d}}t}= \underbrace{\frac{1}{V}\int_\varOmega -{\langle u^{\prime}v^{\prime}\rangle}_{x,z}\, \frac{\partial U}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{uv}} + \underbrace{\frac{1}{V} \int_\varOmega -{\langle v^{\prime}w^{\prime} \rangle}_{x,z}\, \frac{\partial W}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{vw}} - \underbrace{\frac{\nu}{V} \int_\varOmega \langle (\boldsymbol{\nabla} \boldsymbol{u^{\prime}})^{2} \rangle_{x,z}\,{{\rm d}}V}_{\mathcal{D}_{\boldsymbol{u}}}, \end{equation}

with $E_{\boldsymbol {u}}=E_{uu}+E_{vv}+E_{ww}$ from (2.5). Here, the first two terms, $\mathcal {P}_{uv}$ and $\mathcal {P}_{vw}$, are identical to those of (3.3a), but with opposite signs, emerging as the rate of turbulence production by streamwise and spanwise mean shear, and the last term, ${\mathcal {D}_{\boldsymbol {u}}}$, is the TKE dissipation. For details about the derivation of (3.3a) and (3.4), the reader may refer to Ricco et al. (Reference Ricco, Ottonelli, Hasegawa and Quadrio2012). Finally, the rate of total energy input and dissipation are given by $\mathcal {I}\equiv \mathcal {I}_U+\mathcal {I}_W$ and $\mathcal {D}\equiv \mathcal {D}_U+\mathcal {D}_W+{\mathcal {D}_{\boldsymbol {u}}}$, respectively.

Two state portraits from the domain $\varOmega _{HKW}$ at $Re=400$ are plotted in the $\mathcal {I}_U$–$\mathcal {D}_{\boldsymbol {u}}$ and $E_{uu}$–$E_{vv}$ planes, respectively, and they are shown in figure 11. Here, the equilibrium solutions and periodic edge state in the presence of control are presented by averaging over a single wall oscillation period, while the turbulent state is shown with its time trajectory. When the control is applied with ${A_w^{+}}=17.7$ and ${T^{+}}=14.3$, the non-zero spanwise energy input $\mathcal {I}_W$ is given for all the solutions (closed symbols). Figure 11(a) shows that the control pushes all the solutions towards the laminar state in the direction of decreasing $\mathcal {I}_U$ and $\mathcal {D}_{\boldsymbol {u}}$. Indeed, at these specific parameters, all the solutions indicate a reduction of the drag because $\mathcal {I}_U$ in (3.3a) is proportional to the streamwise wall shear stress (figure 4). We observe that the upper branches and the mean turbulent state are much more affected by the control than the lower branches. Indeed, the lower branches EQ4$_{L}$ and EQ7$_{L}$ show $7.3\,\%$ and $3.2\,\%$ of $\mathcal {DR}$, respectively; the upper branches EQ4$_{U}$ and EQ7$_{U}$ show $25.8\,\%$ and $33.7\,\%$ of $\mathcal{DR}$, respectively, and the mean turbulent state shows $10.9\,\%$ of $\mathcal {DR}$. The state portrait of the streamwise kinetic energy fluctuation $E_{uu}$ against the wall-normal kinetic energy fluctuation $E_{vv}$ shown in figure 11(b) indicates similar results. The control is seen to be responsible for a reduction in the wall-normal velocity fluctuation, which results in a decrease in $E_{vv}$ and pushes the solutions towards the laminar state. The state portraits indicate that the region of the state space where the turbulent state is organised is reduced by the spanwise wall oscillation. Furthermore, the state portraits also show that the state space distances between the upper and lower branches of each solution pair presented at least in figure 11 are reduced, indicating stabilisation of turbulence.

Figure 11.

