3.1. Spanwise enstrophy transport

Figure 2 shows the response of turbulent enstrophy for all the cases listed in table 1. In canonical wall-bounded turbulent flows, the spanwise turbulent vorticity $\omega _z$ is dominant next to the wall. It is approximately equivalent to the uniform streamwise fluctuating stress $\tau ^\prime$ up to $y=3$, i.e. $\omega _z\approx -\partial u/\partial y$. In the large amplitude STW cases, $\omega _z$ is entirely annihilated up to $y=8$, with a negligible turbulent activity at the wall (i.e. $\tau ^\prime \approx 0$), and its peak is pushed towards the high buffer layer at $y=20$. This can be attributed to the strong damping of the near-wall streaks of streamwise fluctuating velocity. This is one of the outstanding effects of STW control compared with the HWO control – the profile of which collapses rather with the small amplitude STW case $A0.30$ (figure 2c). In the HWO case $\overline {\omega _z\omega _z}$ is damped in the viscous sublayer with a maximum at approximately $y=10$.

Figure 2. Mean profiles of turbulent enstrophy in each direction for uncontrolled and controlled cases: (a) streamwise $\overline {\omega _x\omega _x}$; (b) wall-normal $\overline {\omega _y\omega _y}$; and (c) spanwise $\overline {\omega _z\omega _z}$. Note that all the profiles are scaled with the local friction velocities of the drag reduced flows. Refer to table 1 for markers corresponding to different cases.

The production term for the mean spanwise turbulent enstrophy $\overline {\omega _z\omega _z}$ reads

(3.1)\begin{align} \overline{ P_{\omega_z\omega_z}} &= \underbrace{ \overline{2\widetilde{\varOmega_x}\left\langle \omega_z\frac{\partial w}{\partial x}\right\rangle} }_{\overline{ P^{1}_{\omega_z\omega_z} }}+ \underbrace{ \overline{2\widetilde{\varOmega_y}\left\langle \omega_z\frac{\partial w}{\partial y}\right\rangle} }_{\overline{ P^{2}_{\omega_z\omega_z} }}+ \underbrace{ \overline{2\widetilde{\varOmega_z}\left\langle \omega_z\frac{\partial w}{\partial z}\right\rangle} }_{\overline{ P^{3}_{\omega_z\omega_z} }}+ \underbrace{ 2\overline{\varOmega_z}\overline{\omega_z\frac{\partial w}{\partial z}} }_{\overline{ P^{4}_{\omega_z\omega_z} }}+ \underbrace{ 2\overline{\omega_x\omega_z\frac{\partial w}{\partial x}} }_{\overline{ P^{5}_{\omega_z\omega_z} }} \nonumber\\ &\quad +\underbrace{ 2\overline{\omega_y\omega_z\frac{\partial w}{\partial y}} }_{\overline{ P^{6}_{\omega_z\omega_z} }}+ \underbrace{ 2\overline{\omega_z\omega_z\frac{\partial w}{\partial z}} }_{\overline{ P^{7}_{\omega_z\omega_z} }}+ \underbrace{ 2\overline{\langle\omega_x\omega_z\rangle\frac{\partial\widetilde {W}}{\partial x}} }_{\overline{ P^{8}_{\omega_z\omega_z} }}+ \underbrace{ 2\overline{\langle\omega_y\omega_z\rangle\frac{\partial\widetilde {W}}{\partial y}} }_{\overline{ P^{9}_{\omega_z\omega_z} }}. \end{align}

The complete transport equation for each component can be found in Appendix A. The terms with ‘$\sim$’ are purely a consequence of the periodic forcing in the form of spanwise wall oscillations. In the canonical turbulent channel flows, the mean production term $\overline {P_{\omega _z\omega _z}}$ peaks at approximately $y=4$ within the viscous sublayer, and is roughly in equilibrium with the dissipation, as shown by the profiles in broken lines in figure 3(a). The viscous and turbulent diffusion terms are negligible except next to the wall at which they equilibrate mutually as expected. In the controlled cases, however, all the transport terms are shifted away from the viscous sublayer. In the HWO and low amplitude STW $(A<0.50)$ cases, the total mean production term peaks at approximately $y=9$, and it is even further shifted towards the middle of the buffer layer at $y\approx 20$ for the large amplitude STW cases (figure 3b). A similar attenuation and shift can also be observed in the dissipation $\overline {\epsilon _{\omega _z\omega _z}}$ (figure 4).

Figure 3. Budget of spanwise turbulent enstrophy transport for (a) HWO and (b) $A1.25$ cases. The budget for the reference (uncontrolled) canonical case is also included in (a) in grey broken lines for comparison. The mean advection term $\overline {A_{\omega _z\omega _z}}$ is negligible, and hence not displayed in the figures to avoid cluttering.

Figure 4. Dissipation term $(\overline {\epsilon _{\omega _z\omega _z})}$ for the spanwise turbulent enstrophy. Refer to table 1 for markers corresponding to different cases.

