3.1. Global drag change

The global drag change $\Delta D$ is defined as the change in friction coefficient $c_{f}$ under control

(3.1) \begin{equation} \Delta D = 1 - \underbrace {\frac {(c_{f})_{c}}{(c_{f})_{0}}}_{\equiv \xi }; \quad c_{f} = \frac {{\tau }^{\star }_w}{\frac {1}{2} {\rho }^{\star } ({U}^{\star }_b)^2} .\end{equation}

Positive values of $\Delta D$ indicate drag reduction, while negative values represent drag increase. In order to characterise the drag change, it is sufficient to keep track of the friction coefficient ratio $\xi$ , which is interpreted as

(3.2) \begin{equation} \xi \in \begin{cases} [0, 1) & \text{drag reduction (smaller is better)} ,\\ (1, \infty ) & \text{drag increase (larger is worse)}, \end{cases} \end{equation}

and the drag is unchanged if $\xi = 1$ . We use $\Delta D$ and $\xi$ interchangeably throughout this paper, with an emphasis on $\xi$ for qualitative visualisations and $\Delta D$ for quantitative statements. It is important to point out that $\Delta D$ does not account for the work done by the actuation. Equation (3.1) does therefore not quantify energetic savings, but rather categorises the flow response into two broad classes, namely a drag-reducing ( $\xi \lt 1)$ and a drag-increasing ( $\xi \gt 1$ ) one.

Figure 3.

Drag reduction of varying-phase opposition control with sensors located at $y_d^+ = 15$ . Filled symbols inside the green-shaded area indicate drag reduction ( $\xi \lt 1$ ), while open symbols in the red region indicate unchanged or increasing drag ( $\xi \geqslant 1$ ). The area inside the black square in figure 3(a) is magnified in figure 3(b).

Varying-phase opposition control with $\angle \hat {A}_d$ according to (2.5) has two parameters, the sensor location ( $y_d$ ) and phase shift ( $\angle \hat {A}_d$ ). The present study focuses on the role of the phase shift. We therefore fix $y_d^+ = 15$ , which corresponds to a sensor location with a strong flow response (see Toedtli et al. Reference Toedtli, Luhar and McKeon2019a for details) and vary the phase shift over its entire range $\angle \hat {A}_d \in [-\pi , +\pi ]$ . The associated drag change in terms of $\xi$ is shown by the orange circles in figure 3(a) (entire parameter range) and figure 3(b) (detailed view of the black square in figure 3 a). Filled symbols in the green-shaded area indicate drag reduction ( $\xi \lt 1$ ), while open symbols in the red-shaded region denote unchanged or increasing drag ( $\xi \geqslant 1$ ). The data clearly show that the drag change strongly depends on the phase shift of the wall transpiration. Small negative phase shifts lead to drag reduction, while positive or large negative phase shifts result in drag increase. In particular, the drag increase at $\angle \hat {A}_d = \pm \pi$ (which corresponds to positive control feedback) was so pronounced that the simulations diverged at the current resolution and, therefore, no data point is plotted at this phase shift.

In the following sections, we analyse the three example controllers listed in table 1 in detail and use their labels (e.g. N25) to refer to a specific configuration. Controller N25 leads to 21 % drag reduction, which is the largest drag reduction over the $\{y_d, \angle \hat {A}_d\}$ parameter range considered by Toedtli et al. (Reference Toedtli, Luhar and McKeon2019a ). The remaining two example configurations lead to a drag increase of 113 % (N75) and 180 % (P50), respectively. Two drag-increasing example controllers are chosen to investigate if the physical mechanisms that drive the flow response are different for positive and negative $\angle \hat {A}_d$ .

Table 1.

Parameters for example controllers. Positive values of $\Delta D$ indicate drag reduction (DR), while negative values represent drag increase (DI). In all cases, the sensors are located at $y_d^+ = 15$ .

Figure 4.

(a,c,e) Instantaneous spatial structure of the transpiration $v^+(y_w)$ at the wall. (b,d,f) Time-averaged actuation spectrum $\varPhi _{vv}^+(y_w)$ . The green contour lines denote $\varPhi _{{vv}}^+ / \max (\varPhi _{{vv}}^+) = (0.15, 0.3, 0.45)$ and the blue and red squares indicate the example spatial scales $\boldsymbol{\kappa }_s$ and $\boldsymbol{\kappa }_r$ , respectively.

3.2. Wall transpiration structure and scale families

Figures 4(a,c,e) show representative instantaneous snapshots of the wall transpiration for each example controller. These qualitative flow visualisations highlight two important aspects. First, the magnitude of the transpiration varies significantly with $\angle \hat {A}_d$ . For example, the control input of the drag-reducing configuration N25 is an order of magnitude smaller compared with the drag-increasing controllers N75 and P50 (note the logarithmic colour scale). A larger wall transpiration is therefore not necessarily beneficial in terms of drag reduction. The second observation is the different spatial structure of the transpiration for positive and negative phase shifts. The control input of controllers N25 and N75 consists of streamwise-elongated streaky structures and we observe larger, more coherent structures in the N25 case. In contrast, the transpiration of controller P50 exhibits little coherence in the streamwise direction and coherent patches instead occur at an angle to the mean flow or even oriented along the span.

The power spectra in figure 4(b,d,f) provide a statistical characterisation of the transpiration structure and confirm the observations from the instantaneous flow fields. The most energetic scales in the spectrum of N25 (figure 4 d) are streamwise-elongated ( $k_x$ small and $k_x \lt k_z$ ) and the peak occurs at $\boldsymbol{\kappa }_s = (k_x = 0.5, k_z = 11)^{\intercal }$ (or, equivalently, $\boldsymbol{\lambda }_s^+ = (\lambda _x^+ \approx 2260, \lambda _z^+ \approx 100)^{\intercal }$ ), which is indicated by the blue square. Scales centred around $\boldsymbol{\kappa }_s$ are also the most active in configuration N75 (figure 4 b), but the energetic region extends to larger $k_x$ and $k_z$ , which confirms that this transpiration is more multi-scale than N25. The streamwise-elongated scales that dominate the actuator input at negative phase shifts define a first scale family, termed streak scales, with scale $\boldsymbol{\kappa }_s$ as a representative example. A scale family is loosely defined as a collection of spatial scales that lie close in spectral space and respond similarly to $\angle \hat {A}_d$ . The streak scales coincide with the near-wall cycle at the present ${\textit{Re}}_{\tau }$ , but their scaling with Reynolds number remains an open question.

In contrast, the most active scales in the spectrum of P50 (figure 4 f) are short in the streamwise direction and wide in the span. The peak occurs at $\boldsymbol{\kappa }_r = (k_x = 6.5, k_z = 0)^{\intercal }$ (or $\boldsymbol{\lambda }_r^+ = (\lambda _x^+ \approx 170, \lambda _z^+ \rightarrow \infty )^{\intercal }$ ), which is marked by the red square. The short and wide scales that dominate the actuator input at positive phase shifts define a second scale family, termed roller scales, and $\boldsymbol{\kappa }_r$ is a representative example of this scale family. Conditional averaging shows that the short and wide scales correspond to spanwise rollers, which extend from the wall to $y^+ \approx 40$ . The eigenspectrum of the unforced Navier–Stokes equations can predict the onset and spatial structure of the spanwise rollers and suggests that they convect at ${c}^{\star } \approx 0.4 ( {U}^{\star }_c )_0 \approx 7.6 ( {u}^{\star }_{\tau } )_0$ (Toedtli et al. Reference Toedtli, Yu and McKeon2020). The spatial scale and convection speed of the rollers are comparable to those observed over riblets ( $\lambda _x^+ \approx 150$ , $c^+ \approx 6-8$ ; García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021), but smaller than those in the porous wall case ( $\lambda _x^+ \approx 900$ , $c^+ \approx 5-8$ ; Jiménez et al. Reference Jiménez, Uhlmann, Pinelli and Kawahara2001). The streak scales observed in cases N25 and N75 are not very pronounced in configuration P50, which underscores that the transpiration structure at positive and negative phase shifts is markedly different.

