4.1 One-dimensional spectra

Streamwise and spanwise 1-D spectra of the decomposed velocity fluctuations ( $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ ) are performed to statistically quantify the scales of the turbulent motions represented by modes. In order to provide a compact analysis, we report only the results for the two higher-Reynolds-number cases (Re480 and Re810). For case Re180, the spectral content of LSMs overlaps with the near-wall cycle (Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010; Smits et al. Reference Smits, McKeon and Marusic2011), and inner and outer scales are not clearly separated. The 1-D spectra for case Re180 can be found in appendix C. They will be used in §§ 4.2 and 5 to investigate the relationship between modes and attached eddies and the Reynolds shear stress carried by eddies with different scales, respectively.

The following discussions are organized around the mode number. For the first two modes, lengths are normalized using inner scaling ( $\unicode[STIX]{x1D6FF}_{v}=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ ) denoted by $^{+}$ . For the third and fourth modes, lengths are scaled in outer units ( $\unicode[STIX]{x1D6FF}=h$ ).

4.1.1 First mode

The spanwise premultiplied 1-D spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ for the first mode are displayed in figure 3. Figure 3(a)–(c) show the spanwise spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for case Re480, and figure 3(d)–(f) show the spanwise spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for case Re810. The contour levels of the spectra shown in figure 3 and the following figures 4–11 indicate 0.17, 0.33, 0.50, 0.67 and 0.84 of their maximum, respectively. The results reveal that the spectra of $u_{d}^{\prime }$ and $w_{d}^{\prime }$ are located within the near-wall region and peak at $y^{+}\approx 12$ , whereas the spectra of $v_{d}^{\prime }$ are more intense across the buffer layer $y^{+}\approx 45$ . The spanwise length scale $\unicode[STIX]{x1D706}_{z}^{+}$ of all decomposed velocity fluctuations peaks at $\unicode[STIX]{x1D706}_{z}^{+}\approx 100$ , which coincides with near-wall coherent structures reported in previous studies (Smith & Metzler Reference Smith and Metzler1983; Kim et al. Reference Kim, Moin and Moser1987). Similar features are observed in the Re180 case, as shown in appendix C.

Figure 3.

The spanwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the first mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

Figure 4.

The streamwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the first mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

The streamwise premultiplied 1-D spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ for the first mode are displayed in figure 4, which are equivalent to figure 3. For $u_{d}^{\prime }$ , the streamwise spectra span a wide range of scales in the streamwise direction $\unicode[STIX]{x1D706}_{x}^{+}\approx 200{-}700$ and peak at $y^{+}\approx 12$ . On the contrary, the spectra for $v_{d}^{\prime }$ and $w_{d}^{\prime }$ cover a shorter range of scales $\unicode[STIX]{x1D706}_{x}^{+}\approx 150{-}200$ . The statistical characteristics of $u_{d}^{\prime }$ obtained for the first mode are consistent with previous studies (Jiménez & Moin Reference Jiménez and Moin1991; Jiménez & Pinelli Reference Jiménez and Pinelli1999) and the results by Hwang (Reference Hwang2013), who conducted a numerical experiment to remove all the turbulent motions wider than $\unicode[STIX]{x1D706}_{z}^{+}\approx 100$ and found that the streamwise spectra of $u^{\prime }$ have the largest value at $y^{+}\approx 10{-}20$ and $\unicode[STIX]{x1D706}_{x}^{+}\approx 600$ .

Figure 5.

The spanwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the second mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

Figure 6.

The streamwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the second mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

Therefore, the turbulent motions identified by the first mode are representative of the near-wall coherent structures. The streamwise spectra of $u_{d}^{\prime }$ agree with the known properties of the near-wall streaks, whereas the spectra of $v_{d}^{\prime }$ and $w_{d}^{\prime }$ mainly characterize quasi-streamwise vortices. This will be further discussed in § 4.1.2.

