3.1. The control effects

Using the database, we first show the distribution of the skin-friction coefficients and their variation with regard to the reference cases on the suction side of an NACA4412 wing section in figure 1. It can be easily found that uniform blowing causes friction-drag reduction whereas suction causes friction-drag increase, regardless of the Reynolds number and streamwise position on the control surface. A stronger intensity of blowing/suction leads to a larger drag-reduction/increase rate, as expected. Such phenomena are consistent with previous studies (Kametani & Fukagata Reference Kametani and Fukagata2011; Kametani et al. Reference Kametani, Fukagata, Örlü and Schlatter2015; Atzori et al. Reference Atzori, Vinuesa, Fahland, Stroh, Gatti, Frohnapfel and Schlatter2020). The mechanisms of the drag reduction/increase by blowing/suction can be possibly classified as: (i) direct effect by the excess/defect of displacement thickness arising from the wall flux; (ii) indirect effect associated with the interactions between the ‘cross-stream’ and quasi-streamwise vortical structures in the near-wall region, which probably enhance/dampen the behaviour of sweep/ejection events and yield modifications of the mean velocity profiles (Park & Choi Reference Park and Choi1999; Kim et al. Reference Kim, Sung and Chung2002; Hwang Reference Hwang2004; Kornilov Reference Kornilov2015). The variation of turbulent dynamics in the near-wall region leads to a redistribution of the turbulence kinetic energy and alteration of the turbulent momentum transport across the wall layer. The vortical structures in the outer layer will also be influenced by the near-wall blowing/suction, especially in the downstream of the control surface. Consequently, the generation of the skin-friction drag, which is linked to the turbulence statistics across the wall layer, will be correspondingly changed.

Figure 1.

Skin-friction coefficients on the suction side of an NACA4412 wing section at ($a$) $Re_c=200\,000$ and ($b$) $Re_c=400\,000$. Variation of $C_f$ with regards to the reference case on the suction side of an NACA4412 wing section at ($c$) $Re_c=200\,000$ and ($d$) $Re_c=400\,000$. (The superscript of ‘ref ’ represents the reference case without control.).

To clarify such control effects, we conduct the decomposition of skin-friction coefficients on the suction side (within $0.2\le x/c \le 0.85$) of NACA4412 by the RD identity. Note that the relative errors, $(C_{f,V}+C_{f,T}+C_{f,G}-C_f)/C_f$, where $C_f$ is directly calculated with the normal gradient of tangential velocity at the wall surface (i.e. $C_f=(\mu \partial U/\partial y)|_{wall}/(0.5\rho U_e^2)$), are well confined within $\pm 0.12\,\%$ for all cases considered, which confirms the reliability of the decomposition method.

Figure 2 shows the variations of $C_{f,V}$, $C_{f,T}$ and $C_{f,G}$ induced by uniform blowing and suction with regard to the reference case, at $Re_c=200\,000$ (figure 2a–c) and $Re_c=400\,000$ (figure 2d–f). With blowing, the friction contribution of direct viscous dissipation ($C_{f,V}$) is reduced at both Reynolds numbers. Possibly, both the displacement-thickness increase and the outward wall-normal convection may be responsible for the $C_{f,V}$ reduction. With blowing, an increase of the generation of turbulence kinetic energy production ($C_{f,T}$) is found in figures 2(b) and 2(e). The variations of $C_{f,V}$ and $C_{f,T}$ are essentially associated with the influences on the wall-normal profiles of mean viscous shear stress and Reynolds shear stress across the wall layer.

Figure 2.

Variation of ($a$,$d$) $C_{f,V}$, ($b$,$e$) $C_{f,T}$ and ($c$,$f$) $C_{f,G}$ with regard to the reference case on the suction side of an NACA4412 wing section at ($a$–$c$) $Re_c=200\,000$ and ($d$–$f$) $Re_c=400\,000$. (The legend in panel (a) refers to panels ($a$–$c$), while that in panel (d) refers to panels ($d$–$f$).).