Additional visualisations and a more in-depth discussion of the controlled flow structure and scale families can be found in Toedtli et al. (Reference Toedtli, Yu and McKeon2019b ), Toedtli (Reference Toedtli2021). A key takeaway from these references is that the flow structures of the three example controllers differ not only near the wall, but throughout the entire channel.

3.3. Scale-by-scale drag change

The friction coefficient $c_f$ can be expressed as the sum of a laminar ( $c_{f,l}$ ) and a turbulent contribution ( $c_{f,t}$ ) (see (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002) for details). The turbulent contribution results from a weighted wall-normal integral of the Reynolds stresses and can be further decomposed into scale-by-scale contributions $c_{f,t\boldsymbol{\kappa }}$ :

(3.3) \begin{equation} c_f = \underbrace {\frac {12}{{\textit{Re}}_b}}_{=c_{f,l}} + \underbrace {12 \int _{-1}^{1} y (\overline {{u}^{\prime } {v}^{\prime }}) \, {\textrm{d}} y}_{=c_{f,t}} = \frac {12}{{\textit{Re}}_b} + \sum _{k_x} \sum _{k_z} \underbrace {12 \int _{-1}^1 y \, {\textrm{Re}} \{ \hat {u}_{\boldsymbol{\kappa }} \hat {v}^{\ast }_{\boldsymbol{\kappa }} \} \, {\textrm{d}}y}_{=c_{f,t \boldsymbol{\kappa }}} ,\end{equation}

where ${\textrm{Re}}$ denotes the real part of a complex-valued quantity. This decomposition is valid as long as the assumptions outlined in Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002) are met and the velocities are non-dimensionalised by twice the bulk velocity. The sign of $c_{f, t \boldsymbol{\kappa }}$ results from the phase relation between $\hat {u}_{\boldsymbol{\kappa }}$ and $\hat {v}_{\boldsymbol{\kappa }}$ across $y$ . In a canonical turbulent channel flow, $\hat {u}_{\boldsymbol{\kappa }}$ and $\hat {v}_{\boldsymbol{\kappa }}$ are out-of-phase on average for $y\leqslant 0$ (and vice versa for $y\gt 0$ ), so that the scale-by-scale turbulent drag contribution is positive. Control alters $\hat {u}_{\boldsymbol{\kappa }}$ and $\hat {v}_{\boldsymbol{\kappa }}$ and possibly their relative arrangement, so that $c_{f, t \boldsymbol{\kappa }}$ of a controlled flow can in principle have either sign. A negative sign would imply that control fundamentally alters the phase relation between $\hat {u}_{\boldsymbol{\kappa }}$ and $\hat {v}_{\boldsymbol{\kappa }}$ so that the Reynolds stresses become an accelerating force.

Equation (3.3) can be used to decompose the drag change into scale-by-scale contributions $\Delta D_{\boldsymbol{\kappa }}$ , i.e.

(3.4) \begin{equation} \Delta D = \sum _{k_x} \sum _{k_z} \Delta D_{\boldsymbol{\kappa }};\! \quad \!\Delta D_{\boldsymbol{\kappa }} = \frac {( c_{f, t\boldsymbol{\kappa }})_0 - ( c_{f, t\boldsymbol{\kappa }})_c}{( c_f )_0} ,\end{equation}

where the laminar contribution $c_{f, l}$ of the uncontrolled and controlled flow cancel because the bulk Reynolds number is held constant. The scale-by-scale drag change provides an objective metric for scale suppression and amplification under control: a scale is said to be suppressed if $\Delta D_{\boldsymbol{\kappa }} \gt 0$ (i.e. the turbulent drag contribution decreases) and amplified if $\Delta D_{\boldsymbol{\kappa }} \lt 0$ (turbulent drag contribution increases). The top row of figure 5 shows the scale-by-scale drag change for the three example controllers, along with the spectral content of the actuation (solid green contours) and the uncontrolled turbulent drag (dashed black contour). The spectral regions of drag change and actuator input show significant overlap, which may suggest a correlation between the two. In configurations N25 and N75 the drag change is primarily driven by the streak scales, which carry significant turbulent drag in the uncontrolled flow. These streak scales are suppressed by controller N25 and amplified by N75. Note that the spectral signature of the actuation is similar in both cases, which highlights that the phase shift (streamwise position relative to the background flow) has a profound impact on the control response. In contrast, controller P50 amplifies both streak and roller scales.

Figure 5.

Scale-by-scale drag change integrated over the entire channel height (top row), inner region (defined as $|y^+ - y_w^+| \leqslant 40$ , centre row), and outer region ( $|y^+ - y_w^+| \gt 40$ , bottom row). The solid green contour lines denote $\varPhi _{{vv}}^+ / \max (\varPhi _{{vv}}^+) = (0.15, 0.3, 0.45)$ and are carried over from figure 4, while the black dashed contour outlines $(c_{f,t\boldsymbol{\kappa }})_0 / \max ((c_{f,t\boldsymbol{\kappa }})_0) = 0.15$ .

The scale-by-scale drag change spectra also provide insight into the challenges of achieving drag reduction. We found that $(c_{f, t \boldsymbol{\kappa }})_c \geqslant 0$ in configuration N25, which indicates that the present transpiration reduces or at best annihilates the scale-by-scale drag contribution and limits the achievable drag reduction. On the other hand, $(c_{f, t \boldsymbol{\kappa }})_c$ can be a multiple of $(c_{f, t \boldsymbol{\kappa }})_0$ in drag-increased flows and there is no apparent limit on how much control can amplify a scale. For example, the roller scales, which do not carry significant turbulent drag in a canonical flow, can be amplified by more than an order of magnitude. In addition, the effect of control is often non-uniform in spectral space (see, e.g. figure 5 a) and a highly amplified scale can cancel the effect of many weakly suppressed scales. For all these reasons, it is much more difficult to reduce drag than to increase it.

Thus far, the integral in the scale-by-scale turbulent drag reduction has been evaluated over the entire channel height. To highlight the near-wall effect of control, we split the integral into two parts: an inner part, which includes the region $|y^+ - y_w^+| \leqslant 40$ , and an outer part, which comprises the rest of the channel. This specific threshold is motivated by the earlier observation that spanwise rollers extend up to $y^+ \approx 40$ . The drag change spectra for the inner and outer region are shown in the centre and bottom row of figure 5. Across all controllers, the inner region accounts for approximately 53 % of the total drag change, while the outer region contributes 47 %. In the inner region, the drag change is most pronounced in spectral regions of large actuator input. For negative phase shifts, the inner-region drag change is similar to the global drag change. However, configuration P50 reveals distinct near-wall characteristics: the signature of the spanwise rollers is more pronounced, and control suppresses large streamwise scales at $k_x \approx 0.5$ . In contrast, the drag change in the outer region correlates less with actuator input and is confined to large streamwise and spanwise scales, which likely have wall-normal support throughout the channel. Interestingly, the drag change in the inner and outer region can oppose each other at specific scales. This is most apparent in configuration N25, where opposing trends weaken the net drag change at the example scale $ \boldsymbol{\kappa} _s$ , and in configuration P50, where large streamwise scales are suppressed in the inner region, but amplified in the outer.

3.4. Connection between transpiration phase shift and scale response

The drag change spectra of figure 5 suggest a correlation between actuator input and scale suppression or amplification. However, the drag change at scale $\boldsymbol{\kappa }$ cannot be directly attributed to the transpiration at the same scale, due to the nonlinear governing equations. Such a connection can only be established if possible nonlinear control effects are minimised. This can be achieved by restricting control to a single wavenumber, or to a small set of wavenumbers that are not triadically consistent. We refer to these controllers as scale-restricted controllers, and have analysed the flow response to two scale-restricted controllers: the first controller generates a transpiration at a small number of triadically inconsistent streak scales and the second one generates a transpiration with a single roller scale. The details of the scale-restricted controllers are summarised in Appendix A.