4.1.2 Second mode

The spanwise and streamwise spectra of all decomposed velocity fluctuations for the second mode are shown in figures 5 and 6, respectively. The spectra of the second mode follow a distribution similar to the first mode except for their locations and length scales, which are slightly larger and located further from the wall. The spectra for $u_{d}^{\prime }$ peaks at $y^{+}\approx 16{-}20$ with a wide range of streamwise length scales from $\unicode[STIX]{x1D706}_{x}^{+}\approx 200$ to $1100$ , while the spanwise sizes remain shorter, $\unicode[STIX]{x1D706}_{z}^{+}\approx 150{-}180$ . The sizes are again consistent with the length scale of the near-wall streaks ( $\unicode[STIX]{x1D706}_{x}^{+}\approx 1000$ ) (Hwang Reference Hwang2013). The spectra of $v_{d}^{\prime }$ and $w_{d}^{\prime }$ are more isotropic with peaks located at $\unicode[STIX]{x1D706}_{x}^{+}\approx 180{-}250$ , $\unicode[STIX]{x1D706}_{z}^{+}\approx 150{-}200$ and $y^{+}\approx 40{-}85$ , in agreement with the observations of near-wall quasi-streamwise vortices by Kim et al. (Reference Kim, Moin and Moser1987) and Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997), among others. The spectra show a weak dependence on $Re_{\unicode[STIX]{x1D70F}}$ , which is acceptable considering, first, that BEMD is a data-driven method which does not contain any explicit information regarding inner or outer units, and, second, the modulation effect of the large scales on the near-wall region for increasing Reynolds number as argued by Mathis, Hutchins & Marusic (Reference Mathis, Hutchins and Marusic2009, Reference Mathis, Hutchins and Marusic2011) and Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). Although both the first and second modes share similar scale and shape, their spanwise length scales are slightly different, $\unicode[STIX]{x1D706}_{z}^{+}=100$ and $\unicode[STIX]{x1D706}_{z}^{+}=150{-}200$ , respectively, which could be indicative of different dynamical structures involved in near-wall cycles within $y^{+}<100$ .

The message from the analysis of the first two modes should be that they characterize velocity fluctuations within the buffer layer, and that their sizes scale in wall units. Both the first and second modes represent long streaky structures for $u_{d}$ , and short vortex packets for $v_{d}$ and $w_{d}$ . The main difference between the two modes is that the first mode captures the flow structure associated with the near-wall streaks at $y^{+}\approx 13$ , while the second mode contains larger eddies, although still confined within the buffer layer. The results are consistent with the bimodal behaviour of near-wall velocity components (Jiménez, Del Álamo & Flores Reference Jiménez, Del Álamo and Flores2004).

Figure 7.

The spanwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the third mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

Figure 8.

The streamwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the third mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ , and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

4.1.3 Third mode

The spanwise and streamwise spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ for the third mode are shown in figures 7 and 8, respectively. The spectral peaks of $u_{d}^{\prime }$ and $w_{d}^{\prime }$ are still located within the buffer layer at $y^{+}\approx 15$ for $u_{d}^{\prime }$ and $y^{+}\approx 40{-}50$ for $w_{d}^{\prime }$ . However, the top parts of both spectra penetrate into the log layer ( $y^{+}>100$ ) while the bottom parts remain attached to the wall ( $y^{+}<15$ ). On the contrary, the spectral peak of $v_{d}^{\prime }$ can be found in the outer region ( $y\approx 0.25h$ ) and the $v_{d}^{\prime }$ spectra remain detached from the wall. It was argued by Townsend (Reference Townsend1976) that the only limiting mechanism of the size of energy-containing structures is the presence of the wall, and that the largest energetic eddies would grow until they were wall-attached. On the other hand, the impermeability condition prevents the wall-normal velocity from extending close to $y/h=0$ , hence remaining detached. Our outcome is consistent with Townsend’s argument, and with the work by Jiménez (Reference Jiménez2013), who showed that the spectra of the wall-parallel velocities and the pressure are attached, while the wall-normal velocity and the tangential Reynolds stress are detached. Hwang (Reference Hwang2015) also argued that the penetration of $u_{d}^{\prime }$ close to the wall is accompanied by a non-negligible $w_{d}^{\prime }$ in the same region due to continuity.

Analogously to the results reported for the first two modes, all decomposed velocity fluctuations share comparable spanwise wavelengths, although in this case the characteristic lengths scale in outer units rather than inner units, i.e. $\unicode[STIX]{x1D706}_{z}\approx 0.5h$ . Similarly, in spite of the elongated spectral tail of $u_{d}^{\prime }$ in the streamwise direction, the spectra of the three velocities exhibit streamwise maxima close to $\unicode[STIX]{x1D706}_{x}\approx (0.5{-}0.67)h$ . Therefore, the energy-containing motions represented by the third mode are characterized by $\unicode[STIX]{x1D706}_{x}\sim \unicode[STIX]{x1D706}_{z}$ .