As for the generation of the spatial growth ($C_{f,G}$), it is decreased by the blowing and the influence on $C_{f,G}$ is stronger than that on $C_{f,V}$ and $C_{f,T}$. To further clarify the cause of such variation, we trace back to its sub-constituents in (2.2) and plot the variations of $C_{f,C}$, $C_{f,D}$ and $C_{f,P}$ in figure 3.

Figure 3.

Variation of ($a$,$d$) $C_{f,C}$, ($b,e$) $C_{f,D}$ and ($c$,$f$) $C_{f,P}$ with regard to the reference case on the suction side of an NACA4412 wing section at ($a$–$c$) $Re_c=200\,000$ and ($d$–$f$) $Re_c=400\,000$.

In the case with blowing, the generation of wall-normal convection ($C_{f,C}$) is significantly decreased, owing to the amplified wall-normal velocity $V$ in the $C_{f,C}$ integrand. Vinuesa & Schlatter (Reference Vinuesa and Schlatter2017) pointed out that the effects of uniform blowing on the inner-scaled velocity profiles are similar to those of APG enhancement, which lead to a higher wall-normal velocity and stronger wall-normal convection in the wake region (Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018). In the meanwhile, in the inner region, the wall-normal velocity is also increased as a direct consequence of the wall-normal mass flux. In the absolute reference frame with the local mean velocity being $U-U_e$, the work done by the wall-normal convection of $V\partial U/\partial y$ is negative. The enhanced wall-normal-velocity profile by blowing yields a negative (i.e. decreased) contribution to the generation of skin friction. Such a phenomenon is also observed as the magnitude of the pressure gradient increases, as shown in figure 12 in Senthil et al. (Reference Senthil, Kitsios, Sekimoto, Atkinson and Soria2020). In figure 3(b,e), the contribution by streamwise gradients is shown to be increased by blowing, which results from the stronger streamwise development of boundary layer thickness. This is consistent with the von Kármán integral momentum equation (von Kármán Reference von Kármán1921; Goldschmied Reference Goldschmied1951; Stroh et al. Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016), where the streamwise derivative of the momentum thickness acts as a positive contribution to $C_f$ and will increase $C_f$ when it is increased by blowing. At last, $C_{f,P}$, which plays a significant role and contributes negatively to $C_f$ in strong APG-TBLs (Senthil et al. Reference Senthil, Kitsios, Sekimoto, Atkinson and Soria2020), is observed to be decreased by blowing, owing to the direct effect of an increased adverse pressure gradient. A slight increase of $C_{f,P}$ is observed near the trailing edge for the blowing cases (see figure 3c), which is associated with the fact that the boundary layer is approaching the condition of mean separation (Atzori et al. Reference Atzori, Vinuesa, Fahland, Stroh, Gatti, Frohnapfel and Schlatter2020). Generally, the positive variation of $C_{f,D}$ is overcome by the negative influence on $C_{f,C}$ and $C_{f,P}$, which consequently leads to the overall reduction of $C_{f,G}$ by blowing (Mahfoze et al. Reference Mahfoze, Moody, Wynn, Whalley and Laizet2019), as shown in figures 2(c) and 2(f).

In § 3.2, we will further analyse the wall-normal distributions of these friction constituents to better relate them to control-induced changes of the boundary layer properties. With mass suction, the control effects on the $C_f$ constituents shown in figures 2 and 3 are opposite to those with blowing. Results of control effects on the contribution of $C_{f,V}$, $C_{f,T}$, $C_{f,G}$ and its constituents are summarized in table 2. Here we did not add more discussions on the suction cases for brevity.

Table 2.

Summary of control effects on the generation of skin friction and its decomposed constituents on the suction side of the NACA4412 case.