In the case of scale-restricted controllers, which minimise nonlinear control effects, the drag change at a controlled scale can be directly attributed to the transpiration at that scale. The controlled scales also drive the global drag change of the scale-restricted controllers, which is shown by the blue triangles (streak scales) and red squares (roller scale) in figure 3. Depending on the phase shift, scale-restricted varying-phase opposition control suppresses ( $\lvert \angle \hat {A}_d \rvert \lesssim \pi / 4$ ) or amplifies the streak scales ( $\lvert \angle \hat {A}_d \rvert \gtrsim \pi / 2$ ), which results in global drag reduction and increase, respectively. Note that the controller with $\angle \hat {A}_d = \pm \pi$ also diverged in this case. In contrast, scale-restricted varying-phase opposition control leaves the roller scale unaffected at negative phase shifts but significantly amplifies it at $\angle \hat {A}_d \in [+\pi / 4, +3 \pi / 4]$ , which leads to drag increase. These scale-restricted control experiments confirm that scale suppression and amplification are to a significant part tied to the transpiration and $\angle \hat {A}_d$ at that scale.

A comparison between the drag change of the scale-restricted and the full controller further suggests that the flow response to varying-phase opposition control can be understood from a superposition of the two scale families. The roller scales are inactive for $\angle \hat {A}_d \in [-\pi , +\pi / 4)$ and the flow response is fully determined by the streak scales, which reduce (small phase shifts) or increase (large positive or negative $\angle \hat {A}_d$ ) the drag. Both scale families contribute to the drag increase at $\angle \hat {A}_d \in [+\pi / 4, 3 \pi / 4]$ , but the roller scales dominate due to the stronger flow response (compare the drag increase due to the two scale-restricted controllers in figure 3). The dominance of the roller scales for positive phase shifts is further consistent with the actuation spectra of figure 4. It is important to emphasise that the scale-restricted controllers act only on a subset of each scale family. The remaining members of the scale family contribute to the drag change of the full control scheme as well, and consequently, the drag change produced is larger compared with the scale-restricted controllers.

4.1. Wall pressure structure

The spatial structure of the wall pressure for the three example controllers is depicted in figures 6(a), 6(c) and 6(e). The time instant and location are identical to the earlier visualisations of the transpiration (figures 4 a, 4 c and 4 e). Similar to the earlier discussion of the transpiration, we observe differences in both magnitude and structure across the three example wall pressure fields. The wall pressure of the drag-reducing controller N25 (figure 6 c) features large-scale patches of positive and negative fluctuations, while that of the drag-increasing controllers figures 6(a) and 6(e) is more fragmented. The wall pressure of the drag-increasing controllers is also an order of magnitude larger compared with the drag-reducing case.

Figures 6(b), 6(d) and 6(f) shows the time-averaged power spectra of the wall pressure, with distinct levels outlined by the dashed pink lines. In addition, the green contour lines represent the spectral content of the actuator input (see figure 4) and the blue and red squares indicate the representative streak ( $\boldsymbol{\kappa }_s$ ) and roller scales ( $\boldsymbol{\kappa }_r$ ), respectively. The overall structure of the three wall pressure spectra is similar, with the most energetic region located at small streamwise and spanwise wavenumbers. The magnitude of the scale-by-scale contribution and the extent of the energetic scales depends strongly on $\angle \hat {A}_d$ , with drag-increasing configurations leading to more energetic wall pressure fluctuations (note the logarithmic colour scale) over a wider range of scales. This is, on average, the spectral analogue of the more intense and fragmented instantaneous wall pressure structures observed in figures 6(a) and 6(e).

The Greens function solution, (2.10) and (2.11), provides some insight into the structure of the wall pressure spectra and their relation to the transpiration. The Green’s function kernel $G(y,a)$ is real-valued and only depends on the magnitude of the wavenumber vector, $\kappa$ . Different combinations of $k_x$ and $k_z$ with the same $\kappa$ thus have an identical $G(y,a)$ and example kernels for the wall pressure at various $\kappa$ are shown in figure 7. For increasing $\kappa$ , the wall-normal support of the Green’s function decreases and the maximum amplitude decreases like $1/\kappa$ . It is important to reiterate the observation of Kim (Reference Kim1989) that the Green’s function kernels for small $\kappa$ decrease slowly with $y$ and can be non-zero throughout the channel (see, e.g. the profile $\kappa = 1$ in figure 7). The slow decay implies a large domain of dependence in $y$ and can even introduce a mutual influence of the two walls. Similar observations apply to the Stokes pressure as well, but the details are omitted for the sake of brevity.

Figure 7.

Example Green’s function kernels for the wall pressure ( $y = -1$ ) at various values of $\kappa$ . The Green’s function is plotted versus the integration variable $a$ .

The wall pressure spectra peak at low $k_{x}$ and $k_{z}$ , where the Green’s function kernels have the largest $y$ support. Regions far from the wall can thus potentially contribute to these maxima. Interestingly, the wall pressure spectra are a function of $k_x$ and $k_z$ , which is different from the $\kappa$ dependence of the Green’s function kernel. This dependence on $(k_x, k_z)$ must arise from the source terms (slow and fast pressure) and from the boundary conditions (Stokes pressure). One aspect that certainly contributes to the observed asymmetry is the fast pressure, whose source term contains an $x$ derivative that amplifies smaller streamwise scales (large $k_x$ ) and damps larger ones (small $k_x$ ). The edge case is $k_{x} = 0$ , where the fast pressure disappears all together. Consistent with this argument, the wall pressure spectra become less energetic as $k_x \rightarrow 0$ .

Likely for the same reason, actuation with streak scales does not generate a strong pressure response at low $k_x$ (figures 6 b and 6 d). In contrast, actuation with roller scales goes hand-in-hand with an increase in pressure signature at small $k_z$ , where the fast pressure source term is not damped. It is also interesting to note that the drag-increasing controllers increase the wall pressure in spectral regions beyond the actuator input. This hints at the importance of nonlinear interactions, either through the slow pressure or energy transfer across velocity scales.

4.2. Relative importance of the pressure components

Figure 8.

Total variance (a,b,c) and example scale contribution (d,e,f) to the fast (dotted orange line), slow (dashed blue line) and Stokes pressure (dash-dotted purple line). The grey lines in figure 8(a) outline the reference profiles for a canonical turbulent channel flow at $({\textit{Re}}_{\tau })_0 = 180$ .

The discussion thus far has considered the total wall pressure field. Subsequent analyses will consider the pressure components (fast, slow and Stokes pressure) individually. Interpretation of these more finely partitioned data requires quantification of the relative importance of the pressure components, as shown in figure 8. The coloured lines in figure 8(a–c) show the total pressure component variance of the controlled flows, while the grey lines in figure 8(a) outline the reference profiles for a canonical turbulent channel flow without transpiration. The sum of the component variances at $y_w$ represents a subset of the terms in the Parseval identity and is thus related to the wall pressure spectra of figure 6. The relative magnitude of the pressure component fluctuations underscore again the earlier observations that drag-increasing controllers lead to larger pressure fluctuations at the wall compared with the drag-reducing configuration.

It is also apparent that wall transpiration changes the relative importance of the pressure components compared with a canonical turbulent channel flow. In the absence of transpiration, the slow pressure is larger than the fast one, except very near the wall, and the Stokes pressure is the smallest of the three components. The Stokes pressure further scales as $1/{\textit{Re}}_b$ and is therefore assumed negligible at high Reynolds numbers (see (2.10) and discussion in Kim (Reference Kim1989) for details). In contrast, the magnitude of the Stokes pressure becomes comparable to the other components close to the wall when the no-throughflow condition is relaxed, regardless of whether the drag is reduced or increased. This increase in magnitude occurs because the transpiration adds a leading-order inertial term (see (2.10)). The Stokes pressure in transpiration-based controlled flows can thus be leading order, possibly even at high Reynolds numbers. Moreover, the relative magnitude of the slow and fast pressure can change with transpiration. For example, the relative magnitude of the fast pressure decreases when the drag is reduced (figure 8 b) and increases when the drag is increased, especially near the wall. This is likely a consequence of control reducing or amplifying the wall-normal velocity (see spectra in figure 4), which enters the source term of the fast pressure.

