4.1 Triple decomposition

In view of the steady control flow, different from both the mean flow and turbulent fluctuations, it seems natural to interpret the total flow from the perspective of the following three components by using the triple decomposition (Reynolds & Hussain Reference Reynolds and Hussain1972):

(4.1) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(x,y,z,t)=\boldsymbol{U}(y)+\underbrace{\tilde{\boldsymbol{u}}(y,z)+\boldsymbol{u}^{\prime \prime }(x,y,z,t)}_{\boldsymbol{u}^{\prime }}. & & \displaystyle\end{eqnarray}$$

Here, $\boldsymbol{u}$ is the instantaneous total velocity field; $\boldsymbol{U}=\langle \overline{\boldsymbol{u}}\rangle$ is the mean velocity, where the overbar indicates averaging in time and in the streamwise direction – and also includes averaging in the spanwise direction if more than one spanwise wavelength is contained in the computational domain; and the bracket $\langle \boldsymbol{\cdot }\rangle$ indicates spanwise averaging over one period; $\boldsymbol{u}^{\prime }=\boldsymbol{u}-\boldsymbol{U}$ is the total fluctuation; $\tilde{\boldsymbol{u}}=\overline{\boldsymbol{u}}-\boldsymbol{U}$ is the so-called ‘organized field’ representing the coherent motion induced by the swirls; and $\boldsymbol{u}^{\prime \prime }=\boldsymbol{u}-\overline{\boldsymbol{u}}$ is the random turbulent fluctuation.

Figure 4.

Forcing amplitude ( $A^{+}$ ) dependence: (a) the maximum mean spanwise velocity $|\tilde{w}|_{m}$ and (b) $\Vert \tilde{\boldsymbol{u}}\Vert$ at $Re_{\unicode[STIX]{x1D70F}}=180$ . Superscript $+$ indicates scaling with $u_{\unicode[STIX]{x1D70F}}$ of uncontrolled flow. Different symbols represent different controlling wavenumbers $\unicode[STIX]{x1D706}^{+}$ .

Since the forcing amplitude $A^{+}$ does not correspond to a directly measurable quantity in the flow, two quantities, based on $\tilde{\boldsymbol{u}}$ , are employed to characterize the flow response to the forcing. The first one is the maximum mean spanwise velocity $|\tilde{w}|_{m}=\max \{|\tilde{w}|\}$ , which characterizes the strength of the spanwise wall-jet motion. The second one is chosen to be the norm of $\tilde{\boldsymbol{u}}$ ,

(4.2) $$\begin{eqnarray}\displaystyle \Vert \tilde{\boldsymbol{u}}\Vert =\sqrt{\frac{1}{\unicode[STIX]{x1D6FA}}\int _{\unicode[STIX]{x1D6FA}}\tilde{\boldsymbol{u}}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}\,\text{d}\unicode[STIX]{x1D6FA}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}$ is the total volume of the computational domain. $\Vert \tilde{\boldsymbol{u}}\Vert$ represents the volume-averaged kinetic energy of the induced motion. The variations of the two quantities as a function of the forcing amplitude are shown in figure 4 for three different $\unicode[STIX]{x1D706}^{+}$ values. As expected, both $|\tilde{w}|_{m}$ and $\Vert \tilde{\boldsymbol{u}}\Vert$ increase with increasing $A^{+}$ and $\unicode[STIX]{x1D706}^{+}$ . In addition, we also find that, at a given $A^{+}$ , the increase of $|\tilde{w}|_{m}$ with the increase of $\unicode[STIX]{x1D706}^{+}$ becomes slower, in comparison to the faster increase of $\Vert \tilde{\boldsymbol{u}}\Vert$ with $\unicode[STIX]{x1D706}^{+}$ , which implies the ineffectiveness of a large forcing spacing, since more energy input is required to generate similar strengths of the spanwise wall-jet velocity as with small spacing. Interestingly, for high ${\mathcal{R}}$ cases, $|\tilde{w}|_{m}\approx 2\sim 4u_{\unicode[STIX]{x1D70F},0}$ , which is very close to the optimal spanwise perturbation velocity introduced by plasma actuators in TBL control (Thomas et al. Reference Thomas, Corke, Hussain, Duong, McGowan, Jasinski and Simmons2016).

Figure 5.

Cross-stream ( $y{-}z$ ) plane view of mean velocity field ( $\overline{\boldsymbol{u}}$ ) with the contours indicating the mean streamwise velocity (normalized by the bulk velocity $U_{b}$ ), and the vectors denoting the wall-normal and spanwise velocities for (a) $A^{+}=0.015$ ; (b) $A^{+}=0.06$ with $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and $y_{c}^{+}=30$ .

