1. Introduction
The friction drag generated by intense turbulent wall-shear stresses exists in numerous engineering and environmental wall-bounded flows, leading to immense economic costs. For instance, a 1 % drag reduction for a long-range transport aircraft could increase the payload by 91 kg at the cruise condition (Meredith Reference Meredith1993; Fan & Li Reference Fan and Li2019). Therefore, the reduction of turbulent friction drag is of great importance for energy saving and commercial value. Strategies to reduce the turbulent friction drag have long been pursued in industrial designs, which require profound understanding on the generation mechanisms and contributing factors of the friction drag in wall-bounded turbulence.
By definition, the mean skin-friction drag is the mean shear stress on the wall surface. In wall-bounded turbulent flows, the near-wall turbulent events are pronouncedly impacted by the motions in the logarithmic and outer region, especially the large-scale motions (LSMs) and very-large-scale motions (VLSMs) by means of superposition and modulation (Hoyas & Jiménez Reference Hoyas and Jiménez2006; Mathis et al. Reference Mathis, Monty, Hutchins and Marusic2009). For this reason, it is reasonable that the generation of mean skin-friction drag is related to the flow motions and quantities across the whole layer. The first breakthrough towards this theme was made by Fukagata, Iwamoto & Kasagi (Reference Fukagata, Iwamoto and Kasagi2002), who derived a concise identity (referred to as FIK identity) of skin-friction coefficient 
$C_f$ by performing triple integration on the streamwise Reynolds-averaged Navier–Stokes (RANS) equation. For fully developed turbulent channel flows, on the premise of homogeneity in streamwise and spanwise directions, the identity is simplified as
\begin{equation} C_f=\underbrace{6\int_0^1(1-y)\langle-u'v'\rangle\,\mathrm{d} y}_{C_{f1,{FIK}}}+\underbrace{\dfrac{6}{Re_b}}_{C_{f2,{FIK}}}. \end{equation}
Herein, the streamwise, wall-normal and spanwise directions are denoted by 
$x$, 
$y$ and 
$z$, respectively, and the corresponding velocity components are 
$u$, 
$v$ and 
$w$. All quantities in (1.1) are normalized with 
$u_b^*$ and 
$h^*$, where 
$u_b^*$ is bulk velocity and 
$h^*$ is the half-channel height. Additionally, 
$Re_b= u_b^*h^*/\nu ^*$ is the bulk Reynolds number and 
$\nu ^*$ is kinematic viscosity. The superscript 
$^*$ denotes a dimensional variable. The angle bracket 
$\langle \rangle$ represents averaging over homogeneous directions (streamwise and spanwise directions) and time, and primed quantities refer to the fluctuations. In the FIK identity, 
$C_{f1,{FIK}}$ is the integral of the Reynolds stress weighted by 
$(1-y)$, which measures contribution from turbulent motions, and 
$C_{f2,{FIK}}$ is identical to the well-known laminar solution. Renard & Deck (Reference Renard and Deck2016) provided another identity to decompose the skin friction based on the mean streamwise kinetic-energy budget in an absolute reference frame (named as RD identity). In their framework, 
$C_f$ in turbulent channel flows is attributed to the molecular-viscous-dissipation contribution and the production of the turbulent kinetic energy.
The FIK and RD identity have been popularly utilized, and alternative forms and modifications have emerged gradually in the literature. Gomez, Flutet & Sagaut (Reference Gomez, Flutet and Sagaut2009) derived a compressible form of the FIK identity to investigate the compressibility effects on skin-friction drag generation. Bannier, Garnier & Sagaut (Reference Bannier, Garnier and Sagaut2015) extended the FIK identity to complex wall surfaces and investigated the drag-reduction mechanism of riblet-mounted surfaces. Kametani & Fukagata (Reference Kametani and Fukagata2011) and Kametani et al. (Reference Kametani, Fukagata, Örlü and Schlatter2015) investigated the turbulent boundary layer with wall blowing/suction and evaluated contributions of each term in the FIK identity. Ricco & Skote (Reference Ricco and Skote2022) proposed several alternative integral relations and discussed the feasibility of an arbitrary-fold multiple integral. Wenzel, Gibis & Kloker (Reference Wenzel, Gibis and Kloker2022) employed a twofold integral to obtain improved FIK-like decomposition identities of 
$C_f$ and 
$C_h$ (the heat-transfer coefficient). Analogously, Xu, Wang & Chen (Reference Xu, Wang and Chen2022) applied an FIK-like decomposition to investigate the streamwise evolutions of 
$C_f$ and 
$C_h$ in hypersonic transitional and turbulent boundary layers. Other applications and modifications of the FIK identity can be found from Mehdi & White (Reference Mehdi and White2011), Mehdi et al. (Reference Mehdi, Johansson, White and Naughton2013), Peet & Sagaut (Reference Peet and Sagaut2009) and Modesti et al. (Reference Modesti, Pirozzoli, Orlandi and Grasso2018), to name a few. When it comes to the RD identity, it gives an edge in interpreting skin-friction generation from the view of energy balance and has also been further developed in recent years. Wei (Reference Wei2018) verified it in three canonical turbulent wall-bounded flows using direct numerical simulation (DNS) and experimental data. Li et al. (Reference Li, Fan, Modesti and Cheng2019) and Fan, Li & Pirozzoli (Reference Fan, Li and Pirozzoli2019b), Fan et al. (Reference Fan, Li, Atzori, Pozuelo, Schlatter and Vinuesa2020) extended it to compressible turbulent channel flows, zero- and adverse-pressure-gradient boundary layers. Additionally, Fan et al. (Reference Fan, Atzori, Vinuesa, Gatti, Schlatter and Li2022) employed it to investigate the effects of uniform blowing and suction on the generation of skin friction over an airfoil.
It is noteworthy that Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016) proposed a velocity–vorticity-based method of skin-friction decomposition by performing triple integration on the mean spanwise vorticity transport equation. For turbulent channel flows, it is expressed as
\begin{align} C_f=\underbrace{\int_0^12(1-y)\langle v'\omega_z'\rangle\,\mathrm{d} y}_{C_{f1,{Yoon}}}+\underbrace{\int_0^12(1-y)\langle-w'\omega_y'\rangle\,\mathrm{d} y}_{C_{f2,{Yoon}}}+\underbrace{\left.\dfrac{1}{Re_b}\dfrac{\mathrm{d} \langle\omega_z\rangle}{\mathrm{d} y}\right|_{y=0}}_{C_{f3,{Yoon}}}\underbrace{-\dfrac{1}{Re_b} \int_0^12\langle\omega_z\rangle\,\mathrm{d} y}_{C_{f4,{Yoon}}}, \end{align}
where vorticities in streamwise, wall-normal and spanwise directions are denoted by 
$\omega _x$, 
$\omega _y$ and 
$\omega _z$. This identity provides a tool to understand the mechanism of skin-friction generation from the perspective of vortical motions, yielding contributions of advective vorticity transport (
$C_{f1,{Yoon}}$), vortex stretching (
$C_{f2,{Yoon}}$) by integrals of velocity-vorticity correlation, and the viscous effects of the mean vorticity (
$C_{f3,{Yoon}}$ and 
$C_{f4,{Yoon}}$). A relevant further application of this identity was carried out in adverse-pressure-gradient turbulent boundary layers to identify contributions from the large-scale velocity–vorticity correlated motions to the skin-friction drag (Yoon, Hwang & Sung Reference Yoon, Hwang and Sung2018).
Actually, one can perform a fourfold integration 
$\int _0^1\int _0^y\int _0^{y_3}\int _0^{y_2}\,\mathrm {d} y_1\,\mathrm {d} y_2\,\mathrm {d} y_3\,\mathrm {d} y$ (where 
$y$, 
$y_1$, 
$y_2$ and 
$y_3$ are all wall-normal distances) on the mean spanwise vorticity equation, to reduce Yoon's identity to the FIK identity (see Appendix A). However, it is argued that the triple integral applied in the derivation of the FIK identity has no clear physical interpretations, including a dimensional problem and a weight factor 
$(1-y)$ of Reynolds stress (Renard & Deck Reference Renard and Deck2016; Wenzel et al. Reference Wenzel, Gibis and Kloker2022). On one hand, the second integral multiplies a force (from the first integral) by length and this should have the dimension of an energy, rather than a velocity as stated by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002). On the other hand, the weight factor 
$(1-y)$ in 
$C_{f1,{FIK}}$ results from the mathematical derivation (the third integral) and has no clear explanation in terms of physical processes (Deck et al. Reference Deck, Renard, Laraufie and Weiss2014; Renard & Deck Reference Renard and Deck2016; Fan, Cheng & Li Reference Fan, Cheng and Li2019a; Li et al. Reference Li, Fan, Modesti and Cheng2019; Fan et al. Reference Fan, Li, Atzori, Pozuelo, Schlatter and Vinuesa2020). It magnifies the weight of flow information in the near-wall region and attenuates that in the outer layer, which may require high spatial resolution and accuracy in the vicinity of wall-surface for both experimental measurements and numerical simulations (Xia, Zhang & Yang Reference Xia, Zhang and Yang2021). Moreover, Wenzel et al. (Reference Wenzel, Gibis and Kloker2022) declared that an arbitrary-fold integral would result in an FIK-like identity according to Cauchy's formula for repeated integration. Zhang & Xia (Reference Zhang and Xia2020) and Wenzel et al. (Reference Wenzel, Gibis and Kloker2022) pointed out that the twofold integral has a more intuitive physical interpretability than the threefold one on account of the absence of the weight factor 
$(1-y)$ in channel flows. In (1.2), the ill-defined weight factor 
$(1-y)$ also appears in the first two terms, which is consistent with the FIK identity due to the analogous integration performed. Therefore, in this study, we give an alternative form of skin-friction drag decomposition based on the velocity–vorticity correlation. Although it resembles the identity proposed by Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016), to some extent, better physics-informed interpretation could be inferred in this form since it does not suffer from the weight factor 
$(1-y)$ in the integrand.