The spatially averaged variances discussed thus far represent a sum over all spatial scales, but the relative importance of the pressure components can in principle vary from scale to scale. This aspect is explored in figure 8(d–f), which show the variance contributions of the relevant example scale (streak scale $\boldsymbol{\kappa }_s$ for negative $\angle \hat {A}_d$ and roller scale $\boldsymbol{\kappa }_r$ for positive $\angle \hat {A}_d$ ) for each controller. A comparison between the top and bottom row demonstrates that the relative magnitude is indeed scale-dependent. For example, the fast pressure contributes most to the wall pressure when summed over all scales of configuration N75, but it is the smallest component at scale $\boldsymbol{\kappa }_s$ .

The scale-by-scale analysis further shows that the dominant pressure components change with $\angle \hat {A}_d$ and $\boldsymbol{\kappa }$ . For negative $\angle \hat {A}_d$ , the slow and Stokes pressure at $\boldsymbol{\kappa }_s$ are almost an order of magnitude larger at the wall compared with the fast pressure. It can therefore be expected that they determine the overall wall pressure signature at this scale. On the other hand, for positive $\angle \hat {A}_d$ , the fast and Stokes pressure are dominant at the scale $\boldsymbol{\kappa }_r$ . Figures 8(d) and 8(f) further show that the different relative importance occurs even when the transpiration magnitude is comparable. This suggests that the different drag increase mechanisms, amplification of the near-wall cycle for $\boldsymbol{\kappa }_s$ and generation of spanwise rollers for $\boldsymbol{\kappa }_r$ , lead to fundamentally different source terms for the pressure components and, therefore, to different relative magnitudes. Some aspects of this will be further explored in § 5.2.

5.1. Phase of wall pressure components: domain of dependence

The present section analyses the structure of the Greens function kernel and the analytical Stokes pressure solution. The discussion focuses on the wall pressure and, in particular, on how its phase is established through the domain of dependence in the wall-normal direction and in spectral space. These observations apply to turbulent channel flows in general and will be applied to the example controlled flows in § 5.2 to make sense of the observed statistics.

The phase of the fast and slow wall pressure results from a convolution of the Green’s function kernel with the associated source term

(5.2) \begin{equation} \begin{aligned} \angle \hat {p}_{\boldsymbol{\kappa }, \{f/s\}}(y_w, t) &= \angle \left \lbrace \int _{-1}^{1} G(y_w, a) \hat {f}_{\{f/s\}}(a, t) \, {\textrm{d}} a \right \rbrace \\ &= \angle \left \lbrace \int _{-1}^{1} \lvert G(y_w, a) \hat {f}_{\{f/s\}}(a, t) \rvert e^{i (\angle \hat {f}_{\{f/s\}}(a,t) + \pi) } {\textrm{d}} a \right \rbrace\! .\end{aligned} \end{equation}

Recall from (2.12) that the Green’s function kernel $G(y,a)$ is real-valued and negative throughout the channel, which is reflected in the additional $\pi$ phase above. The integral expression in (5.2) shows that the phases of the fast and slow wall pressure are weighted wall-normal averages of $\angle \hat {f}_{\{f/s\}}(y)$ . The weight function is given by $\lvert G(y_w, a) \hat {f}_{\{f/s\}}(a) \rvert$ , and differs for the fast and slow pressure due to the different source terms (see (2.9)). Both weight functions have compact support in $y$ for sufficiently large $\kappa$ , since $G(y_w, a)$ is localised (see figure 7). The wall-normal support of the weight function for the fast pressure may be further restricted by the mean shear $S$ , which is part of the source term. This is especially relevant at low $\kappa$ when the magnitude of $G(y_w, a)$ decays slowly (see, e.g. the profile for $\kappa = 1$ in figure 7).

Despite their similarities in the wall-normal domain of dependence, the fast and slow pressure have very different spectral dependencies. The source term of the fast pressure is linear, which implies that $\hat {p}_{\boldsymbol{\kappa }, f}(y_w)$ and its phase only depend on the flow state at that same $\boldsymbol{\kappa }$ . Conversely, the source term of the slow pressure is nonlinear, so that its value and phase at the wall do not depend on the flow state at $\boldsymbol{\kappa }$ itself, but on all wavenumbers that are triadically consistent with $\boldsymbol{\kappa }$ . Approximately steady phase relations between wall transpiration and fast or slow pressure can result if the pressure source term correlates with $\hat {v}_{\boldsymbol{\kappa }}(y_w)$ over the wall-normal layer where the weight function is non-zero.

The Stokes pressure on the other hand is forced by the boundary condition. The relation is linear, which implies that $\hat {p}_{St,\boldsymbol{\kappa }}$ only depends on the flow state at that same wavenumber. For sufficiently large $\kappa$ , when the influence of one wall is negligible at the other, the Stokes pressure further depends on local wall information only. In general, the Stokes pressure is a complex function of the wall transpiration gradients (see (2.9) and (2.10)), which makes its analysis challenging. Substantial simplifications are possible if $\kappa$ is large enough, so that the two walls are approximately independent, and if the rate of change of the transpiration is large relative to the viscous terms. These assumptions are reasonable for a realistic flow and enable approximating the Stokes pressure at the lower wall as

(5.3) \begin{equation} \hat {p}_{\boldsymbol{\kappa },St}(y=-1, t) \approx - \frac {\hat {P}_{b}(t)}{\kappa \tanh (2 \kappa )} \approx \frac {1}{\kappa \tanh (2 \kappa )} \frac {\partial \hat {v}_{\boldsymbol{\kappa }}}{\partial t}(y=-1,t), \end{equation}

and a similar expression can be derived for the upper wall as well. The phase of the Stokes pressure then depends on the temporal frequency content of $\partial \hat {v}_{\boldsymbol{\kappa }}/ \partial t$ , i.e.

(5.4) \begin{equation} \angle \hat {p}_{\boldsymbol{\kappa }, St}(y_w, t) = \angle \left \lbrace \int _{-\infty }^{\infty } -i \omega \hat {v}_{\boldsymbol{\kappa }}(y_w, \omega ) \, e^{-i \omega t} \, {\textrm{d}} \omega \right \rbrace ,\end{equation}

where the the sign of the complex exponential in (5.4) is inverted relative to the Fourier series in (2.2), as is common practice in the stability literature. Approximately steady phase relations between wall transpiration and Stokes pressure can result if the temporal frequency content in (5.4) is sparse.

5.2. Phase difference at example scales

Figure 10.

Histograms of the phase difference between wall pressure and transpiration. Each row corresponds to a different pressure component, and each column represents a specific combination of phase shift and scale. The red symbol indicates the circular mean.

This section focuses on spatial scales $\boldsymbol{\kappa }_s$ and $\boldsymbol{\kappa }_r$ in the regime where they are dynamically relevant. Select histograms of the phase difference $\Delta \theta _{\boldsymbol{\kappa }}$ are shown in figure 10. Each column represents a different control configuration and scale, while each row shows a different pressure component. The bin count of the phase difference is represented by the segment radius and the circular mean is indicated by the red circle. The histograms further give a sense for the spread about the mean value, even if the circular variance is not explicitly quantified.

