3.1 Turbulence statistics

The mean velocity profiles are shown in figure 3(a) along with the DNS data for the ZPG TBL ( $Re_{\unicode[STIX]{x1D70F}}=830$ ) reported by Schlatter & Örlü (Reference Schlatter and Örlü2010). The red and black lines are for the APG and ZPG TBLs, respectively. The logarithmic law of the APG TBL is slightly shifted downward, and there is a large wake region above $y^{+}\approx 300$ . Note that the turbulence statistics obtained in the present study were averaged over the region $x^{\prime }=x_{r}\pm 0.5\unicode[STIX]{x1D6FF}$ , where $x_{r}$ is the reference position at $Re_{\unicode[STIX]{x1D70F}}\approx 835$ . Figure 3(b) shows the defect velocity profiles ( $U_{\infty }$ – $U$ ) $^{+}$ versus $y/\unicode[STIX]{x1D6E5}$ for the equilibrium region in the APG TBL. The grey and blue lines represent the profiles at $Re_{\unicode[STIX]{x1D703}}=2100$ –3900 and at $Re_{\unicode[STIX]{x1D703}}=4100$ –5300 within the equilibrium region, respectively, for increments of $Re_{\unicode[STIX]{x1D703}}=300$ . As shown in the inset of figure 3(b), all the defect velocity profiles (blue) collapse onto a single curve, indicating that the flows in the equilibrium region satisfy self-similarity (Henkes Reference Henkes1998; Skote et al. Reference Skote, Henningson and Henkes1998; Lee Reference Lee2017).

Figure 3.

(a) Mean velocity profiles in wall units: red, the APG TBL at $Re_{\unicode[STIX]{x1D70F}}=834$ ( $Re_{\unicode[STIX]{x1D703}}=5400$ ); black, the ZPG TBL at $Re_{\unicode[STIX]{x1D70F}}=837$ . Dashed lines indicate the linear law and the logarithmic law, where $\unicode[STIX]{x1D705}$ is the von Kármán constant of 0.41 and $C$ is constant value of 5.1. Defect velocity profiles normalized by (b) the friction velocity $u_{\unicode[STIX]{x1D70F}}$ and the Rotta–Clauser length scale $\unicode[STIX]{x1D6E5}$ : grey, $Re_{\unicode[STIX]{x1D703}}=2100$ –3900; blue, $Re_{\unicode[STIX]{x1D703}}=4100$ –5300 for increments of $Re_{\unicode[STIX]{x1D703}}=300$ .

Figure 4(ai–di) shows the Reynolds stresses (i.e. $\langle uu\rangle$ , $\langle vv\rangle$ , $\langle ww\rangle$ and $\langle -uv\rangle$ ), which have a clear peak at $y^{+}\approx 300$ due to the strong turbulence in the wake region. All the Reynolds stresses are higher for all wall-normal locations in the APG TBL, especially in the outer region. In particular, the value of $\langle ww\rangle$ is significantly higher below $y^{+}=20$ ; there is a plateau in the range $20<y^{+}<80$ and also a clear outer peak. Given that the wall-parallel components carried by the large scales were non-zero close to the wall in the sense of Townsend’s attached eddy hypothesis (Townsend Reference Townsend1976), the difference in the wall-parallel components ( $\langle uu\rangle$ and $\langle ww\rangle$ ) close to the wall could imply the influence of enhanced large scales in the outer region; this point will be discussed in detail in §§ 3.2 and 3.4.

Figure 4.

The Reynolds stresses: (a) $\langle uu\rangle$ , (b) $\langle vv\rangle$ , (c) $\langle ww\rangle$ and (d) $\langle -uv\rangle$ . The quantities in (i) were normalized by the wall units, in (ii) were normalized by the reference velocity $U_{e}$ and the displacement thickness $\unicode[STIX]{x1D6FF}^{\ast \ast }$ , and in (iii) were normalized by the friction velocity $u_{\unicode[STIX]{x1D70F}}$ and the Rotta–Clauser thickness $\unicode[STIX]{x1D6E5}$ . Grey and blue lines in (iii) are the same as those in figure 3(b).

The Reynolds stresses scaled with the reference velocity ( $U_{e}$ ) and the displacement thickness ( $\unicode[STIX]{x1D6FF}^{\ast \ast }$ ) similar to those of Spalart & Watmuff (Reference Spalart and Watmuff1993) are shown in figure 4(aii–dii), where red circles indicate DNS data of a self-similar APG TBL with $\unicode[STIX]{x1D6FD}=1.00$ at $Re_{\unicode[STIX]{x1D703}}=4150$ (Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017). Note that $U_{e}$ is defined as $U_{e}(x)=-\int _{0}^{y_{th}(x)}\unicode[STIX]{x1D6FA}_{z}(x,y)\,\text{d}y$ and $\unicode[STIX]{x1D6FF}^{\ast \ast }$ can be calculated as $\unicode[STIX]{x1D6FF}^{\ast \ast }(x)=-U_{e}^{-1}(x)\int _{0}^{y_{th}(x)}y\unicode[STIX]{x1D6FA}_{z}(x,y)\,\text{d}y$ , where $y_{th}(x)$ is the wall-normal position at 0.2 % of $\unicode[STIX]{x1D6FA}_{z}(x)|_{y=0}$ (Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017). As shown in figure 4(aii–dii), all the Reynolds stresses are in good agreement with the data of Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017). Figure 4(aiii–diii) shows the Reynolds stresses with respect to $y/\unicode[STIX]{x1D6E5}$ , where the grey and blue lines are the same as those in figure 3. As shown, the profiles in the equilibrium region fall onto a single curve, representing the self-similarity in the equilibrium region of the present APG TBL.