Additionally, the velocity–vorticity correlations appearing in (1.2), 
$\langle v'\omega _z'\rangle$ and 
$\langle -w'\omega _y'\rangle$, play important roles in the dynamics of vortical motions. Tennekes & Lumley (Reference Tennekes and Lumley1972) first uncovered that 
$\langle v'\omega _z'\rangle$ is associated with the vortex transport and 
$\langle -w'\omega _y'\rangle$ with the vortex stretching. Simultaneously, 
$\langle -w'\omega _y'\rangle$ is hypothesized to be the key factor driving the energy cascade and angular momentum transferring from large eddies to small ones (Tennekes & Lumley Reference Tennekes and Lumley1972; Priyadarshana et al. Reference Priyadarshana, Klewicki, Treat and Foss2007). In incompressible turbulent flows, there exists a relationship to combine 
$\langle v'\omega _z'\rangle$ and 
$\langle -w'\omega _y'\rangle$ into the turbulent inertia (TI), i.e.
\begin{equation} \dfrac{\mathrm{d}\langle-u'v'\rangle}{\mathrm{d} y}=\langle v'\omega_z'\rangle+\langle-w'\omega_y'\rangle. \end{equation}
TI is the gradient of the Reynolds shear stress in the wall-normal direction. It derives from the time rate of change of momentum (i.e. inertial) part of the RANS equation (Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). In the mean momentum equation, TI acts as the momentum source in the near-wall region and the momentum sink in the outer layer. Equation (1.3) connects the mean wall-normal transport of Reynolds shear stress with the vorticity field (Klewicki Reference Klewicki1989) and provides a way to understand the generation of TI (Guerrero, Lambert & Chin Reference Guerrero, Lambert and Chin2022). Klewicki (Reference Klewicki2013) evidenced the coexistence of dynamical and vortical processes, in which the wall-ward momentum transport occur simultaneously with the outward vorticity transport (dominated by mean spanwise vorticity 
$\langle \omega _z\rangle$) under the same physical mechanisms. Three-dimensional vortical structures populate through the whole flow field especially in the near-wall region, which indicates that the generation of skin-friction drag is closely related to the vortical motions. In this sense, Yoon's identity in (1.2) indeed provides an effective idea to investigate their relationship.
Another objective in this study is to investigate the Reynolds number effects on the skin-friction-drag generation from the perspective of vortex dynamics. As the Reynolds number increases, the outer scales (characterized by 
$h^*$ and 
$u_b^*$) are increasingly larger than the inner scales (characterized by viscous length scale 
$\delta _\nu ^*$ and friction velocity 
$u_\tau ^*$) (Priyadarshana et al. Reference Priyadarshana, Klewicki, Treat and Foss2007). The contribution of the near-wall small-scale structures to the skin-friction generation is attenuated, while that of the large-scale motions is enhanced due to the more intense amplitude modulation and energy superposition effects (Hwang Reference Hwang2013). Fan et al. (Reference Fan, Cheng and Li2019a) studied the Reynolds number effects on the terms in the FIK and RD identity in turbulent channel flows up to 
$Re_\tau =5200$ (where 
$Re_\tau =u_\tau ^*h^*/\nu ^*$ is the friction Reynolds number). Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014) demonstrated that the streamwise pre-multiplied co-spectra 
$\phi _{v'\omega _z'}$ scaled in inner units exhibits invariance with 
$Re_\tau$, whereas 
$\phi _{-w'\omega _y'}$ shows non-negligible variations with 
$Re_\tau$.
Based on the aforementioned issues, we are motivated to give a skin-friction drag decomposition method with more intuitive physical interpretability using velocity–vorticity correlations, and quantify the contribution as well as investigate the Reynolds number effects of each individual term. The rest of this paper is organized as follows. Mathematical derivation and physical explanations are introduced in § 2. Details of direct numerical simulations of five turbulent channel cases are described in § 3. In § 4, statistical properties of the decomposed components, including absolute contribution values and proportions, local large- and small-scale motions contributions, and quadrant characteristics, are investigated. Particular attention is paid to the effects of Reynolds number on the velocity–vorticity integral terms. Finally, concluding remarks are summarized in § 5.
2. Mathematical approach
The equation of the streamwise mean momentum balance for fully developed incompressible turbulent channel flows is expressed as
\begin{equation} \left.\dfrac{\mathrm{d}\langle-u'v'\rangle}{\mathrm{d} y}+\dfrac{1}{Re_b}\dfrac{\mathrm{d}^2\langle u\rangle}{\mathrm{d} y^2}+\dfrac{1}{Re_b}\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}\right|_{y=0}=0. \end{equation}
Hereafter, variables without the superscript 
$^*$ represent non-dimensional quantities normalized by bulk mean velocity 
$u_b^*$ and half-channel height 
$h^*$, and variables with the superscript ‘
$+$’ represent non-dimensional quantities normalized by friction velocity 
$u_\tau ^*$ and viscous length scale 
$\delta _\nu ^*$, unless otherwise mentioned. With 
$\langle v\omega _z\rangle =\langle v'\omega _z'\rangle$ and 
$\langle -w\omega _y\rangle =\langle -w'\omega _y'\rangle$, (1.3) can be re-expressed as
\begin{equation} \dfrac{\mathrm{d}\langle-u'v'\rangle}{\mathrm{d} y}=\langle v\omega_z\rangle+\langle-w\omega_y\rangle. \end{equation}
Substituting (2.2) into (2.1) and integrating in the wall-normal direction from 0 (wall surface) to an arbitrary height 
$y\in (0,1]$, we get
\begin{equation} \int_0^y\langle v\omega_z\rangle\,\mathrm{d} y+\int_0^y\langle-w\omega_y\rangle\,\mathrm{d} y=-\dfrac{1}{Re_b}\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}+\left.\dfrac{1}{Re_b}\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}\right|_{y=0}-\left.\dfrac{1}{Re_b}y\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}\right|_{y=0}. \end{equation} The skin-friction coefficient is defined as 
$C_f=\langle \tau _{w}^*\rangle /(\frac {1}{2}\rho ^*u_b^{*2})$, where 
$\tau _{w}^*=\mu ^*({\partial u^*}/{\partial y^*})|_{y=0}$ is the wall-shear stress, 
$\mu ^*$ is the dynamic viscosity and 
$\rho ^*$ is the density. It has an alternative form 
$C_f=2({\mathrm {d}\langle u\rangle }/{\mathrm {d} y})|_{y=0}/Re_b$. Substituting it into (2.3), we have
\begin{equation} C_f=\underbrace{2\int_0^y\langle v\omega_z\rangle\,\mathrm{d} y}_{C_{f1}}+\underbrace{2\int_0^y\langle-w\omega_y\rangle\,\mathrm{d} y}_{C_{f2}}+\underbrace{\dfrac{2}{Re_b}\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}}_{C_{f3}}+\underbrace{\left.\dfrac{2}{Re_b}y\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}\right|_{y=0}}_{C_{f4}}. \end{equation}
The first two terms 
$C_{f1}$ and 
$C_{f2}$ measure the contribution from advective vorticity transport and vortex stretching throughout the height interval 
$[0,y]$ uniformly. Additionally, 
$C_{f3}$ represents the non-dimensional mean streamwise shear stress at height 
$y$, i.e. 
$\langle \tau ^*\rangle (y)/(\frac {1}{2}\rho ^*u_b^{*2})$, and 
$C_{f4}$ is simply 
$yC_f$ that will be discussed later. Equation (2.4) can be generalized to the cases of turbulent pipe and boundary layer flows, which are introduced in Appendices C and D, respectively.
Several remarks about (2.4) need to be elucidated. Initially, the first two terms on the right-hand side, 
$C_{f1}$ and 
$C_{f2}$, are exactly equivalent to the Reynolds shear stress (two times) at height 
$y$. Decomposing the Reynolds stress into 
$C_{f1}$ and 
$C_{f2}$ provides a way to understand how the vortical motions fundamentally contribute to the skin friction. For the integral in 
$C_{f1}$ and 
$C_{f2}$, the upper bound is not necessarily 1 (Mehdi et al. Reference Mehdi, Johansson, White and Naughton2013) and even the lower bound is not necessarily 0 (Xia et al. Reference Xia, Zhang and Yang2021). In this study, integration from the wall surface to an arbitrary height is applied to weight all contributions in this interval.
Additionally, by multiplying 
$(1-y)$ and integrating in 
$y$ over 
$[0,1]$, (2.4) is transformed to the standard FIK identity (1.1) (see Appendix B), indicating that they are intrinsically consistent. The typical difference between (2.4) and other classic skin-friction coefficient identifiers, such as proposed by Mehdi et al. (Reference Mehdi, Johansson, White and Naughton2013), Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016) and Ricco & Skote (Reference Ricco and Skote2022), is the absence of integrand factor 
$(1-y)$, or equivalently, one less performance of integration over 
$[0,1]$ (Ricco & Skote Reference Ricco and Skote2022; Wenzel et al. Reference Wenzel, Gibis and Kloker2022). As mentioned in the introduction, these FIK-like identities suffer from the ill-defined physical interpretation caused by the factor 
$(1-y)$ in the integrand. Here, (2.4) without 
$(1-y)$ evaluates contributions at each height uniformly.