The findings above suggest that the third mode is representative of self-similar Townsend eddies. Our results are consistent with the studies of Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017), who showed that the momentum-carrying eddies in the log layer have essentially isotropic sizes, and that these are composed of a cross-flow roll ( $v,w$ ) associated with a longer streak ( $u$ ) whose strong part is still comparable to the roll size. The results also agree with Hwang (Reference Hwang2015) and de Giovanetti et al. (Reference de Giovanetti, Hwang and Choi2016), who interpreted the long streaky motions as an ‘inactive’ part of the attached eddies transporting streamwise momentum to the region close to the wall. Note that only one single mode was identified in the log layer. The total number of the BEMD modes is intimately related to the Reynolds number. Increasing the Reynolds number would increase the number of BEMD modes in the log layer. Recently, Agostini & Leschziner (Reference Agostini and Leschziner2018) adopted BEMD to decomposed the streamwise velocity fluctuations at $Re_{\unicode[STIX]{x1D70F}}=1020$ and six modes were identified.

Figure 9.

The spanwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the fourth mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

4.1.4 Fourth mode

Figure 9 shows the spanwise spectra of the decomposed velocity fluctuations for the fourth mode, computed as the final residual of the BEMD. A striking phenomenon observed in figure 9 is that the spanwise spectra of $u_{d}^{\prime }$ peak at two wall-normal locations with approximately the same spanwise length scale, i.e. $\unicode[STIX]{x1D706}_{z}\approx 1h$ , which is characteristic of the LSMs and VLSMs as documented by Del Álamo et al. (Reference Del Álamo, Jiménez, Zandonade and Moser2004), Tomkins & Adrian (Reference Tomkins and Adrian2005) and Hutchins & Marusic (Reference Hutchins and Marusic2007). The first peak is located in the near-wall region ( $y^{+}\approx 15$ ), whereas the second peak appears in the outer region ( $y>0.2h$ ). The latter becomes more discernible as the Reynolds number increases, and it is not observed for case Re180 (figure 23 g). This may indicate that the energy-containing motions in the outer region become more energetic as the Reynolds number increases. As observed for the third mode, the spectra of $u_{d}^{\prime }$ and $w_{d}^{\prime }$ reach deep into the near-wall region, but the spectra of $v_{d}^{\prime }$ are mostly detached from the wall.

Figure 10 shows the streamwise spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ for the fourth mode. The spectral peaks of $v_{d}^{\prime }$ and $w_{d}^{\prime }$ are located at $\unicode[STIX]{x1D706}_{x}\approx (1.2{-}1.4)h$ , which coincides with the typical sizes of LSMs (Lee et al. Reference Lee, Lee, Choi and Sung2014; de Giovanetti, Sung & Hwang Reference de Giovanetti, Sung and Hwang2017). On the other hand, the streamwise spectra of $u_{d}^{\prime }$ have their maxima at $\unicode[STIX]{x1D706}_{x}\approx 6h$ , which is indicative of long streaky structures carrying streamwise turbulent kinetic energy (Lee et al. Reference Lee, Lee, Choi and Sung2014). This last observation is consistent with VLSMs, which are known to prevail over a wide range of wall-normal locations, from near-wall to outer regions (Kim Reference Kim1999; Jin, Ahn & Sung Reference Jin, Ahn and Sung2015). From the above discussions, we may conclude that the fourth mode characterizes LSMs and VLSMs. Finally, some wiggles can be observed in the spectra of streamwise velocity in figure 10. Since the last mode is obtained as the final residual of the decomposition, these wiggles may be the natural outcome of the BEMD procedure.

Figure 10.

The streamwise premultiplied 1-D spectra of the decomposed velocity fluctuations in the fourth mode: (a–c) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re480 case; (d–f) spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ , respectively, for the Re810 case.