Figure 5(a,b) shows the cross-streamwise view ( $y{-}z$ plane) of contours of the streamwise velocity and the cross-plane velocity vectors of $\overline{\boldsymbol{u}}$ at two different forcing amplitudes $A^{+}=0.015$ (corresponding to the optimal control case) and $A^{+}=0.06$ (corresponding to a drag increase case) with $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and $y_{c}^{+}=30$ . For $A^{+}=0.015$ , the topmost low-speed streak envelope, which has only one peak, is due to the lift-up effect of the counter-rotating control swirls, and extends up to the outer flow region ( $y/h>0.6$ ). Near the wall, however, the streak has dual peaks, which are due to the secondary swirls. As the forcing amplitude increases to $A^{+}=0.06$ , the lift-up effect of the control swirls becomes even stronger; and the topmost low-speed streak extends up to the half-channel height. In addition, the near-wall streaks fall apart more due to the effect of stronger secondary swirls.

To compare the turbulent fluctuations with those in the uncontrolled case, figure 6 shows the spanwise-averaged velocity $\{\langle \overline{u_{i}^{\prime \prime 2}}\rangle ^{1/2}\}^{\ast }$ and vorticity $\{\langle \overline{\unicode[STIX]{x1D714}_{i}^{\prime \prime 2}}\rangle ^{1/2}\}^{\ast }$ fluctuations at two forcing magnitudes $A^{+}=0.015$ and $0.06$ (the superscript $\ast$ indicates non-dimensionalization by the wall units of the controlled flow). All of the velocity and vorticity fluctuations are suppressed near the wall due to the merger of numerous streaks and hence fewer streamwise vortices are regenerated under control; however, they are all enhanced in the outer region ( $y/h>0.6$ ), mainly due to the advection of the large-scale control swirls, which transport the near-wall turbulence and vortical structures to the outer flow (see § 4.3). Notable differences are also observed by comparing $A^{+}=0.015$ and $0.06$ . In the outer region, all the turbulent fluctuations are further increased for the $A^{+}=0.06$ case, which is, as expected, due to the even stronger advection effect of the control swirls. In the near wall, the root mean square (r.m.s.) of $v^{\prime \prime }$ , $w^{\prime \prime }$ and $\unicode[STIX]{x1D714}_{x}^{\prime \prime }$ are increased, and $u^{\prime \prime }$ , $\unicode[STIX]{x1D714}_{y}^{\prime \prime }$ and $\unicode[STIX]{x1D714}_{z}^{\prime \prime }$ are decreased, which is consistent with figure 13(e) where more streamwise aligned vortices are observed near the collision region of two swirls.

Figure 6.

Spanwise-averaged turbulent velocity (top row) and vorticity (bottom row) fluctuations with various forcing amplitudes for $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and $y_{c}^{+}=30$ . Solid line: uncontrolled case; dashed line: drag reduction with $A^{+}=0.015$ ; dash-dotted line: drag increase with $A^{+}=0.06$ .

4.2 Scales of drag production

To analyse the contribution of the drag coefficient, the FIK (Fukagata, Iwamoto, Kasagi) identity (Fukagata et al. Reference Fukagata, Iwamoto and Kasagi2002; Rastegari & Akhavan Reference Rastegari and Akhavan2015), which links the skin friction with the total Reynolds shear stress $\langle \overline{u^{\prime }v^{\prime }}\rangle$ , is invoked:

(4.3) $$\begin{eqnarray}\displaystyle C_{f}=\underbrace{\frac{12}{Re_{b}}}_{C_{f}^{L}}+\underbrace{12\int _{0}^{2}(1-y)\left(-\frac{\langle \overline{u^{\prime }v^{\prime }}\rangle }{4U_{b}^{2}}\right)\,\text{d}y}_{C_{f}^{T}}. & & \displaystyle\end{eqnarray}$$

The drag coefficient $C_{f}$ thus has an ‘equivalent’ laminar part $C_{f}^{L}=12/Re_{b}$ , and a turbulent part $C_{f}^{T}$ represented by the weighted integration of the total Reynolds shear stress (RSS) $\langle \overline{u^{\prime }v^{\prime }}\rangle$ . As the latter is composed of a coherent part $\langle \tilde{u} \tilde{v}\rangle$ and a random part $\langle \overline{u^{\prime \prime }v^{\prime \prime }}\rangle$ , the FIK identity can be further decomposed as

(4.4) $$\begin{eqnarray}\displaystyle C_{f}=\underbrace{\frac{12}{Re_{b}}}_{C_{f}^{L}}+\underbrace{12\int _{0}^{2}(1-y)\left(-\frac{\langle \tilde{u} \tilde{v}\rangle }{4U_{b}^{2}}\right)\,\text{d}y}_{C_{f}^{C}}+\underbrace{12\int _{0}^{2}(1-y)\left(-\frac{\langle \overline{u^{\prime \prime }v^{\prime \prime }}\rangle }{4U_{b}^{2}}\right)\,\text{d}y}_{C_{f}^{R}}. & & \displaystyle\end{eqnarray}$$