In the canonical turbulent channel flows, the main contribution to the total production of $\omega _z$ comes from $\overline {P^{4}_{\omega _z\omega _z}}$ which emanates from the stretching of the spanwise vorticity interacting with the mean shear $\overline {\varOmega _z}$. It peaks at roughly $y=5$ in the viscous sublayer (figure 5c). For the large amplitude STW cases, it is almost annihilated up to $y=10$ and there is a shift in its peak of approximately 10 wall units. There is a difference in its maximum of approximately 20 % for the A1.25 case compared with the HWO case.

Figure 5. Production terms (a) $\overline {P^{1}_{\omega _z\omega _z}}$, (b) $\overline {P^{2}_{\omega _z\omega _z}}$, (c) $\overline {P^{4}_{\omega _z\omega _z}}$, (d) $\overline {P^{6}_{\omega _z\omega _z}}$, (e) $\overline {P^{9}_{\omega _z\omega _z}}$ and (f) $\overline {P^{1*}_{\omega _z\omega _z}}$. Refer to table 1 for markers corresponding to different cases.

The Stokes straining production terms $\overline {P^1_{\omega _z\omega _z}}$ and $\overline {P^2_{\omega _z\omega _z}}$ act to destroy the production of $\omega _z$ (figure 5a,b). The former originates from the twisting of $\widetilde {\varOmega _x}$ by the local $\partial w/\partial x$ gradient, while the latter due to the tilting of $\widetilde {\varOmega _y}$ by the local $\partial w/\partial y$ gradient. In the case of HWO, $\overline {P^1_{\omega _z\omega _z}}$ is almost negligible and $\overline {P^2_{\omega _z\omega _z}}$ is obviously absent (as $\partial \widetilde {W}/\partial x=0$), suggesting that both of these terms are a result of particular effects of STW.

There is another Stokes straining term, denoted by $\overline {P^9_{\omega _z\omega _z}}$, that is significant in both the HWO and STW cases. It results from the tilting of $\omega _y$ by the Stokes strain $\partial \widetilde {W}/\partial y$, and is large next to the edge of the viscous sublayer (figure 5e). The terms $\overline {P^{1}_{\omega _z\omega _z}}$ and $\overline {P^{9}_{\omega _z\omega _z}}$ can be combined to give

(3.2)\begin{equation} \overline{P^{1*}_{\omega_z\omega_z}} = \overline{P^{1}_{\omega_z\omega_z}} + \overline{P^{9}_{\omega_z\omega_z}} = 2\overline{\frac{\partial \widetilde {W}}{\partial y} \left\langle \omega_z\frac{\partial u}{\partial z} \right\rangle}. \end{equation}

The resulting term $\overline {P^{1*}_{\omega _z\omega _z}}$ now has a different physical meaning. As $\langle \omega _z {\partial u}/{\partial z}\rangle$ represents twisting of $\omega _x$ in its transport equation, $\overline {P^{1*}_{\omega _z\omega _z}}$ therefore represents the Stokes straining of $\langle \omega _z {\partial u}/{\partial z}\rangle$ by the deterministic streamwise vorticity $\widetilde {\varOmega _x}$. Figure 5(f) clearly shows that $\overline {P^{1*}_{\omega _z\omega _z}}$ is strongly attenuated in the STW cases of large amplitudes $(A>0.5)$, and its peak is further shifted away towards the buffer layer compared with HWO. The shift in $\overline {P^{1*}_{\omega _z\omega _z}}$ is as large as twice the thickness of the viscous sublayer for $A1.25$ case with respect to the HWO case.

A closer look at figure 6 reveals another peculiar behaviour of the $\overline {\omega _z\omega _z}$ production mechanism next to the wall. The production terms $\overline {P^{2}_{\omega _z\omega _z}}$ and $\overline {P^{6}_{\omega _z\omega _z}}$ cancel each other in the viscous sublayer and low buffer layer up to $y=10$. Figure 6(c) shows how well their phase averages coincide in a somewhat unexpected way at $y=2$. It is important to note that there is no direct link between $\overline {P^{2}_{\omega _z\omega _z}}$ and $\overline {P^{6}_{\omega _z\omega _z}}$. The former results from Stokes straining, while the latter comes from the local tilting term $\langle \omega _y{\partial w}/{\partial y} \rangle$. Next to the wall, $\overline {P^{6}_{\omega _z\omega _z}}$ reduces to the triple correlation $\overline {\omega _x\omega _y\omega _z}$, and can hardly be connected to $\overline {P^{2}_{\omega _z\omega _z}}$. The deterministic $\partial \langle W\rangle /\partial x$ reaches very large values at large imposed amplitudes next to the wall, and correlates almost perfectly with $\langle \omega _z\partial w/\partial y \rangle$ (figure 6d).

Figure 6. Phase wise variations of (a) $\langle P^{2}_{\omega _z\omega _z} \rangle$, (b) $\langle P^{6}_{\omega _z\omega _z} \rangle$ at different wall-normal locations, (c) $\langle P^{2}_{\omega _z\omega _z} \rangle$ and $\langle P^{6}_{\omega _z\omega _z} \rangle$ and (d) $\langle \partial W/\partial x\rangle$ and $\langle \omega _z\partial w/\partial y\rangle$ for $A0.95$ case at $y=2$, respectively. The values of $\langle \omega _z\partial w/\partial y\rangle$ are multiplied by a factor of 25.