4.2 Relationship between modes and attached eddies

The velocity fluctuations are decomposed into several modes with typical spanwise length scales. Figure 11 shows the spanwise spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ contained in each mode for Re810, compared with the full spectra calculated from full data. It is evident that the original spectra are covered by modes at different spanwise length scales and wall-normal positions, suggesting that the energy-containing motions are organized into several groups with different characteristic length scales. It is interesting that the decomposed results are consistent with the work of De Silva, Marusic & Hutchins (Reference De Silva, Marusic and Hutchins2016), who proposed that the velocity field at $Re_{\unicode[STIX]{x1D70F}}=800$ can be synthesized by $\unicode[STIX]{x1D6EC}$ -eddies with four hierarchies.

Figure 11.

The spanwise spectra of $u_{d}^{\prime },~v_{d}^{\prime }$ and $w_{d}^{\prime }$ in modes together with the full spectra calculated from full data for Re810: (a) spectra of $u_{d}^{\prime }$ , (b) spectra of $v_{d}^{\prime }$ , and (c) spectra of $w_{d}^{\prime }$ . Black line, full spectra; red line, mode 1; green line, mode 2; blue line, mode 3; magenta line, mode 4.

The 1-D spectra shown in the present study indicate that the modes of the fluctuating velocities represent a set of flow motions with very similar spanwise sizes scaling in either inner or outer units. For each mode, we have argued that $v_{d}^{\prime }$ and $w_{d}^{\prime }$ can be assumed to represent the short and tall vortical structures, while $u_{d}^{\prime }$ is mainly linked to long streaky structures, although with some contribution from vortex packets. The first two modes characterize the near-wall streaks and quasi-streamwise vortices, and the third and fourth modes represent larger streaky structures and vortex packets spanning the logarithmic and outer regions. These suggest that each BEMD mode represents a typical attached eddy characterized by its spanwise length scale.

Figure 12.

Profiles of the normalized root mean squares of the decomposed velocity fluctuations as a function of wall-normal distances rescaled with the spanwise length $L_{z}$ of each mode: (a) streamwise component, (b) wall-normal component, and (c) spanwise component.

Figure 12 shows the root mean square (r.m.s.) of the decomposed velocity fluctuations ( $u_{d,rms}^{\prime }$ , $v_{d,rms}^{\prime }$ and $w_{d,rms}^{\prime }$ ) for the three cases. The velocity fluctuations are normalized with their peak values, and the wall-normal distances are rescaled by the value of $\unicode[STIX]{x1D706}_{z}$ where the spectrum peaks for each individual mode. The normalization is consistent with idealized Townsend eddies as illustrated in Townsend (Reference Townsend1976, p. 155), which shows that the second-order moments of attached eddies should be self-similar with respect to the wall-normal distance $y^{\ast }=y/y_{a}$ , where $y_{a}$ is the location of the maximum intensity of the eddy. Figure 12(a) shows the profiles of $u_{d,rms}^{\prime }$ . The first two modes collapse for $y\leqslant 0.2L_{z}$ , but the last mode for case Re180 and the last two modes in the Re480 and Re810 cases do not. This lack of self-similarity for $u_{d,rms}^{\prime }$ is consistent with the classic inner scaling ascribed to the influence of viscosity close to the wall. Figure 13 displays $u_{d,rms}^{\prime }$ scaled by inner units and shows that the profiles collapse below $y^{+}\approx 15$ , confirming that the near-wall part of $u_{d,rms}^{\prime }$ in each mode is dominated by viscous effects. However, recent studies have revealed that the near-wall footprint of LSMs scales in inner units (Hwang Reference Hwang2016; Cho, Hwang & Choi Reference Cho, Hwang and Choi2018; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). This may not be evident from figure 13, which is probably dominated by smaller-scale contributions.

Figure 13.

Profiles of the normalized root mean squares of the streamwise velocity fluctuations as a function of wall-normal distances rescaled with the inner scale $\unicode[STIX]{x1D6FF}_{v}$ of each mode. Lines and symbols are as in figure 12.

Figure 12(b,c) display the profiles of $v_{d,rms}^{\prime }$ and $w_{d,rms}^{\prime }$ , respectively. Excluding the last mode of the case Re180 and Re480, all profiles are self-similar within $y\leqslant 0.5L_{z}$ for $v_{d,rms}^{\prime }$ , and $y\leqslant 0.6L_{z}$ for $w_{d,rms}^{\prime }$ . The breakdown of self-similarity beyond these heights could be induced by the interaction between turbulent motions characterized by different length scales. Moreover, the deviation observed for the last mode is probably caused by the fact that it is obtained as the final residual in the BEMD procedure and, therefore, it is prone to mode mixing and statistical errors. Additionally, previous works have shown that large-scale structures are mainly ‘inactive’ (Perry & Chong Reference Perry and Chong1982; Perry et al. Reference Perry, Henbest and Chong1986; Hwang & Sung Reference Hwang and Sung2018), which could also contribute to the breakdown of self-similarity reported above.