3.2 Large scales in the pre-multiplied spanwise spectra

To further explore the contribution of scale to the differences between two TBLs in $\langle uu\rangle$ and $\langle ww\rangle$ , the pre-multiplied spanwise energy spectra of the streamwise and spanwise velocity fluctuations ( $k_{z}\unicode[STIX]{x1D719}_{uu}$ and $k_{z}\unicode[STIX]{x1D719}_{ww}$ ) are displayed in figure 5(c,d). Here, $k_{z}$ ( $=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z}$ ) is the spanwise wavenumber. The profiles of $\langle uu\rangle$ and $\langle ww\rangle$ in figure 5(a,b) can be obtained by integrating $k_{z}\unicode[STIX]{x1D719}_{uu}$ and $k_{z}\unicode[STIX]{x1D719}_{ww}$ with respect to $k_{z}$ . The uu energy spectrum of the APG TBL contains a strong peak in the outer region ( $y^{+}\approx 300$ ), which results from the energy carried by the long-wavelength motions $(\unicode[STIX]{x1D706}_{z}^{+}\approx 630)$ . The value of this outer peak is 2.7 times larger than that of the ZPG TBL. In the present study, the cutoff spanwise wavelength of $\unicode[STIX]{x1D706}_{z,c}^{+}=400(\unicode[STIX]{x1D706}_{z,c}/\unicode[STIX]{x1D6FF}\approx 0.5)$ was selected to separate the large- and small-scale streamwise velocity fluctuations (Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011; Ahn et al. Reference Ahn, Lee, Jang and Sung2013; Hwang & Sung Reference Hwang and Sung2017). The uu energy in the inner region is slightly lower than that of the ZPG TBL, which is related to the self-sustaining processes of small scales (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997). As shown in figure 5(c), however, the value of the inner peak of $\langle uu\rangle$ for the APG TBL is larger than that for the ZPG TBL, which indicates that the superimposed LSMs $(\unicode[STIX]{x1D706}_{z}^{+}\geqslant 400)$ in the near-wall region make the dominant contribution to the increase in the value of the inner peak of $\langle uu\rangle$ . In other words, the contribution of the LSMs to $\langle uu\rangle$ is enhanced not only in the outer region but also in the near-wall region. The enhanced footprints of the LSMs in the APG TBL are likely to have greater influence on the near-wall turbulence than those in the ZPG TBL.

As shown in figure 4(ci), the value of $\langle ww\rangle$ for the APG TBL is much higher than that for the ZPG TBL in both the inner and outer regions. The ww energy spectrum contains an outer peak ( $y^{+}\approx 330$ ) that results from the outer long-wavelength motions with $\unicode[STIX]{x1D706}_{z}^{+}\approx 830\;(\unicode[STIX]{x1D706}_{z}\approx \unicode[STIX]{x1D6FF})$ , which is the spanwise characteristic length scale of $w$ structures (Kim & Adrian Reference Kim and Adrian1999). The value of the outer peak of the ww energy spectrum for the APG TBL is 1.6 times higher than that for the ZPG TBL. Similar to $k_{z}\unicode[STIX]{x1D719}_{uu}$ , the large-scale ww energy is extended to the near-wall region. In contrast to the trend in $k_{z}\unicode[STIX]{x1D719}_{uu}$ , the small-scale energy of $k_{z}\unicode[STIX]{x1D719}_{ww}$ near the wall increases significantly, which implies that the AM influences of the LSMs on the small-scale spanwise velocity fluctuations near the wall are enhanced in the APG TBL. The enhanced LSMs result in the increased AM influence on the small-scale streamwise velocity fluctuations (Harun et al. Reference Harun, Monty, Mathis and Marusic2013). Besides, the AM effects on other velocity components (Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014) and on the swirling strengths (Hwang & Sung Reference Hwang and Sung2017) were reported in ZPG TBLs. Thus, the AM influence of the LSMs on the vortical motions could be greater in the APG TBL than in the ZPG TBL.

Figure 5.

Profiles of (a) $\langle uu\rangle$ and (b) $\langle ww\rangle$ . Pre-multiplied spanwise energy spectra of (c) the streamwise and (d) the spanwise velocity fluctuations. The white dashed line indicates the cutoff wavelength, $\unicode[STIX]{x1D706}_{z,c}^{+}=400$ .

Figure 6(a–c) shows the r.m.s. values of the streamwise, wall-normal and spanwise vorticity fluctuations ( $\unicode[STIX]{x1D714}_{x,rms}$ , $\unicode[STIX]{x1D714}_{y,rms}$ and $\unicode[STIX]{x1D714}_{z,rms}$ ). In the near-wall region ( $y^{+}<15$ ), the higher r.m.s. values are observed in the APG TBL, and in particular the difference is more prominent for the streamwise and spanwise components. To further examine the difference, the pre-multiplied spanwise energy spectra of the vorticity fluctuations in the streamwise, wall-normal and spanwise directions ( $k_{z}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D714}_{x}\unicode[STIX]{x1D714}_{x}},k_{z}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D714}_{y}\unicode[STIX]{x1D714}_{y}}$ and $k_{z}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D714}_{z}\unicode[STIX]{x1D714}_{z}}$ ) are shown in figure 6(d–f). The peaks are observed at $\unicode[STIX]{x1D706}_{z}^{+}\approx 100$ and the energy contained around the peak is enhanced, indicating that the vortical motions are more active due to the presence of the APG. In addition, the large scales $(\unicode[STIX]{x1D706}_{z}^{+}\geqslant 400)$ of $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{z}$ are more pronounced in the near-wall region. The intense r.m.s. values of the vorticity fluctuations (figure 6 a–c) arise from the enhanced small-scale energy, and also those of $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{z}$ result from the superposition of large scales. This means that the enhanced large scales are connected to the active vortical motions, and thus the increase in the small scales near the wall might be attributed to the AM effects of the LSMs.

Figure 6.

Profiles of (a) $\unicode[STIX]{x1D714}_{x,rms}$ , (b) $\unicode[STIX]{x1D714}_{y,rms}$ and (c) $\unicode[STIX]{x1D714}_{z,rms}$ . Pre-multiplied spanwise energy spectra of (d) the streamwise, (e) wall-normal and (f) spanwise vorticity fluctuations.