To summarize, figure 6 recapitulates the major production terms of the $\overline {\omega _z\omega _z}$ transport equations, and figure 7 summarizes its main characteristics. There are finally two terms, namely, $\overline {P^{4}_{\omega _z\omega _z}}$ which is the major term in the uncontrolled flow and $\overline {P^{1*}_{\omega _z\omega _z}}$ which is specific to HWO and STW. All these production terms are the consequences of different mechanisms. Basically, the proper (direct) effect of STW is to reduce the intensity of the total production by Stokes straining, consequently its peak is shifted away from the wall to the buffer layer by approximately 10 wall units for large amplitude STW with respect to the HWO case.

Figure 7. Summary of main characteristics of the effect of control on the spanwise turbulent enstrophy production process.

3.2. Streamwise enstrophy transport

The production term for the streamwise turbulent enstrophy is given as

(3.3)\begin{align} \overline{ P_{\omega_x\omega_x}} &= \underbrace{ \overline{2\widetilde{\varOmega_x}\left\langle \omega_x\frac{\partial u}{\partial x}\right\rangle} }_{\overline{ P^{1}_{\omega_x\omega_x}}}+ \underbrace{ \overline{2\widetilde{\varOmega_y}\left\langle \omega_x\frac{\partial u}{\partial y}\right\rangle} }_{\overline{ P^{2}_{\omega_x\omega_x}}}+ \underbrace{ \overline{2\widetilde{\varOmega_z}\left\langle \omega_x\frac{\partial u}{\partial z}\right\rangle} }_{\overline{ P^{3}_{\omega_x\omega_x}}}- \underbrace{ 2\overline{\omega_x\frac{\partial w}{\partial x}}\frac{{\rm d}\overline {U}}{{\rm d}y} }_{\overline{ P^{4}_{\omega_x\omega_x}}} \nonumber\\ &\quad +\underbrace{ \overline{2\omega_x\omega_x\frac{\partial u}{\partial x}} }_{\overline{ P^{5}_{\omega_x\omega_x}}}+ \underbrace{ \overline{2\omega_x\omega_y\frac{\partial u}{\partial y}} }_{\overline{ P^{6}_{\omega_x\omega_x}}}+ \underbrace{ \overline{2\omega_x\omega_z\frac{\partial u}{\partial z}} }_{\overline{ P^{7}_{\omega_x\omega_x}}}+ \underbrace{ \overline{2\langle\omega_x\omega_x\rangle\frac{\partial\widetilde {U}}{\partial x}} }_{\overline{ P^{8}_{\omega_x\omega_x}}}+ \underbrace{ \overline{2\langle\omega_x\omega_y\rangle\frac{\partial\widetilde {U}}{\partial y}} }_{\overline{ P^{9}_{\omega_x\omega_x}}}. \end{align}

A careful analysis reveals that most of the terms in (3.3) are either negligible or cancel each other. Thus, the total production term $\overline {P_{\omega _x\omega _x}}$ effectively reduces to

(3.4)\begin{equation} \overline{P_{\omega_x\omega_x}} \approx \underbrace{2\overline{\widetilde{\varOmega_x} \left\langle \omega_x\frac{\partial u}{\partial x} \right\rangle}}_{\overline{P^1_{\omega_x\omega_x}}} \underbrace{- 2\overline{\omega_x\frac{\partial w}{\partial x}} \frac{{\rm d}\overline {U}}{{\rm d}y}}_{\overline{P^4_{\omega_x\omega_x}}}. \end{equation}

The production term $\overline {P^1_{\omega _x\omega _x}}$ represents the interaction of the Stokes shear $\partial \widetilde {W}/\partial y$ with the stretching of the streamwise vorticity $\langle \omega _x\partial u/\partial x \rangle$. This term is a direct consequence of forcing, and hence is absent in the canonical case. It increases with the amplitude of the STW, reaching large values for the $A1.25$ case, and peaks within the viscous sublayer at $y=5$ (figure 8a). Whereas in the case of HWO, it is negligible compared with the large amplitude STW cases.

Figure 8. Production terms (a) $\overline {P^1_{\omega _x\omega _x}}$ and (b) $\overline {P^4_{\omega _x\omega _x}}$. Refer to table 1 for markers corresponding to different cases.

The response of the streamwise turbulent enstrophy $\overline {\omega _x\omega _x}$ to the STW forcing is shown in figure 2(a). The local minimum and maximum in the profiles of $\overline {\omega _x\omega _x}$ are attributed to the streamwise vortices in the near-wall region (Kim, Moin & Moser Reference Kim, Moin and Moser1987). The local minimum increases with the amplitude of the STW, under the effect of Stokes shear $\partial \widetilde {W}/\partial y$ induced production $\overline {P^1_{\omega _x\omega _x}}$ which weakens the signature of the QSVs near the wall under large amplitude STW. Note that there is also a significant undermining of $\overline {\omega _x\omega _x}$ variation in the viscous sublayer for STW cases of $A>0.50$. For the $A1.25$ case, there is only a slight variation of $\overline {\omega _x\omega _x}$ between the local minimum and the wall, whereas in the case of uncontrolled flow there is an approximately $400\,\%$ increase at the same range of wall-normal distance. This is related to the lack of velocity-pressure gradient correlation in the spanwise velocity transport equation in the near-wall region (Umair et al. Reference Umair, Tardu and Doche2022).

