2.1. Turbulence statistics

Figure 2 shows the turbulence statistics of turbulent channel flows with slip and no-slip conditions. The quantities are normalized by the wall units of each case. The magnitude of streamwise mean velocity $({U^ + })$ is larger than that of the no-slip case due to the streamwise slip (figure 2a). The dashed line in figure 2(a) represents the mean velocity relative to the wall $({U^ + } - U_S^ + )$ for the slip case, which coincides with the profile of ${U^ + }$ for the no-slip case. The mean shear in the wall units $(\textrm{d}{U^ + }\textrm{/d}{y^ + })$ is insensitive to the streamwise slip. The streamwise slip induces a virtual origin in U ⁺ at ${y^ + } =-L_S^ +$, leading to the upward shift of U ⁺ as $\Delta {U^ + } = U_S^ + = L_S^ + $ in the entire region (García-Mayoral, Gómez-de-Segura & Fairhall Reference García-Mayoral, Gómez-de-Segura and Fairhall2019). The streamwise turbulence intensity $(u_{rms}^ + )$ is amplified with respect to that of the no-slip case below y ⁺ = 10, especially very close to the wall (figure 2c), due to the streamwise slip. A virtual origin for turbulence is absent due to the combined influences of wall-normal and spanwise slip lengths (Ibrahim et al. Reference Ibrahim, Gómez-de-Segura, Chung and García-Mayoral2021), showing almost the same inner peaks of $u_{rms}^ +$ (Kim, Moin & Moser Reference Kim, Moin and Moser1987). In contrast, a profile of Reynolds shear stress $({\langle - uv\rangle ^ + })$ is well matched with that for the no-slip case in the inner region as a result of an impermeable condition at the wall. For outer scaling, the profiles of $U_c^ +{-} {U^ + }$ (figure 2b), $u_{rms}^ +$ and ${\langle - uv\rangle ^ + }$ (figure 2d) for both cases collapse well in the outer region, supporting Townsend's outer-layer similarity hypothesis and in agreement with the results of Flores & Jiménez (Reference Flores and Jiménez2006) and Chung et al. (Reference Chung, Monty and Ooi2014). Here, U_c is the mean velocity at the channel centre.

Figure 2. Profiles of (a) mean velocity (U ⁺) and (b) defect form $(U_c^ +-{U^ + })$ of mean velocity, where U_c is the mean velocity at the channel centre. A dashed line in panel (a) represents mean velocity relative to wall $({U^ + }-U_S^ + )$ for the slip case. Profiles of streamwise turbulence intensity $(u_{rms}^ + )$ and Reynolds shear stress $({\langle - uv\rangle ^ + })$ for (c) inner scaling and (d) outer scaling.

3.1. Identification of coherent structures

The 3-D u clusters in instantaneous flow fields can be defined as the groups of connected points satisfying $u(\boldsymbol{x},t) \ge \alpha {u_{rms}}(y)$ and $u(\boldsymbol{x},t) \le - \alpha {u_{rms}}(y)$. Here, α is the threshold, which is selected from the percolation diagram in figure 3(a), and x and t represent the spatial vector and time, respectively. The total number (N) and total volume (V) of u clusters at a given α are normalized by the maximum N (N_max) and maximum V (V_max), respectively. As α decreases, new clusters appear or some adjacent clusters merge. The peak in N/N_max at α = 1.6 results from the tradeoff between these two influences; the former is dominant for α > 1.6 and vice versa. In addition, V/V_max increases as α decreases with strong variation in the region of 1.4 < α < 1.8, where the percolation transition occurs. Hence, we chose α = 1.6 in the present study.

Figure 3. (a) Percolation diagram for the detected u clusters. Variations with α in the total volume (V) and total number (N) of clusters. (b) Number of u clusters per unit wall-parallel area (n*) with respect to $y_{min}^ +$ and $y_{max}^ +$.

Based on this condition, each u cluster can be detected by using the connectivity of six-orthogonal grids at a given node in Cartesian coordinates (Moisy & Jiménez Reference Moisy and Jiménez2004; del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Hwang & Sung Reference Hwang and Sung2018; Lozano-Durán & Bae Reference Lozano-Durán and Bae2019; Hwang et al. Reference Hwang, Lee and Sung2020; Yoon et al. Reference Yoon, Hwang, Yang and Sung2020). As a result, the spatial information for each u cluster can be obtained. The u clusters with volumes less than 30³ wall units are discarded to avoid grid resolution (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006). Figure 3(b) shows the population density (n*) of all the identified u clusters as functions of y_max and y_min, which are the maximum and minimum distances from the wall, respectively. Here, n* is defined as the number of u clusters (n) per unit wall-parallel area (A_xz = L_xL_z), i.e. ${n^\ast } = n({y_{max}},{y_{min}})/(m{A_{xz}})$, where m is the number of snapshots (m = 1164 for the slip case and m = 1144 for the no-slip case). Two distinct groups are evident at $y_{min}^ +\approx 0$ and $y_{min}^ + > 0$, where the former are wall-attached structures and the latter are wall-detached structures (Hwang & Sung Reference Hwang and Sung2018; Yoon et al. Reference Yoon, Hwang, Yang and Sung2020).