The total pressure is the measurable quantity in experiments and real-world applications, which makes the associated phase difference $\Delta \theta _{\boldsymbol{\kappa }}$ the most relevant one. This phase difference is shown in the first row (yellow segments) of figure 10 and varies significantly across control configurations and spatial scales. For example, the mean phase difference at scale $\boldsymbol{\kappa}_s$ changes from a value of approximately $+\pi / 2$ for configuration N75 (left column) to $0$ (i.e. in-phase) for controller N25 (centre column). On the other hand, $v$ and $p$ in configuration P50 are out-of-phase at scale $\boldsymbol{\kappa}_r$ (right column). The spread about the circular mean is also different in each case. The spread is small for N25 and P50, making conclusions for these configurations statistically strong. On the other hand, the spread for N75 is much larger, which renders the interpretation of this controller more ambiguous. To understand the differences in mean values and spread between configurations, it is helpful to decompose the pressure into its components and study the associated phase relation separately. For each component, we can combine pressure source term data with the analytical insights from § 5.1 to explain the observed phase relations.

The second row of figure 10 (orange segments) shows the phase difference between fast pressure and wall transpiration $\Delta \theta _{\boldsymbol{\kappa },f}$ . An approximately steady phase difference with little temporal spread is established in all three cases and the mean phase difference appears to shift in tandem with $\angle \hat {A}_d$ . For example, $\overline {\Delta \theta }_{\boldsymbol{\kappa },f} \approx 0$ for controller N25 and increases to $\overline {\Delta \theta }_{\boldsymbol{\kappa },f} \approx 3\pi / 8$ for N75. It is also evident that all values in the time series are clustered around the circular mean, with the greatest spread observed for configuration N25. This robust phase relation is an immediate consequence of the control and the structure of the fast pressure solution: the opposition control scheme, (2.3), relates wall transpiration to sensor measurement and, thus, constrains the wall-normal profile of $\hat {v}_{\boldsymbol{\kappa }}$ between $y_w$ and $y_d$ . The phase of the fast pressure at the wall corresponds to a weighted average of $\angle \hat {v}_{\boldsymbol{\kappa }}$ according to (5.2). Since the weighting function decays quickly away from the wall, the phase of the fast wall pressure is determined by the wall-normal velocity in the constrained region, and a robust phase relation $\Delta \theta _{\boldsymbol{\kappa },f}$ can be established.

This point is illustrated in figure 11 for scale $\boldsymbol{\kappa }_s$ and controller N75 (small spread about circular mean, see figure 10 a) and N25 (larger spread, figure 10 b). For controller N75, the normalised weighting function (grey curve) is indeed close to zero for $y\gt y_d$ , demonstrating that the phase of the fast pressure at the wall is determined by $y \in [y_w, y_d]$ where $\angle \hat {v}_{\boldsymbol{\kappa }}$ is closely constrained. In contrast, the normalised weighting function of controller N25 is larger for $y\gt y_d$ , allowing the fast pressure at the wall to acquire phase from regions where $\angle \hat {v}_{\boldsymbol{\kappa }}$ is unconstrained. This leads to the larger spread observed in figure 10. It is important to note that both plots in figure 11 show the same spatial scale $\boldsymbol{\kappa }_s$ at two different $\angle \hat {A}_d$ . The Green’s function kernel is identical for the two cases and the difference in the weighting function comes from the mean shear, which is greater in the drag-increasing case. The phase relation between fast wall pressure and transpiration is thus particularly robust for small scales (due to the fast decay of the Green’s function kernel) or drag-increased flows (due to the large mean shear close to the wall), as confirmed by the histogram for scale $ \boldsymbol{\kappa} _r$ and controller P50.

Figure 11.

Representative instantaneous phase profile $\angle \hat {v}$ (solid black line) and product of Green’s function kernel $G$ and mean shear $S$ normalised by the maximum value (solid grey line). The two plots show the same spatial scale $\boldsymbol{\kappa }_s$ at different $\angle \hat {A}_d$ .

The phase difference between slow pressure and transpiration $\Delta \theta _{\boldsymbol{\kappa }, s}$ (blue segments in the third row) shows significant variation in all cases. The three circular means have similar values, but their significance is likely reduced due to the large variance. This large variance is again a consequence of the Green’s function structure. The source term of the slow pressure is the divergence of the nonlinear advection term. The nonlinearity can further be split into an irrotational and solenoidal part, each of which has a distinct dynamical role (Chorin & Marsden (Reference Chorin and Marsden1993). Only the irrotational component enters the source term of the slow pressure and determines the nonlinear effects in the pressure field. The solenoidal part, on the other hand, governs the nonlinear effects in the evolution of the velocity field and represents the forcing in input--output formulations of the Navier–Stokes equations (McKeon & Sharma Reference McKeon and Sharma2010). Due to the quadratic form of the nonlinearity, the source term of the slow pressure depends on all scales that are triadically consistent with $\boldsymbol{\kappa }$ , but not on $\boldsymbol{\kappa }$ itself. With disjoint spectral domains of dependence, it is unlikely that the weighted integral of the source term is strongly correlated with $\hat {v}_{\boldsymbol{\kappa }}(y_w)$ and that a robust phase relation can be established. The large variance in $\Delta \theta _{\boldsymbol{\kappa }, s}$ , as shown in the histograms of figure 10, supports this interpretation. The phase difference between transpiration and slow pressure can possibly be modelled as a stochastic process, but the sample size is too small to draw meaningful conclusions.

The last row of figure 10 (purple segments) completes the picture and shows the phase difference between the Stokes pressure and the transpiration $\Delta \theta _{\boldsymbol{\kappa },St}$ . The phase difference for negative $\angle \hat {A}_d$ shows considerable spread around the mean, whereas all values are tightly clustered near the mean for controller P50. Equation (5.4) suggests that $\Delta \theta _{\boldsymbol{\kappa }, s}$ is approximately determined by the temporal frequency content of $\partial \hat {v}_{\boldsymbol{\kappa }} / \partial t$ and that a robust phase relation can be established when the frequency content is sparse. To validate this, we analyse the temporal frequency content of two example control configurations. In this specific instance, the DNS is run at a constant time step to enforce a uniform sampling rate and the temporal frequency content is estimated using Welch’s method. Each time series is divided into segments of $M$ samples, which are multiplied by a Hamming window

(5.5) \begin{equation} w[n] = 0.54 - 0.46 \cos \left ( \frac {2 \pi n}{M-1} \right )\!, \quad 0 \leqslant n \leqslant M-1 ,\end{equation}

to enforce periodicity. Consecutive segments have a $50\,\%$ overlap and the robustness of the results was assessed through a convergence study. The results for the two example configurations are shown in figure 12: panel (a) shows the temporal frequency content of $\partial \hat {v}_{\boldsymbol{\kappa }} / \partial t$ at scale $\boldsymbol{\kappa }_s$ for controller N25 (larger variance in $\Delta \theta _{\boldsymbol{\kappa }, St}$ ), while panel (b) presents scale $\boldsymbol{\kappa }_r$ and controller P50 (small variance). Each of the power spectra is normalised with its maximum value and only a subset of the resolved phase speeds $c = \omega / k_x$ is shown for clarity.

Figure 12.

Power spectral density estimate ${\mathbb{E}} (\varPhi)$ of $\partial \hat {v}_{\boldsymbol{\kappa }} / \partial t$ for two different scales and phase shifts. Both spectra are normalised by their respective maximum values. The dotted vertical line in figure 12(b) indicates the wavespeed of the amplified eigenmode reported in Toedtli et al. (Reference Toedtli, Yu and McKeon2020).

For controller N25, the spectral content of $\partial \hat {v}_{\boldsymbol{\kappa }} / \partial t$ is distributed over a large range of positive phase speeds. The rate of change of the wall transpiration is not restricted to wavespeeds below the centreline velocity, as is typically the case for the velocity fluctuations in a canonical turbulent channel flow (Bourguignon et al. Reference Bourguignon, Tropp, Sharma and McKeon2014). This difference can likely be attributed to the time derivative, which amplifies higher frequencies, and to the closed loop control, which introduces additional dynamics. The temporal frequency content is broad-band and $\angle \hat{p}_{\boldsymbol{\kappa}, \textit{St}}$ therefore depends on multiple temporal frequencies. This can decrease the correlation to $\hat {v}_{\boldsymbol{\kappa }}$ and leads to the broader range of phase differences observed in figure 10.