The vortical motions are closely related to the ejection and sweep motions, which are major contributors to the Reynolds shear stress (Robinson Reference Robinson1991). The wall-normal derivative of the Reynolds shear stress $(\unicode[STIX]{x2202}\langle -uv\rangle /\unicode[STIX]{x2202}y)$ is referred as the net mean effect of the turbulent inertia that acts as a momentum source or sink. In addition, $\unicode[STIX]{x2202}\langle -uv\rangle /\unicode[STIX]{x2202}y$ is directly related to $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ corresponding to the advective transport and the vortex stretching, respectively. To further explore the vortical motions under the presence of the APG, the velocity–vorticity correlations and their spanwise co-spectra are shown in figure 7. Figure 7(a,b) shows the profiles of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ . The zero-crossing location arises at $y^{+}\approx 8$ in the profile of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ , and $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ passes through a positive peak at $y^{+}\approx 5$ and a negative peak at $y^{+}\approx 18$ . The positive $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ near the wall can be interpreted physically as the vertical advection of the sublayer streaks, and the negative $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ in the buffer region indicates the outward motions of the hairpin vortex heads (Klewicki, Murray & Falco Reference Klewicki, Murray and Falco1994; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ has positive values for all wall-normal locations with a positive peak at $y^{+}\approx 10$ . The positive values of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ correspond physically to a simultaneous collapse of two adjacent hairpin vortex legs resulted from the stretching of hairpin vortex (Eyink Reference Eyink2008; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). The peak locations of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ are similar, but the magnitude of the peaks for the APG TBL is greater than that for the ZPG TBL, indicating that the near-wall vortical motions are enhanced in the APG TBL. This observation accords with the increase in near-wall energy of the vorticities (figure 6) and in the r.m.s. values of the swirling strength in the APG TBL (Lee et al. Reference Lee, Lee, Lee and Sung2010).

Figure 7.

Profiles of (a) $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and (b) $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ . Pre-multiplied spanwise co-spectra of (c) $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and (d) $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ .

The two velocity–vorticity correlations can be expressed as the pre-multiplied spanwise co-spectra of $\langle v\unicode[STIX]{x1D714}_{z}\rangle (k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}})$ and of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle (k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}})$ by using the Fourier transform,

(3.1a,b ) $$\begin{eqnarray}\langle v\unicode[STIX]{x1D714}_{z}\rangle =\int _{0}^{\infty }k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}\,\text{d}\ln k_{z},\quad \langle -w\unicode[STIX]{x1D714}_{y}\rangle =\int _{0}^{\infty }k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}\,\text{d}\ln k_{z}.\end{eqnarray}$$

Hence, the co-spectra represent the contributions from different spanwise wavelengths ( $\unicode[STIX]{x1D706}_{z}$ ). Figure 7(c,d) shows the pre-multiplied spanwise co-spectra $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ and $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ , where the small scales $(\unicode[STIX]{x1D706}_{z}^{+}<400)$ below $y^{+}=100$ are dominant in both co-spectra. The values of the positive and negative peaks of $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ are greater in the APG TBL, although the small-scale energy of $k_{z}\unicode[STIX]{x1D719}_{uu}$ near the wall is attenuated (figure 5 c). This implies that the AM influences of the LSMs on the vortical motions are related to the increase in the intensity of $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ in the near-wall region. In addition, the positive values of the large scales $(\unicode[STIX]{x1D706}_{z}^{+}\geqslant 400)$ of $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ increase around $60<y^{+}<300$ , and the negative values of those emerge near the boundary edge (figure 7 a). The enhanced positive values of $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ by the LSMs contribute to the increase in the skin friction induced by $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ (§ 3.3). The vortical motions of the large-scale $v\unicode[STIX]{x1D714}_{z}$ in the outer region are different from those of the small-scale $v\unicode[STIX]{x1D714}_{z}$ in the inner region. The physical interpretation of these results can be found in § 3.5.

Similar to $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ , the value of the positive peak of $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ is significantly greater in the APG TBL. Near the boundary edge, a negative value of $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ is observed above $\unicode[STIX]{x1D706}_{z}^{+}=400$ . Furthermore, the contribution of the large scales $(\unicode[STIX]{x1D706}_{z}^{+}\geqslant 400)$ to $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ is greater than that of the ZPG TBL. The value of the LSMs of $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ increases near the wall, and their contribution to $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ is extended to the viscous sublayer ( $y^{+}<5$ ), representing that the superposition effects of the LSMs on the vortical motions of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ are enhanced in the APG TBL. In particular, the large scales of $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ are remarkable near $y^{+}=200$ and observed up to $y^{+}=240$ .

The large scales above $y^{+}=100$ in the pre-multiplied streamwise co-spectra of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle (k_{x}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}})$ emerge and increase as the increase in the Reynolds number (Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014), and those contribute to the increase in the large-scale accelerating motions $(\unicode[STIX]{x2202}\langle -uv\rangle /\unicode[STIX]{x2202}y>0)$ at higher Reynolds number. According to the scaling of Tennekes & Lumley (Reference Tennekes and Lumley1972), $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ can be expressed as $u^{\ast }(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)(\unicode[STIX]{x2202}l/\unicode[STIX]{x2202}y)$ , where $u^{\ast }$ and $l$ indicate the characteristic velocity and length scales and is associated with the change of eddy size ( $\unicode[STIX]{x2202}l/\unicode[STIX]{x2202}y$ ) with the vorticity of order $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y$ . When the vortices are stretched, the energy of the mean flow is transferred to eddies, and then the velocity fluctuations perpendicular to the vorticity vector become strong. Hence, enhanced large scales over $y^{+}=100$ in $k_{x}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ can be interpreted as the amplitude modulation (change of scale effect) of the small scales by the LSMs (Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016b ). The presence of the APG results in the increase in the AM influences of the LSMs on the small scales (Harun et al. Reference Harun, Monty, Mathis and Marusic2013). Thus, the increase in the large scales of $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ near $y^{+}=200$ could be attributed to the increase in the AM influences of the LSMs on the small scales.