One may be confused by 
$C_{f4}$ in (2.4), as it is simply 
$yC_f$. An explicit expression could be obtained if we move 
$C_{f4}$ to the left-hand side of (2.4). However, this will cause singularity at 
$y = 1$. According to the derivation in Appendix A, we find an equality relation: 
$({\mathrm {d}\langle u\rangle }/{\mathrm {d} y})|_{y=0}=({\mathrm {d}\langle \omega _z\rangle }/{\mathrm {d} y})|_{y=0}$. The last term in (2.4) can be rewritten as 
$C_{f4}=({2}/{Re_b})y({\mathrm {d}\langle \omega _z\rangle }/{\mathrm {d} y})|_{y=0}$. In this sense, 
$C_{f4}$ can be explained as the weighted mean vorticity gradient at the wall. The weight factor 
$y$ is the consequence of wall-normal integration to an arbitrary height. Therefore, 
$C_{f4}$ could also be interpreted as the accumulative effect of the wall-surface mean vorticity gradient below 
$y$. The expression 
$C_{f4}=({2}/{Re_b})y({\mathrm {d}\langle \omega _z\rangle }/{\mathrm {d} y})|_{y=0}$ is akin to 
$C_{f3,{Yoon}}=({1}/{Re_b})({\mathrm {d}\langle \omega _z\rangle }/{\mathrm {d} y})|_{y=0}$ in the study by Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016). They interpreted 
$C_{f3,{Yoon}}$ as ‘the viscous effects of the mean vorticity, i.e. the molecular diffusion at the wall’. Term 
$C_{f3,{Yoon}}$ is not weighted by 
$y$, as the outermost integral of the triple integration was performed from the wall to the central plane.
As a final note about (2.4), the derivation of (2.4) requires: (i) the homogeneity in the streamwise and spanwise directions; and (ii) the zero mean of 
$v$ and 
$w$ at each height. Therefore, (2.4) is unable to handle cases that do not meet these two requirements, for instance, turbulent flows with wall blowing/suction or riblet-mounted surfaces.
3. DNS database
We use the DNS database of turbulent channel flows built by Jiménez (Del Álamo & Jiménez Reference Del Álamo and Jiménez2003; Del Álamo et al. Reference Del Álamo, Jiménez, Zandonade and Moser2004; Hoyas & Jiménez Reference Hoyas and Jiménez2006). Five cases covering a relatively wide Reynolds number range (
$Re_\tau =186-2003$) are taken into account. Simulation parameters are summarized in table 1. It has been demonstrated that the scale of VLSMs is no larger than around 
$2h^*$ in the spanwise direction and 
$10h^*$ in the streamwise direction (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). Thus, the sizes of the computational domains are large enough to capture VLSMs and relatively smaller scale motions.
Table 1. Simulation parameters of the DNS database. Here, 
$N_x$, 
$N_z$ and 
$N_y$ are grid numbers in the streamwise, spanwise and wall-normal direction, respectively; 
$L_x^*$, 
$L_z^*$ and 
$L_y^*$ are sizes of the computation domain in three directions. The 
$\Delta y_{min}^+$ and 
$\Delta y_{max}^+$ are the finest and coarsest grid spacings in the wall-normal direction.
4. Results and discussion
Detailed descriptions and interpretations of the results are presented in this section. Profiles of 
$C_{f1}$ and 
$C_{f2}$ and their proportions based on (2.4) are given in § 4.1. Section 4.2 provides the spanwise co-spectra of velocity–vorticity correlations and corresponding scale-dependence of the contributing terms at different Reynolds numbers. Section 4.3 describes the quadrant decomposition of the velocity–vorticity correlations and their relations with the skin-friction drag generation.
4.1. Statistics of the decomposition
Figure 1(a) shows the variation of 
$C_f$ calculated by (2.4) as a function of 
$y$, compared with the reference ones directly calculated by 
$C_{f,{ref}}=\langle \tau _{w}^*\rangle /(\frac {1}{2}\rho ^*u_b^{*2})=2Re_\tau ^2/Re_b^2$. Theoretically, the 
$C_f$ calculated by (2.4) should remain constant regardless of the wall-normal location and match its reference value. Figure 1(b) shows the relative errors between 
$C_f$ and 
$C_{f,{ref}}$, 
$E_{C_f}(y)=[C_f(y)-C_{f,{ref}}]/C_{f,{ref}}$. It can be seen that the errors are limited within 
$-2.5\,\%\le E_{C_f}\le 1\,\%$. Overall, the relative errors are acceptable, which validates the skin-friction drag decomposition method in (2.4) and lays the foundation for the further discussions.
Figure 1. (a) Profiles of 
$C_f$ estimated by (2.4) and 
$C_{f,{ref}}$ (nearest dashed lines) directly calculated by 
$C_{f,{ref}}=2Re_\tau ^2/Re_b^2$. (b) Relative errors between 
$C_f$ and 
$C_{f,{ref}}$.
Figure 2 shows the proportion profiles of the terms in (2.4) as a function of 
$y^+$ for the 
$Re$950 case. Similar trends are observed in other cases and not shown here for the sake of simplicity. Here, 
$C_{f3}$ dominates the contributions in the viscous sublayer and decreases rapidly away from the wall. It is quite natural since 
$C_{f3}$ equals to the non-dimensional mean streamwise shear stress 
$\langle \tau ^*\rangle (y)/(\frac {1}{2}\rho ^*u_b^{*2})$, which goes to 
$C_f$ when 
$y\to 0$ and descends fast when 
$y$ goes out of the near-wall region. Furthermore, 
$C_{f4}$ is mathematically equal to 
$yC_f$, so its proportion scales up from 0 to 100 % and exhibits an exponential curve in the single logarithmic coordinate. Regarding 
$C_{f1}$ and 
$C_{f2}$, the primary focuses in this study, we will present their specific trends and Reynolds number effects in the following.
Figure 2. Proportion profiles of the contributing terms in (2.4) for case 
$Re$950.
Figure 3 illustrates the value profiles of 
$C_{f1}$ and 
$C_{f2}$ as a function of inner scale 
$y^+$. In figure 3(a), 
$C_{f1}$ holds negative values for all cases at most wall-normal distances except for the near-wall region where there appears a merely positive peak. Figure 3(b) reveals that 
$C_{f2}$ is totally positive. It seems that at 
$y=1$, either 
$C_{f1}$ or 
$C_{f2}$ at all Reynolds numbers converge to an approximately constant value, but opposite, i.e. 
$-0.008$ for 
$C_{f1}$ and 
$0.008$ for 
$C_{f2}$. According to (2.2), we have 
$C_{f1}+C_{f2}=2\langle -u'v'\rangle$ and 
$\langle -u'v'\rangle =0$ at 
$y=1$, so it is a direct consequence that the converged value of 
$C_{f1}$ equals to that of 
$-C_{f2}$. The nearly constant converged value for all cases may indicate that the accumulated intensities, to some extent, are not sensitive to Reynolds numbers (at least for the five cases considered in this study).
Figure 3. Value profiles of (a) 
$C_{f1}$ and (b) 
$C_{f2}$ versus 
$y^+$.
Moreover, both 
$C_{f1}$ and 
$C_{f2}$ exhibit more gentle slopes at higher 
$Re_\tau$, which indicates that the integrands of 
$C_{f1}$ and 
$C_{f2}$, i.e. the velocity–vorticity correlations, tend to be less intense at higher 
$Re_\tau$, as shown in figure 4. Integration of each curve returns to the corresponding individual skin-friction drag coefficient term. Profiles of all cases reach local peaks at the same wall-normal distance, and the peak values are inclined to be smaller at higher Reynolds numbers. Tennekes & Lumley (Reference Tennekes and Lumley1972) stated that 
$\langle v\omega _z\rangle$ and 
$\langle -w\omega _y\rangle$ are body forces in the streamwise direction associated with transport and stretching of vorticity, respectively. This was in tune with another understanding that they are the streamwise component of the Lamb vector. As shown in figure 4(a), the negative 
$\langle v\omega _z\rangle$ in most of the wall-normal distance apart from the near-wall region represents the streamwise deceleration arising from the vortex transport. The streamwise deceleration contributes negatively to the generation of skin friction. Meanwhile, the positive 
$\langle -w\omega _y\rangle$ (figure 4b) reflects the streamwise acceleration that contributes positively to the skin friction. This coheres with the four-layer model of the dynamical and vortical processes proposed by Klewicki (Reference Klewicki2013), in which the stretching vortices were surmised to act as a momentum source in the inner region and the advecting vortices as the momentum sink above the inner region. Although the positive value concentrates in the region where 
$y<0.3$ (not marked in this figure) for all Reynolds numbers, it makes the decisive contribution on the overall positive 
$C_{f2}$. In other words, the streamwise acceleration below 
$y=0.3$ caused by the vortex stretching is the critical factor contributing to the skin friction.
Figure 4. Profiles of (a) 
$y\langle v\omega _z\rangle$ and (b) 
$y \langle -w\omega _y\rangle$ versus 
$y^+$.
Figure 5 shows the proportions of 
$C_{f1}$ and 
$C_{f2}$ to 
$C_f$, which measure their relative contributions. In figure 5(a), a plausible collapse emerges among curves of 
$C_{f1}/C_f$ versus 
$y^+$. The values at 
$y=1$ decrease with 
$Re_\tau$, owing to the nearly constant 
$C_{f1}$ there and the descent of 
$C_f$ with the increase of 
$Re_\tau$. A similar trend of 
$C_{f2}/C_f$ at 
$y=1$ is observed in figure 5(b). Higher proportion at high 
$Re_\tau$ suggests relatively more active vortical contributions of 
$C_{f1}$ and 
$C_{f2}$.
Figure 5. Proportion profiles of (a) 
$C_{f1}/C_f$ and (b) 
$C_{f2}/C_f$ versus 
$y^+$.