For all the cases, the main production of $\overline {\omega _x\omega _x}$ comes from the tilting of $\omega _y$ by the mean shear ${\rm d}\overline {U}/{{\rm d}y}$ (figure 8b). This term, denoted by $\overline {P^4_{\omega _x\omega _x}}$ in (3.3), peaks roughly at $y=10\unicode{x2013}12.5$ for all the cases. Note that the profile of $\overline {P^4_{\omega _x\omega _x}}$ in HWO case collapse almost perfectly with the STW case $A0.30$. This strengthens again the observation made by Umair et al. (Reference Umair, Tardu and Doche2022) that the direct effects of STW mainly appear at $A>0.30$.

The turbulent transport (or turbulent diffusion) term in the uncontrolled flow is $\overline {T_{\omega _x\omega _x}}=-{\rm d}\overline {\omega _x\omega _xv}/{{\rm d}y}$, and is globally negligible compared with the other terms appearing in the transport equation of $\overline {\omega _x\omega _x}$. However, two additional transport terms emerge directly from the imposed unsteadiness in the STW cases. They are, respectively, given as $\overline {T^1_{\omega _x\omega _x}}=-2\overline {\langle \omega _x u\rangle \partial \widetilde {\varOmega _x}/\partial x}$ and $\overline {T^2_{\omega _x\omega _x}}=-2\overline {\langle \omega _x v\rangle \partial \widetilde {\varOmega _x}/\partial y}$ (Appendix A). Figure 9(b) shows that $\overline {T^2_{\omega _x\omega _x}}$ attains large values in the large amplitude STW cases. Recall that the turbulent transport terms correspond to the spatial redistribution of $\omega _x$. These nonlinear terms neither create nor destroy enstrophy, but act to simply redistribute it in space. A movie is attached showing that the production and transport are closely associated next to the wall at approximately $y=10$. The important point here is that the turbulent transport reaches large values comparable to the production (figure 10b). The direct consequence is the setup of excessive dissipation, which is almost twice the production near $y=10$. As mentioned before, the turbulent transport term is negligible for the uncontrolled case (represented by broken lines in figure 10a). In the HWO case, it is relatively smaller, and it is restricted to $y<10$ (figure 10a).

Figure 9. Turbulent transport (or turbulent diffusion) terms (a) $\overline {T^1_{\omega _x\omega _x}}$ and (b) $\overline {T^2_{\omega _x\omega _x}}$. Refer to table 1 for markers corresponding to different cases.

Figure 10. Budget of streamwise turbulent enstrophy transport for (a) HWO and (b) $A1.25$ cases. The budget for the reference (uncontrolled) canonical case is also included in (a) in grey broken lines for comparison. The mean advection term $\overline {A_{\omega _x\omega _x}}$ is negligible, and hence not displayed in the figures to avoid cluttering.

To partially resume, $\overline {\omega _x\omega _x}$ attains large values close to the wall compared with the uncontrolled case. For large amplitude STW, the near-wall variation from the location of local minima and the wall is almost flattened as a consequence of the lack of a velocity-pressure gradient term in the transport equation of spanwise turbulent intensity $\overline {ww}$. The streamwise vorticity layers in the STW cases with large amplitudes are simultaneously produced and transported in space by the turbulent diffusion, but at the same time dissipate quickly and hence do not contribute actively to the formation of the near-wall QSVs.

3.3. Wall-normal enstrophy transport

The peculiar behaviour of the $\partial w/\partial x$ and $\partial u/\partial z$ shear layers, constituting $\omega _y$ at large amplitude STW, will be discussed in § 5.3 in detail. The maximum of $\overline {\omega _y\omega _y}$ is at $y\approx 10$ in the uncontrolled flow, and is shifted towards $y\approx 20$ in the large amplitude STW cases (figure 2b). Globally, all the transport terms in the transport equation of $\overline {\omega _y\omega _y}$ are weakened and shifted towards the middle of the buffer layer. This is clearly seen in the dissipation $(\overline {\epsilon _{\omega _y\omega _y}})$ profiles shown in figure 11(c). It shows also the suppression of $\omega _y$ activity in the viscous sublayer with $\overline {\epsilon _{\omega _y\omega _y}}\approx 0$ at the wall.

Figure 11. The main transport terms appearing in the transport equation of wall-normal turbulent enstrophy $\overline {\omega _y\omega _y}$: (a) the production term $\overline {P^4_{\omega _y\omega _y}}$; (b) the turbulent diffusion term $\overline {T^3_{\omega _y\omega _y}}$; and (c) the dissipation term $\overline {\epsilon _{\omega _y\omega _y}}$. Refer to table 1 for markers corresponding to different cases.