Since $Re_{b}$ is held constant in all simulations under different control parameters, only changes in the other two parts (coherent $C_{f}^{C}$ and random $C_{f}^{R}$ ) affect the drag. Figure 7(a) shows the profiles of the total weighted Reynolds shear stress (WRSS) $-(1-y)\langle \overline{u^{\prime }v^{\prime }}\rangle$ , and figure 7(b) shows the coherent part $-(1-y)\langle \tilde{u} \tilde{v}\rangle$ (denoted as C-WRSS) and the random part $-(1-y)\langle \overline{u^{\prime \prime }v^{\prime \prime }}\rangle$ (denoted as R-WRSS). For the low amplitude $A^{+}=0.015$ case, the drag reduction, following (4.3), is due to the alteration of the Reynolds shear stress throughout the whole range of $y$ . Further, from figure 7(b), we find that the random turbulent Reynolds shear stress is significantly suppressed, and the coherent part contributes the most to the total Reynolds shear stress. In contrast, at a very high amplitude ( $A^{+}=0.06$ ), the drag increment is due to the enhancement of the coherent Reynolds shear stress away from the wall ( $y^{+}>20$ ). This also can be clearly shown through the decomposed total skin friction (normalized by the uncontrolled total skin friction) in figure 8(a). For the uncontrolled case (no coherent part), the laminar part contributes 25.0 % to the skin friction, and the turbulent part contributes 75.0 %. For $A^{+}=0.015$ , the random part contribution is reduced to 22.6 %, and the coherent part increased to 33.8 %; hence the net skin friction is reduced by 18.6 %. In contrast, for a higher $A^{+}=0.06$ , although the random turbulent part is reduced to approximately 15.5 %, the coherent part is increased to 63.6 %, hence leading to the net skin friction increase of 4.1 %. (Note that figure 8(b) shows a similar analysis of $C_{f}$ for the effect of different $y_{c}^{+}$ , to be discussed in § 4.4.) Therefore, effective drag reduction results from a compensative effect of suppressing random turbulence (via suppression of the streaks as well as the streamwise vortices) countered by enhancing the coherent part induced by the energetic forcing.

Figure 7.

(a) Total and (b) decomposed weighted Reynolds shear stress (WRSS) distributions for different control amplitudes $A^{+}$ with $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and $y_{c}^{+}=30$ at $Re_{\unicode[STIX]{x1D70F}}=180$ .

To quantify the contributions to the skin friction from different scales, the FIK identity in combination with the spanwise wavenumber spectrum of the Reynolds shear stress is also employed (see also de Giovanetti, Hwang & Choi (Reference de Giovanetti, Hwang and Choi2016)). (These contributions do not exist independently because different scales results from nonlinear interactions among them.) The total Reynolds shear stress can be expressed in terms of the one-dimensional spanwise Reynolds shear stress spectral density $\unicode[STIX]{x1D6F7}^{T}(y;k_{z})$ , i.e.

(4.5) $$\begin{eqnarray}\displaystyle -\langle \overline{u^{\prime }v^{\prime }}\rangle =\int _{-\infty }^{\infty }\unicode[STIX]{x1D6F7}^{T}(y;k_{z})\,\text{d}k_{z}, & & \displaystyle\end{eqnarray}$$

where $k_{z}(=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z})$ is the spanwise wavenumber. Then, the Reynolds shear stress contributed by the motions with spanwise size $\unicode[STIX]{x1D706}_{z}$ smaller than a given cutoff wavelength $\unicode[STIX]{x1D706}_{z,c}$ is:

(4.6) $$\begin{eqnarray}\displaystyle -\langle \overline{u^{\prime }v^{\prime }}\rangle _{c}(y;\unicode[STIX]{x1D706}_{z,c})=2\int _{k_{z,c}}^{\infty }\unicode[STIX]{x1D6F7}^{T}(y;k_{z})\,\text{d}k_{z}, & & \displaystyle\end{eqnarray}$$

where the cutoff wavenumber is $k_{z,c}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z,c}$ . The above spanwise spectrum analysis can be similarly applied to the coherent Reynolds shear stress $-\langle \tilde{u} \tilde{v}\rangle _{c}=2\int _{k_{z,c}}^{\infty }\unicode[STIX]{x1D6F7}^{C}(y;k_{z})\,\text{d}k_{z}$ and the random turbulent stress $-\langle \overline{u^{\prime \prime }v^{\prime \prime }}\rangle _{c}=2\int _{k_{z,c}}^{\infty }\unicode[STIX]{x1D6F7}^{R}(y;k_{z})\,\text{d}k_{z}$ . Therefore, one obtains the skin friction coefficient generated by the motions with  $\unicode[STIX]{x1D706}_{z}<\unicode[STIX]{x1D706}_{z,c}$ :