To assess the control effects on the streamwise integrated skin-friction drag over the control surface, a parameter $\bar {D}_{f,i}$ is introduced:

(3.1)\begin{equation} \bar{D}_{f,i}=\int_{\varOmega_{ctr}}{\tau_{w,i}}(\boldsymbol{t} \boldsymbol{\cdot} \boldsymbol{k})\,\textrm{d}\xi, \end{equation}

where ${\tau _{w,i}}=C_{f,i}\cdot (0.5\rho U_e^2)$ is the decomposed component of wall-shear stress, with the subscript ‘i’ working as a label to denote each $C_f$ constituent, i.e. ‘V ’, ‘T’, ‘G’, ‘C’, ‘D’ and ‘P’ as mentioned in (2.1) and (2.2), $\boldsymbol {t}$ and $\boldsymbol {k}$ denote the unit vectors tangential to the airfoil surface and along the free-stream direction, respectively, $\xi$ is the curvilinear coordinate along the airfoil surface and $\varOmega _{ctr}$ represents the area of the control surface. The change rate of $\bar {D}_{f,i}$, with respect to the uncontrolled reference case, is then defined as

(3.2)\begin{equation} R_i=\frac{(\bar{D}_{f,i}-\bar{D}_{f,i}^{ref})}{\bar{D}_{f}^{ref}}. \end{equation}

Figure 4 shows the result of $R_i$ under different control schemes at $Re_c=200\,000$ and $400\,000$. It can be seen that $R_i$ appears to scale linearly, for the chosen low actuation amplitudes within $-40.2\,\% \leq V_{wall}/U_\infty \leq 0.2\,\%$, with the control intensity of the blowing/suction. However, note that this might not always be true when $V_{wall}/U_\infty$ becomes much larger and needs to be validated in the future work. Among the decomposed constituents, the most significant control effect lies on the friction constituent of spatial growth of the flow ($C_{f,G}$), with its sub-constituents primarily correlated with the convection, streamwise growth and pressure gradient in the outer region (Fan et al. Reference Fan, Li, Atzori, Pozuelo, Schlatter and Vinuesa2020a). This reveals that the drag control via blowing and suction significantly influences the turbulence properties in the outer layer. Moreover, weak-Reynolds-number effects are found, especially for $R_V$ and $R_G$, where a stronger control effect is achieved at higher Reynolds number which, to some extent, validates the theoretical estimation by Kametani & Fukagata (Reference Kametani and Fukagata2011).

Figure 4.

The change rate of streamwise integrated friction drag over the control surface, as a function of control intensity ($V_{wall}/U_\infty$), on the suction side of an NACA4412 wing section.

3.2. Wall-normal distributions of the $C_f$ constituents

To answer the question about how uniform blowing/suction specifically influences the sources of skin-friction generation, the wall-normal distributions of the decomposed $C_f$ constituents across the boundary layer are investigated. As $C_f$ and its constituents vary along the streamwise direction on the wing surface, we only discuss the wall-normal contributions of the $C_f$ constituents at $x/c \approx 0.75$, where the friction Reynolds numbers are $Re_\tau \approx 224$, $205$, $180$, $237$, $245$, $362$, $332$ and $387$ in the cases of ‘$Re200k, ss, ref$’, ‘$Re200k, ss, blw1$’, ‘$Re200k, ss, blw2$’, ‘$Re200k, ss, sct1$’, ‘$Re200k, ss, sct2$’, ‘$Re400k, ss, ref$’, ‘$Re400k, ss, blw1$’ and ‘$Re400k, ss, sct1$’, respectively. Similar conclusions can be drawn at other positions within $0.2\le x/c \le 0.85$, and the results are not shown here for simplicity.

The $C_f$ constituents are expressed in inner scales as

(3.3)$$\begin{gather} {C_{f,V}} = 2/U_e^{{+}3}\int_{0}^{\infty}\left(\frac{\partial U^+}{\partial y^+}\right)^2\textrm{d} y^+, \end{gather}$$

(3.4)$$\begin{gather}{C_{f,T}} = 2/U_e^{{+}3} \int_{0}^{\infty}\langle -u'v'\rangle ^+\frac{\partial U^+}{\partial y^+}\,\textrm{d} y^+, \end{gather}$$

(3.5)$$\begin{gather}{C_{f,G}} = 2/U_e^{{+}3} \int_{0}^{\infty}(U^+{-}U_e^+)\frac{\partial}{\partial y^+}\left(\frac{\partial U^+}{\partial y^+}-\langle u'v'\rangle ^+\right)\textrm{d} y^+, \end{gather}$$

where the superscript $+$ denotes normalization by viscous units, i.e. friction velocity $u_\tau$ and viscous length scale $\delta _\nu =\nu /u_\tau$.