The 3-D iso-surfaces of u of wall-attached and wall-detached structures are shown in figure 4(a) for the slip case and in figure 4(b) for the no-slip case. Blue and red correspond to negative u and positive u, respectively. The characteristic length scales of each u cluster can be defined in terms of the dimensions of the box circumscribing the object. The inset in figure 4(a) shows a wall-attached structure and its length scales, i.e. l_x, l_z and l_y, which are the streamwise and spanwise sizes and the height, respectively. As can be seen in figure 4, the wall-attached u structures are coherent in the streamwise direction and aligned side by side in the spanwise direction. At first sight, those for the slip case appear similar respectively to those for the no-slip case. However, upon closer inspection, differences in terms of the population density of the wall-attached structures are observed. The wall-attached structures are sparsely distributed for the slip case. The streamwise slip induces streamwise velocities at the wall (figure 2a) and enhances the magnitude of $u_{rms}^ +$ near the wall (figure 2c). We can observe sparse wall-attached structures in an instantaneous flow field despite the presence of slip at the wall. The wall-detached structures are less coherent than the wall-attached structures and are randomly distributed in the domain.

Figure 4. 3-D iso-surfaces of the wall-attached and wall-detached u structures in an instantaneous flow field for (a) the slip case and (b) the no-slip case.

3.2. Influences of streamwise slip on wall-attached structures

Figure 5 shows the joint probability density functions (JPDF) of $l_x^ +$ and $l_z^ +$ with respect to $l_y^ +$, which is the height of wall-attached structures (l_y = y_max, $l_y^ +\approx {y^ + }$ because of y_min ≈ 0). Colour and line contours apply to the slip and no-slip cases, respectively. Circles correspond to the mean lengths for the slip (red) and no-slip (black) cases.

Figure 5. Contours of JPDF of (a) $l_x^ +$ and $l_y^ +$ and of (b) $l_z^ +$ and $l_y^ +$ of wall-attached u structures. The circles are the mean lengths. The green lines in panels (a) and (b) are $\langle {l_x}\rangle \sim l_y^{0.74}$ and $\langle {l_z}\rangle \sim {l_y}$, respectively.

As shown in figure 5(a), the mean $l_x^ + (\langle l_x^ + \rangle )$ is proportional to $l_y^ +$ with the power of 0.74 (Hwang & Sung Reference Hwang and Sung2018) in the region of $l_y^ + \ge 70 \approx 3Re_\tau ^{0.5}$, whereas not proportional to $l_y^ + (l_y^ + < 70)$. Here, $3Re_\tau ^{0.5}$ is the lower limit of the logarithmic region (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Hwang & Sung Reference Hwang and Sung2019), which accords with approximately 70ν/u_τ at the present Reynolds number (Re_τ ≈ 500). In addition, there is a linear relationship between the mean $l_z^ + (\langle l_z^ + \rangle )$ and $l_y^ + (l_y^ + \ge 70)$ (figure 5b), which shows that the spanwise size of wall-attached u structures is proportional to the distance from the wall (Tomkins & Adrian Reference Tomkins and Adrian2003; del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Hwang Reference Hwang2015). In particular, the JPDF of $l_x^ +$ and $l_z^ +$ with $l_y^ +$ for both cases coincide in the entire $l_y^ +$ region, especially $l_y^ + \ge 70$, in agreement with the results of Lozano-Durán & Bae (Reference Lozano-Durán and Bae2019). Accordingly, we divide wall-attached structures into non-self-similar $(l_y^ + < 70)$ and self-similar $(l_y^ + \ge 70)$ structures.

The population density $(n_a^\ast )$ of wall-attached structures with respect to $l_y^ +$ is shown in figure 6(a). Here, n_a is the number of wall-attached structures as a function of their height l_y, and the population density $(n_a^\ast )$ of wall-attached structures is defined as n_a per unit wall-parallel area, i.e. $n_a^\ast = {n_a}/(m{A_{xz}})$. The magnitude of $n_a^\ast $ is smaller than that for the no-slip case in the entire region. The wall-attached structures with the height of $l_y^ + = O(10)$ are related to near-wall streaks. As shown in figure 2(a), the mean shear is preserved near the wall for the slip case, sustaining turbulence via the formation process of streaky structures (Hamilton, Kim & Waleffe Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997). The streamwise slip attenuates streamwise vortices near the wall, creating streaks sparsely (Min & Kim Reference Min and Kim2004; Kim Reference Kim2011). The lack of streamwise vortices leads to reduced populations of wall-attached structures near the wall.