(4.7) $$\begin{eqnarray}\displaystyle C_{f,c}(\unicode[STIX]{x1D706}_{z,c}) & = & \displaystyle \frac{12}{Re_{b}}+\underbrace{12\int _{0}^{2}(1-y)\left(-\frac{\langle \overline{u^{\prime }v^{\prime }}\rangle _{c}}{4U_{b}^{2}}\right)\text{d}y}_{C_{f,c}^{T}}\nonumber\\ \displaystyle & = & \displaystyle \frac{12}{Re_{b}}+\underbrace{12\int _{0}^{2}(1-y)\left(-\frac{\langle \tilde{u} \tilde{v}\rangle _{c}}{4U_{b}^{2}}\right)\text{d}y}_{C_{f,c}^{C}}+\underbrace{12\int _{0}^{2}(1-y)\left(-\frac{\langle \overline{u^{\prime \prime }v^{\prime \prime }}\rangle _{c}}{4U_{b}^{2}}\right)\text{d}y}_{C_{f,c}^{R}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

which provides a tool to estimate the amount of drag contributed by coherent motions and random turbulent fluctuations of different scales.

Figure 8.

The decomposed drag ( $C_{f}^{L}$ , $C_{f}^{C}$ , $C_{f}^{R}$ ) for (a) different $A^{+}$ and (b) different $y_{c}^{+}$  at $Re_{\unicode[STIX]{x1D70F}}=180$ .

Figure 9.

Premultiplied spanwise wavenumber spectra of the total Reynolds shear stress $k_{z}\unicode[STIX]{x1D6F7}_{uv}^{T}$ for (a) uncontrolled flow; (b) $A^{+}=0.015$ ; and (d) $A^{+}=0.06$ , and the coherent Reynolds shear stress $k_{z}\unicode[STIX]{x1D6F7}_{uv}^{C}$ for (c) $A^{+}=0.015$ , and (e) $A^{+}=0.06$ . The contour labels indicate 0.2, 0.4, 0.6 and 0.8 of the maximum value of the uncontrolled case, and dashed lines for negative values.

The premultiplied spanwise wavenumber spectrum of the Reynolds shear stress $k_{z}\unicode[STIX]{x1D6F7}^{T}$ for the uncontrolled case is displayed in figure 9(a). For $Re_{\unicode[STIX]{x1D70F}}=180$ , it shows that the energy containing motions are mainly concentrated among $0.5\lesssim \unicode[STIX]{x1D706}/h\lesssim 1.1$ (or $100\lesssim \unicode[STIX]{x1D706}_{z}^{+}\lesssim 200$ ) and $0.1\lesssim y/h\lesssim 0.3$ (or $20\lesssim y^{+}\lesssim 50$ ) – these motions presumably correspond to quasi-streamwise vortices. This also explains the parametric study such that when setting $y_{c}^{+}$ within the buffer layer (figure 3 c), the energy containing motions are effectively weakened and hence significant drag reductions can be achieved. The effects of control at $A^{+}=0.015$ on the spanwise wavenumber spectrum of the total ( $k_{z}\unicode[STIX]{x1D6F7}^{T}$ ) and coherent ( $k_{z}\unicode[STIX]{x1D6F7}^{C}$ ) Reynolds shear stresses are shown in figure 9(b,c), respectively. The difference of the two spectra represents the random turbulent contribution ( $k_{z}\unicode[STIX]{x1D6F7}^{R}$ ), which is induced mostly by the near-wall small-scale motions ( $y<0.3h$ and $\unicode[STIX]{x1D706}_{z}<2h$ ). It is obvious that the contribution to the total Reynolds stress is mainly due to the coherent motions, consistent with figure 8(a). Compared with the uncontrolled case, the contribution from the near-wall region is effectively suppressed; the energy containing part is shifted outward, while $\unicode[STIX]{x1D706}_{z}$ is also changed from $0.5h$ to approximately $3h$ , the latter being close to the spacing of two adjacent control swirls. This suppression of structures at small scales is consistent with the observations of de Giovanetti, Sung & Hwang (Reference de Giovanetti, Sung and Hwang2017) in their study of streak instability for large-scale motions. Note that, as the forcing amplitude increases to $A^{+}=0.06$ , figure 9(c,d) shows that the Reynolds stress due to the coherent part (located in the outer region) further increases, while that from the random part is very small. Interestingly, there appears a positive Reynolds shear stress region (i.e. $1<\unicode[STIX]{x1D706}_{z}/h<2$ ) marked by the dashed contours in figure 9(d,e), which actually results from the notable fluctuations in the Q1 $(u^{\prime }>0,v^{\prime }>0)$ and Q3 $(u^{\prime }<0,v^{\prime }<0)$ quadrants shown by the instantaneous velocity field in figure 11(f).

Figure 10.