3.3 Superposition effects of the large scales on the skin friction

In this section, we investigate the large-scale influences on the skin friction by quantifying the contributions of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ . Applying triple integration to the spanwise component of the mean vorticity equation, $C_{f}$ can be decomposed into five terms as follows (Yoon et al. Reference Yoon, Ahn, Hwang and Sung2016a ):

(3.2) $$\begin{eqnarray}\displaystyle C_{f} & = & \displaystyle \underbrace{2\int _{0}^{1}\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\langle v\unicode[STIX]{x1D714}_{z}\rangle }{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}\,\text{d}\frac{y}{\unicode[STIX]{x1D6FF}}}_{C_{f,1}}+\underbrace{2\int _{0}^{1}\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\langle -w\unicode[STIX]{x1D714}_{y}\rangle }{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}\,\text{d}\frac{y}{\unicode[STIX]{x1D6FF}}}_{C_{f,2}}\nonumber\\ \displaystyle & & \displaystyle +\underbrace{\frac{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}}{U_{\infty }^{2}}\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{z}}{\unicode[STIX]{x2202}y}\right|_{y=0}-\frac{2\unicode[STIX]{x1D708}}{U_{\infty }^{2}}\int _{0}^{1}\unicode[STIX]{x1D6FA}_{z}\,\text{d}\frac{y}{\unicode[STIX]{x1D6FF}}+\int _{0}^{1}\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)^{2}\frac{\langle I_{x}\rangle }{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}^{2}}\,\text{d}\frac{y}{\unicode[STIX]{x1D6FF}}}_{C_{f,others}},\end{eqnarray}$$

where $\langle I_{x}\rangle =\unicode[STIX]{x2202}(U\unicode[STIX]{x1D6FA}_{z}+\langle u\unicode[STIX]{x1D714}_{z}\rangle -\langle w\unicode[STIX]{x1D714}_{x}\rangle )/\unicode[STIX]{x2202}x+\unicode[STIX]{x2202}(V\unicode[STIX]{x1D6FA}_{z})/\unicode[STIX]{x2202}y-\unicode[STIX]{x1D708}\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FA}_{z}/\unicode[STIX]{x2202}x^{2}$ is a streamwise-inhomogeneous term. The first and second terms ( $C_{f,1}$ and $C_{f,2}$ ) are the contributions of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ to the skin friction, respectively. The third and fourth terms are the contributions of the molecular diffusion at the wall and the molecular transfer due to the mean vorticity, respectively. The fifth term is the contribution of the spatial development in the streamwise direction. Note that $C_{f,others}$ in (3.2) is the sum of the last three terms.

Figure 8 shows the decomposition of the skin friction coefficient according to (3.2). We found that $C_{f,1}$ and $C_{f,2}$ constitute the majority of the total $C_{f}$ . Accordingly, we focused on $C_{f,1}$ and $C_{f,2}$ in the present study. $C_{f,1}$ contributes to the decrease in the total skin friction ( $C_{f,total}$ ), while $C_{f,2}$ increases $C_{f,total}$ . The negative contribution of $C_{f,1}$ is lower in the APG TBL, whereas the positive contribution of $C_{f,2}$ is higher. This variation in $C_{f,1}$ and $C_{f,2}$ increases the total $C_{f}$ , but the total $C_{f}$ in the APG TBL is approximately 28 % lower than that in the ZPG TBL due to the negative contribution of $C_{f,others}$ . The third term $(\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FF}/U_{\infty }^{2})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{z}/\unicode[STIX]{x2202}y)|_{y=0}$ is the main contributor to the negative value of $C_{f,others}$ . At the wall, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{z}/\unicode[STIX]{x2202}y$   ( $=-\unicode[STIX]{x2202}^{2}U/\unicode[STIX]{x2202}y^{2}$ ) is negative in the APG TBL due to the inflection point of the mean velocity profile close to the wall.

Figure 8.

Decomposition of the skin friction coefficient.

To further explore the contributions of turbulent motions to $C_{f,1}$ and $C_{f,2}$ , we examined the pre-multiplied spanwise co-spectra of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ with the decomposition method of Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016a ). The two velocity–vorticity correlations in (3.2) are expressed in terms of the pre-multiplied spanwise co-spectra $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ and $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ as shown in  (3.1). Thus, $C_{f,1}$ and $C_{f,2}$ in (3.2) can be expressed as the spanwise co-spectra $\unicode[STIX]{x1D719}_{f,1}$ and $\unicode[STIX]{x1D719}_{f,2}$ ,

(3.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle C_{f,1}=\int _{-\infty }^{0}\int _{0}^{\infty }yk_{z}\underbrace{2\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}}{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}}_{=\,\unicode[STIX]{x1D719}_{f,1}(k_{z},y)}\,\text{d}\ln k_{z}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}},\\ \displaystyle C_{f,2}=\int _{-\infty }^{0}\int _{0}^{\infty }yk_{z}\underbrace{2\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}}{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}}_{=\,\unicode[STIX]{x1D719}_{f,2}(k_{z},y)}\,\text{d}\ln k_{z}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}}.\end{array}\!\!\right\}\end{eqnarray}$$

In addition, the difference in $yk_{z}\unicode[STIX]{x1D719}_{f,1}$ between the APG and ZPG TBLs can be calculated as $yk_{z}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{f,1}=yk_{z}\unicode[STIX]{x1D719}_{f,1}|_{APG}-yk_{z}\unicode[STIX]{x1D719}_{f,1}|_{ZPG}$ , and $yk_{z}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{f,2}=yk_{z}\unicode[STIX]{x1D719}_{f,2}|_{APG}-yk_{z}\unicode[STIX]{x1D719}_{f,2}|_{ZPG}$ for $C_{f,2}$ .

Figure 9(ai,bi) shows the pre-multiplied spanwise co-spectra $yk_{z}\unicode[STIX]{x1D719}_{f,1}$ and $yk_{z}\unicode[STIX]{x1D719}_{f,2}$ in the $\unicode[STIX]{x1D706}_{z}$ – $y$ plane on a logarithmic scale along the spanwise wavelength and the wall-normal location. Hence, these spectra show the local contribution of a given spanwise length scale to the skin friction. As shown in figure 7, the small scales are dominant in $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ and $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ below $y^{+}=100$ . However, the high value of the small scales is not connected to their dominant contribution to the skin friction. In figure 9(ai), the positive contribution of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ to the skin friction near the wall ( $y^{+}\approx 6$ ) is significantly reduced compared to $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ , while the contribution of the outer large scales to the skin friction increases. The negative values of $yk_{z}\unicode[STIX]{x1D719}_{f,1}$ carried by the small scales with $\unicode[STIX]{x1D706}_{z}^{+}\approx 90$ near $y^{+}=25$ and the large scales with $\unicode[STIX]{x1D706}_{z}^{+}\approx 600$ near $y^{+}=400$ induce the negative contribution to $C_{f,1}$ . For $yk_{z}\unicode[STIX]{x1D719}_{f,2}$ , the small scales with $\unicode[STIX]{x1D706}_{z}^{+}\approx 100$ near $y^{+}=15$ and the large scales with $\unicode[STIX]{x1D706}_{z}^{+}\approx 600$ near $y^{+}=100$ make dominant contribution to the increase in $C_{f,2}$ .