Most of the terms appearing in the production term of the wall-normal enstrophy (Appendix A) either vanish or are negligible. Among the first three terms involving the Stokes strain, the only term which differs from zero is $\overline {P^1_{\omega _y\omega _y}}$, but it is also found to be negligible (not shown). Hence, the major production of $\overline {\omega _y\omega _y}$ for the controlled cases still comes from the tilting of the wall-normal turbulent vorticity $\omega _y$ by the mean shear, i.e. the term $\overline {P^4_{\omega _y\omega _y}}$, as in the case of the uncontrolled flow. The maximum of the production term $\overline {P^4_{\omega _y\omega _y}}$ is shifted towards $y\approx 20$ in the large amplitude STW cases (figure 11a), which explains the shift in the maximum of $\overline {\omega _y\omega _y}$ in figure 2(b). The Stokes straining turbulent transport terms $\overline {T^1_{\omega _y\omega _y}}=-2\overline {\langle \omega _y u\rangle \partial \widetilde {\varOmega _y}/\partial x}$ and $\overline {T^2_{\omega _y\omega _y}}=-2\overline {\langle \omega _y v\rangle \partial \widetilde {\varOmega _y}/\partial y}$ are both nearly zero (not shown). The third turbulent transport term $\overline {T^3_{\omega _y\omega _y}}=-2\,{\rm d}\overline {\omega _y\omega _yv}/{{\rm d}y}$ has some importance at $y<20$ in the uncontrolled flow, but becomes insignificant in the STW cases (figure 11b).

5.1. Drifts of the near-wall QSVs

Quasistreamwise vortices are a prominent feature of the near-wall turbulent flow field. The QSVs have been recognized to play a significant role in the regeneration cycle of near-wall turbulence (Hamilton, Kim & Waleffe Reference Hamilton, Kim and Waleffe1995). They facilitate the exchange of momentum and energy in the near-wall region, and contribute significantly to the generation of Reynolds shear-stress by inducing ejection and sweep events.

Earlier studies have reported a drastic suppression of the near-wall quasistreamwise vortices due to the spanwise wall oscillations control. Yakeno, Hasegawa & Kasagi (Reference Yakeno, Hasegawa and Kasagi2014) studied the effect of homogeneous spanwise wall oscillations on the structural modifications of QSVs and report that the wall-normal location of the peak population density of QSVs is not affected by the presence of control. Extending their analysis to the STW control, Gallorini, Quadrio & Gatti (Reference Gallorini, Quadrio and Gatti2022) also report little to no shift in the position of QSVs. However, their analysis includes only one STW case that produces a DR margin of approximately 36 %. Umair et al. (Reference Umair, Tardu and Doche2022), on the other hand, point out that the QSVs reach their full maturity well above $y=20$ for the large amplitude STW. This point, indeed, needs to be clarified whether there is a shift in the position of the near-wall QSVs because of the imposed control.

To clarify this point, we examine the effect of STW actuation on the near-wall QSVs, focusing on detecting the drift $(\Delta d_s)$ in QSVs because of the control, if any. We employ the classical eduction scheme proposed by Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997) to detect the near-wall QSVs. First, we detect the vortex cores where the value of $\lambda _2\le -0.02$ directly from the 3-D instantaneous velocity fields, where $\lambda _2$ being the second-largest eigenvalue of the symmetric and antisymmetric parts of the velocity gradient tensor. Then we identify and count only the structures that have streamwise extent greater than or equal to 150 wall units with inclination and titling angles in the range $-30^{\circ }$ to $+30^{\circ }$, respectively. The centre of each individual structure is obtained by locating the point where $\lambda _2$ attains its local minimum value within the structure. For all the cases studied here, we analysed an ensemble of at least 50 independent full 3-D instantaneous fields separated by roughly one full-through time unit $(20h/U_c)$. Since the criteria we chose to select the relevant QSVs is quite strict, as a consequence only few structures qualify to be considered as relevant candidates. For example, in the uncontrolled case, only approximately 90 structures met the imposed criteria in one snapshot of the instantaneous 3-D field. Figure 15 shows the $\lambda _2$ structures for a few of the cases studied here, and figure 16 shows their respective wall-normal distribution of the number of vortices, normalized by the maximum number of vortices.

Figure 15. Quasistreamwise vortical structures identified using $\lambda _2=-0.02$ along with the instantaneous field of streamwise velocity fluctuations $u$ (red, $u>0$; blue, $u<0$) on a wall-parallel plane at $y=15$ for (a) uncontrolled, (b) HWO, (c) $A0.50$ and (d) $A1.25$ case, respectively. The centres of the active QSVs educed using the criteria mentioned in § 5.1 are marked with yellow dots. The schematic diagram on the top shows the drift $(\Delta d_s)$ in the QSVs with respect to the canonical uncontrolled case. (HSS: high-speed streaks; LSS: low-speed streaks.)

Figure 16. The distribution of QSVs for the uncontrolled, HWO, $A0.50$ and $A1.25$ cases, respectively. The colours and markers in the plot correspond to those presented in figure 2.