Cumulative friction coefficient of $C_{f,c}^{T}$ in (a), and of its coherent $C_{f,c}^{C}$ and random part $C_{f,c}^{C}$ in (b), varying with the cutoff spanwise wavelength $\unicode[STIX]{x1D706}_{z,c}$ . Note that the laminar part is subtracted from all of the plots.

Finally, figure 10 shows the cumulative skin friction due to the turbulent motions of $\unicode[STIX]{x1D706}_{z}<\unicode[STIX]{x1D706}_{z,c}$ for different $A^{+}$ values. The laminar part, identical for all cases, is subtracted from the plot so that all of the cumulative $C_{f}$ values start from zero. For the uncontrolled case, motions with a spanwise scale in the range of $0.5h\leqslant \unicode[STIX]{x1D706}_{z}\leqslant 1.5h$ are responsible for the majority of the drag (approximately $54\,\%$ ), consistent with the observation by de Giovanetti et al. (Reference de Giovanetti, Hwang and Choi2016). For controlled cases, the cumulative $C_{f}$ contributed by the total Reynolds stress is plotted in figure 10(a), and that by the coherent and random Reynolds stresses are plotted in figure 10(b). Taking $A^{+}=0.015$ in figure 10(b) for example, the random Reynolds shear stress is mainly induced by small-scale motions ( $\unicode[STIX]{x1D706}_{z}<2h$ ) and contributes to approximately 20 % of the total drag. In contrast, the coherent part is negligible for $\unicode[STIX]{x1D706}_{z}<2h$ and becomes important only when $\unicode[STIX]{x1D706}_{z}>2h$ , beyond which the random part is almost unchanged, and the coherent part alone contributes to the total cumulative drag increase. In this case, the effect of suppressing the small scales exceeds that of enhancing the large scales, leading to net drag decrease. As $A^{+}$ increases to $0.06$ , the random turbulent part due to the small scales further decreases. Note particularly the dip of the coherent part near $\unicode[STIX]{x1D706}_{z}\approx 2h$ is due the positive Reynolds shear stress identified in figure 9(e). Above this wavelength, the cumulative drag increases again due to the increasing contribution from the large-scale coherent motions. Thus, opposite to the $A^{+}=0.015$ case, the effect of excitation of the large scales at $A^{+}=0.06$ exceeds that of weakening the small-scale motions, hence leading to net drag increase.

4.3 Coherent structures

Figure 11.

Contour of streamwise velocity fluctuations in the ( $x{-}z$ ) plane at $y^{\ast }=20$ : (a) no control; (b) $u^{\prime }$ for $A^{+}=0.015$ ; (c) $u^{\prime \prime }$ for $A^{+}=0.015$ ; (d) $u^{\prime }$ for $A^{+}=0.06$ ; and (e) $u^{\prime \prime }$ for $A^{+}=0.06$ . The forcing spacing is $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and control height is $y_{c}^{+}=30$ .

Figure 11 shows the contours of the low-speed streaks for both $u^{\prime }$ and $u^{\prime \prime }$ at $y^{\ast }=20$ for different forcing amplitudes (uncontrolled $A^{+}=0$ , low $A^{+}=0.015$ and high $A^{+}=0.06$ ). While typical meandering low-speed streaky structures are observed aligned along the streamwise direction in the uncontrolled case (figure 11 a), the number of streaks is decreased for the controlled cases. In the presence of control, the basic near-wall streaks of $u^{\prime }$ are destroyed (figure 11 b,d), and the near-wall flow becomes a single very wide meandering streak extending the entire flow length, as expected from the merger of the numerous background streaks forced by the two wall jets. Note that such a widening and weakening of near-wall streaks has also been found in the drag reduction by spanwise wall oscillations (Jung et al. Reference Jung, Mangiavacchi and Akhavan1992; Akhavan, Jung & Mangiavacchi Reference Akhavan, Jung and Mangiavacchi1993), which presumably plays a similar role as SOJF merging many streaks together. The streaks are located essentially in the collision region – particularly clear in the random part $u^{\prime \prime }$ (figure 11 c) when the coherent flow $\tilde{u}$ is subtracted from the total fluctuation velocity $u^{\prime }$ . As the forcing amplitude increases, more random streaks appear (figure 11 e), which is consistent with the greater number of vortical structures observed in figure 13(e).

Figure 12.

Vector plots of $(v,w)$ in the ( $y{-}z$ ) plane located at $x=L_{x}/3$ : (a) no control; (b) $A^{+}=0.015$ ; and (c) $A^{+}=0.06$ . Light (yellow) and dark (blue) colours indicate positive and negative streamwise velocity fluctuations $u^{\prime }$ . The forcing spacing is $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and control height is $y_{c}^{+}=30$ .

Figure 13.