Moreover, end points in figure 5(b) appear to be aligned on the semi-logarithmic coordinate, then a natural inference is that 
$1/C_f$ may be linearly related to 
$\lg Re _\tau$. Figure 6(a) presents the value of 
$1/C_f$ versus 
$Re_\tau$ in log scale. A well-fitted relation is obtained as
\begin{equation} C_f=\frac{1}{110.62\lg Re_\tau-130.63}. \end{equation}
The 
$R$-squared value is 0.9955, suggesting a good fitting effect. Another classic prediction of 
$C_f$ as a function of 
$Re_b$ (Dean Reference Dean1978), 
$C_f=0.073\times (2Re_b)^{-0.25}$, is also plotted in figure 6(b), with the 
$R$-squared value being 0.9752, slightly lower than that of the fitted line (4.1).
Figure 6. Variation of 
$C_f$ as a function of (a) 
$Re_\tau$ and (b) 
$Re_b$. (
$\blacksquare$) Actual computed value for each case. (a) Fitted line (– –) by the five cases; (b) reference line (– –) of Dean (Reference Dean1978), 
$C_f=0.073\times (2Re_b)^{-0.25}$.
4.2. Pre-multiplied spanwise co-spectra and positive/negative structure contributions
As the Reynolds number increases, the inner and outer scale separation becomes more profound in wall-bounded turbulent flows (Jiménez Reference Jiménez2018). The scale separation is associated with the population of energy-containing eddies in the logarithmic region (Lozano-Durán, Bae & Encinar Reference Lozano-Durán, Bae and Encinar2020), and the generation of LSMs and VLSMs in the logarithmic and outer region (Del Álamo et al. Reference Del Álamo, Jiménez, Zandonade and Moser2004; Hutchins & Marusic Reference Hutchins and Marusic2007; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013). The LSMs and VLSMs exert footprints and modulations on the near-wall flow quantities and further on the skin-friction drag. Therefore, extracting large-scale motions is significant to investigate the generation mechanism of skin-friction drag.
To divide the small- and large-scale structures with the low-pass spatial filtering, an appropriate cutoff wavelength needs to be determined. Actually, there was no strict standard for selecting the cutoff wavelength. The criterion 
$\lambda _z^*=\alpha h^*$ is widely adopted to investigate the velocity structures, where 
$\alpha$ is a parameter of order 
$O(1)$ selected differently in previous studies (Hutchins & Marusic Reference Hutchins and Marusic2007; Lee & Moser Reference Lee and Moser2019; Doohan, Willis & Hwang Reference Doohan, Willis and Hwang2021). With respect to the co-spectra of velocity–vorticity correlations, coupling the dynamics of velocity and vorticity structures, the low-pass filter with this cutoff wavelength fails to properly extract the small- and large-scale motions, see Appendix E.
Here, we attempt to use an alternative filtering criterion to separate the scales of velocity–vorticity correlations based on the spanwise pre-multiplied co-spectra of the turbulent inertia, 
$yk_z\phi _{TI}$. Figure 7 displays 
$yk_z\phi _{TI}$ for all cases, where 
$k_z=2{\rm \pi} /\lambda _z$ is the spanwise wavenumber, 
$\lambda _z$ is the spanwise wavelength and 
$\phi _{TI}$ is the spanwise co-spectra calculated by 
$\phi _{TI}=\partial \textrm{Re} (-\overline {\widehat {u'}}\widehat {v'})/\partial y$. The variables with a hat represent their Fourier transform, the complex number with an overbar represents its conjugate and 
$\textrm{Re}$ means taking the real part. Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014) studied the streamwise co-spectra of the turbulent inertia and separated the large-/small-scale motions by streamwise scale. However, if the streamwise Fourier mode is used, it is difficult to distinguish the inter-scale energy transfer (like the scale interaction) from the intra-scale one (like the self-sustaining process). The spanwise Fourier mode could avoid this problem (Hwang Reference Hwang2015; Cho, Hwang & Choi Reference Cho, Hwang and Choi2018), so we use the spanwise co-spectra instead of the streamwise ones here.
Figure 7. Spanwise pre-multiplied co-spectra of the turbulent inertia 
$yk_z\phi _{TI}$ for increasing Reynolds number in panels (a–e). The dashed line indicates 
$\lambda _z^+=3.75y^+$.
In figure 7, it is interesting to find that the spectral space is clearly divided into two individual negative and positive regions, with a linear relationship of 
$\lambda _z^+=3.75y^+$ (the dashed lines). The positive and negative regions also appear to be scaled linearly with distance from the wall. These features do not vary with Reynolds number, and the linear relationship of 
$\lambda _z^+=3.75y^+$ seems to be universal and not sensitive to the Reynolds number.
To analyse the connection between the turbulent inertia and Reynolds stress, we show their co-spectra, i.e. 
$yk_z\phi _{TI}$ and 
$k_z\phi _{-u'v'}$, in figure 8 for case 
$Re 2000$. By definition, 
$yk_z\phi _{TI}$ is the wall-normal gradient of 
$k_z\phi _{-u'v'}$ (with a logarithmic scale in 
$y^+$). Therefore, the positive region in 
$yk_z\phi _{TI}$ means that the Reynolds stress at the given spanwise scale is amplified in the wall-normal direction, and vice versa. The dividing line 
$\lambda _z^+=3.75y^+$ in 
$yk_z\phi _{TI}$ corresponds to the ridge line in 
$k_z\phi _{-u'v'}$, indicating the linear spanwise scale growth of the Reynolds stress with the distance from the wall.
Figure 8. Spanwise pre-multiplied co-spectra of (a) the Reynolds stress 
$k_z\phi _{-u'v'}$ and (b) the turbulent inertia 
$yk_z\phi _{TI}$ for case 
$Re 2000$. The dashed diagonal indicates 
$\lambda _z^+=3.75y^+$.
In the study of De Giovanetti, Hwang & Choi (Reference De Giovanetti, Hwang and Choi2016), three energy-containing motions were clearly identified from the Reynolds stress co-spectra. (i) The motions at 
$\lambda _z^+\approx 100$ are composed of the near-wall streaks and the quasi-streamwise vortices. The inner positive and negative peaks of the TI co-spectra are also at the scale 
$\lambda _z^+\approx 100$. (ii) The motions at the outer length scale 
$\lambda _z\approx 1.5$ are represented by the VLSMs (outer streaks) and the LSMs (outer streamwise vortical structures). (iii) At the log-layer length scale between 
$\lambda _z^+\approx 100$ and 
$\lambda _z\approx 1.5$, the motions with 
$\lambda _z$ growing linearly with 
$y$ are self-similar to one another, in accordance with the attached eddy hypothesis proposed by Townsend (Reference Townsend1976). These features can also be identified in the TI co-spectra. The scales of the energy-containing motions are universal at different Reynolds numbers, thus not shown for brevity.
Considering that 
$\lambda _{z}^+=3.75y^+$ is not sensitive to the Reynolds number and the turbulent inertia combines the dynamics of velocity–vorticity correlations (see (1.3)), we employ 
$\lambda _{z}^+=3.75y^+$ as the filtering criterion to extract the structures associated with the positive and negative contributions to the skin-friction drag generation. A similar criterion with a varying cutoff wavelength is also applied by Chan, Schlatter & Chin (Reference Chan, Schlatter and Chin2021) to distinguish the large- and small-scale components of inter-scale transport of Reynolds shear stress. In this work, we address these two separated regions as positive structures (PSs) and negative structures (NSs), and they also imply the local large- and small-scale structures, respectively. Since the turbulent inertia indicates the streamwise body force, it could be inferred that the PSs lead to the streamwise acceleration and the NSs are associated with the streamwise retardation.
Recall from (2.2), the spanwise pre-multiplied co-spectra of the turbulent inertia can also be decomposed into that of 
$v\omega _z$ and 
$-w\omega _y$, i.e. 
$yk_z\phi _{v\omega _z}$ and 
$yk_z\phi _{-w\omega _y}$, which are displayed in figure 9 for the cases of 
$Re 180$, 
$Re 950$ and 
$Re 2000$. Similar features are observed in the other two cases, but not shown here for brevity. We find that 
$yk_z\phi _{v\omega _z}$ is dominated by negative values, with a peak at 
$y^+=10\unicode{x2013}30$ and 
$\lambda _z^+=100$, regardless of the Reynolds numbers. The peak of 
$yk_z\phi _{-w\omega _y}$ locates at the same wall-normal position and spanwise wavelength as 
$yk_z\phi _{v\omega _z}$, but with opposite sign. The filtering criterion 
$\lambda _z^+=3.75y^+$ is plotted in figure 9 as dashed lines. Several noticeable features can be observed by using this varying cutoff wavelength: (i) in terms of 
$yk_z\phi _{v\omega _z}$, the cutoff line roughly goes through the negative peak and a spoon-handle-shaped extension of negative motions in the outer region distributes along this line; (ii) two weak positive areas are assigned in PSs as well as a negative part populated at 
$y^+=10\unicode{x2013}50$ (which presents a bit of an extension towards local larger scale); (iii) negative part of 
$yk_z\phi _{-w\omega _y}$ is assigned in NSs, while motions in PSs contribute positively to skin-friction drag apart from a negative part in the outer region above the line. The dominant negative 
$yk_z\phi _{v\omega _z}$ and positive 
$yk_z\phi _{-w\omega _y}$ also support the four-layer model given by Klewicki (Reference Klewicki2013).