State portraits for controlled (${A_w^{+}}=17.7$, ${T^{+}}=14.3$) and uncontrolled states at $Re=400$ in (a) the $\mathcal {I}_U$–$\mathcal {D}_{\boldsymbol {u}}$ plane (with $\mathcal {I}_r$ and $\mathcal {D}_r$ the energy input and rate of total dissipation of the uncontrolled laminar state, respectively), and (b) the $E_{uu}$–$E_{vv}$ plane. Here, black dashed and cyan solid lines ($Tur$ in the figure's legend) indicate uncontrolled and controlled turbulent states, with their means ($\overline{Tur}$ in the figure's legend) represented by open and closed circles. The uncontrolled and controlled equilibrium states are shown by open and closed symbols, respectively. The controlled equilibrium states are presented by averaging over the wall oscillation period.

The stability of the exact coherent states is also examined. Since the exact coherent states in the present study are stationary in space and are periodic in time under the spanwise wall oscillation, the normal-mode ansatz for the Floquet analysis yields the following form for a small perturbation around them:

(3.5)\begin{equation} \boldsymbol{u}(x,y,z,t)= \boldsymbol{Q}(x,y,z,t)\,\hat{\boldsymbol{u}}(x,y,z)\,{\rm e}^{\lambda t}+c.c, \end{equation}

where $\lambda$ is the complex Floquet exponent (eigenvalue), $\hat {\boldsymbol {u}}(x,y,z)$ is the corresponding complex Floquet eigenmode, $\boldsymbol {Q}(x,y,z,t)=\boldsymbol {Q}(x,y,z,t+T)$ and $\boldsymbol {Q}(x,y,z,0)=\boldsymbol {I}$, with $\boldsymbol {I}$ being the identity operator. We note that (3.5) is extendable for the stability of the uncontrolled steady solutions with $\boldsymbol {Q}(x,y,z,t)=\boldsymbol {I}$. Figure 12 shows the eigenspectra of the lower and upper branches of EQ4 and EQ7, with the real and imaginary parts representing the growth rate and frequency of the eigenmodes, respectively. We observe that for the upper-branch solutions of both cases, where the control yields a drag reduction, the growth rates of the eigenmodes are also decreased, indicating stabilisation. On the contrary, the lower-branch solutions show much smaller drag reduction (or even drag increase), and a marginal change of the growth rates of the leading eigenmode is observed, indicating a weak effect of the applied control. These observations are consistent with the reduction in size of the state space region of turbulent state in figure 11.

Figure 12.

Eigenspectra at $Re=400$ compared between uncontrolled and controlled (${A_w^{+}}=17.7$, ${T^{+}}=14.3$) states for: (a,b) EQ4, and (c,d) EQ7. The lower-branch solutions are shown in (a,c), and the upper-branch in (b,d).

It is well understood that the uncontrolled turbulent state at $Re=400$ in the given computational domain is a ‘chaotic saddle’, which is characterised by the finite lifetime of the chaotic dynamics. The lifetime of the uncontrolled turbulent state also increases very rapidly with the Reynolds number (Eckhardt et al. Reference Eckhardt, Schneider, Hof and Westerweel2007). The state space characteristics discussed in figures 11 and 12 suggest that the effect of the spanwise wall oscillation is similar to that of lowering the Reynolds number, indicating a reduction of turbulence lifetime due to the wall oscillation. To evaluate the turbulence lifetime statistics, we performed groups of 100 controlled and uncontrolled DNS at $Re=400$. The initial conditions of the simulations were obtained from 100 states computed at a slightly higher Reynolds number, $Re=415$. The lifetime is evaluated as a probability distribution for the amplitude ${A_w^{+}} = 1.2$ and different oscillation periods ${T^{+}}=57.2$, $71.5$, $85.8$, $114.4$ and $172$. To monitor the lifetime of each simulation, we define a threshold of the turbulent energy fluctuation $E_{vv} + E_{ww}<10^{-5}$, below which turbulence will decay to laminar flow. The probability for turbulence to live up to a given time $t$ as a function of ${A_w^{+}}$ and ${T^{+}}$ is reported in figure 13. The exponential distribution of the lifetime implies the memoryless nature of the decay process for different initial conditions (see the linear red dashed lines in figure 13), as shown by Eckhardt et al. (Reference Eckhardt, Schneider, Hof and Westerweel2007). The lifetime probability decreases with ${T^{+}}$ at ${A_w^{+}}=1.2$. It is clear that the control tends to relaminarise the flow faster, as the wall oscillation period is increased. Importantly, the exponential lifetime distributions obtained by varying the oscillation parameters suggest that the way that the flow is relaminarised faster by the control is similar to that by reducing the Reynolds number, as the reduction of the lifetime maintaining the memoryless nature has been observed as the effect of the Reynolds number (e.g. Eckhardt et al. Reference Eckhardt, Schneider, Hof and Westerweel2007).