Iso-surfaces of $\unicode[STIX]{x1D706}_{2}=-1$ (shaded with the wall-normal distance from the bottom wall): (a) no control; (c) $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and $A^{+}=0.015$ ; (e) $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ and $A^{+}=0.06$ ; (b,d) and (f) are the corresponding skin friction normalized by the uncontrolled case on the bottom wall.

The flow field difference among the three different amplitudes is further examined through the velocity vectors in the cross-stream ( $y{-}z$ ) plane, in figure 12. Here, the light (yellow) colour indicates positive streamwise velocity fluctuation $u^{\prime }$ and dark (blue) colour indicates negative streamwise velocity fluctuation $u^{\prime }$ (only the left half-domain $z/\unicode[STIX]{x1D706}\leqslant 0.5$ is displayed, since the left and right parts are almost symmetric). Compared with the chaotic feature of the uncontrolled case (figure 12 a), the velocity vectors near the wall are more parallel along the spanwise direction for $A^{+}=0.015$ , indicating organized motions induced by SOJF similar to that of figure 5(a). Moreover, the downwash by the near-wall streamwise vortices observed in the uncontrolled case is significantly reduced by the SOJF. At higher $A^{+}=0.06$ , the control flow is clearly much more dominant and organized. As marked in figure 12(c), a strong Q2 event (negative $u^{\prime }$ and positive $v^{\prime }$ associated with the lift up of the large-scale swirls) occurs in the region $0.3\lesssim z/\unicode[STIX]{x1D706}\lesssim 0.5$ , contributing to the increase of Reynolds shear stress in the bulk flow (figure 8) and hence drag increment. In addition, notable Q1 and Q3 events are also observed (located between Q2 and Q4 events), which cause the alternative positive/negative Reynolds shear stress contours in figures 9(d,e).

We then display the $\unicode[STIX]{x1D706}_{2}$ vortices (Jeong & Hussain Reference Jeong and Hussain1995) (calculated based on the random turbulent fluctuation velocity $\boldsymbol{u}^{\prime \prime }$ ) and the skin friction at the bottom wall in figure 13. Note that numerous slender vortices are distributed throughout the wall region in the uncontrolled case (figure 13 a), lying directly above the high skin friction regions (figure 13 b); and hairpins are few and far between, consistent with the observations of Schlatter et al. (Reference Schlatter, Li, Örlü, Hussain and Henningson2014). At $A^{+}=0.015$ , most of the near-wall streamwise vortices disappear except near the collision region, just above the merged streak envelope (figure 13 c); this disappearance is consistent with the suppression of $\unicode[STIX]{x1D714}_{x,rms}^{\prime \prime }$ in figure 6(d). From figure 13, we observe that the vortex structures mostly lie on top of the low skin friction region, which would appear to contradict the well-accepted notion that peak skin friction underlines streamwise vortices. This apparent enigma is reconciled below.

Figure 14.

(a,b) Cross-section view of the instantaneous $\unicode[STIX]{x1D706}_{2}$ structures; (c,d) low-speed streaks; and (e,f) spanwise distribution of the wall shear stress at streamwise location $x=L_{x}/3$ . Note that the dashed line in the bottom row is the streamwise- and time-averaged wall shear stress. The left and right columns are for $A^{+}=0.015$ ; and $A^{+}=0.06$ , respectively.

Figure 14 shows the instantaneous cross-section view of $\unicode[STIX]{x1D706}_{2}$ contours at a specific streamwise location $x=L_{x}/3$ , together with the associated low-speed streaks and the spanwise variation of wall shear stress. The strength of combined large streak envelope, although below the critical value, is so only on an average basis, some individual streaks survive on an instantaneous basis with very high flank angles (figure 14 c,d). Although it may not be obvious, instantaneously the vortical structures, the streaks and wall shear stress are quite uniquely related; there are instances and locations of high instantaneous wall shear stresses, which can be uniquely identified with instantaneous streamwise vortical structures. However, after ensemble averaging, the spiked wall shear stress is no longer apparent, due to the intermittent feature of these vortices.

Figure 15.

Contours (red lines) of (a) $\unicode[STIX]{x1D714}_{x}$ ; (b) forcing generation; (c) the $-\unicode[STIX]{x2202}_{x}w\unicode[STIX]{x2202}_{y}u$ (wall-normal vorticity) tilting; (d) the $\unicode[STIX]{x2202}_{x}v\unicode[STIX]{x2202}_{z}u$ (spanwise vorticity) tilting; (e) vortex stretching; (f) the cross-stream ( $y{-}z$ ) advection. The dash line in (a) indicates the low-speed streak in figure 14(a); (b–f) identify the positive $\unicode[STIX]{x1D714}_{x}$ boundaries as shown in (a). Also in (a–f), solid (dot-dashed) lines indicate positive (negative) magnitudes of each term.