Figure 9. Pre-multiplied spanwise co-spectra (a–c) 
$yk_z\phi _{v\omega _z}$ and (d–f) 
$yk_z\phi _{-w\omega _y}$. (a,d) 
$Re$180; (b,e) 
$Re$950; (c,f) 
$Re$2000. The dashed line indicates 
$\lambda _z^+=3.75y^+$.
Priyadarshana et al. (Reference Priyadarshana, Klewicki, Treat and Foss2007) stated that the co-spectra of velocity–vorticity correlations tend to gain peaks at wavelengths near the peaks of individual components (velocity and vorticity itself) at a fixed wall-normal distance. According to this, we can interpret the attribution of the inner peak and outer extension of velocity–vorticity co-spectra. On one hand, the inner peak of the co-spectra with approximately constant position and length scale at different Reynolds numbers, as displayed in figure 9, is probably related to the vorticity component, since vortical motions approximately maintain a relatively small scale as Reynolds number increases. On the other hand, as Reynolds number increases, the co-spectra extend to the outer region with considerable energy. This may be attributed to the velocity structures, in terms of LSMs and VLSMs populating in the outer region, especially at high Reynolds numbers.
At a fixed 
$y^+$, 
$\langle v\omega _z\rangle$ and 
$\langle -w\omega _y\rangle$ of PSs and NSs can be obtained by integrating the respective co-spectra along 
$\lambda _{z}^+>3.75y^+$ and 
$\lambda _{z}^+<3.75y^+$. Their profiles multiplied by 
$y$ are shown in figure 10. Two positive peaks of 
$y\langle v\omega _z\rangle _{PS}$ in figure 10(a) are associated with the two weak positive areas in the near-wall and logarithmic region, respectively; whereas the intense minimum at 
$y^+\approx 20$ is related to the extension of local negative events towards larger scale, see figures 9(a)–9(c). In figure 10(b), 
$y\langle v\omega _z\rangle _{NS}$ stays negative now that the positive areas are all assigned in PSs, and the local minimum at 
$y^+=42$ is relevant to the negative peak of 
$yk_z\phi _{v\omega _z}$. Likewise, the positions of the maximum in figures 10(c) and 10(d) are also associated with the local inner peaks of 
$yk_z\phi _{-w\omega _y}$, and almost positive values in figure 10(c) correspond to aforementioned feature (iii). Each curve reaches the maximum or minimum at a constant wall-normal position in the near-wall region in figure 10, indicating that the structures are well scaled by inner units. Additionally, profiles in figures 10(b) and 10(d) at higher Reynolds number appear to form plateaus in part of the logarithmic region (referenced by the red dashed line), implying that 
$\langle v\omega _z\rangle _{NS}$ and 
$\langle -w\omega _y\rangle _{NS}$ are inversely proportional to the wall-normal distance.
Figure 10. Profiles of (a) 
$y\langle v\omega _z\rangle _{PS}$, (b) 
$y\langle v\omega _z\rangle _{NS}$, (c) 
$y\langle -w\omega _y\rangle _{PS}$ and (d) 
$y\langle -w\omega _y\rangle _{NS}$ versus 
$y^+$. Red dashed line in panels (b) and (d) is a reference line for plateaus.
Overall, it is clear that lower Reynolds number leads to a greater absolute value in inner peaks and outer region (versus 
$y$, not shown here). As mentioned in § 4.1, 
$C_{f1}$ and 
$C_{f2}$ converge to the approximately same respective value for all cases, which means double integrals of the pre-multiplied co-spectra 
$yk_z\phi _{v\omega _z}$ (figures 9a–9c) and 
$yk_z\phi _{-w\omega _y}$ (figures 9d–9f) on the full plane are respectively equal. Nevertheless, with Reynolds number increasing, more larger-scale motions, whether positive or negative, occur and manifest as the extension in the logarithm region, taking up a certain proportion and hence draining peaks in the inner and outer region.
Regarding 
$C_{f1}$ and 
$C_{f2}$, they can be further divided into two parts by integrating the velocity–vorticity correlations in PSs and NSs, viz.,
$$\begin{gather} {C_{f1}^{PS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{>}3.75y^+}yk_z \phi_{v\omega_z}\,\mathrm{d}\ln \lambda_z^+\mathrm{d}\ln y^+, \end{gather}$$
$$\begin{gather}{C_{f1}^{NS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{<}3.75y^+}yk_z\phi_{v\omega_z}\,\mathrm{d} \ln \lambda_z^+\mathrm{d}\ln y^+, \end{gather}$$
$$\begin{gather}{C_{f2}^{PS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{>}3.75y^+}yk_z\phi_{-w\omega_y}\, \mathrm{d}\ln \lambda_z^+\mathrm{d}\ln y^+, \end{gather}$$
$$\begin{gather}{C_{f2}^{NS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{<}3.75y^+}yk_z\phi_{-w\omega_y}\,\mathrm{d}\ln \lambda_z^+\mathrm{d}\ln y^+. \end{gather}$$ Profiles of 
$C_{fi}^{PS}$ and 
$C_{fi}^{NS}$ (
$i=1, 2$) can be found in Appendix F. Here, the contributions are divided by 
$C_f$ to discuss the Reynolds number effects in the proportion of individual contribution. Figure 11 shows the profiles of 
$C_{fi}^{PS}/C_{f}$ and 
$C_{fi}^{NS}/C_{f}$ as a function of 
$y^+$.
Figure 11. Proportion profiles of (a) 
$C_{f1}^{PS}$, (b) 
$C_{f1}^{NS}$, (c) 
$C_{f2}^{PS}$ and (d) 
$C_{f2}^{NS}$ in total 
$C_f$ versus 
$y^+$. Red dashed line in panels (b) and (d) is a reference line for log-law.
As a consequence of the dominant position of 
$C_{f1}^{NS}$ in 
$C_{f1}$ and 
$C_{f2}^{PS}$ in 
$C_{f2}$, proportion 
$C_{f1}^{NS}/C_{f}$ and 
$C_{f2}^{PS}/C_{f}$ collapse better with 
$C_{f1}/C_{f}$ and 
$C_{f2}/C_{f}$, respectively. Compared with figure 5, figure 11(b) reveals less distinctions (which are caused by 
$C_{f1}^{PS}$) at different Reynolds numbers, which means that the contribution proportion of local small-scale advective vorticity transport on the skin friction is less influenced by Reynolds number in inner units. Moreover, a log-law occurs in the logarithmic region (referenced by the nearby red dashed line). However, according to the plateaus in figure 10(b), the better log-law only occurs at higher Reynolds number, and the worse fitting in lower Reynolds number is weakened by overlapped curves. An analogous log-law appears in 
$C_{f2}/C_{f}$ (figure 11d) at higher Reynolds number.
In the inner region, compared with the distinction peaks of 
$C_{fi}^{PS}$ and 
$C_{fi}^{NS}$ (
$i=1, 2$), especially 
$C_{f1}^{PS}$ and 
$C_{f2}^{NS}$ (see Appendix F), their proportions to 
$C_f$ appear to collapse well, suggesting the insensitivity to Reynolds number. In the outer region, although almost all the four components in figure 11 occupy a higher proportion with the increase of Reynolds number, the total 
$C_f$ presents an opposite trend, indicating that the local small-/large-scale advective vorticity transport and vortex stretching cancel each other out to a larger extent.
4.3. Quadrant analyses
Quadrant decomposition of the velocity–vorticity correlations (
$v'\omega _z'$ and 
$w'\omega _y'$) is performed to further investigate the friction drag generation linked to different quadrant motions. To provide a visual perspective of the quadrant motions, the joint probability distribution functions (JPDFs) of 
$v'\omega _z'$ and 
$w'\omega _y'$ are investigated at some wall-normal distances. Note that the components in the velocity–vorticity correlations are rewritten into corresponding fluctuations, the reason for which will be discussed later. We use the velocity component as the horizontal axis and vorticity component as the vertical axis to identify the quadrant.
Based on the peaks in figure 4, three typical locations at 
$y^+=6, 17$ and 
$25$ are selected to check the JPDFs of 
$v'\omega _z'$ and 
$w'\omega _y'$, as displayed in figure 12. Figure 12(a) shows JPDFs of 
$v'\omega _z'$ at 
$y^+=6$ for all cases. The contour lines are normalized by each maximum value, and the axes are normalized by the root mean square (r.m.s.) of corresponding fluctuations. The normalized JPDFs contours have a nice collapse at different Reynolds numbers and are ellipse-like in shape whose major axes distribute along the first and third quadrants, leading to the JPDFs extension and dominantly distributing in these two quadrants (especially the first quadrant). Figure 12(a) reveals that the small positive 
$v'\omega _z'$ at 
$y^+=6$ is mainly dominated by the upward anticlockwise-rotating motions (
$v'>0$ and 
$\omega _z'>0$). A similar conclusion can be found by Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014), in which both JPDFs and weighted JPDFs (figures 6 and 7 in Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014)) were investigated to demonstrate this point. However, something wrong might exist in figure 6 of Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014), for instance, where the core of JPDF occurred at the third quadrant and the PDF of 
$\omega _z'$ was positively skewed. The sign of 
$\omega _z'$ is consistent with that of 
$-\partial u'/\partial y$ in the near-wall region, as 
$\omega _z'=\partial v'/\partial x-\partial u'/\partial y$. Generally, a positively skewed distribution of 
$\tau _{w}'$ is observed (e.g. Cheng et al. Reference Cheng, Li, Lozano-Durán and Liu2020), which suggests that the PDF of 
$\omega _z'$ is negatively skewed, as shown in the right panel of figure 12(a). The PDF of 
$v'$ is also plotted in the top panel of figure 12(a), showing its relatively symmetric feature.