Figure 5 shows the wall-normal distributions of the pre-multiplied integrand of $C_{f,V}$, $C_{f,T}$ and $C_{f,G}$ in (3.3)–(3.5) as a function of $y^+$. The semi-logarithmic plots retain the advantage that the areas beneath the curves directly yield the total generation of the constituents. Comparisons among figures 5(a)–5(c) for the cases at $Re_c=200\,000$ and 5(d)–5(f) for the cases at $Re_c=400\,000$ confirm that the Reynolds number variation does not change the conclusions, which will be presented in the following from a qualitative perspective.

Figure 5.

Pre-multiplied integrands of ($a$,$d$) $C_{f,V}$, ($b$,$e$) $C_{f,T}$ and ($c$,$f$) $C_{f,G}$ at $x/c\approx 0.75$ as a function of $y^+$ on the suction side of an NACA4412 wing section at ($a$–$c$) $Re_c=200\,000$ and ($d$–$f$) $Re_c=400\,000$.

For the distribution of $C_{f,V}$ contributions, two peaks are respectively observed in the near-wall and outer region of the APG-TBLs. Most of the $C_{f,V}$ contributions come from the inner region ($y^+<30$), which indicates that the viscous dissipation is mostly concentrated in the near-wall region, as expected. In the meantime, a secondary peak appears in the outer region, which probably arises from the energy enhancement by APG (Sanmiguel Vila et al. Reference Sanmiguel Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2020; Tanarro et al. Reference Tanarro, Vinuesa and Schlatter2020). The secondary peak is absent in the ZPG-TBLs even at higher friction Reynolds number up to $Re_\tau =1270$ (Fan et al. Reference Fan, Li and Pirozzoli2019b). When uniform blowing/suction is applied, the locations of the inner peaks are fixed at a wall-normal distance of $y^+\approx 5.0 - 6.0$, regardless of the control scheme. In the blowing cases, the inner peak of $C_{f,V}$ contributions is reduced whereas the outer peak is increased, suggesting a lowered mean shear in the near-wall region while this is enhanced in the outer region owing to the lifting-up of the boundary layer (Kornilov Reference Kornilov2015), which is also validated by showing the wall-normal gradient of the tangential velocity in figures 6(a) and 6(c). This reveals that blowing has different actions in different sub-layers, namely inhibiting the contribution of inner-layer dynamics to skin-friction generation while promoting that of outer-layer dynamics. When the blowing intensity is increased up to 0.2 %, the outer-layer contributions seem to be comparable to the inner-layer contributions. However, suction behaves quite the opposite for all constituents, which will not be repeated hereafter.

Figure 6.

Profiles of ($a$,$c$) wall-normal velocity gradient and ($b$,$d$) Reynolds shear stress on the suction side of an NACA4412 wing section at ($a$,$b$) $Re_c=200\,000$ and ($c$,$d$) $Re_c=400\,000$.

Similar inner and outer peaks are also observed in the pre-multiplied distribution of $C_{f,T}$ contributions, as shown in figures 5(b) and 5(e), with the former well collapsed at the inner-scaled wall-normal distance $y^+\approx 16.0 - 17.0$. It can be found that the outer-layer motions dominate the contributions of $C_{f,T}$, although the $Re_\tau$ is lower than $400$, which is very different from the features in the ZPG-TBLs (Fan et al. Reference Fan, Li and Pirozzoli2019b). The prominent peak in the outer region suggests the energization of large-scale outer motions by APGs (Harun et al. Reference Harun, Monty, Mathis and Marusic2013). When blowing is applied, the inner peak of the $C_{f,T}$ contributions is reduced, while the outer peak is increased. This phenomenon is linked to the wall-normal distributions of the wall-normal velocity gradient and Reynolds shear stress, as shown in figure 6. With blowing, the wall-normal velocity gradients are suppressed significantly in the near-wall region, which probably results from the lifting-up of the boundary layer. In the outer region, the amplified Reynolds shear stress is observed by blowing, which partially arises from the process where the near-wall small-scale structures are convected to the outer layer (Tanarro et al. Reference Tanarro, Vinuesa and Schlatter2020). However, the mean shear in the outer region is increased, which also contributes to the production of Reynolds stress, because the velocity difference across the mean shear can basically yield coherent structures that are larger than the Corrsin scale and thus promotes the kinetic energy of turbulence fluctuations (Corrsin Reference Corrsin1958; Dong et al. Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017; Jiménez Reference Jiménez2018). These two actions of mean velocity and Reynolds stress are consequently responsible for the changes of the inner and outer peaks in the distribution of $C_{f,T}$ contributions.