The negative value of $yk_{z}\unicode[STIX]{x1D719}_{f,1}$ at $\unicode[STIX]{x1D706}_{z}^{+}=90$ and $y^{+}=25$ and the positive value of $yk_{z}\unicode[STIX]{x1D719}_{f,2}$ at $\unicode[STIX]{x1D706}_{z}^{+}=100$ and $y^{+}=15$ are 21.0 % and 17.6 % lower than those in the ZPG TBL. As shown in figure 9(ai), the largest positive and negative values of $yk_{z}\unicode[STIX]{x1D719}_{f,1}$ are observed at $\unicode[STIX]{x1D706}_{z}^{+}=640$ and $y^{+}=160$ and at $\unicode[STIX]{x1D706}_{z}^{+}=615$ and $y^{+}=420$ , respectively, and their values are 10.8 and 2.0 times larger than those for the ZPG TBL. For $yk_{z}\unicode[STIX]{x1D719}_{f,2}$ , an outer peak with a positive value arises at $\unicode[STIX]{x1D706}_{z}^{+}=620$ and $y^{+}=150$ in the APG TBL, whereas no outer peak is observed in the ZPG TBL. The presence of the outer peak in $yk_{z}\unicode[STIX]{x1D719}_{f,2}$ results from the increase in the large scales and in the extension of the large-scale influences of $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ as shown in figure 7. The superposition effects of the large scales on the skin friction are stronger in the APG TBL, especially on the skin friction induced by $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ . Figure 9(aii,bii) shows the difference in $yk_{z}\unicode[STIX]{x1D719}_{f,1}$ and $yk_{z}\unicode[STIX]{x1D719}_{f,2}$ . A positive region at $\unicode[STIX]{x1D706}_{z}^{+}\approx 600$ in the outer region indicates that the large scales contribute to the increase in $C_{f,1}$ and $C_{f,2}$ . In particular, the positive region in figure 9(bii) is extended up to $y^{+}=20$ , representing that the superposition effects of the large scales on the skin friction are dominant in $C_{f,2}$ .

Figure 9.

Pre-multiplied spanwise co-spectra of the weighted velocity–vorticity correlations for (ai) $C_{f,1}$ and (bi) $C_{f,2}$ . (aii) $yk_{z}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{f,1}=yk_{z}\unicode[STIX]{x1D719}_{f,1}|_{APG}-yk_{z}\unicode[STIX]{x1D719}_{f,1}|_{ZPG}$ and (bii) $yk_{z}\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{f,2}=yk_{z}\unicode[STIX]{x1D719}_{f,2}|_{APG}-yk_{z}\unicode[STIX]{x1D719}_{f,2}|_{ZPG}$ . The black dashed line indicates the cutoff wavelength, $\unicode[STIX]{x1D706}_{z,c}^{+}=400$ .

To further explore the large- and small-scale influences of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ on the skin friction, $C_{f,1}$ and $C_{f,2}$ were decomposed into large- and small-scale contributions by integration of the enclosed area of each pre-multiplied spanwise co-spectrum in figure 9(ai,bi),

(3.4) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle C_{f,1}^{L}=\int _{-\infty }^{0}\int _{0}^{k_{z,c}}2y\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}}{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}\,\text{d}k_{z}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}},\\ \displaystyle C_{f,1}^{S}=\int _{-\infty }^{0}\int _{k_{z,c}}^{\infty }2y\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}}{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}\,\text{d}k_{z}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}},\end{array}\!\!\right\} & & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle C_{f,2}^{L}=\int _{-\infty }^{0}\int _{0}^{k_{z,c}}2y\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}}{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}\,\text{d}k_{z}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}},\\ \displaystyle C_{f,2}^{S}=\int _{-\infty }^{0}\int _{k_{z,c}}^{\infty }2y\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}}{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}\,\text{d}k_{z}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}},\end{array}\!\!\right\} & & \displaystyle\end{eqnarray}$$

where $k_{z,c}$ is a cutoff wavenumber $(k_{z,c}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z,c},\unicode[STIX]{x1D706}_{z,c}^{+}=400)$ . Figure 10 illustrates bar graphs representing the decomposition of $C_{f,1}$ and $C_{f,2}$ into large and small scales. The skin friction coefficient induced by $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ ( $C_{f,1}$ ) can be obtained as the sum of $C_{f,1}^{L}$ and $C_{f,1}^{S}$ , likewise for $C_{f,2}$ . $C_{f,1}$ mostly comes from the small scales, which are dominant in the buffer region and the logarithmic region, due to the cancellation of the positive and negative contributions of the large scales (figure 9 ai). In the APG TBL $,C_{f,1}^{L}$ has a positive value resulted from the enhanced large scales $(\unicode[STIX]{x1D706}_{z}^{+}\geqslant 400)$ , which have a positive value near $y^{+}=100$ (figure 9 ai). $C_{f,2}^{L}$ and $C_{f,2}^{S}$ are the main contributors to the positive values of $C_{f,2}$ in the inner and outer regions, respectively (figure 9 bi).