Figure 12. JPDFs normalized by each maximum of 
$v'\omega _z'$ at (a) 
$y^+=6$ and (b) 
$y^+=25$ and 
$w'\omega _y'$ at (c) 
$y^+=17$ with each axis normalized by its r.m.s. for all Reynolds numbers. Three clusters of contours in each panel represent 0.9, 0.6 and 0.3 from inside out. The top plot in each panel presents the marginal distributions (normalized by each maximum) of corresponding r.m.s.-normalized velocity fluctuation, and the right plot in each panel presents those of r.m.s.-normalized vorticity fluctuation.
Figure 12(b) shows JPDFs of 
$v'\omega _z'$ at 
$y^+=25$. There exist the core and prominent extension in the second quadrant (
$v'<0$ & 
$\omega _z'>0$) which is connected with the downward anticlockwise-rotating motions contributing to the negative peak of 
$v'\omega _z'$ most. Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014) pointed out that it could be viewed as the head of hairpin vortices advecting outwards, but this interpretation was based on the improper JPDF that the sign of the axes was opposite. The negative skewness of 
$\omega _z'$ is still obvious in the right plot, while the distribution of 
$v'$ is relatively symmetric.
The JPDFs of 
$w'\omega _y'$ at 
$y^+=17$ are plotted in figure 12(c). Nice centre symmetry about the original point is observed. The dominant second and fourth quadrant motions are related to a couple of counter-rotating normal vortices who move oppositely in the spanwise direction. In addition, compared with 
$\omega _z'$, the distribution of 
$\omega _y'(=\partial u'/\partial z-\partial w'/\partial x)$ presents good symmetry owing to the homogeneity in the streamwise and spanwise directions.
Having investigated the schematic JPDFs of 
$v'\omega _z'$ and 
$w'\omega _y'$ at specific positions, we then focus on the quantitative intensity of 
$v'\omega _z'$ and 
$w'\omega _y'$ in each quadrant, i.e. 
$\langle v'\omega _z'\rangle _{Qi}$ and 
$\langle -w'\omega _y'\rangle _{Qi}$, where 
$i=1\unicode{x2013}4$ and 
$\langle \varphi \rangle _{Qi}=\sum _{j}\varphi _j|_{\varphi _j\in Qi}/N$ (
$N$ is total mesh number at each height). The benefit of this definition of quadrant ensemble average rather than divided by 
$N|_{\varphi _j\in Qi}$ is to ensure 
$\langle \varphi \rangle =\sum _{i}\langle \varphi \rangle _{Qi}$. The reason why we adopt the velocity–vorticity correlations with fluctuation components is that the quadrant analyses are based on the concept of joint distribution of fluctuations and 
$\langle v\omega _z\rangle _{Qi}\neq \langle v'\omega _z'\rangle _{Qi}$ on account of 
$\bar {\omega }_z\neq 0$. Corresponding 
$C_{f1}^{Qi}$ and 
$C_{f2}^{Qi}$ are defined as
$$\begin{gather} C_{f1}^{Qi}=\int_{0}^{y}y\langle v'\omega_z'\rangle_{Qi}\, \mathrm{d}\ln y, \end{gather}$$
$$\begin{gather}C_{f2}^{Qi}=\int_{0}^{y}y\langle-w'\omega_y'\rangle_{Qi}\, \mathrm{d}\ln y. \end{gather}$$ Figure 13 shows the profiles of 
$\langle v'\omega _z'\rangle _{Qi}$ and 
$\langle -w'\omega _y'\rangle _{Qi}$ multiplied by 
$y$ for case 
$Re 950$. The results of other cases are not shown here because of high resemblance. All pre-multiplied 
$\langle v'\omega _z'\rangle _{Qi}$ (figure 13a) reach a mutual inhibition and consequently, by contrast, pre-multiplied 
$\langle v'\omega _z'\rangle$ is of relatively small amplitude above the inner region, where 
$v'\omega _z'$ in 
$Q2$ and 
$Q4$ (especially 
$Q4$) seems to be slightly in the lead compared with 
$Q3$ and 
$Q1$, promoting 
$C_{f1}$ to be negative. Here, 
$Q2$ and 
$Q4$ correspond to downward anticlockwise-rotating motions and upward clockwise-rotating motions, respectively, with the latter inhibiting skin-friction drag most.
Figure 13. Profiles of (a) 
$y\langle v'\omega _z'\rangle _{Qi}$ and (b) 
$y\langle -w'\omega _y'\rangle _{Qi}$ (
$i=1-4$) versus 
$y$ for case 
$Re 950$.
In terms of 
$\langle -w'\omega _y'\rangle _{Qi}$ (figure 13b), it is obvious that 
$\langle -w'\omega _y'\rangle _{Q2}$ and 
$\langle -w'\omega _y'\rangle _{Q4}$ collapse perfectly, and so do 
$\langle -w'\omega _y'\rangle _{Q1}$ and 
$\langle -w'\omega _y'\rangle _{Q3}$. This collapse is due to the symmetry of the value distribution of 
$w'$ and 
$\omega _y'$, as shown in the top and right plots of figure 12(c), and this symmetry holds throughout the whole channel height. Here, 
$C_{f2}^{Qi}$ is the integral of 
$\langle -w'\omega _y'\rangle _{Qi}$, and accordingly, it collapses perfectly as well, which is not observed in 
$C_{f1}$ on account of the asymmetry of 
$\omega _z'$. Additionally, despite symmetry of 
$w'$ and 
$\omega _y'$, they are not necessarily independent and their joint distributions exhibit a central symmetry but not necessarily a uniform symmetry. That is to say, 
$Q2$ and 
$Q4$ distribute necessarily symmetrically, and so do 
$Q1$ and 
$Q3$, but 
$Q1$ and 
$Q2$ are not necessarily symmetric. For example, there is a degree of skewness where 
$Q2$ and 
$Q4$ are a bit more intense than 
$Q1$ and 
$Q3$ in the region where 
$y<0.3$, leading to the positive 
$\langle -w'\omega _y'\rangle$, especially in the near-wall region (as shown in figure 12c). The skewness is approximately negligible in the vicinity of the zero point (
$y=0.3$) of 
$\langle -w'\omega _y'\rangle$ and the JPDF is relatively uniform there.
Profiles of pre-multiplied 
$\langle v'\omega _z'\rangle _{Qi}$ and 
$\langle -w'\omega _y'\rangle _{Qi}$ as well as corresponding proportion of 
$C_{f1}^{Qi}$ and 
$C_{f2}^{Qi}$ to 
$C_f$ are shown for all Reynolds numbers in figure 14. By reason of the symmetry of the distributions of 
$w'$ and 
$\omega _y'$ mentioned above, profiles of 
$\langle -w'\omega _y'\rangle _{Qi}$ gain excellent collapse in diagonal quadrants as shown in figures 14(b-i,b-iii), 14(b-ii,b-iv), 14(d-i,d-iii) and 14(d-ii,d-iv). This is not remarkable in 
$v'\omega _z'$. Additionally, it is obvious that all quantities are amplified with the increase of Reynolds number (see figure 14a,b), of which the root cause is that the high Reynolds number excites more intense fluctuations, thus 
$\langle v'\omega _z'\rangle _{Qi}$ and 
$\langle -w'\omega _y'\rangle _{Qi}$ could reach larger values. Furthermore, the proportion profiles of 
$C_{f1}^{Qi}$ and 
$C_{f2}^{Qi}$ follow the same rule (see figure 14c,d).
Figure 14. Profiles of (a-i) 
$y\langle v'\omega _z'\rangle _{Qi}$, (b-i) 
$y\langle -w'\omega _y'\rangle _{Qi}$, (c-i) 
$C_{f1}^{Qi}/C_f$ and (d-i) 
$C_{f2}^{Qi}/C_f$ (
$i=1-4$) versus 
$y$ for all five cases: (
$\square$) 
$Re 180$, (
$\triangle$) 
$Re 350$, (
$\times$) 
$Re 550$, (
$+$) 
$Re 950$, (
$\circ$) 
$Re 2000$. Arrows represent that the Reynolds number increases.
It seems that all pre-multiplied quadrant velocity–vorticity correlations in figure 14 reach the extreme value at the position around 
$y=0.48$ at all Reynolds numbers, suggesting that near this position, all quadrant motions are scaled with the outer units and approximately contribute most to the skin-friction drag, but as shown in figure 13, all quadrant motions reach a balance here and consequently no prominent feature occurs for the total pre-multiplied correlations at this position. In addition, according to figure 3, total 
$C_{f1}$ and 
$C_{f2}$ converge to the same respective value at 
$y=1$ for all cases, indicating that amplifications of quadrant quantities at higher Reynolds number almost cancel each other out in the vicinity of the central plane. Figure 15 displays 
$C_{f1}^{Qi}$ and 
$C_{f2}^{Qi}$ (
$i=1-4$) at 
$y=1$ for all cases. Each profile exhibits the approximately linear growth as the increase of Reynolds number. Because of the nearly constant convergent value of 
$C_{f1}$ and 
$C_{f2}$, the summation of all profiles in each of figures 15(a) and 15(b) is constant.
Figure 15. Profiles of (a) 
$C_{f1}^{Qi}$ and (b) 
$C_{f2}^{Qi}$ (
$i=1-4$) at 
$y=1$ versus 
$Re_\tau$.
5. Concluding remarks
In the present study, we have given a form of skin-friction drag decomposition (2.4) based on the velocity–vorticity correlations. Better physics-informed interpretation could be inferred from this form since it does not suffer from the weight factor 
$(1-y)$ appearing in other FIK-like identities. By using cases of incompressible turbulent channel flows at friction Reynolds numbers from 186 to 2003, we investigate the Reynolds number effects on the contributing terms, their scale-dependence and quadrant characteristics. The key findings are summarized as follows.