To conclude the section, it should also be noted that detached flow motions are not energetic enough to be captured by the velocity spectra, although they might be contained in the first EMD mode. As shown by Del Álamo et al. (Reference Del Álamo, Jiménez, Zandonade and Moser2006) and Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012), the sizes of detached eddies are roughly $20\unicode[STIX]{x1D702}$ , where $\unicode[STIX]{x1D702}$ is the Kolmogorov length scale, and their streamwise and spanwise length scales are smaller than 100 inner units. They are responsible for the energy cascade and dissipation processes, but they do not carry energy nor contribute to the bulk momentum transfer. The characteristics discussed above support the modes obtained with the BEMD as representative of Townsend attached eddies.

4.3 Geometrical properties of attached eddies

The geometrical properties of attached eddies can be estimated by the location of the maximum in the premultiplied spectra (see appendix D). Considering that a single attached eddy is composed of two elements, i.e. vortex packets and streaky structures, the sizes of the former may be approximated from the spectra of $v_{d}^{\prime }$ as

(4.1a,b ) $$\begin{eqnarray}\displaystyle y\approx (0.3{-}0.6)\unicode[STIX]{x1D706}_{z}\quad \text{and}\quad \unicode[STIX]{x1D706}_{x}\approx (1{-}2)\unicode[STIX]{x1D706}_{z}, & & \displaystyle\end{eqnarray}$$

where $y$ and $\unicode[STIX]{x1D706}_{x}$ are measured with respect to the peak of $\unicode[STIX]{x1D706}_{z}$ for each eddy. The estimation is roughly consistent with the result of Hwang (Reference Hwang2015) ( $y\approx (0.5{-}0.7)\unicode[STIX]{x1D706}_{z}$ and $\unicode[STIX]{x1D706}_{x}\approx (2{-}3)\unicode[STIX]{x1D706}_{z}$ ). Del Álamo et al. (Reference Del Álamo, Jiménez, Zandonade and Moser2006) reported a similar relation for the attached clusters ( $\unicode[STIX]{x1D706}_{x}\approx 2\unicode[STIX]{x1D706}_{z}$ ). Analogously, the geometry of the streaky structures can be represented by

(4.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{x}\approx (1{-}7)\unicode[STIX]{x1D706}_{z}. & & \displaystyle\end{eqnarray}$$

The geometrical properties of the longest near-wall streaks ( $\unicode[STIX]{x1D706}_{x}\approx 7\unicode[STIX]{x1D706}_{z}$ ) estimated in the present study are comparable to the results given by Hwang (Reference Hwang2015) ( $\unicode[STIX]{x1D706}_{x}\approx 10\unicode[STIX]{x1D706}_{z}$ ) and nearly identical to the value proposed by Jiménez (Reference Jiménez2018) ( $\unicode[STIX]{x1D706}_{x}\approx 8\unicode[STIX]{x1D706}_{z}$ ), providing further evidence for the connection between attached eddies and BEMD modes.

4.4 Near-wall streamwise turbulence intensity

Based on the idea that the attached eddy motions in the logarithmic and outer regions impose a forcing on the buffer and viscous sublayer, Marusic & Kunkel (Reference Marusic and Kunkel2003) proposed a new formulation to describe the similarity of the streamwise turbulence intensity for a smooth-wall zero-pressure-gradient turbulent boundary layer. The formulation reads as $u_{rms}/u_{\unicode[STIX]{x1D70F}}=\sqrt{f_{1}(y^{+})\,f_{T}(y^{+},Re_{\unicode[STIX]{x1D70F}})}$ , where $f_{1}(y^{+})$ is a universal function representing the turbulence fluctuations taking place in the inner region of the boundary layer, and $f_{T}(y^{+},Re_{\unicode[STIX]{x1D70F}})$ denotes a force function accounting for the influence of the outer layer. The empirical curve for $f_{1}(y^{+})$ is given by