In contrast, the temporal frequency content of $\partial \hat {v}_{\boldsymbol{\kappa }} / \partial t$ for configuration P50 and scale $\boldsymbol{\kappa }_r$ has a pronounced peak. Equation (5.4) suggests that the phase difference between Stokes pressure and transpiration is $-\pi / 2 + n 2\pi$ when a single frequency dominates, consistent with the histogram in figure 10. The dominant wavespeed roughly aligns with the phase speed of the amplified eigenmode that corresponds to the spanwise roller (green dotted line), which Toedtli et al. (Reference Toedtli, Yu and McKeon2020) computed for this configuration under idealised conditions (uncontrolled mean, single mode in isolation). The temporal frequency content thus provides further evidence that the amplified eigenmode drives the flow response of configuration P50 at scale $\boldsymbol{\kappa }_r$ and leaves a clear imprint not only in the velocity field, but also in the Stokes pressure.

Figures 8 and 10 together offer insights into how the phase difference between total pressure and transpiration ( $\Delta \theta _{\boldsymbol{\kappa }}$ ) is established. Consider, for example, scale $\boldsymbol{\kappa }_r$ in control configuration P50. The dominant Stokes and fast pressure (see figure 8 f) add to a resultant vector in the complex plane that is close to out-of-phase with the transpiration. The slow pressure adds a stochastic element to the resultant, but its magnitude is too small to significantly change the orientation of the pressure vector. The fast and Stokes pressure thus establish the out-of-phase relationship observed for the total pressure in this case. In contrast, the slow and Stokes pressure are dominant at $\boldsymbol{\kappa }_s$ and negative phase shifts and imprint the overall phase relation between transpiration and wall pressure. Given the significant variation in the direction of the slow pressure, it remains somewhat unclear how controller N25 establishes a robust overall phase relationship.

5.3. Phase difference at all scales

We next discuss the mean phase difference $\overline {\Delta \theta }_{\boldsymbol{\kappa }}$ between total pressure and transpiration across all scales, which is shown in figure 13(a,c,e). The interpretation of these spectra requires care: the phase of a Fourier coefficient with (close to) zero magnitude is ill-defined, which implies that the phase difference at scale $\boldsymbol{\kappa }$ is only meaningful if both $\hat {p}_{\boldsymbol{\kappa }}$ and $\hat {v}_{\boldsymbol{\kappa }}$ are non-zero. The spectra in figure 13 are therefore only shown for scales that satisfy $\varPhi _{{vv}} \varPhi _{pp} / \max (\varPhi _{{vv}} \varPhi _{pp}) \geqslant 10^{-2}$ . Scales that do not satisfy this condition are blanked out.

Figure 13.

(a,c,e) Phase difference spectra $\overline {\Delta \theta }_{\boldsymbol{\kappa }}/\pi$ and (b,d,f) drag change spectra. Both quantities are only shown for the scales with $\varPhi _{{vv}} \varPhi _{pp} / \max (\varPhi _{{vv}} \varPhi _{pp}) \geqslant 10^{-2}$ . The green contour lines are the same as in figure 4 and label $\varPhi _{{vv}}^+ / \max (\varPhi _{{vv}}^+) = (0.15, 0.3, 0.45)$ .

The average phase difference for the drag-reducing controller N25 is shown in figure 13(c). The highlighted entry at scale $\boldsymbol{\kappa }_s$ corresponds to the circular average of the histogram shown in figure 10(b), and similar histograms could be constructed for all other scales. Analogous to scale $\boldsymbol{\kappa }_s$ , the phase difference at scales with large control input is approximately zero, indicating that transpiration and pressure at the streak scales are in-phase for controller N25. On the other hand, the phase relation between $v$ and  $p$ for controller N75 (figure 13 a) is less clear. The average phase difference for the energetic streak scales is between $0$ and $+\pi / 2$ , with significant variability from scale to scale. As shown in figure 10(a), the circular variance for this control configuration is larger compared with the other two configurations, which makes the value of the circular mean less conclusive and possibly contributes to the observed spread across $\boldsymbol{\kappa }$ . Finally, transpiration and pressure at the roller scales are approximately out-of-phase for P50, as can be seen in figure 13(e). This is again consistent with the earlier observations for example scale $\boldsymbol{\kappa }_r$ .

The above discussion highlights that the example scales $\boldsymbol{\kappa }_s$ and $\boldsymbol{\kappa }_r$ are representative of their scale families also in terms of the mean phase difference. This suggests that the relative importance of the pressure components and the robustness of the phase relations through the domain of dependence apply broadly and underlie the observed phase difference spectra of each scale family.

5.4. Connection to drag change

We conclude this section by investigating whether specific values of $\overline {\Delta \theta }_{\boldsymbol{\kappa }}$ correlate with scale amplification and suppression. Figure 13(b,d,f) displays the scale-by-scale drag change, which was shown earlier in figure 5, but is now restricted to regions with non-zero $\hat {p}_{\boldsymbol{\kappa }}$ at $\hat {v}_{\boldsymbol{\kappa }}$ at the wall. A comparison between Figures 13(a,c,e) and 13(b,d,f) suggests a correlation between velocity-pressure phase difference and scale amplification. In the drag-reducing configuration N25, transpiration is in-phase with the wall pressure and the streak scales are suppressed. In contrast, the drag-increasing configuration P50 generates a transpiration that is out-of-phase with the wall pressure and amplifies the roller scales. The data for configuration N75 are less conclusive: the streak scales are amplified and the phase difference between transpiration and wall pressure is positive, but different from $\pi$ and shows considerable scatter. It is also important to remember that the variance in the phase difference was largest in case N75, which reduces the confidence in the exact value of the circular mean. One may speculate that the phase relation between $\hat {v}_{\boldsymbol{\kappa }}$ and $\hat {p}_{\boldsymbol{\kappa }}$ at the streak scales varies smoothly with $\angle \hat {A}_d$ . The two are in-phase at $\angle \hat {A}_d = - \pi / 4$ (configuration N25), when the drag is maximally reduced, and might be out-of-phase at $\angle \hat {A}_d = \pm \pi$ , when the drag increase is largest and the DNS diverges. From this perspective, configuration N75 could represent an intermediate state of positive phase shift as the scales approach the out-of-phase relation.

These observations suggest that the phase difference between transpiration and wall pressure is an important parameter of the problem. Drag reduction with streak scales coincides with actuation and wall pressure being in-phase, while drag increase with roller scales occurs if the two are out-of-phase. It is therefore plausible that the wall pressure encodes the state of the background flow, relative to which the transpiration is shifted by means of $\angle \hat {A}_d$ . The non-local domain of dependence (§ 5.1) is further consistent with an interpretation of the wall pressure as an averaged background flow state.

6.1. Summary of results

In this study we conduced DNS of low-Reynolds-number turbulent channel flows with wall transpiration and investigated the wall pressure field and its phase relation to the transpiration. The analysis considered the Fourier domain in the wall-parallel $\{x, z\}$ directions and focused on distinct spatial scales identified by the wavenumber combination $\boldsymbol{\kappa } = (k_x, k_z)^{\intercal }$ . The wall-normal and temporal coordinate were retained in physical space so that the Fourier coefficients depended on $y$ and $t$ .

The transpiration was generated by the varying-phase opposition control scheme, which measures the Fourier coefficient of the wall-normal velocity $\hat {v}_{\boldsymbol{\kappa }}(y_d, t)$ at a distance $y_d$ from the wall, and generates a wall transpiration $\hat {v}_{\boldsymbol{\kappa }}(y_w, t) = - \hat {A}_d \hat {v}_{\boldsymbol{\kappa }}(y_d, t)$ . The complex-valued controller gain $\hat {A}_d$ allowed shifting the phase of the transpiration ( $\angle \hat {v}_{\boldsymbol{\kappa }}(y_w, t)$ ) relative to the sensor measurement and the background flow. The gain was carefully defined so that $\angle \hat {A}_d$ corresponded to a shift in the streamwise direction. The sensor location was fixed at $y_d^+ = 15$ and $\angle \hat {A}_d$ was varied over the entire parameter range $[-\pi , +\pi ]$ . Our analysis focused on three example controllers: N75 ( $\angle \hat {A}_d = -3 \pi / 4$ ), N25 ( $\angle \hat {A}_d = -\pi / 4$ ) and P50 ( $\angle \hat {A}_d = +\pi / 2$ ).