(i)
$C_{f1}$, which is generated from the advective vorticity transport ($v\omega _z$), contributes negatively to the skin friction except in the near-wall region; whereas $C_{f2}$, which is associated with the vortex stretching ($-w\omega _y$), is overall positive. The negative $\langle v\omega _z\rangle$ characterizes the streamwise deceleration (or the momentum sink) arising from the vortex transport. Reversely, the positive $\langle -w\omega _y\rangle$ reflects the streamwise acceleration (or the momentum source). The pre-multiplied intensities of $\langle v\omega _z\rangle$ and $\langle -w\omega _y\rangle$ reach peaks at positions that are independent on the Reynolds number, where they locally contribute most to $C_{f1}$ and $C_{f2}$, respectively. At $y=1$, either $C_{f1}$ or $C_{f2}$ converges to an approximately constant value regardless of the Reynolds number.(ii) It is found that the spanwise pre-multiplied co-spectra of the turbulent inertia is clearly divided into individual positive and negative regions, with a universal linear relationship of
$\lambda _z^+=3.75y^+$ for all cases. The scaling law could be observed in the co-spectra of both the Reynolds stress and the turbulent inertia, indicating the connection with the self-similar energy-containing motions predicted by the attached eddy hypothesis. We adopt this varying cutoff wavelength as a scale separation strategy, to decompose the turbulent motions into positive (PSs) and negative (NSs) structures. Results show that the advective vorticity transport is principally responsible for NSs acting as the momentum sink, while the vortex stretching is the main cause of PSs acting as the momentum source. From another perspective, NSs and PSs also represent the local small- and large-scale motions, respectively, suggesting that the advective vorticity transport is statistically in smaller scale, while the vortex stretching process is quite the contrary. Consequently, NSs and PSs play the dominant role in the generation of $C_{f1}$ and $C_{f2}$, respectively. Extreme points of pre-multiplied $\langle v\omega _z\rangle$ and $\langle -w\omega _y\rangle$ of PSs and NSs are connected with peaks in the corresponding spanwise pre-multiplied co-spectra in the inner region and consistent at different Reynolds numbers. Moreover, a log-law occurs in profiles of $C_{f1}^{NS}$ and $C_{f2}^{NS}$ in the logarithmic region only at high Reynolds numbers.(iii) The joint probability distribution functions (JPDFs) of the velocity–vorticity correlations (
$v'\omega _z'$ and $w'\omega _y'$) in the inner region exhibit skewness in the quadrants, which is directly linked to the inner peaks of correlations. The second and fourth quadrant motions of $w'\omega _y'$ are dominant at $y^+=17$, which is related to the generation mechanism of vortex stretching. As the wall-normal distance increases, JPDFs tend to be more uniform, especially for $w'\omega _y'$. Additionally, $\langle v'\omega _z'\rangle _{Qi}$ and $\langle -w'\omega _y'\rangle _{Qi}$ are enhanced with the increase of Reynolds number by reason that the high Reynolds number excites more intense fluctuations. Nevertheless, intensities of all quadrants reach a mutual inhibition and consequently their summations vary in relatively small amplitudes. Their integrals, i.e. $C_{f1}^{Qi}$ and $C_{f2}^{Qi}$, and proportions to $C_f$ also share the same conclusion.
Acknowledgements
Special thanks to Dr C. Cheng at The Hong Kong University of Science and Technology for his utmost assistance.
Funding
The funding support of the National Natural Science Foundation of China (under the grant nos. 12372221 and 92052101) is acknowledged.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Reduce Yoon's identity to the FIK identity
One can perform a fourfold integration 
$\int _0^1\int _0^y\int _0^{y_3}\int _0^{y_2}\,\mathrm {d} y_1\,\mathrm {d} y_2\,\mathrm {d} y_3\,\mathrm {d} y$ (where 
$y$, 
$y_1$, 
$y_2$ and 
$y_3$ are all wall-normal distances) to the mean spanwise vorticity equation, to reduce Yoon's identity to the FIK identity. We give a brief proof here.
The mean spanwise vorticity equation is expressed as
\begin{equation} \dfrac{\mathrm{d}\langle v\omega_z\rangle}{\mathrm{d} y}+\dfrac{\mathrm{d}\langle-w\omega_y\rangle}{\mathrm{d} y}-\dfrac{1}{Re_b}\dfrac{\mathrm{d}^2\langle\omega_z\rangle}{\mathrm{d} y^2}=0. \end{equation}
Performing the first integration 
$\int _0^{y_2}\,\mathrm {d} y_1$ to (A1), we obtain
\begin{equation} \langle v\omega_z\rangle+\langle-w\omega_y\rangle-\dfrac{1}{Re_b} \dfrac{\mathrm{d}\langle\omega_z\rangle}{\mathrm{d} y}+\left.\dfrac{1}{Re_b}\dfrac{\mathrm{d}\langle\omega_z\rangle}{\mathrm{d} y}\right|_{y=0}=0. \end{equation}
Substituting the turbulent inertia decomposition (2.2) as well as 
$\langle \omega _z\rangle =-{\mathrm {d}\langle u\rangle }/{\mathrm {d} y}$ into (A2), we have
\begin{equation} \left.\dfrac{\mathrm{d}\langle-u'v'\rangle}{\mathrm{d} y}+ \dfrac{1}{Re_b}\dfrac{\mathrm{d}^2\langle u\rangle}{\mathrm{d} y^2}+ \dfrac{1}{Re_b}\dfrac{\mathrm{d}\langle\omega_z\rangle}{\mathrm{d} y}\right|_{y=0}=0. \end{equation}Comparing (A3) with the equation of the streamwise mean momentum balance (2.1), we can find that 
$({\mathrm {d}\langle u\rangle }/{\mathrm {d} y})|_{y=0}=({\mathrm {d}\langle \omega _z\rangle }/{\mathrm {d} y})|_{y=0}$. This indicates that (A3) (i.e. the integral of (A1)) is equivalent to (2.1). Applying the remaining triple integration 
$\int _0^1\int _0^y\int _0^{y_3}\,\mathrm {d} y_2\,\mathrm{d} y_3\,\mathrm {d} y$ to (A3) is also equivalent to applying that to (2.1), which is the exact derivation of the FIK identity by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002). Consequently, the fourfold integration of the mean vorticity equation reduces Yoon's identity to the FIK identity.
Appendix B. Reduce identity (2.4) to the FIK identity
To transform (2.4) to the FIK identity, we first rewrite (2.4) (by substituting (2.2)) as
\begin{equation} C_f=2\langle-u'v'\rangle+\dfrac{2}{Re_b}\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}+yC_f. \end{equation}
Multiplying (B1) by 
$(1-y)$ and integrating it in 
$y$ over 
$[0,1]$, we obtain
\begin{equation} \int_{0}^{1}(1-y)^2C_f\mathrm{d} y=2\int_{0}^{1}(1-y)\langle-u'v'\rangle\,\mathrm{d} y+\dfrac{2}{Re_b}\int_{0}^{1}(1-y)\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}\,\mathrm{d} y. \end{equation}Note that the second term in right-hand side of (B2) can be computed by integration by parts:
\begin{equation} \left.\int_{0}^{1}(1-y)\dfrac{\mathrm{d}\langle u\rangle}{\mathrm{d} y}\,\mathrm{d} y=(1-y)\langle u\rangle\right|_{y=0}^{y=1}+\int_{0}^{1}\langle u\rangle\,\mathrm{d} y=1. \end{equation}Then (B2) is simplified to
\begin{equation} \dfrac{1}{3}C_f=2\int_{0}^{1}(1-y)\langle-u'v'\rangle\,\mathrm{d} y+\dfrac{2}{Re_b}, \end{equation}which is identical to the FIK identity (1.1).
Appendix C. Form of (2.4) in the case of turbulent pipe flows
A similar identity can be derived also in the case of turbulent pipe flows. In this appendix, some notation and symbols are different from the main text. We use 
$x$, 
$r$ and 
$\theta$ to represent the streamwise, radial and azimuthal directions, respectively. Velocities in these three directions are 
$u_x$, 
$u_r$ and 
$u_\theta$. Corresponding vorticities are 
$\omega _x$, 
$\omega _r$ and 
$\omega _\theta$. The angle bracket 
$\langle \rangle$ represents averaging over homogeneous directions (streamwise and azimuthal directions) and time. Here, 
$R^*$ is the radius of the pipe and 
$Re_b=u_b^*R^*/\nu ^*$ is the bulk Reynolds number. Variables without the superscript 
$^*$ represent non-dimensional quantities normalized by 
$u_b^*$ and 
$R^*$. Wall-shear stress is written as 
$\tau _{w}^*=-\mu ^*({\partial u_x^*}/{\partial r^*})|_{r=1}$. Notation and symbols without mentions are consistent with those in the main text.