To measure how much each scale contributes to the difference in $C_{f,1}$ or $C_{f,2}$ between the APG TBL and ZPG TBL, $\unicode[STIX]{x0394}C_{f,i}^{L}$ and $\unicode[STIX]{x0394}C_{f,i}^{S}$ are calculated in percentage form and presented in figure 10,

(3.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x0394}C_{f,i}^{L}[\%]=\frac{C_{f,i}^{L}|_{APG}-C_{f,i}^{L}|_{ZPG}}{C_{f,i}|_{APG}-C_{f,i}|_{ZPG}}\times 100,\\ \displaystyle \unicode[STIX]{x0394}C_{f,i}^{S}[\%]=\frac{C_{f,i}^{S}|_{APG}-C_{f,i}^{S}|_{ZPG}}{C_{f,i}|_{APG}-C_{f,i}|_{ZPG}}\times 100\;(i=1,2).\end{array}\!\!\right\}\end{eqnarray}$$

The contributions of the large and small scales to the difference in $C_{f,1}$ between the APG and ZPG TBLs ( $\unicode[STIX]{x0394}C_{f,1}^{L}$ and $\unicode[STIX]{x0394}C_{f,1}^{S}$ ) are 41.8 % and 58.2 %, respectively. On the other hand, the large scales contribute dominantly to the difference in $C_{f,2}\;(\unicode[STIX]{x0394}C_{f,2}^{L}=80.2\,\%)$ , which shows the dominance of the superimposed large scales on the skin friction via $C_{f,2}$ .

Figure 10.

Decomposition of $C_{f,1}$ and $C_{f,2}$ into the large and small scales.

3.4 Conditional sampling of the large-scale motions

To examine the variation in the AM of the vortical motions with the strength of the LSMs, the procedures used in Ganapathisubramani et al. (Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012) and Hwang & Sung (Reference Hwang and Sung2017) were adopted to perform the conditional sampling of the amplitudes of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ . Figure 11 shows an example of the signals at $y^{+}=15$ . First, the large-scale streamwise velocity fluctuations ( $u_{L}$ ) were separated from the streamwise velocity fluctuations ( $u$ ) by using a long-wavelength-pass filter with $\unicode[STIX]{x1D706}_{z,c}^{+}=400$ . Figure 11(a,b) shows the signals for $u$ and $u_{L}$ , respectively. Second, fluctuating signals were divided into individual segments of $400\unicode[STIX]{x1D6FF}^{+}$ , which corresponds to the cutoff wavelength. The vertical dashed lines mark the boundaries of each segment. The circle inserted in figure 11(b) indicates the value of $u_{L}^{+}$ at the centre of each segment, which is the representative of each segment. The bin size of $u_{L}^{+}$ in the conditional sampling of the LSMs was set to 0.2 (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012; Hwang & Sung Reference Hwang and Sung2017). Third, the mean values of all signals were averaged over each segment as functions of $u_{L}^{+}$ and $y$ . Instantaneous fluctuating signals of $v\unicode[STIX]{x1D714}_{z}$ and $-w\unicode[STIX]{x1D714}_{y}$ are shown in figure 11(c,d). Here, the inserted numbers are the r.m.s. values of the velocity–vorticity correlations for each segment. The r.m.s. values of $v\unicode[STIX]{x1D714}_{z}$ and $-w\unicode[STIX]{x1D714}_{y}$ are lower under the negative- $u_{L}$ events than under the positive- $u_{L}$ events, representing that the LSMs could modulate the amplitude of the velocity–vorticity correlations in the near-wall region.

Figure 11.

Signals for (a) the streamwise velocity fluctuations, (b) the large-scale streamwise velocity fluctuations ( $u_{L}$ ) and the velocity–vorticity correlations of (c) the wall-normal velocity and the spanwise vorticity fluctuations $(v\unicode[STIX]{x1D714}_{z})$ , and of (d) the spanwise velocity and the wall-normal vorticity fluctuations $(-w\unicode[STIX]{x1D714}_{y})$ at $y^{+}=15$ . The vertical dashed lines show the segment boundaries. The circles in (b) indicate representative values of $u_{L}^{+}$ in each segment, and the numbers in (c,d) indicate the r.m.s. values averaged in each segment.

First of all, we examine the population of $u_{L}$ by determining the probability density function (PDF) of $u_{L}$ across the $y$ direction, $N(u_{L},y)/\int N(u_{L},y)\,\text{d}u_{L}$ , where $N(u_{L},y)$ is the number of samples at given $u_{L}$ and $y$ . Figure 12(a,b) displays the PDFs of $u_{L}$ in the APG and ZPG TBLs, respectively. Near the wall and the boundary layer edge, high probability values are evident in the vicinity of $u_{L}^{+}=0$ , which indicates that the distribution of $u_{L}$ is narrow. Small-scale fluctuations are dominant in the near-wall region and near the boundary layer edge. The PDFs of $u_{L}$ for both TBLs contain a peak in the vicinity of $u_{L}^{+}=0$ and decrease with increasing $|u_{L}^{+}|$ . For the APG TBL, the PDF of $u_{L}$ is widely distributed with a relatively higher magnitude of $|u_{L}^{+}|$ . As shown in figure 12(a,b), strong LSMs with $|u_{L}^{+}|=6$ are observed only in the APG TBL. These are consistent with the isosurfaces of $u$ in figure 2 and the pre-multiplied spanwise energy spectra of $u$ in figure 5. In addition, the population of $u_{L}$ near the wall is wider in the APG TBL, indicating that the footprints of the LSMs are dominant in the APG TBL.

Figure 12(c) shows the averaged PDF of $u_{L}$ at six wall-normal locations: $y^{+}=15$ , 30, 55, 100, 200 and 400. The PDFs of $u_{L}$ for both TBLs are almost symmetric with respect to $u_{L}^{+}=0$ . The vertical dashed lines in figure 12(c) mark $u_{L}^{+}=\pm 2$ , where the PDF of $u_{L}$ is half of the value at $u_{L}^{+}=0$ . In particular, the PDFs of $u_{L}$ cross at $u_{L}^{+}=\pm \,2$ , indicating that intense negative- and positive- $u_{L}$ events ( $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ ) frequently occur in the APG TBL whereas the population of relatively weak $u_{L}$ events $(-2<u_{L}^{+}<+2)$ is reduced. The higher values of the PDFs for $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ are the reminiscent of the increase in the outer large-scale energy in the pre-multiplied spanwise energy spectra of $u$ (figure 5).

Figure 12.

The PDFs of $u_{L}$ with respect to $y^{+}$ and strength of $u_{L}$ in (a) the APG TBL and (b) the ZPG TBL. (c) Averaged profiles of the population of $u_{L}$ . Vertical dashed lines indicate $u_{L}^{+}=\pm 2$ .