As for the distribution of $C_{f,G}$ contributions in figures 5(c) and 5(f), negative contributions are observed in the region $y^+\lesssim 80$ for the uncontrolled reference case at $Re_c=200\,000$ ($y^+\lesssim 130$ at $Re_c=400\,000$) and positive values are observed beyond this region. This differs from the result in the ZPG-TBLs, for which $C_{f,G}$ always remains positive across the wall layer (Fan et al. Reference Fan, Li, Atzori, Pozuelo, Schlatter and Vinuesa2020a). Blowing enhances both the negative and positive distributions, as the strengthened adverse pressure gradient promotes a more pronounced growth of the boundary layer and a more prominent outer region (Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018).

In contrast to the wall-normal distributions as a function of $y^+$ shown in figure 5, figure 7 plots their profiles as a function of $y/\delta _{99}$. The outer-peak locations of $C_{f,V}$, $C_{f,T}$ and $C_{f,G}$ contributions normalized by the outer scale are well collapsed at $y/\delta _{99}\approx 0.7$, $0.53$ and $0.65$, respectively, as marked with vertical dashed lines in figure 7, regardless of the control scheme and Reynolds number. These phenomena are consistent with our previous finding (Fan et al. Reference Fan, Li, Atzori, Pozuelo, Schlatter and Vinuesa2020a), i.e. the inner-peak locations (in $C_{f,V}$ and $C_{f,T}$ contributions) exhibit good scaling in the inner unit ($\delta _\nu$), and the outer-peak locations in the outer unit ($\delta _{99}$), regardless of the friction Reynolds number, the magnitude of the APG and its development history. This finding suggests that self-similarity is exhibited in the inner or outer scales for the turbulence statistics associated with the friction-drag generation.

Figure 7.

Pre-multiplied integrands of ($a$,$d$) $C_{f,V}$, ($b$,$e$) $C_{f,T}$ and ($c$,$f$) $C_{f,G}$ as a function of $y/\delta _{99}$, on the suction side of an NACA4412 wing section at ($a$–$c$) $Re_c=200\,000$ and ($d$–$f$) $Re_c=400\,000$.

The generation of $C_{f,G}$ results from a counterbalance between the negative work done by $C_{f,C}$ and $C_{f,P}$ and the positive work by $C_{f,D}$. Figure 8 quantifies their wall-normal distributions and only the outer scaling by $\delta _{99}$ is applied herein. Good collapses of the peak locations are also observed at $y/\delta _{99}\approx 0.59$, $0.56$ and $0.43$ for $C_{f,C}$, $C_{f,D}$ and $C_{f,P}$ contributions, respectively. In the blowing cases, the APG effects are strengthened, a fact that promotes the population/energization of outer-layer structures (Harun et al. Reference Harun, Monty, Mathis and Marusic2013). Meanwhile, the wall-normal convection and the streamwise boundary-layer growth are intensified in the outer region (Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018). Therefore, in an absolute sense, the generations of the components ($C_{f,C}$, $C_{f,D}$ and $C_{f,P}$) are all enhanced. Thereinto, as the convection and pressure gradient do negative work for the friction-drag generation, blowing acts to reduce the $C_{f,C}$ and $C_{f,P}$ contributions. However, the positive contribution of $C_{f,D}$ is increased, as shown in figures 8(b) and 8(e). These three components counterbalance each other, as summarized in table 3. Consequently, the negative $C_{f,C}$ and $C_{f,P}$ contributions are responsible for the $C_{f,G}$ reduction in the near-wall region, whereas the positive $C_{f,D}$ contributions are responsible for the increase of $C_{f,G}$ in the outer region, as seen in figures 5(c) and 5(f).