(4.3) $$\begin{eqnarray}\displaystyle f_{1}(y^{+})=\frac{a_{1}(y^{+})^{2}}{(1+a_{2}(y^{+})^{2})^{1/2}(1+a_{3}(y^{+})^{2a_{4}})^{1/2}}, & & \displaystyle\end{eqnarray}$$

where the fitting constants are $a_{1}=0.16$ , $a_{2}=0.008$ , $a_{3}=0.115$ and $a_{4}=1.6$ (Marusic & Kunkel Reference Marusic and Kunkel2003). By numerically removing turbulent motions wider than $\unicode[STIX]{x1D706}_{z}^{+}\approx 100$ , Hwang (Reference Hwang2013) proposed a new set of fitting constants, i.e. $a_{1}=0.113$ , $a_{2}=0.008$ , $a_{3}=0.10$ and $a_{4}=1.9$ .

In the present study, we have shown that the first mode characterizes the smallest attached eddies in the near-wall region with $\unicode[STIX]{x1D706}_{z}^{+}\approx 100$ . Then, $u_{d,rms}^{\prime }$ associated with the first mode can be used to tune the empirical curve (4.3). Here, we propose new fitting constants based on the BEMD results given by

(4.4a-d ) $$\begin{eqnarray}\displaystyle a_{1}=0.122,\quad a_{2}=0.046,\quad a_{3}=0.078,\quad a_{4}=2.213. & & \displaystyle\end{eqnarray}$$

Figure 14 shows that the fitting curve using the constants in (4.4) is in good agreement with $u_{d,rms}^{\prime }$ of the first mode over a wide range of wall-normal distances. The amplitude of the curve proposed is smaller than that in Marusic & Kunkel (Reference Marusic and Kunkel2003) and Hwang (Reference Hwang2013). Note that the curve in Marusic & Kunkel (Reference Marusic and Kunkel2003) contains the outer-flow effects, and the curve in Hwang (Reference Hwang2013) relies on the ‘artificial’ numerical experiments with enforced scale-based truncation, in which the interaction between energy-containing motions near the cutoff wavelength may not be accurately resolved (de Giovanetti et al. Reference de Giovanetti, Hwang and Choi2016).

Figure 14.

Streamwise velocity fluctuations in the first mode and fitting curves.

5.1 FIK identity and scale-targeted Reynolds shear stress

The FIK identity, first proposed by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002), decomposes the mean skin-friction drag coefficient into a laminar and a turbulent component. For turbulent channel flows, it is expressed as

(5.1) $$\begin{eqnarray}\displaystyle C_{f,FIK}=\underbrace{\frac{12}{Re_{m}}}_{C_{f1}}+\underbrace{6\int _{0}^{h}\left(1-\frac{y}{h}\right)\left(-\frac{\overline{u^{\prime }v^{\prime }}}{U_{m}^{2}}\right)\text{d}y}_{C_{f2}}, & & \displaystyle\end{eqnarray}$$

where $U_{m}$ is the bulk velocity, and $Re_{m}=2U_{m}h/\unicode[STIX]{x1D708}$ . The first term $C_{f1}$ is the laminar drag at the same Reynolds number, and thus it is referred to as the laminar contribution. The second term $C_{f2}$ accounts for the drag by the turbulent fluctuations, i.e. the integrated Reynolds stress over the boundary layer thickness, and consequently it is termed the turbulent contribution. The FIK identity has been adopted in previous works to investigate the contribution of large-scale structures to the skin-friction generation (Deck et al. Reference Deck, Renard, Laraufie and Weiss2014; de Giovanetti et al. Reference de Giovanetti, Hwang and Choi2016).

The mean skin-friction drag coefficient estimated from (5.1) is compared with the exact values and Dean’s empirical correlation ( $C_{f}=0.073Re_{m}^{-0.25}$ ) (Dean Reference Dean1978) in figure 15. The exact $C_{f}$ is directly calculated from the normal gradient of the mean streamwise velocity $\langle U\rangle$ at the wall, $C_{f}=(2/Re_{m})(\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}y|_{y=0})$ . The agreement with Dean’s empirical correlation is reasonable, and $C_{f,FIK}$ matches the exact $C_{f}$ within 4 % error.

Figure 15.