3.5 The influence of the large scales on the skin friction

To assess the influences of the LSMs on $C_{f,1}$ and $C_{f,2}$ , $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ were conditionally averaged as functions of $u_{L}$ and $y$ ,

(3.7a,b ) $$\begin{eqnarray}\langle v\unicode[STIX]{x1D714}_{z}|_{u_{L}}\rangle =\left\langle \sum v\unicode[STIX]{x1D714}_{z}|_{u_{L}}\right\rangle \bigg/N_{u_{L}}(u_{L},y),\quad \langle -w\unicode[STIX]{x1D714}_{y}|_{u_{L}}\rangle =\left\langle \sum -w\unicode[STIX]{x1D714}_{y}|_{u_{L}}\right\rangle \bigg/N_{u_{L}}(u_{L},y).\end{eqnarray}$$

To enhance the convergence of the conditional average, we discarded the data for any bin of $u_{L}^{+}$ if the number of samples in that bin did not exceed a minimum standard. This was set as 750 samples per bin for the APG TBL and as 600 samples per bin for the ZPG TBL. The difference between the minimum standards of the APG and ZPG TBLs arises from the difference between their number of grid points in $x^{\prime }$ . The contributions of the modulated vortical motions to the skin frictions $\langle C_{f,1}|_{u_{L}}\rangle$ and $\langle C_{f,2}|_{u_{L}}\rangle$ were obtained through the integration of (3.2) and (3.7) on a semi-log scale,

(3.8) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \langle C_{f,1}|_{u_{L}}\rangle =\int _{0}^{1}\underbrace{2y\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\langle v\unicode[STIX]{x1D714}_{z}|_{u_{L}}\rangle }{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}}_{\unicode[STIX]{x1D701}_{f,1}(u_{L},y)}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}},\\ \displaystyle \langle C_{f,2}|_{u_{L}}\rangle =\int _{0}^{1}\underbrace{2y\left(1-\frac{y}{\unicode[STIX]{x1D6FF}}\right)\frac{\langle -w\unicode[STIX]{x1D714}_{y}|_{u_{L}}\rangle }{U_{\infty }^{2}/\unicode[STIX]{x1D6FF}}}_{\unicode[STIX]{x1D701}_{f,2}(u_{L},y)}\,\text{d}\ln \frac{y}{\unicode[STIX]{x1D6FF}}.\end{array}\!\!\right\}\end{eqnarray}$$

Hence, the summation of $\langle C_{f,i}|_{u_{L}}\rangle$ over $u_{L}$ is equal to $C_{f,i}$ in figure 8, $\sum _{u_{L}}\langle C_{f,i}|_{u_{L}}\rangle =C_{f,i}$ ( $i=1$ , 2).

Figure 13.

Two-dimensional contour maps of (a) $\unicode[STIX]{x1D701}_{f,1}$ and (b) $\unicode[STIX]{x1D701}_{f,2}$ on the semi-log scale for the wall-normal direction. Profiles of (c) $\sum \langle C_{f,1}|_{u_{L}}\rangle$ and (d) $\sum \langle C_{f,2}|_{u_{L}}\rangle$ . Vertical dashed lines indicate $u_{L}^{+}=\pm 2$ .

Figure 13(a,b) displays the two-dimensional contour plots for $\unicode[STIX]{x1D701}_{f,1}$ and $\unicode[STIX]{x1D701}_{f,2}$ , which directly represent the contributions of the advective transport and the vortex stretching to the skin friction with the strength of $u_{L}$ , respectively. In these plots of $\unicode[STIX]{x1D701}_{f,1}$ , there are three distinct regions with respect to the wall-normal location. First, positive values of $\unicode[STIX]{x1D701}_{f,1}$ are observed in the vicinity of $u_{L}^{+}=0$ near the wall ( $y^{+}<10$ ), which contribute to the increase in $C_{f,1}$ . The first quadrant motions of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ ( $v>0$ and $\unicode[STIX]{x1D714}_{z}>0$ ), induced by lifting sublayer streaks, are dominant in this region. Second, negative values of $\unicode[STIX]{x1D701}_{f,1}$ contribute to the decrease in $C_{f,1}$ between $y^{+}=10$ and 80. The second quadrant motions of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ ( $v>0$ and $\unicode[STIX]{x1D714}_{z}<0$ ), induced by the lifting of the hairpin vortex heads, are dominant. Third, above $y^{+}=80$ for $u_{L}^{+}\leqslant -2$ , positive and negative values of $\unicode[STIX]{x1D701}_{f,1}$ are dominant under the negative- and positive- $u_{L}$ events, respectively. Figure 13(b,d), positive values of $\unicode[STIX]{x1D701}_{f,2}$ are dominant for all $u_{L}$ below $y^{+}=100$ , which is due to the simultaneous collapse of the two adjacent hairpin vortex legs resulted from the vortex stretching. Above that, negative and positive values of $\unicode[STIX]{x1D701}_{f,2}$ are observed for $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ , respectively. As shown in the pre-multiplied spanwise co-spectra $k_{z}\unicode[STIX]{x1D719}_{v\unicode[STIX]{x1D714}_{z}}$ and $k_{z}\unicode[STIX]{x1D719}_{-w\unicode[STIX]{x1D714}_{y}}$ in figure 7, the small/large scales of $\unicode[STIX]{x1D701}_{f,1}$ and $\unicode[STIX]{x1D701}_{f,2}$ are expected to be remarkable near the wall (below $y^{+}=10$ )/away from the wall (above $y^{+}=80$ ). $\unicode[STIX]{x1D701}_{f,1}$ and $\unicode[STIX]{x1D701}_{f,2}$ are biased toward positive $u_{L}$ , which means that the positive- $u_{L}$ events enhance the vortical motions and make dominant contribution to the skin friction.