The structure and magnitude of the wall pressure as well as the relative importance of its components (fast, slow and Stokes) changed with $\angle \hat {A}_d$ of the transpiration. Notably, the transpiration added an inertial forcing term to the Stokes pressure and made its magnitude comparable to the fast and slow pressure. The Stokes pressure is therefore not necessarily negligible in flows with wall transpiration.

We further analysed the structure of the pressure Green’s function solution. Specifically, we showed how the phase of the wall pressure is established and formulated conditions for robust phase relations between $\hat {p}_{\boldsymbol{\kappa }}$ and $\hat {v}_{\boldsymbol{\kappa }}$ . The phase relation between transpiration and fast and slow pressure was determined by a weighted wall-normal integral and a robust relation resulted if the integrand was correlated to $\hat {v}_{\boldsymbol{\kappa }}(y_w)$ over the wall-normal extent where the weight function was non-zero. This was typically the case for the fast pressure but not for the slow pressure. The difference is due to the spectral dependence of the two source terms: the source term of the fast pressure depends on $\hat {v}_{\boldsymbol{\kappa }}$ at the same wavenumber, which is constrained by the varying-phase opposition control scheme. On the other hand, the nonlinear source term of the slow pressure depends on all wavenumbers that are triadically consistent with $\boldsymbol{\kappa }$ and is typically not strongly correlated to $\hat {v}_{\boldsymbol{\kappa }}(y_w)$ . The phase relation of the Stokes pressure depended on the temporal frequency content of the transpiration dynamics at the wall. A robust phase relation resulted for approximately sparse frequency content, which was observed when the eigenspectrum of the linearised Navier–Stokes operator had an amplified eigenvalue.

Most of our subsequent analysis focused on the phase difference (relative streamwise arrangement) between wall pressure and transpiration, which changed as a function of  $\angle \hat {A}_d$ . We considered scales at distinct $\boldsymbol{\kappa }$ , which belonged to one of two families: streak scales ( $k_x$ small and $k_x \lt k_z$ , associated with the near-wall cycle at the current ${\textit{Re}}_{\tau }$ ) and roller scales ( $k_z$ small and $k_x \gt k_z$ ). A significant contribution from the roller scales was only observed in configuration P50. For such positive $\angle \hat {A}_d$ , wall pressure and transpiration were approximately out-of-phase ( $\overline {\Delta \theta }_{\boldsymbol{\kappa }} \approx \pi$ ) and control amplified the roller scales, which lead to a large drag increase. Earlier studies showed that the drag increase coincides with the appearance of an amplified eigenvalue in the linearised Navier–Stokes operator and the presence of spanwise rollers in the DNS (Toedtli et al. Reference Toedtli, Yu and McKeon2020). On the other hand, the streak scales were most active for $\angle \hat {A}_d \lesssim 0$ . For configuration N25, wall pressure and transpiration at the streak scales were in-phase ( $\overline {\Delta \theta }_{\boldsymbol{\kappa }} \approx 0$ ) and transpiration suppressed the near-wall cycle and resulted in maximum drag reduction. The phase difference for more negative $\angle \hat {A}_d$ was positive, with large variance at each $\boldsymbol{\kappa }$ and significant variation across scales. In this case, transpiration amplified the near-wall cycle and lead to a drag increase.

To our knowledge, this study presented the first direct evidence that drag reduction correlates with $\Delta \theta _{\boldsymbol{\kappa }} \approx 0$ in the full nonlinear flow, confirming the hypothesis of Xu et al. (Reference Xu, Rempfer and Lumley2003). Similarly, the present study provided evidence that $\Delta \theta _{\boldsymbol{\kappa }} \approx \pm \pi$ correlates with drag increase and the appearance of spanwise rollers, which to our knowledge had not been appreciated thus far. It is possible that $\Delta \theta _{\boldsymbol{\kappa }} \approx \pm \pi$ can induce other flow structures as well (we hypothesised that this phase relation could also be established at $\angle \hat {A}_d = \pm \pi$ ) but this aspect remained speculative. At its core, the present study made a case that the phase relation between wall transpiration and pressure deserves more attention in the flow control community.

6.2. Connection to tailored surfaces and implications for modelling and control

We next connect the varying-phase opposition control results to tailored surfaces. As described in the introduction, the occurrence of spanwise rollers is well documented for a range of tailored surfaces including porous (Jiménez et al. Reference Jiménez, Uhlmann, Pinelli and Kawahara2001) and permeable walls (see, for example, Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006; Efstathiou & Luhar Reference Efstathiou and Luhar2018; Gómez-de Segura & García-Mayoral Reference Gómez-de Segura and García-Mayoral2019) and riblets past the viscous breakdown (García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011). The appearance of spanwise rollers coincides with a significant drag increase and, in many cases, with the presence of an amplified eigenvalue in the spectrum of the linearised Navier–Stokes operator. A number of pressure-driven tailored surfaces further impose a phase relation between pressure and transpiration at the wall. For example, Jiménez et al. (Reference Jiménez, Uhlmann, Pinelli and Kawahara2001) modelled their porous wall as

(6.1) \begin{equation} v(x, y_w , z, t) = -\beta p^{\prime }(x, y_w, z, t), \end{equation}

where $\beta \geqslant 0$ is a material parameter and $p^{\prime }$ denotes the pressure fluctuations above the background pressure and mean gradients. By construction, wall pressure and transpiration are out-of-phase in this configuration.

The flow response to varying-phase opposition control with roller scales and positive $\angle \hat {A}_d$ leads to the same observations: the transpiration induces spanwise rollers near the wall and causes a large drag increase; the onset of the rollers and their wall-normal structure can be predicted from the eigenspectrum of the linearised Navier–Stokes equations; and wall transpiration and pressure are out-of-phase. The close correspondence of flow responses suggest that transpiration with the roller scales and $\angle \hat {A}_d \gt 0$ is dynamically similar to this class of tailored surfaces. We use the term dynamical similarity to indicate that the overall flow response is analogous, even if finer details are different. For example, figure 6(f) suggests that wall pressure and transpiration of configuration P50 are not simply related by a constant, which is different from the boundary condition (6.1). To further test the dynamical equivalence, it would be interesting to verify if $\hat {v}_{\boldsymbol{\kappa }}$ and $\hat {p}_{\boldsymbol{\kappa }}$ are out-of-phase when other configurations, such as riblets, generate spanwise rollers. As mentioned in the introduction, the notion of transpiration in this case would apply to a suitably chosen plain inside the flow domain.

Tailored surfaces also interact with the near-wall cycle. For example, riblets in the viscous regime suppress the near-wall cycle and reduce drag, while riblets with spacings past the optimum can amplify these scales and increase the drag (Chavarin & Luhar Reference Chavarin and Luhar2020). Varying-phase opposition control with streak scales, which are associated with the near-wall cycle at the present ${\textit{Re}}_{\tau }$ , leads to the same observations: a transpiration with small negative $\angle \hat {A}_d$ suppresses the near-wall cycle and leads to drag reduction, analogous to riblets in the viscous regime. On the other hand, transpiration with large negative or positive $\angle \hat {A}_d$ amplifies the near-wall cycle and leads to a drag increase, analogous to the flow response to riblets past the viscous regime. The close correspondence of flow responses again suggests that transpiration with streak scales is dynamically similar to tailored surfaces interacting with the near-wall cycle. It would be interesting to verify also in this case if wall pressure and transpiration are in-phase when riblets suppress the near-wall cycle. A further analogy between the two control methods occurs for transpiration with $\angle \hat {A}_d \gt 0$ and the riblets past the viscous breakdown: both configurations amplify the near-wall cycle and induce spanwise rollers, but the amplification of the rollers is much stronger so that they dominate the flow response and the drag increase (see figures 3 and 4 f and García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011).