The above investigation triggers another important question, i.e. where do these vortical structures above the streak envelope come from? Do they get pushed outward from the wall or freshly created locally? Note that since the vortices are predominantly streamwise (figure 13 c), the evolutionary dynamics of $\unicode[STIX]{x1D714}_{x}$ is further investigated here. The inviscid evolution of $\unicode[STIX]{x1D714}_{x}$ is governed by

(4.8) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}t}=\underbrace{\unicode[STIX]{x1D714}_{x}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}}_{\text{I}}-\underbrace{\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}}_{\text{IIa}}+\underbrace{\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}}_{\text{IIb}}-\underbrace{u\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}x}}_{\text{III}}-\underbrace{v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}y}-w\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}z}}_{\text{IV}}+\underbrace{\frac{\unicode[STIX]{x2202}F_{z}}{\unicode[STIX]{x2202}y}}_{\text{V}}. & & \displaystyle\end{eqnarray}$$

The terms on the right-hand side are: (I) vortex stretching, (IIa,b) vortex tilting, (III) streamwise advection, (IV) cross-stream advection and (V) vorticity generation due to forcing.

In order to get some clear understanding of the flow physics, it is necessary to evaluate the role of each term in the vorticity equation (4.8). Note that for simplicity, we focus on the structures with positive streamwise vorticity $+\unicode[STIX]{x1D714}_{x}$ (SP) and their generation, ignoring the comparable numbers of negative streamwise vorticity $-\unicode[STIX]{x1D714}_{x}$ (SN) structures. Figure 15 shows a small domain around the mid-span of figure 14: (a)  $\unicode[STIX]{x1D714}_{x}$ associated with SPs, along with the low-speed streak boundary from figure 14(c); (b)  $\unicode[STIX]{x1D714}_{x}$ generation by the forcing term (V); (c)  $\unicode[STIX]{x1D714}_{x}$ generation by wall-normal vorticity titling ( $-\unicode[STIX]{x2202}_{x}w\unicode[STIX]{x2202}_{y}u$ , IIa); (d) $\unicode[STIX]{x1D714}_{x}$ generation by spanwise vorticity titling ( $\unicode[STIX]{x2202}_{x}v\unicode[STIX]{x2202}_{z}u$ , IIb); (e) streamwise vortex stretching (I); and (f) cross-stream advection (IV). Note that the key $+\unicode[STIX]{x1D714}_{x}$ contours (SP structures) from figure 15(a) are repeated in figures 15(b–f) in order to provide a common spatial reference. Generation terms which strengthen SP are indicated by positive values (solid lines), while negative values indicate local weakening of SP (dash-dot lines).

At location $A$ in figure 15(b), one SP is generated by the shearing induced by forcing. Most of the SPs are located in the middle region ( $z/\unicode[STIX]{x1D706}\sim 0.5$ ), where the forcing generation term is small. The source of generation of these SPs with $+\unicode[STIX]{x1D714}_{x}$ is mainly the tilting of spanwise and wall-normal vorticity by the streamwise flow induced shear. For example, in the near-wall region ( $y^{\ast }<50$ , $B$ in figure 15 c), we find important vorticity generation due to the wall-normal vorticity tilting $-\unicode[STIX]{x2202}_{x}w\unicode[STIX]{x2202}_{y}u$ by the streamwise flow, where the instantaneous streak has a steep flank $\unicode[STIX]{x1D714}_{y}$ (figure 15 c). In the bulk flow (at point $C$ in figure 15 d where $y^{\ast }\sim 50$ ), the spanwise vorticity tilting by streamwise velocity enhances the SP due to the presence of highly lifted-up low-speed streak (having strong $\unicode[STIX]{x2202}_{z}u$ ). Note that once SPs are generated, they are further amplified by the direct stretching due to the streak waviness $\unicode[STIX]{x2202}_{x}u$ (similar to that observed by SH1), evident from the strong regions of $+\unicode[STIX]{x1D714}_{x}\unicode[STIX]{x2202}_{x}u$ embedded in SP (region $D$ in figure 15 e). Note that the intense vortex stretching (producing $+\unicode[STIX]{x1D714}_{x}$ at point $E$ in figure 15 e), does not result in SP, due to the simultaneous counter-effect of the tilting (producing $-\unicode[STIX]{x1D714}_{x}$ ) in figure 15(c). Finally, the generated SP structures are further transported outward by the swirls as also downstream by the base flow, evident from the positive amplitude of the advection term on top of the SP in $F$ in figure 15(f).

For the higher forcing amplitude $A^{+}=0.06$ case, the skin friction near the wall-jet sides is significantly increased (figure 14 f) due to the squashing of the wall jet, and hence increased $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y$ (figure 11 f). Although numerous streaks merge into a large streak envelope, the streak strength is larger near the collision plane (figure 14 d) compared with the small amplitude case $A^{+}=0.015$ , and hence leads to the generation of more vortex structures, as seen in figure 14(b). In addition, the downwash of the secondary swirls is larger and stronger at high $A^{+}$ (i.e. $=0.06$ ), which causes a local high skin friction at the mid-span (i.e. $z/\unicode[STIX]{x1D706}\approx 0.5$ in figure 14 f).