Comparison of $C_{f,FIK}$ with exact $C_{f}$ and Dean’s empirical correlation.

In § 4, we have identified attached eddies by applying BEMD to the full turbulent flow. Based on this scale decomposition of attached eddies, the streamwise and wall-normal velocity fluctuations of the flow fields can be expressed as

(5.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}u^{\prime }=u_{t}^{\prime }+u_{o}^{\prime },\\ v^{\prime }=v_{t}^{\prime }+v_{o}^{\prime },\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where the subscript $t$ stands for attached eddies with a targeted length scale $\unicode[STIX]{x1D706}_{z}$ , and the subscript $o$ stands for the remaining attached eddies. Then, the Reynolds shear stress can be written as

(5.3) $$\begin{eqnarray}\displaystyle -\overline{u^{\prime }v^{\prime }}=-\overline{(u_{t}^{\prime }+u_{o}^{\prime })(v_{t}^{\prime }+v_{o}^{\prime })}. & & \displaystyle\end{eqnarray}$$

Using the Reynolds averaging operator, equation (5.3) can be expressed as

(5.4) $$\begin{eqnarray}\displaystyle -\overline{u^{\prime }v^{\prime }}=\underbrace{-\overline{u_{t}^{\prime }v_{t}^{\prime }}}_{tt}\underbrace{-\overline{u_{o}^{\prime }v_{o}^{\prime }}}_{oo}\underbrace{-\overline{u_{t}^{\prime }v_{o}^{\prime }}}_{to}\underbrace{-\overline{u_{o}^{\prime }v_{t}^{\prime }}}_{ot}, & & \displaystyle\end{eqnarray}$$

splitting the Reynolds shear stress into four different contributions. The first term on the right-hand side of (5.4) represents the Reynolds stress produced by the targeted attached eddy, the second term indicates the Reynolds shear stress produced by attached eddies excluding the targeted one, and the last two terms are the Reynolds shear stress generated by their interactions. The decomposition above allows for the evaluation of the skin friction generated by different scale-targeted Reynolds shear stresses.

5.2 Contribution of near-wall eddies to the skin friction

Figure 16.

Variation of the four contributing terms in (5.4) when the near-wall attached eddies are set as the targeted eddy: (a) Re180; (b) Re480; (c) Re810.

We assess the contribution to the skin-friction generation of the near-wall attached eddies within the buffer layer. These eddies are characterized by $\unicode[STIX]{x1D706}_{z}^{+}\approx 100{-}200$ and are represented by the first two modes as shown in § 4. The analysis is relevant since most of the drag-reduction approaches, such as spanwise-oscillating walls (Jung, Mangiavacchi & Akhavan Reference Jung, Mangiavacchi and Akhavan1992), near-wall blowing and suction (Choi, Moin & Kim Reference Choi, Moin and Kim1994), and micro-grooved structures or riblets (Choi, Moin & Kim Reference Choi, Moin and Kim1993), attempt to manipulate turbulent structures within the near-wall region.

Figure 16 shows the variation of the four different terms in (5.4). The results are normalized using $u_{\unicode[STIX]{x1D70F}}$ . As expected, the sum of the four terms shows excellent agreement with the full Reynolds shear stress. The Reynolds shear stress generated by the near-wall (targeted) attached eddies reaches its maximum at $y^{+}\approx 20$ , regardless of the Reynolds number. The remaining contribution is dominated by eddies in the logarithmic and outer regions, consistent with the observations of the 1-D spectra in § 4.

The corresponding skin-friction generation obtained by integrating the various Reynolds shear stress components over the boundary layer is illustrated in figure 17. As the Reynolds number increases, the skin friction generated by the smallest attached eddies decreases significantly from 52.4 % in case Re180 to 34.3 % in case Re810, whereas other attached eddies play a more prominent role, contributing 12 %, 40 % and 43.7 % of the total skin friction for cases Re180, Re480 and Re810, respectively. This implies that special attention should be paid to the energy-containing motions residing in the logarithmic and outer regions for turbulent drag reduction at high Reynolds numbers. With respect to the contribution of interacting eddies, their values are not negligible and they still account for approximately 10 % of the skin-friction generation.

Figure 17.

Decomposition of $C_{f}$ when the near-wall eddies are set as the target eddy: (a) Re180; (b) Re480; (c) Re810. (d) The contributions of each term as a function of $Re_{\unicode[STIX]{x1D70F}}$ .