For the canonical turbulent channel flow, the majority of the mature QSVs are located at $y\approx 24$, which is in agreement with Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997). For the STW cases, however, we observe a systematic drift of QSVs away from the wall (figure 16). The drift is approximately six wall units for the $A0.50$ case, and 10 wall units for the $A1.25$ case. This is in disagreement with Gallorini et al. (Reference Gallorini, Quadrio and Gatti2022). These authors consider a single DR case with control parameters different from us. The closest case we have to them in terms of DR is $A0.30$ for which we observe a drift of approximately 4.5 wall units, while they report little to no drift. This discrepancy is plausibly coming from the criteria used to select the vortices. There are two points to consider: first, a ‘vortex’ has to be a ‘developed compact object’ of sufficient length to exist and be dynamically significant (mature). In the present work, we exclude vortices with streamwise extents shorter than 150 wall units in the same way as in Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997), whereas Gallorini et al. (Reference Gallorini, Quadrio and Gatti2022) use rather a lower threshold length of 50 wall units, which means in their case a significant number of the detected vortices are immature low buffer layer structures. That indeed results in a twice smaller tilt angle of the structures they detect compared with Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997). Second, in their controlled case they ‘opted to discard the same percentage of candidate vortices considered in the reference case, to avoid the assumption that control does not affect the length of QSV’, but forcing may affect the related probability density functions, and rejecting the same percentage as in the reference case may induce some bias. Whereas, in the present work, the criteria used to select the mature vortices is unique and consistent for all the cases. The capacity of near-wall QSVs to regenerate new structures is proportional to their intensity and inversely proportional to the square of their distances $(d_s)$ from the wall, as discussed in the following subsection. In the drag reduction scenarios, one would expect an increase in $d_s$, as observed in the present study. It may, however, happen that $d_s$ remains unaffected, but the intensity of the structures or their population density decreases.

In the subsection below, we estimate the drift directly from the instantaneous visualizations of the near-wall velocity streaks, and show that the drifts estimated from these two independent methods are in close agreement. Moreover, we show that the drift we observe combined with the rest of the results give us a reasonable estimate of the DR margin, thus increasing the quality of these results.

5.2. Near-wall streaks

The drift $\Delta d_s$ of the active streamwise $\omega _x$ vorticity layers decreases the drag according to the conceptual model of Jiménez (Reference Jiménez1994) and also discussed in some detail in Tardu (Reference Tardu1995). By active $\omega _x$ layers, we mean those layers from which the Reynolds shear stress producing QSVs emerge. Given the complexity of the wall response to the large amplitude STW discussed before, it is difficult to directly determine the drift. One somewhat qualitative, yet objective, way is to investigate the near-wall streaks in order to estimate the drift. The near-wall streaks in the canonical turbulent wall flows are the footprints of the QSVs, and are at best detected by the $\partial u/\partial z$ shear layers near the wall (Tardu Reference Tardu2022). The $\partial u/\partial z$ shear layers are the thin wall-normal turbulent vorticity $\omega _y$ layers separating the low- and high-speed streaks. Figure 17 shows a snapshot of $\partial u/\partial z$ in the uncontrolled, HWO and STW $A1.25$ case, respectively. The near-wall long streaks in the canonical flow are already detectable at the edge of the viscous sublayer with a streak spacing of approximately 80 wall units, in agreement with previously published results (Tardu Reference Tardu2014). The footprints of QSVs are weak at $y=5$ under HWO, and they are clearly discernible only at $y\approx 12$, with slightly larger spacing of approximately 100 wall units.

Figure 17. Instantaneous fields of $\partial u/\partial z$ shear layers at $y=5, 10, 12, 15\ \text {and}\ 20$ for the uncontrolled, HWO and $A1.25$ case, respectively. The blue colour represents the region where $\partial u/\partial z < 0$, while the red colour represents the region where $\partial u/\partial z > 0$. The contours are in the range $-0.2$ to $+0.2$.

On comparing figure 17(o,k), we see that in the case of large amplitude STW $(A1.25)$, we have to go even farther in the buffer layer, towards $y\approx 20$, to detect the near-wall streaks of reasonable coherence. Let us suppose that the drift $\Delta d_s$ in coherent $\partial u/\partial z$ layers can be related to the drift $\Delta d_s$ of the active streamwise vorticity layers from which Reynolds shear stress producing eddies emanate. Then figure 17 suggests that $\Delta d_s\approx 7$ in the HWO case, because the distribution of $\partial u/\partial z$ shear layers at $y\approx 12$ under HWO becomes comparable to that of the uncontrolled flow at $y=5$. Similarly, comparing figure 17(l,a) suggests that $\Delta d_s\approx 10$ for STW $A1.25$ case. These observations are in agreement with the previous discussion based on figure 16.

The flow is turbulent in all the controlled cases here, even though the laminar limit is closely approached for the $A1.25$ case. Thus, we have on purpose chosen to use the local wall units to scale the turbulent quantities, the main aim being here is to determine the proper structural modifications. All the quantities scaled by the local inner variables can easily be transformed to the quantities scaled with respect to those based on the friction velocity of the uncontrolled case. Figure 18(a) shows the wall-normal distribution of the streamwise turbulent enstrophy $\overline {\omega _x\omega _{x0}}$ and its main production term, $\overline {P^4_{\omega _x\omega _{x0}}}$ in figure 18(b), both scaled with the wall units based on the friction velocity of the uncontrolled flow. It is seen that $\overline {\omega _x\omega _{x0}}$ decreases systematically in the controlled cases, while the peak location $y_0$ is not significantly altered and is at approximately 20 wall units.

Figure 18. Wall-normal distribution of the (a) streamwise turbulent enstrophy $\overline {\omega _x\omega _{x0}}$, and (b) its main production term $\overline {P^4_{\omega _x\omega _{x0}}}$, scaled with the wall units based on the friction velocity of the uncontrolled flow. Refer to table 1 for markers corresponding to different cases.