4.4 The effect of $y_{c}^{+}$

Figure 16.

Cross-stream ( $y{-}z$ ) plane view of (a) mean velocity field ( $\overline{\boldsymbol{u}}$ ) with the contours indicating the mean streamwise velocity (normalized by $U_{b}$ ), and the vectors denoting the wall-normal and spanwise velocities for (a) $y_{c}^{+}=5$ ; (b) $y_{c}^{+}=80$ .

Compared with the previous large-scale control methods (Canton et al. Reference Canton, Örlü, Chin, Hutchins, Monty and Schlatter2016a , SH2), where the height of the control swirl was fixed at the centre of the channel, a major difference of the current SOJF method is the introduction of $y_{c}$ to characterize the wall-jet height. To demonstrate the effect of $y_{c}^{+}$ on drag reduction ${\mathcal{R}}$ , two additional cases with $y_{c}^{+}=5$ and $80$ are discussed here (while $A^{+}=0.015$ and $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ remain the same as the optimal for $y_{c}^{+}=30$ ). Compared with the case at $y_{c}^{+}=30$ (producing $18.6\,\%\pm 0.9\,\%$ of ${\mathcal{R}}$ ), ${\mathcal{R}}$ becomes $1.4\,\%\pm 0.7\,\%$ and $13.5\,\%\pm 1.1\,\%$ for $y_{c}^{+}=5$ and $80$ , respectively. Figure 16 shows the cross-streamwise view (the ( $y{-}z$ ) plane) of contours of the streamwise velocity and the cross-plane velocity vectors of $\overline{\boldsymbol{u}}$ for the two different forcing height cases. For the low $y_{c}^{+}$ case, the wall jet is too thin, and limited very close to the bottom wall, which hardly has any effect on the suppression of the streaks, whilst at high $y_{c}^{+}$ , the energetic, counter-rotating swirls well organize the flow, and the squashing effect along the sides of the swirls also becomes stronger. Compared to $y_{c}^{+}=30$ , the secondary swirls at $y_{c}^{+}=80$ near the collision plane are much larger, separating the dual peaks further.

Figure 17 further shows the spanwise-averaged turbulent fluctuations $\{\langle \overline{u_{i}^{\prime \prime 2}}\rangle ^{1/2}\}^{\ast }$ . For $y_{c}^{+}=5$ , turbulent fluctuations are almost unchanged, as extremely near-wall forcing is ineffective, as discussed above, also consistent with figure 8(b) where $C_{f}^{C}$ is rather small and $C_{f}^{T}$ is nearly unchanged. For $y_{c}^{+}=80$ , the turbulent fluctuations are suppressed near the wall (due to the merger of streaks), while they are increased in the outer flow region (due to upward advecting of the structures). Since coherent motions are more pronounced at $y_{c}^{+}=80$ compared to $y_{c}^{+}=5$ (figure 16), a larger $C_{f}^{C}$ is observed for $y_{c}^{+}=80$ , as shown in figure 8(b). $C_{f}^{C}$ is also $5\,\%$ larger at $y_{c}^{+}=80$ compared to that of $y_{c}^{+}=30$ , leading to the degradation of ${\mathcal{R}}$ as $y_{c}^{+}$ increases.

Figure 17.

Spanwise-averaged turbulent velocity fluctuations for various forcing heights $y_{c}^{+}$ at $A^{+}=0.015$ and $\unicode[STIX]{x1D706}^{+}=360\unicode[STIX]{x03C0}$ .

Figure 18.

Iso-surfaces of $\unicode[STIX]{x1D706}_{2}=-1$ associated with the contours of the corresponding skin friction on the bottom walls normalized by the uncontrolled case: (a) $y_{c}^{+}=5$ ; (b) $y_{c}^{+}=80$ at $Re_{\unicode[STIX]{x1D70F}}=180$ .

Similar to figure 13, the $\unicode[STIX]{x1D706}_{2}$ vortex and the corresponding skin friction at the bottom wall are displayed in figure 18 for $y_{c}^{+}=5$ and $80$ . It is obvious that the near-wall structures are barely suppressed at $y_{c}^{+}=5$ compared with the uncontrolled case. At $y_{c}^{+}=80$ , much fewer vortex structures survive near the mid-span. In addition, there are two distinct low skin friction regions lying along the streamwise direction, caused by the near-wall lifting up and separated by the larger secondary swirls. The downwash motions associated with the larger secondary swirls would cause drag increase and should be suppressed also. In summary, it is clear that $y_{c}$ has to have an optimal value – neither too small nor too large.