Figure 8.

Pre-multiplied integrands of ($a$,$d$) $C_{f,C}$, ($b$,$e$) $C_{f,D}$ and ($c$,$f$) $C_{f,P}$ as a function of $y/\delta _{99}$, on the suction side of an NACA4412 wing section at ($a$–$c$) $Re_c=200\,000$ and ($d$–$f$) $Re_c=400\,000$.

Table 3.

Summary of control effects on the wall-normal distribution of $C_{f,G}$ and its constituents.

3.3. Contributions of small- and large-scale structures to the friction-drag generation

First, we use EMD (Huang et al. Reference Huang, Shen, Long, Wu, Shih, Zheng, Yen, Tung and Liu1998) to identity the small- and large-scale turbulence structures. EMD is an adaptive mode-decomposition technique, which extracts characteristic wavelengths of non-stationary signals automatically without a priori basis functions. It has been applied for wall-bounded turbulence (Agostini & Leschziner Reference Agostini and Leschziner2014, Reference Agostini and Leschziner2019; Cheng et al. Reference Cheng, Li, Lozano-Durán and Liu2019; Dogan et al. Reference Dogan, Örlü, Gatti, Vinuesa and Schlatter2019) and details of the methodology of EMD can be found in Huang et al. (Reference Huang, Shen, Long, Wu, Shih, Zheng, Yen, Tung and Liu1998). Here we just describe EMD very briefly.

With EMD, a raw temporal or spatial signal $f(t)$ is decomposed into a sum of multiple intrinsic mode functions (IMFs) with a residual $R(t)$:

(3.6)\begin{equation} f(t)=\sum_{i=1}^{m}\textrm{IMF}_i(t)+R(t), \end{equation}

where $m$ is the number of IMFs. The IMFs are obtained through a sifting procedure in a data-driven manner. They represent components with different wavelengths or scales in the full field. In this study, the velocity fluctuations ($u'$ and $v'$) at the streamwise location $x/a\approx 0.75$ are decomposed into four modes (three IMFs with a final residual). The first two modes represent the small-scale structures and the others characterize the large scales, which is justified based on a preliminary analysis (similar to the studies of Agostini & Leschziner Reference Agostini and Leschziner2014, Reference Agostini and Leschziner2019).

The contours of spanwise pre-multiplied spectra of $u'$ and $v'$ scaled with friction velocities are plotted in figure 9 for the $Re_c=400\,000$ cases. Similar features can also be found in the low-Reynolds-number ($Re_c=200\,000$) cases, which are not shown here for brevity. In figure 9, the iso-contour levels marked with dashed and solid lines represent the spectra of small- and large-scale velocity fluctuations, respectively. These contour lines indicate 0.12, 0.42 and 0.72 of their maxima, from outside to inside. For the reference case, as shown in figure 9(a), the spectra of small-scale $u'$ structures peak at the wall-normal distance $y^+\approx 10$ with a spanwise wavelength $\lambda _z^+\approx 80$, and those of large-scale $u'$ structures peak at $y^+\approx 100$ ($y/\delta _{99}\approx 0.28$) with $\lambda _z^+\approx 310$ ($\lambda _z / \delta _{99}\approx 0.85$). This observation is consistent with the study of Cheng et al. (Reference Cheng, Li, Lozano-Durán and Liu2019), where small-scale $u'$ structures identified by EMD are representative of the near-wall coherent motions, whereas the large-scale $u'$ structures characterize large-scale motions.

Figure 9.

Spanwise pre-multiplied spectra of the tangential and normal velocity fluctuations on the suction side of an NACA4412 wing section at $Re_c=400\,000$ ($a$,$d$) without control, with ($b$,$e$) uniform blowing and ($c$,$f$) uniform suction. The nephogram represents the spectra of the full field; the dashed and solid contour lines represent those of small and large scales, respectively.