Orlandi & Jiménez (Reference Orlandi and Jiménez1994) relates the location $d_s$ and intensity of the QSVs to the wall shear by

(5.1)\begin{equation} \tau^*_w \propto \left( \frac{\varGamma^*}{\nu d^{*2}_s} \right)^{1/2}, \end{equation}

in dimensional units. In this relation $\varGamma ^*={\rm \pi} R^{*2}\omega _x^*$ is the mean circulation of the QSVs, $d^*_s$ is the distance to the wall and $R$ is their radius. The qualifier ‘$*$’ here represents quantities in physical dimensional units. This approximate relationship is obtained through a physical argument based on the stagnation flow induced by the QSVs. It gives only a qualitative description of the effect of QSVs on the wall shear, but has the merit to relate intensity and the stand-off distance of the coherent eddies on $\tau _w$. According to (5.1), the rate of change of $\Delta \tau _{w0}$ can be related to

(5.2)\begin{equation} \frac{\Delta \tau_{w0}}{\tau_{w0}} = \frac{1}{2}\frac{\Delta \varGamma_0}{\varGamma_0} - \frac{\Delta d_{s0}}{d_{s0}}, \end{equation}

when adequately scaled with the wall units based on the friction velocity of the uncontrolled flow.

One of the observations emerging from figure 17 and the $\lambda _2$ structures, shown before in figure 15, is that the radius of the QSVs (scaled in local wall units), in a rough sense, remains unaffected in all the controlled cases. Using $R=$ constant and $\overline {\omega _x\omega _x}_0$ profiles (figure 18a), we estimated 42 % of DR for $A0.50$ case from (5.2), which is in close agreement with the value reported in table 1. However, (5.2) gives 80 % of drag reduction at the largest imposed amplitude $A1.25$ case, which is approximately 30 % overestimated. In any case, the $\Delta \varGamma _0/\varGamma _0$ term is found to contribute only 20 % to the DR. Consequently, the drift $\Delta d_{s0}/d_{s0}$ plays a critical role in the drag reduction mechanism investigated here.

Duggleby, Ball & Paul (Reference Duggleby, Ball and Paul2007) examined the effect of spanwise wall oscillations on the dynamics of the near-wall turbulent structures in a turbulent pipe flow using Karhunen–Loève decomposition. They argued that the main effect of the Stokes layer generated by spanwise wall oscillations is to push the structures away from the wall into the region of higher mean velocity by creating a zone where turbulent structures cannot form. As a consequence, the structures are advected faster with less time to interact with the roll modes to transfer energy, resulting in their shorter lifetime, and hence damping of the Reynolds shear stress generating bursting events. Our finding strengthens the arguments presented by them and is in full concordance with their interpretation that although drag reduction results in the decorrelation of the near-wall streaks and the QSVs, but it is the lifting of the turbulent structures away from the wall by the Stokes layer induced by the spanwise wall oscillations that results in drag reduction.

5.3. Shear layers of spanwise fluctuating velocity field

The spanwise fluctuating velocity field plays a prominent role in the near-wall turbulence. Even though they do not implicitly contribute to the production of turbulent kinetic energy, they are, however, linked to the Reynolds shear stress producing events, and hence intrinsically linked to the characteristics of the near-wall coherent structures (Tardu Reference Tardu2016). The connection between the spanwise fluctuating velocity field and the intense Reynolds shear stress generating events become clear if one considers the classical hairpin and horseshoe vortex paradigm. The spanwise fluctuating velocity field is nearly zero between the legs of the hairpin vortices, where the intense sweep or ejection events occur. As mentioned previously, the main production of the turbulent streamwise enstrophy comes from the tilting of the wall-normal vorticity by the mean shear, which reduces to $-2\overline {\omega _x({\partial w}/{\partial x})}({{\rm d}\overline {U}}/{{\rm d}y})$. Hence, the $\partial w/\partial x$ shear layers play a crucial role in the generation of $\omega _x$ prior to their roll up into QSVs (Brooke & Hanratty Reference Brooke and Hanratty1993; Tardu Reference Tardu2008, Reference Tardu2014, Reference Tardu2016). Hence, as we saw earlier, the suppression of the spanwise velocity fluctuations in the viscous sublayer results in significantly larger drag reduction compared with the suppression of the streamwise or wall-normal velocity fluctuations.

In the canonical turbulent channel flow, the $\partial w/\partial x$ shear layers are spotty and slightly stretched in the spanwise direction, as shown in figure 19(a). Using the multiscale approach, Tardu (Reference Tardu2022) showed that some of the $\partial w/\partial x$ shear layers may appear as spanwise streaks. As seen in figure 19(e), the tilting of these shear layers into identifiable elongated $\omega _x$ layers already set up at $y=10$. The morphology of $\partial w/\partial x$ shear layers under the large amplitude STW is strikingly different. The $\partial w/\partial x>0$ and $\partial w/\partial x<0$ are organized into $\varLambda$-shaped Christmas-tree-like structures resulting from a direct effect of STW (figure 19d). The titled $\omega _x$ layers are consequently also $\varLambda$-shaped, and they are being elongated into streamwise structures (figure 19h). It is necessary to go farther away from the wall towards $y\approx 30$ to identify the elongated streamwise vorticity layers (figure 20a,b). It is also roughly at this position that the contours of $\omega _x\partial w/\partial x$, related to the major production term $P^4_{\omega _x\omega _x}$, appear as long coherent streaky-like structures (figure 20c). The $\omega _x$ layers at $y=10$ for $A0.50$ case are organized more clearly into streamwise elongated structures compared with the $A1.25$ case. They achieve their conventional morphology at $y=20$, earlier than the $A1.25$ case.