The positive and negative values of $\unicode[STIX]{x1D701}_{f,1}$ below $y^{+}$ $=$ 80 are lower in the APG TBL than in the ZPG TBL, especially near $u_{L}^{+}=0$ . The near-wall streaks and advective motions of the hairpin vortices for weak $u_{L}$ events $(-2<u_{L}^{+}<+2)$ are attenuated in the APG TBL. This is consistent with the finding of Lee & Sung (Reference Lee and Sung2009), showing a diminution of the sublayer streaks by flow visualization of $u$ at $y^{+}=5.5$ . However, the influences of $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ on $\unicode[STIX]{x1D701}_{f,1}$ are substantially extended above $y^{+}=80$ . The intense negative- and positive- $u_{L}$ events amplify the outer vortical motions of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ , which results in the increase and decrease in $C_{f,1}$ , respectively. Below $y^{+}=100$ , positive values of $\unicode[STIX]{x1D701}_{f,2}$ are reduced for weak $u_{L}$ events, especially in the vicinity of $y^{+}=20$ . Above $y^{+}=20$ , $\unicode[STIX]{x1D701}_{f,2}$ is significantly enhanced for $u_{L}^{+}\geqslant +2$ . The enhanced intense positive- $u_{L}$ events in the outer region induce the strong vortical motions of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ close to the wall and thus result in the increase in $C_{f,2}$ .

Figure 13(c,d) shows cumulative distributions, $\sum \langle C_{f,i}|_{u_{L}}\rangle =\sum _{u_{L}^{+}=-\infty }^{u_{L}^{+}}\langle C_{f,i}|_{u_{L}}\rangle (i=1,2),$ of the profiles of $\langle C_{f,1}|_{u_{L}}\rangle$ and $\langle C_{f,2}|_{u_{L}}\rangle$ . The velocity–vorticity correlation $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ under the negative- $u_{L}$ events enhances the skin friction, whereas under the positive- $u_{L}$ events it predominantly reduces the skin friction (figure 13 c). In the APG TBL, the zero crossing of $\sum \langle C_{f,1}|_{u_{L}}\rangle$ moves from $u_{L}^{+}=-1.4$ (ZPG TBL) to $u_{L}^{+}=0.3$ due to the positive contribution under the intense negative- $u_{L}$ events. The value of the cumulative distribution $\sum \langle C_{f,1}|_{u_{L}}\rangle$ at $u_{L}^{+}=-2$ and $u_{L}^{+}=+2$ of the APG TBL is 8.3 and 2.7 times larger than that of the ZPG TBL. The influences of $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ on $C_{f,1}$ increase in the APG TBL, especially for the negative- $u_{L}$ events. Contrary to $C_{f,1}$ , the contribution of the intense negative- $u_{L}$ events to $C_{f,2}$ is attenuated. The zero crossing of $\sum \langle C_{f,2}|_{u_{L}}\rangle$ slightly moves from $u_{L}^{+}=-2.3$ (ZPG TBL) to $u_{L}^{+}=-1.9$ . The value of $\sum \langle C_{f,2}|_{u_{L}}\rangle$ below $u_{L}^{+}=-2$ is almost zero, representing that the influences of the intense negative- $u_{L}$ events on $C_{f,2}$ are insignificant. For $u_{L}^{+}=+2$ , the value of $\sum \langle C_{f,2}|_{u_{L}}\rangle$ of the APG TBL is 1.75 times larger than that of the ZPG TBL. Thus, the enhanced LSMs in the outer region strongly modulate the vortical motions, leading to a significant contribution to the dependence of the skin friction on the LSMs. The influences of the negative- $u_{L}$ events on $C_{f,1}$ and of the positive- $u_{L}$ events on $C_{f,2}$ are much stronger in the APG TBL.

To further analyse the dominant motions contributed to $C_{f,1}$ under the intense negative- and positive- $u_{L}$ events, the weighted joint probability density functions (JPDFs) of $v$ and $\unicode[STIX]{x1D714}_{z}$ under $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ are shown in figure 14. Here, two wall-normal positions are chosen at $y^{+}=25$ and 300, where the former is the peak of negative $\unicode[STIX]{x1D701}_{f,1}$ in the vicinity of $u_{L}^{+}=0$ and the latter is the high value of $\unicode[STIX]{x1D701}_{f,1}$ under the intense- $u_{L}$ events. Hence, we can determine the quadrant contributions to $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ under the intense- $u_{L}$ events. Note that the weighted JPDFs of $w$ and $\unicode[STIX]{x1D714}_{y}$ are not displayed here, since the pair of two quadrant motions ( $w>0$ and $\unicode[STIX]{x1D714}_{y}<0$ ; $w<0$ and $\unicode[STIX]{x1D714}_{y}>0$ ) is an equally dominant contributor to the positive values of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ .

As shown in figure 14, the quadrant motions are different between the negative- and positive- $u_{L}$ events at the same wall-normal location. At $y^{+}=25$ , the first dominant motion is placed on the fourth quadrant of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ with contributions of approximately 31 %, and the second dominant motion is the first and second quadrants (28 %) for $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ , respectively (figure 14 a,c). The wider negative region in figure 13(a) arises from the vertical advection ( $v>0$ ) of the near-wall streaks with the negative and positive $\unicode[STIX]{x1D714}_{z}$ , which is enhanced under the intense- $u_{L}$ events. At $y^{+}=300$ , the quadrant motions for $u_{L}^{+}\leqslant -2$ and $u_{L}^{+}\geqslant +2$ are symmetric with respect to the transverse axis (figure 14 b,d). For $u_{L}^{+}\leqslant -2$ , the first and second quadrants are dominant, indicating the upward motions of the vortical structures (figure 14 b). On the contrary, the downward motions of the vortices are dominant under $u_{L}^{+}\geqslant +2$ in figure 14(d). These vortical motions are related to the outer ejection and sweep motions carried by the intense negative- and positive- $u_{L}$ events, respectively. In particular, the increase in the magnitude of the positive and negative $\unicode[STIX]{x1D701}_{f,1}$ in the outer region (figure 13 a) arises from the enhanced upwash and downwash motions of the positive $\unicode[STIX]{x1D714}_{z}$ (the first quadrant in figure 14 b and fourth quadrant in figure 14 d), respectively.

Figure 14.

Weighted JPDFs of $v$ and $\unicode[STIX]{x1D714}_{z}$ under (a,b) the intense negative- $u_{L}\;(u_{L}^{+}\leqslant -2)$ and (c,d) positive- $u_{L}$ events $(u_{L}^{+}\geqslant +2)$ at (a,c) $y^{+}=25$ and (b,d) $y^{+}=300$ . The red and blue lines indicate positive and negative values of $v\unicode[STIX]{x1D714}_{z}$ , respectively.