Figure 13.

Probability distribution of sustained turbulence lifetime at $Re = 400$ and ${A_w^{+}}=1.2$ for different spanwise wall oscillation periods ${T^{+}}$. The black arrow indicates the increasing direction of ${T^{+}}$. Here, the red dashed lines indicate a fit obtained using an exponentially decaying function with the characteristic time $\tau ({T^{+}})$ that depends on the oscillation time period.

3.5. Stabilisation mechanism of exact coherent states

Finally, we explore the stabilisation mechanism of the ECS in terms of the self-sustaining process (Hamilton et al. Reference Hamilton, Kim and Waleffe1995). To this end, the velocity fluctuation is decomposed into the streamwise mean and the rest: i.e. $\boldsymbol {u}'=\boldsymbol {u}_1+\boldsymbol {u}_2$, where $\boldsymbol {u}_1=\langle \boldsymbol {u}' \rangle _{x}$. Then $\boldsymbol {u}_1$ describes primarily the streamwise uniform (or elongated) streaks, while $\boldsymbol {u}_2$ depicts the instability wave arising from the streaks and the resulting quasi-streamwise vortices. The equations of motion for $\boldsymbol {u}_1$ and $\boldsymbol {u}_2$ are written as

(3.6a)\begin{align} & \frac{\partial \boldsymbol{u}_1}{\partial t}+(\boldsymbol{{U}}\boldsymbol{\cdot}\boldsymbol{\nabla}) \boldsymbol{u}_1+(\boldsymbol{u}_1\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{{U}} \nonumber\\ &\quad ={-}\frac{1}{\rho}\,\boldsymbol{\nabla} p_1 +\nu\,\nabla^{2}\boldsymbol{u}_1-(\boldsymbol{u}_1\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}_1 -\langle(\boldsymbol{u}_2\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}_2\rangle_{x} +\langle{(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}}\rangle_{x,z}, \end{align}

(3.6b)\begin{gather} \frac{\partial \boldsymbol{u}_2}{\partial t} +[(\boldsymbol{{U}}+\boldsymbol{u}_1)\boldsymbol{\cdot}\boldsymbol{\nabla}]\boldsymbol{u}_2 +(\boldsymbol{u}_2\boldsymbol{\cdot}\boldsymbol{\nabla})(\boldsymbol{{U}}+\boldsymbol{u}_1)={-}\frac{1}{\rho}\,\boldsymbol{\nabla} p_2 +\nabla^{2}\boldsymbol{u}_2+\mathcal{N}, \end{gather}

with

(3.6c)\begin{equation} \mathcal{N}={-}(\boldsymbol{u}_2\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}_2+ \langle(\boldsymbol{u}_2\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}_2\rangle_{x}. \end{equation}

Now, if we multiply $\boldsymbol {u}_1$ by (3.6a) and $\boldsymbol {u}_2$ by (3.6b), and integrate over the computational domain $\varOmega$ and time, the following energy-balance equations are obtained:

(3.7a)\begin{gather} \underbrace{\frac{1}{V}\int_\varOmega -\overline{\langle u_1^{\prime}v_1^{\prime}\rangle}_{x,z}\, \frac{\partial U}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{uv1}} +\underbrace{\frac{1}{V}\int_\varOmega -\overline{\langle v_1^{\prime}w_1^{\prime} \rangle}_{x,z}\, \frac{\partial W}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{vw1}} = \underbrace{\frac{\nu}{V} \int_\varOmega \overline{\langle(\boldsymbol{\nabla} \boldsymbol{u}_1^{\prime})^{2}\rangle}_{x,z}\,{{\rm d}}V}_{\mathcal{D}_{\boldsymbol{u}1}}, \end{gather}

(3.7b)\begin{gather} \underbrace{\frac{1}{V}\int_\varOmega -\overline{\langle u_2^{\prime}v_2^{\prime}\rangle}_{x,z}\, \frac{\partial U}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{uv2}} + \underbrace{\frac{1}{V}\int_\varOmega -\overline{\langle v_2^{\prime}w_2^{\prime} \rangle}_{x,z}\, \frac{\partial W}{\partial y}\,{{\rm d}}V}_{\mathcal{P}_{vw2}} =\underbrace{\frac{\nu}{V} \int_\varOmega \overline{\langle(\boldsymbol{\nabla} \boldsymbol{u}_2^{\prime})^{2}\rangle}_{x,z}\,{{\rm d}}V}_{\mathcal{D}_{\boldsymbol{u}2}}, \end{gather}

where $\mathcal {P}_{uv1}$ is the time-averaged rate of production by the lift-up effect of streaks, $\mathcal {P}_{uv2}$ is the rate of production by the streak instability and breakdown, $\mathcal {P}_{vw1}$ and $\mathcal {P}_{vw2}$ are the productions of streamwise uniform and non-uniform velocity components by the spanwise mean shear from the wall oscillation, and $\mathcal {D}_{\boldsymbol {u}1}$ and $\mathcal {D}_{\boldsymbol {u}2}$ are the rate of dissipation by streamwise uniform and non-uniform velocity components, respectively.

Table 3 lists the values of the energy production terms defined in (3.7) for the upper-branch equilibrium solutions and DNS with/without some suitable spanwise wall oscillations. For all uncontrolled cases, $\mathcal {P}^{+}_{uv2}$ is very small compared to $\mathcal {P}^{+}_{uv1}$, indicating that the key production mechanism lies in the lift-up effect of streaks. When the control is applied, turbulence production of all upper-branch solutions and DNS exhibits a significant reduction. In particular, the reduction in the production appears dominantly in $\mathcal {P}^{+}_{uv1}$, except for EQ4$_U$, which exhibits a significant reduction in $\mathcal {P}^{+}_{uv2}$. This suggests that the central mechanism of the stabilisation of the ECS and turbulence is presumably the inhibition of the lift-up effect of streaks by the spanwise wall oscillation, consistent with the observation in § 3.3 where the drag reduction mechanism was proposed in terms of enhanced mixing between low- and high-speed streaky flow regions by the Stokes layer. This also explains why many previous linear analyses (Duque-Daza et al. Reference Duque-Daza, Baig, Lockerby, Chernyshenko and Davies2012; Moarref & Jovanović Reference Moarref and Jovanović2012; Blesbois et al. Reference Blesbois, Chernyshenko, Touber and Leschziner2013; Hack & Zaki Reference Hack and Zaki2015) have been so successful for a sound description of turbulence suppression and drag reduction by spanwise wall oscillation – one of the main effects that can be quantified by the linear analyses mentioned above is the amplification strength of the streaks by the streamwise vortices (i.e. the lift-up effect). The analysis here also suggests that the nonlinear mechanisms are relatively less important, and they appear to play relatively limited roles in the suppression of turbulence.

Table 3.

The time-averaged rate of productions of TKE with and without spanwise wall oscillation. For further details, see (3.7).