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

(4.1)\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.,

(4.2a)$$\begin{gather} {C_{f1}^{PS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{>}3.75y^+}yk_z \phi_{v\omega_z}\,\mathrm{d}\ln \lambda_z^+\mathrm{d}\ln y^+, \end{gather}$$

(4.2b)$$\begin{gather}{C_{f1}^{NS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{<}3.75y^+}yk_z\phi_{v\omega_z}\,\mathrm{d} \ln \lambda_z^+\mathrm{d}\ln y^+, \end{gather}$$

(4.2c)$$\begin{gather}{C_{f2}^{PS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{>}3.75y^+}yk_z\phi_{-w\omega_y}\, \mathrm{d}\ln \lambda_z^+\mathrm{d}\ln y^+, \end{gather}$$

(4.2d)$$\begin{gather}{C_{f2}^{NS}}(y^+)=\int_0^{y^+}\int_{\lambda_{z}^+{<}3.75y^+}yk_z\phi_{-w\omega_y}\,\mathrm{d}\ln \lambda_z^+\mathrm{d}\ln y^+. \end{gather}$$

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

(4.3a)$$\begin{gather} C_{f1}^{Qi}=\int_{0}^{y}y\langle v'\omega_z'\rangle_{Qi}\, \mathrm{d}\ln y, \end{gather}$$

(4.3b)$$\begin{gather}C_{f2}^{Qi}=\int_{0}^{y}y\langle-w'\omega_y'\rangle_{Qi}\, \mathrm{d}\ln y. \end{gather}$$

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 $Re_\tau$.