When blowing/suction is applied on the airfoil surface, such energy spectra are affected, as seen in figures 9(b) and 9(c). In the case of blowing, the small-scale structures are enhanced and penetrate deeper into the outer region. The peaks of the spectra locate at $(\lambda _z^+,y^+)\approx (80,8)$. Whereas suction has an opposite influence on the amplitude of small scales, with the peak locations only slightly influenced at $(\lambda _z^+,y^+)\approx (90,10)$. As for the large-scale structures, they are enhanced by blowing, which agrees well with the conclusion drawn by Kametani et al. (Reference Kametani, Fukagata, Örlü and Schlatter2015). The peaks of the large-scale spectra locate at $(\lambda _z/\delta _{99},y/\delta _{99})\approx (0.96,0.33)$. As shown in figure 9(c), suction trends to diminish the secondary peak in the outer region with the peak at $(\lambda _z/\delta _{99},y/\delta _{99})\approx (0.93, 0.35)$.

As for the $v'$ structures shown in figures 9(d)–9(f), the small- and large-scale structures have approximately the same spanwise wavelengths as the $u'$ structures, which is consistent with the EMD results of channel flows (Cheng et al. Reference Cheng, Li, Lozano-Durán and Liu2019), suggesting that the decomposed two scales of $u'$ and $v'$ structures can be characterized with the same spanwise wavelength. Both the small- and large-scale $v'$ structures are greatly enhanced by blowing while they are reduced by suction. However, the wall-normal locations of the $v'$ structures are much different from those of $u'$ structures, because the presence of the wall prevents the normal velocity fluctuations from extending close to the near-wall region, in contrast to the wall-parallel component $u'$. As shown in figures 9(d)–9(f), the small-scale spectra of $v'$ are more intense beyond the buffer layer at $y^+\approx 50$, with the spanwise wavelength scale $\lambda _z^+$ peaking at approximately $80$–$100$ for all of the three cases, and the large-scale $v'$ spectra peak at approximately $(\lambda _z/\delta _{99},y/\delta _{99})\approx (0.9,0.35)$.

Hereafter, we denote the small- and large-scale tangential and wall-normal velocity fluctuations as $u_s'$, $u_l'$, $v_s'$ and $v_l'$, respectively. Then the Reynolds stress is decomposed as

(3.7)\begin{equation} -\langle u'v'\rangle ={-}\langle u_s'v_s'\rangle -\langle u_s'v_l'\rangle -\langle u_l'v_s'\rangle -\langle u_l'v_l'\rangle , \end{equation}

where $-\langle u_s'v_s'\rangle$ represents the Reynolds stress carried by small-scale structures, $-\langle u_l'v_l'\rangle$ represents the Reynolds stress associated with large-scale structures, and $-\langle u_s'v_l'\rangle$ and $-\langle u_l'v_s'\rangle$ denote the scale interactions from small- to large- and from large- to small-scale structures, respectively. Their wall-normal profiles, normalized by outer units, are plotted in figure 10. Significant modifications are observed in the outer region, especially for the large-scale motions. Blowing enhances the generation of large-scale motions in the outer region, whereas suction weakens it, in accordance with the phenomenon in figure 9. Meanwhile, the Reynolds stress carried by small scales is also amplified/suppressed by blowing/suction in the outer region. Within the region of $y/\delta _{99}<0.1$, the control impact is hardly recognizable, as directly perceived in figure 6(d).

Figure 10.

Profiles of Reynolds shear stress and its decomposed components in (3.7) on the suction side of an NACA4412 wing section at $Re_c=400\,000$: ($a$) without control; with ($b$) uniform blowing and ($c$) uniform suction.

Substituting these Reynolds-stress components into (3.4), the term of TKE production ($C_{f,T}$) can be further divided into four parts, viz.

(3.8)$$\begin{gather} {C_{f,T,s\text{-}s}} = 2/U_e^{{+}3}\int_{0}^{\infty}\langle -u_s'v_s'\rangle ^+\frac{\partial U^+}{\partial y^+}\,\textrm{d} y^+, \end{gather}$$

(3.9)$$\begin{gather}{C_{f,T,s\text{-}l}} = 2/U_e^{{+}3}\int_{0}^{\infty} \langle -u_s'v_l'\rangle ^+\frac{\partial U^+}{\partial y^+}\,\textrm{d} y^+, \end{gather}$$