Figure 19. Instantaneous field of $\partial w/\partial x$ shear layers and $\omega _x$ layers at $y=10$ for the uncontrolled, HWO, $A0.50$ and $A1.25$ case, respectively.

Figure 20. Instantaneous field of (a) $\partial w/\partial x$ shear layers, (b) $\omega _x$ layers and (c) $\omega _x\partial w/\partial x$ layers at $y=30$ for $A1.25$ case.

The flow under STW control at large amplitudes develops its own structures in the low buffer layer. In the case of HWO, the $\partial w/\partial x$ shear layers are inclined in the streamwise direction, but do not exhibit $\varLambda$-shaped structures that are specific to the STW control. The origin of these structures is tricky to understand. The $\partial w/\partial x$ patterns next to the wall, in the large amplitude STW case, are too regular to be considered as induced by the random turbulence phenomenon. They are indeed related to the large $\partial \widetilde {W}/\partial x$ modulation occurring at the wall. Consider the instantaneous transport equation for the streamwise gradient of the instantaneous spanwise velocity field $(\partial W/\partial x)$ given as

(5.3)\begin{equation} \frac{{\rm D}}{{\rm D} t}\frac{\partial W}{\partial x} = \frac{\partial W}{\partial x} \frac{\partial V}{\partial y} - \frac{\partial V}{\partial x}\frac{\partial W}{\partial y} - \frac{\partial^2 P}{\partial x \partial z} + \nu\nabla^2\frac{\partial W}{\partial x}. \end{equation}

The complete analysis of $({{\rm D}}/{{\rm D} t})({\partial w}/{\partial x})({\partial w}/{\partial x})$, which contains 10 terms on the right-hand side (not shown), is complex, especially, because of the local pressure term appearing on the right-hand side of (5.3), and hence, is out of scope of the present investigation. After applying triple decomposition, several other terms appear on the right-hand side of (5.3), among which the term $({\partial \widetilde {W}}/{\partial x})({\partial v}/{\partial y})$, which represents the stretching of the Stokes shear $\partial \widetilde {W}/\partial x$ by the local $\partial v/\partial y$, is negligible at $y>5$, but reaches large values near the wall at $y\approx 0$. Figure 21(a,b) compares the $({\partial \widetilde {W}}/{\partial x})({\partial v}/{\partial y})$ contours with those of $\partial w/\partial x$ layers at $y=2.5$, respectively. A striking similarity emerges from these two figures. The Stokes shear $\partial \widetilde {W}/\partial x$ induces $\varLambda$-shaped $\partial w/\partial x$ next to the wall. The latter are further stretched, intensified and extended to the viscous sublayer. One quick way to estimate the quantitative importance of $({\partial \widetilde {W}}/{\partial x})({\partial v}/{\partial y})$ is to compare its r.m.s. value with the r.m.s. of $\partial w/\partial x$. A similar procedure has been conducted, for instance, by Brooke & Hanratty (Reference Brooke and Hanratty1993) in the local transport equation ${\rm D}\omega _x/{\rm D} t$. Figure 21(d) shows that the r.m.s. value of $({\partial \widetilde {W}}/{\partial x})({\partial v}/{\partial y})$ is as large as 30 % of the r.m.s. of $\partial w/\partial x$ at $y\approx 1$, and progressively disappears towards the edge of the viscous sublayer. Figure 21(c) shows the normalized cross-correlation coefficient $C$ between $({\partial \widetilde {W}}/{\partial x})({\partial v}/{\partial y})$ and $\partial w/\partial x$ at $y=2.5$. We observe specific cross-correlation patterns that are somewhat similar to the patterns observed in the autocorrelations of $({\partial \widetilde {W}}/{\partial x})({\partial v}/{\partial y})$ and $\partial w/\partial x$ (not shown). The cross-correlations reach values as large as 0.2, which is far from being negligible.

Figure 21. Instantaneous field of (a) $({\partial \widetilde {W}}/{\partial x})({\partial v}/{\partial y})$ shear layers, (b) $\partial w/\partial x$ layers, (c) their cross-correlation coefficient at $y=2.5$ and (d) the wall-normal distribution of the ratio of their root-mean-square (r.m.s.) values for $A1.25$ case.

From the visualizations presented in figure 15, it is clear that the classical topological features of the QSVs are only depicted at $y>20$ for large amplitude STW. Consequently, the peculiar $\varLambda$-shaped $\partial w/\partial x$ shear layers do not roll up into QSVs. Therefore, there is a drift of the active $\omega _x$ layers that lead to Reynolds shear-stress producing eddies at large amplitude STW.