5.3 Contribution of LSMs and VLSMs to the skin friction

As the Reynolds number increases, the thickness of the near-wall layer, and hence the size associated with the structures populating this region, become smaller (Robinson Reference Robinson1991; Gad-el Hak Reference Gad-el Hak1994; Abbassi et al. Reference Abbassi, Baars, Hutchins and Marusic2017). Consequently, the efficiency of near-wall drag-reduction techniques degrades significantly for Reynolds numbers typical of real-world applications (Gatti, Quadrio & Frohnapfel Reference Gatti, Quadrio and Frohnapfel2015; Encinar, Lozano-Durán & Jiménez Reference Encinar, Lozano-Durán and Jiménez2018). Hwang (Reference Hwang2013) pointed out that, for increasing Reynolds numbers, the contribution of the near-wall small-scale structures to the skin friction decays, while the contribution of the LSMs increases due to the amplitude modulation of the small scales and the energy superposition onto the inner region. Thus, evaluating the contribution of the large-scale turbulent motions to the skin-friction generation is necessary to aid the development of new drag-reduction strategies. To that end, we evaluate the drag contribution from LSMs and VLSMs represented by the third BEMD mode in the Re180 case, and the fourth mode in the Re480 and Re810 cases, which are set as targeted eddies.

Figure 18.

Variation of the four contributing terms in (5.4) when the largest attached eddy is set as the target eddy: (a) Re180; (b) Re480; (c) Re810.

Figure 18 shows the variation of the four different terms in (5.4). The Reynolds shear stress generated by LSMs and VLSMs reaches its maximum in the outer region ( $y\approx (0.3{-}0.4)h$ ). The results are consistent with the attached eddy framework; the attached part of the eddies mainly consists of the streamwise and spanwise components, while the wall-normal velocity fluctuations are small due to the constraint of the wall. This makes the attached part of the eddies ‘inactive’, i.e. barely carrying Reynolds stress in the near-wall region (Townsend Reference Townsend1976; Hwang Reference Hwang2015). Recently, de Giovanetti et al. (Reference de Giovanetti, Hwang and Choi2016) suggested that, even if the attached part is ‘inactive’, the streamwise momentum in the logarithmic and outer regions can still be transported to the near-wall region and interact with the wall-detached wall-normal component. This action results in high values of Reynolds shear stress observed far away from the near-wall region. The evidence presented here regarding the distribution of $-\overline{u_{t}^{\prime }v_{t}^{\prime }}$ in figure 18 supports this scenario. The Reynolds shear stress generated by the remaining attached eddies peaks in the buffer layer, i.e. $y^{+}\approx 30$ for all cases.

Figure 19.

Decomposition of $C_{f}$ when the largest attached eddy is set as the target eddy: (a) Re180; (b) Re480; (c) Re810. (d) The contributions of each term with respect to $Re_{\unicode[STIX]{x1D70F}}$ .

Figure 19 displays the result of integrating the Reynolds shear stresses over the boundary layer. It can be seen that the largest attached eddies are responsible for 12 %, 20 % and 23 % of the total skin-friction generation for cases Re180, Re480 and Re810, respectively. The present results are in accordance with de Giovanetti et al. (Reference de Giovanetti, Hwang and Choi2016), who reported that the LSMs and VLSMs are responsible for 20–30 % of the total skin-friction drag. On the other hand, the contribution of other attached eddies is roughly 50 % of the total skin friction. Finally, the interaction between eddies accounts for 10 % of the skin-friction generation, which is smaller than previous contributions but cannot be ignored. The origin of this skin-friction generation due to nonlinear interactions can be found in the second component of the FIK identity. The Reynolds stress in $C_{f2}$ is not only originated from energy-containing motions themselves, but it is also related to the nonlinear interactions among them. Recently, Cho et al. (Reference Cho, Hwang and Choi2018) investigated the turbulent kinetic energy transfer thoroughly and showed that this process is highly associated with the skin-friction generation, consistent with the present results. Additionally, other works such as Mathis et al. (Reference Mathis, Hutchins and Marusic2009) and Mathis et al. (Reference Mathis, Hutchins and Marusic2011) have highlighted the interaction of large-scale with small-scale structures near the wall via amplitude modulation.