(3.10)$$\begin{gather}{C_{f,T,l\text{-}s}} = 2/U_e^{{+}3}\int_{0}^{\infty} \langle -u_l'v_s'\rangle ^+\frac{\partial U^+}{\partial y^+}\,\textrm{d} y^+, \end{gather}$$

(3.11)$$\begin{gather}{C_{f,T,l\text{-}l}} = 2/U_e^{{+}3}\int_{0}^{\infty} \langle -u_l'v_l'\rangle ^+\frac{\partial U^+}{\partial y^+}\,\textrm{d} y^+. \end{gather}$$

Similar to the analysis in § 3.2, we show the wall-normal distributions of the $y^+$-pre-multiplied integrands in (3.8)–(3.11) in figure 11. The wrinkles in figure 11(c) are caused by the limited number of flow snapshots used in EMD. This arises from the data-storage limitations that prevent the acquisition of a full-time series, even though the total simulation time is approximately the same for all cases. For the reference case without control, the small-scale motions play important roles both in the inner and outer regions, as two comparable peaks are respectively observed there, as seen in figure 11(a). This is different from the phenomenon in channel flows (Agostini & Leschziner Reference Agostini and Leschziner2019) and ZPG-TBLs, because the APG strengthens the energy of small-scale structures in the outer region (Tanarro et al. Reference Tanarro, Vinuesa and Schlatter2020). Whereas the large-scale structures dominate the production of $C_{f,T}$ in the outer region, which is associated with the enhanced generation of large-scale motions in APG-TBLs (Lee & Sung Reference Lee and Sung2008; Harun et al. Reference Harun, Monty, Mathis and Marusic2013; Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018). When blowing is applied, the distributions of the $C_{f,T,s\text {-}s}$ contribution and $C_{f,T,l\text {-}l}$ contribution in the outer region are shifted upwards, which suggests that the wall-normal mass flux enhances the outer fluctuations of both small- and large-scale structures. Comparison between $C_{f,T,s\text {-}s}$ and $C_{f,T,l\text {-}l}$ indicates that the blowing raises the relative importance of large scales with respect to the small ones. In addition, the contribution of the scale interaction from small to large scales ($C_{f,T,s\text {-}l}$) is of the same order of that from large to small scales ($C_{f,T,l\text {-}s}$), in general. They are observed to be mainly in the outer region and are not sensitive to the control schemes, which indicates that the explicit interaction between scales does not rely much on the wall mass flux in spite of the significant control impacts on small and large scales. In the study by Cheng et al. (Reference Cheng, Li, Lozano-Durán and Liu2019), they also used EMD to separate the skin-friction contributions by scale-specific motions. It was shown that the terms resulting from the small- and large-scale interactions barely change with the Reynolds number, although the large-scale motions are definitely enhanced as the Reynolds number increases. This is a very interesting phenomenon and further investigations are expected especially on the scale interactions.

Figure 11.

Pre-multiplied integrands of $C_{f,T,i}$ as a function of $y^+$ on the suction side of an NACA4412 wing section at $Re_c=400\,000$: ($a$) without control; with ($b$) uniform blowing and ($c$) uniform suction. Note that the premultiplication factor is $y^+$.

To further check the role of small- and large-scale motions in the generation of TKE production, figure 12 quantifies their integrations normalized by $C_{f,T}$ itself. We can find that the blowing and suction have opposite influences on the contributions of small- and large-scale motions, that is, blowing is able to enhance the contribution of large-scales and to suppress the contribution of small-scales, whereas suction behaves contrarily. As for the scale interactions, i.e. $C_{f,T,s\text {-}l}$ and $C_{f,T,l\text {-}s}$, they account for approximately 20 % of the total $C_{f,T}$ and remain almost unchanged with different control strategies.

Figure 12.

Scale-specific contributions ($C_{f,T,s\text {-}s}$, $C_{f,T,s\text {-}l}$, $C_{f,T,l\text {-}s}$ and $C_{f,T,l\text {-}l}$ normalized by $C_{f,T}$ itself) on the suction side of an NACA4412 wing section at $Re_c=400\,000$: ($a$) without control; with ($b$) uniform blowing and ($c$) uniform suction.