It is also important to highlight that varying-phase opposition control and at least a subset of tailored surfaces, namely the pressure-driven ones, have a complimentary perspective on how the $pv$ phase relation is established: the varying-phase opposition control scheme applies a transpiration with a phase shift $\angle \hat {A}_d$ , but does not explicitly depend on the pressure. Once the transpiration is generated, the pressure field responds and the phase relation is established according to the discussion in §§ 4 and 5. In contrast, a pressure-driven tailored surface generates a transpiration that explicitly depends on the pressure and the phase difference between $p$ and $v$ may be imposed by the boundary condition (see, e.g. (6.1)). In this case, the flow has to adjust the wall-normal profiles of the pressure source term and the spectral content of the Stokes pressure boundary condition to satisfy this phase relation (see § 5.2). This is in some sense a complementary problem to our configuration, but the tools we presented can still be used to analyse the flow changes in detail.

The observed dynamical similarity suggests that transpiration at the two scale families together with the phase difference $\Delta \theta _{\boldsymbol{\kappa }}$ (or, equivalently, $\angle \hat {A}_d$ ) build essential building blocks for flows over tailored surfaces. From a physical perspective, this suggests that the transpiration at the two scale families can provide a suitable diagnostic tool to analyse tailored surfaces and understand their dynamical effect. From a modelling perspective, this provides evidence that tailored surfaces can be represented by velocity boundary conditions and suggests that transpiration has to be a key component of any such dynamical representation, in agreement with the observations of Bottaro (Reference Bottaro2019).

We conclude this section with a few general remarks regarding control. First, the results of this study suggest that drag reduction with passive pressure-driven tailored surfaces is difficult to achieve. The spectral content of the wall pressure and transpiration are reasonably aligned when the drag is increased (see figure 6 f), but not when the drag is reduced (figure 6 d). The pressure therefore likely activates the ‘wrong’ scales if the tailored surface is scale agnostic. Moreover, many passive configurations induce an out-of-phase relation between $v$ and $p$ , which based on the results presented herein does not lead to scale suppression and drag reduction. In order to reduce drag, a passive pressure-driven tailored surface would therefore need to impose (i) a scale-dependent response, and (ii) a different phase relation between $p$ and $v$ . This could perhaps be achieved through an array of suitably placed Helmholtz resonators (see, e.g. Dacome, Siebols & Baars Reference Dacome, Siebols and Baars2023) or a still to be developed meta-material. A permeable substrate with complex-valued transpiration coefficient would be another interesting test case. Gómez-de Segura & García-Mayoral (Reference Gómez-de Segura and García-Mayoral2019) considered real-valued transpiration coefficients, which induced an out-of-phase relation between $p$ and $v$ at the substrate boundary. Drag reduction was observed for small values of the wall-normal permeability, which prevented the formation of spanwise rollers. It would be interesting to see if the spanwise rollers can alternatively be suppressed with complex-valued transpiration coefficients, which induce a different phase relation between $p$ and $v$ .

The second comment regards the pressure as a sensor measurement for control: the present study indicates that the wall pressure signal encodes the background flow and can inform a suitable transpiration. However, this pressure information has to come from the controlled flow itself, since the pressure changes significantly in the presence of transpiration. For example, the Stokes pressure is negligible in the canonical case but can become leading order in the presence of transpiration (see § 4). Control policies or flow estimators based on data from canonical wall-bounded flows are therefore likely not very effective once control is applied. An additional challenge with basing the control input on the wall pressure is the direct coupling between $v$ and $p$ , especially through the Stokes pressure. This is different from, say, opposition control, where the sensor and actuator are separated by a viscous flow region that reduces direct coupling.

6.3. Relevance at higher Reynolds numbers

The results presented in this study were for a low-Reynolds-number turbulent channel flow, which corresponded to $({\textit{Re}}_{\tau })_0 \approx 180$ in the absence of transpiration. Most practical applications operate at significantly higher Reynolds number. An obvious question is whether the scale families and the $pv$ phase relation are relevant at higher ${\textit{Re}}$ as well. We offer a few different perspectives on this question.

The first perspective is a phenomenological one: the two scale families were associated with distinct flow structures. The persistence of these flow structures at higher Reynolds numbers may be taken as an indication that the scale families remain relevant. Following this reasoning, results in the literature suggest that the streak and roller scales and their phase relations remain relevant for at least another decade in ${\textit{Re}}_{\tau }$ . For example, an experimental investigation by Efstathiou & Luhar (Reference Efstathiou and Luhar2018) found evidence for the existence of spanwise rollers over a porous wall at ${\textit{Re}}_{\tau } \approx 1700$ . Similarly, the DNS study of Deng et al. (Reference Deng, Huang and Xu2016) suggests that classical opposition control can reduce drag until at least ${\textit{Re}}_{\tau } \approx 1000$ . Classical opposition control is closely related to $\angle \hat {A}_d = 0$ , at which the streak scales lead to drag reduction. A resolvent study on compliant walls by Luhar et al. (Reference Luhar, Sharma and McKeon2015) at ${\textit{Re}}_{\tau } \approx 2000$ further suggests that the (fast) wall pressure and transpiration have to be in-phase to suppress the near-wall cycle, consistent with the results presented earlier. However, the ${\textit{Re}}$ scaling of the spatial scales remains an open question.

As the Reynolds number increases, large-scale flow structures in the logarithmic region become more energetic and their imprint reaches all the way down to the wall (see, e.g. Hutchins & Marusic Reference Hutchins and Marusic2007b ). A second question is how the results apply to these structures. Substantiated answers for this question are sparse, because effective flow manipulation schemes for large-scale structures are still rare. There has been recent success in suppressing flow structures in the logarithmic region by means of streamwise travelling waves of spanwise wall oscillations (Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), but it remains inconclusive if transpiration-based control can suppress these structures as well (Abbassi et al. Reference Abbassi, Baars, Hutchins and Marusic2017). Opposition control in the logarithmic region has so far lead to drag increase and the generation of spanwise rollers for some configurations (Guseva & Jiménez Reference Guseva and Jiménez2022), which suggests that the roller scales represent a general sensitivity of the flow to a relaxed no-throughflow condition. On the other hand, a different phase relation between $v$ and $p$ may be needed to suppress the large-scale motions in the logarithmic layer. A recent resolvent study suggests that wall pressure and transpiration no longer need to be in-phase to optimally suppress these motions (Luhar et al. Reference Luhar, Sharma and McKeon2015), but the caveat is again that this analysis only captures the fast pressure component. A future study may consider a modified version of the present analysis, with $\hat {A}_d$ according to (2.5) and sensors located in the logarithmic region, to clarify if and how transpiration-based control can increase or decrease the drag and what the $vp$ phase relation in each case is.

Finally, the present work offers a fresh perspective on flow control that can also be applied at higher Reynolds numbers. The phase difference $\Delta \theta _{\boldsymbol{\kappa }}$ is a wall quantity and its strong correlation with drag reduction suggests that flow control can perhaps be understood from the perspective of the wall. Control based on the wall pressure naturally selects scales that have an imprint at the wall and removes the somewhat unsatisfying necessity of identifying target structures inside the flow, at the expense of not being able to target scales that cannot be sensed at the wall. In addition, a wall transpiration with a specific phase relation to the wall pressure systematically alters the scale-by-scale vorticity flux, which represents the dynamical effect of the wall and is perhaps the most natural control target at any Reynolds number. The wall vorticity flux may therefore deserve more attention in the flow control community, as envisioned early on by Koumoutsakos (Reference Koumoutsakos1999).