We start from the dimensional streamwise mean momentum balance for fully developed incompressible turbulent pipe flows:
\begin{equation} \dfrac{\mathrm{d}\langle-u_x'u_r'\rangle^*}{\mathrm{d}r^*}+ \dfrac{\langle-u_x'u_r'\rangle^*}{r^*}-\dfrac{1}{\rho^*}\left\langle\dfrac{\partial p^*}{\partial x^*}\right\rangle+\nu^*\dfrac{1}{r^*}\dfrac{\mathrm{d}}{\mathrm{d}r^*} \left(r^*\dfrac{\mathrm{d}\langle u_x^*\rangle}{\mathrm{d}r^*}\right)=0, \end{equation}
where 
$p^*$ is pressure. By the force balance in the streamwise direction, we have 
$-{\rm \pi} R^{*2}\langle {\partial p^*}/{\partial x^*}\rangle =2{\rm \pi} R^*\langle \tau _{w}^*\rangle$. Substituting this relation into (C1), we obtain the non-dimensional equation
\begin{equation} \dfrac{\mathrm{d}\langle-u_x'u_r'\rangle}{\mathrm{d}r}+\dfrac{\langle-u_x'u_r' \rangle}{r}-\left.\dfrac{2}{Re_b}\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}\right|_{r=1} +\dfrac{1}{Re_b}\left(\dfrac{1}{r}\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r} +\dfrac{\mathrm{d}^2\langle u_x\rangle}{\mathrm{d}r^2}\right)=0. \end{equation}Like (2.2) in turbulent channel flows, there is also a decomposition of the turbulent inertia in the pipe flows:
\begin{equation} \dfrac{\mathrm{d}\langle-u_x'u_r'\rangle}{\mathrm{d}r}+\dfrac{\langle-u_x'u_r'\rangle}{r}=\langle u_r\omega_\theta\rangle+\langle-u_\theta\omega_r\rangle. \end{equation}
Substituting (C3) into (C2) and integrating in the radial direction from an arbitrary radius 
$r\in (0,1)$ to 
$r=1$ (wall surface), we get
\begin{align} &\int_r^1\langle u_r\omega_\theta\rangle\,\mathrm{d}r+\int_r^1\langle-u_\theta\omega_r\rangle\,\mathrm{d}r -\dfrac{1}{Re_b}\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}-\left.\dfrac{2}{Re_b}(1-r)\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}\right|_{r=1}\nonumber\\ &\quad +\dfrac{1}{Re_b}\int_r^1\dfrac{1}{r}\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}\,\mathrm{d}r+\left.\dfrac{1}{Re_b}\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}\right|_{r=1}=0. \end{align}
The skin-friction coefficient can be calculated by 
$C_f=-2({\mathrm {d}\langle u_x\rangle }/{\mathrm {d}r})|_{r=1}/Re_b$. Substituting it into (C4), we have
\begin{align} C_f&=\underbrace{2\int_r^1\langle u_r\omega_\theta\rangle\,\mathrm{d}r}_{C_{f1}}+ \underbrace{2\int_r^1\langle-u_\theta\omega_r\rangle\,\mathrm{d}r}_{C_{f2}}\nonumber\\ & \quad \underbrace{-\dfrac{2}{Re_b}\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}}_{C_{f3}}\underbrace{-\left.\dfrac{4}{Re_b}(1-r)\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}\right|_{r=1}}_{C_{f4}}+\underbrace{\dfrac{2}{Re_b}\int_r^1\dfrac{1}{r}\dfrac{\mathrm{d}\langle u_x\rangle}{\mathrm{d}r}\,\mathrm{d}r}_{C_{f5}}. \end{align}This equation is another form of (2.4) in the case of turbulent pipe flows. The first four terms in the right-hand side of (C5) are similar to those in (2.4) with respect to forms and interpretations. There is an additional term 
$C_{f5}$, whose appearance is due to the Laplacian operator in the cylindrical coordinates, as stated by Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016). By virtue of 
$\langle u_x\rangle$ decreasing along the radial direction, 
$C_{f5}$ contributes negatively to the skin friction. Similar to (2.4), the applicable scope of (C5) should satisfy: (i) the homogeneity in the streamwise and azimuthal directions, and (ii) the zero mean of 
$u_r$ and 
$u_\theta$ at each radius.
Appendix D. Form of (2.4) in the case of turbulent boundary layer flows
In this appendix, we consider the case of turbulent boundary layer flows. The non-dimensional streamwise mean momentum balance for turbulent boundary layer flows is written as
\begin{equation} -\dfrac{\partial\langle u^2\rangle}{\partial x}+\dfrac{\partial\langle -uv\rangle}{\partial y}-\left\langle\dfrac{\partial p}{\partial x}\right\rangle+ \dfrac{1}{Re_\delta}\left(\dfrac{\partial^2\langle u\rangle}{\partial x^2}+ \dfrac{\partial^2\langle u\rangle}{\partial y^2}\right)=0. \end{equation}
The angle bracket 
$\langle \, \rangle$ used here represents averaging over spanwise direction and time. Additionally, 
$p$ is pressure; 
$Re_\delta =u_0^*\delta ^*/\nu ^*$ is the Reynolds number; 
$\delta ^*$ is the boundary-layer thickness (rather than the displacement thickness, as the asterisk means dimensional here) and 
$u_0^*$ is the free stream velocity. Variables without the superscript 
$^*$ represent non-dimensional quantities normalized by 
$u_0^*$ and 
$\delta ^*$. Notation and symbols without mentions are consistent with those in the main text.
The second term in the left-hand side of (D1) is not strictly equal to the turbulent inertia, but the error is negligible. We decompose it like the turbulent inertia, while there is a difference from that in channel flows (2.2) on account of the existence of the streamwise gradient:
\begin{equation} \dfrac{\partial\langle-uv\rangle}{\partial y}=\langle v\omega_z\rangle+\langle-w\omega_y\rangle+\dfrac{1}{2}\dfrac{\partial}{\partial x}\langle u^2-v^2-w^2\rangle. \end{equation}Substituting (D2) into (D1), we get
\begin{equation} \langle v\omega_z\rangle+\langle-w\omega_y\rangle+\dfrac{1}{Re_\delta}\dfrac{\partial^2\langle u\rangle}{\partial y^2}+\left\langle-\dfrac{\partial p}{\partial x}\right\rangle+I_x=0, \end{equation}where
\begin{equation} I_x=\dfrac{1}{Re_\delta}\dfrac{\partial^2\langle u\rangle}{\partial x^2}-\dfrac{\partial\langle E\rangle}{\partial x} \end{equation}
is an extra term owing to inhomogeneity in the streamwise direction, and 
$E=\frac {1}{2}(u^2+v^2+w^2)$ is the kinetic energy. Integrating (D3) in the wall-normal direction from 0 to an arbitrary height 
$y\in (0,1]$ and substituting 
$C_f=2({\partial \langle u\rangle }/{\partial y})|_{y=0}/Re_\delta$, we can obtain
\begin{align} C_f=\underbrace{2\int_0^y\langle v\omega_z\rangle\,\mathrm{d} y}_{C_{f1}}+ \underbrace{2\int_0^y\langle-w\omega_y\rangle\,\mathrm{d} y}_{C_{f2}}+ \underbrace{\dfrac{2}{Re_\delta}\dfrac{\partial\langle u\rangle}{\partial y}}_{C_{f3}} +\underbrace{2\int_0^y\left\langle-\dfrac{\partial p}{\partial x} \right\rangle\,\mathrm{d} y}_{C_{f4}}+\underbrace{2\int_0^yI_x\,\mathrm{d} y}_{C_{f5}}. \end{align}This equation is another form of (2.4) in the case of boundary layer flows. The first three terms in the right-hand side of (D5) are similar to those in (2.4) with respect to forms and interpretations. Here, 
$C_{f4}$ in (D5) represents contributions of pressure gradient to the skin friction and 
$C_{f5}$ measures contributions of the inhomogeneous effect in the streamwise direction. Compared with (2.4), each decomposition term and 
$C_f$ in (D5) are also functions of 
$x$, since we did not take the average over the streamwise direction. The applicable scope of (D5) should satisfy the homogeneity in the spanwise direction.
Appendix E. Filtering strategy using the constant cutoff wavelength
The cutoff wavelength 
$\alpha =0.5$ is plotted as red dashed lines in pre-multiplied spanwise co-spectra of 
$v\omega _z$ and 
$-w\omega _y$, see figure 16. With respect to the co-spectra of velocity–vorticity correlations, coupling the dynamics of velocity and vorticity structures, the low-pass filter with this cutoff wavelength fails to properly extract the small- and large-scale motions, since the proportion of scales larger than 
$\lambda _z^*=\alpha h^*$ appears to merely change while part of the smaller scales keeps growing with the increase of 
$Re_\tau$, regardless of different 
$\alpha$.
Figure 16. Pre-multiplied spanwise co-spectra (a–c) 
$yk_z\phi _{v\omega _z}$ and (d–f) 
$yk_z\phi _{-w\omega _y}$. (a,d) 
$Re$180; (b,e) 
$Re$950; (c,f) 
$Re$2000. The red dashed line indicates 
$\lambda _z^*=0.5h^*$.
Appendix F. Profiles of $C_{fi}^{PS}$ and $C_{fi}^{NS} (i=1, 2)$
Profiles of 
$C_{f1}$ and 
$C_{f2}$ in PSs and NSs are displayed in figure 17. Owing to the counteracting effect of positive and negative events in PSs, the range of 
$C_{f1}^{PS}$ is smaller than 
$C_{f1}^{NS}$, indicating that 
$C_{f1}^{NS}$ contributes to the majority of 
$C_{f1}$. In contrast, 
$C_{f2}^{PS}$ contributes most to 
$C_{f2}$. By definition, NSs actually correspond to the local smaller-scale motions, which principally contribute negatively to 
$C_{f}$. However, it might not be a feasible approach to retard skin-friction drag by attempting to intensify motions in NSs (granted that this could be accomplished), since 
$C_{f1} + C_{f2}$ is essentially the Reynolds stress at height 
$y$ (0 at 
$y=1$) and when NSs are intensified, PSs will also come along to ensure the summation is equal to the Reynolds stress. Now that the drag reduction strategy is not the issue of this work, we will not concentrate on this too much.
Figure 17. Profiles of (a) 
$C_{f1}^{PS}$, (b) 
$C_{f1}^{NS}$, (c) 
$C_{f2}^{PS}$ and (d) 
$C_{f2}^{NS}$ versus 
$y^+$.