Figure 6. (a) Population density $(n_a^\ast )$ and (b) volumes of wall-attached structures per unit domain volume $(V_a^\ast )$ as a function of $l_y^ +$. Green lines denote an inverse power-law distribution of $n_a^\ast $ in panel (a) and linear variation in $V_a^\ast $ in panel (b). Dashed lines in panels (a) and (b) represent $l_y^ + = 1.06{\delta ^ + }$ and $l_y^ + = 1.13{\delta ^ + }$, respectively.

The magnitude of $n_a^\ast $ is inversely proportional to $l_y^ +$ in the region of $250 < l_y^ + < 400$ for both cases, reminiscent of the distribution of hierarchy length scales of attached eddies (Perry & Chong Reference Perry and Chong1982). The inverse power-law dependence arises within the region of $l_y^ + = 0.3{\delta ^ + }-0.6{\delta ^ + }$ for zero-pressure-gradient TBL at Re_τ ≈ 1000 (Hwang & Sung Reference Hwang and Sung2018) and turbulent pipe flows at Re_τ ≈ 1000 and 3000 (Hwang & Sung Reference Hwang and Sung2019). The streamwise slip induces a decrease in the population density of wall-attached structures, but the inverse power law is still valid. The no-slip boundary condition is not indispensable to form hierarchical distributions of wall-attached structures. In addition, a peak in $n_a^\ast $ arises at $l_y^ + = 1.06{\delta ^ + }$ for both cases, which results from the large population of wall-attached δ-height structures (Perry et al. Reference Perry, Henbest and Chong1986).

In contrast to the variation in $n_a^\ast $, the magnitude of $V_a^\ast $ increases with increasing $l_y^ + $ up to $l_y^ +\approx {\delta ^ + }$ (figure 6b). Here, $V_a^\ast $ is defined as the volume of wall-attached structures per unit domain volume, i.e. $V_a^\ast = {V_a}/(m{A_{xz}}\delta )$, where A_xzδ is the domain volume. The profiles of $V_a^\ast $ for both cases collapse well below $l_y^ + = 30$. The magnitude of $V_a^\ast $ is even larger than that for the no-slip case in the region of $l_y^ + = 30-550$, whereas the number of wall-attached structures is smaller than that for the no-slip case (figure 6a). A peak in $V_a^\ast $ arises at $l_y^ + = 1.13{\delta ^ + }$ for both cases; this trend is the same as that in $n_a^\ast $.

Figure 7 shows wall-normal profiles of streamwise Reynolds stresses $(\langle uu\rangle _a^{{\ast}{+} })$ carried by the wall-attached structures with the height of $l_y^ + = 300$ and 1.06δ ⁺,

(3.1)\begin{equation}\langle uu\rangle _a^\ast (y,{l_y}) = \left\langle {{S_a}{{(y,{l_y})}^{ - 1}}\int_{{S_a}} {u(\boldsymbol{x})u(\boldsymbol{x})\,\textrm{d}\kern0.06em x\,\textrm{d}z} } \right\rangle ,\end{equation}

where S_a is the wall-parallel area of wall-attached structures as functions of y and l_y. As shown in figure 7(a), the magnitude of $\langle uu\rangle _a^{{\ast}{+} }$ for $l_y^ + = 300$ is logarithmically proportional to y ⁺ $(\langle uu\rangle _a^{{\ast}{+} }\sim \ln {y^ + })$ in the region of 100 < y ⁺ < 220 (Perry & Chong Reference Perry and Chong1982). Although the present Reynolds number is relatively low (Re_τ ≈ 500), the logarithmic variation is evident in $\langle uu\rangle _a^{{\ast}{+} }$ reconstructed from the wall-attached structures (Hwang & Sung Reference Hwang and Sung2018; Hwang et al. Reference Hwang, Lee and Sung2020; Yoon et al. Reference Yoon, Hwang, Yang and Sung2020). Wall-attached δ-height structures are not self-similar with their height, but contaminate the logarithmic variation in the streamwise Reynolds stress (Hwang et al. Reference Hwang, Lee and Sung2020; Yoon et al. Reference Yoon, Hwang, Yang and Sung2020). Wall-attached δ-height structures are self-similar with l_y (figure 5). The profiles of $\langle uu\rangle _a^{{\ast}{+} }$ for $l_y^ + = 1.06{\delta ^ + }$ are similar to those for $l_y^ + = 300$, and the logarithmic variation is still valid in the region of $l_y^ + = 100-220$ (figure 7b). The logarithmic behaviour of wall-attached structures is sustained regardless of the streamwise slip at the wall.

Figure 7. Streamwise Reynolds stresses $(\langle uu\rangle _a^ + )$ reconstructed by wall-attached structures with the height of (a) $l_y^ + = 300$ and (b) $l_y^ + = 1.06{\delta ^ + }$. Green dashed lines in panel (a) indicate the best fits in the region of 100 < y ⁺ < 220: $\langle uu\rangle _a^{{\ast}{+} } = 35.73 - 5.14\ln {y^ + }$ for the slip case and $\langle uu\rangle _a^{{\ast}{+} } = 36.44 - 5.08\ln {y^ + }$ for the no-slip case, and those in panel (b) are for $\langle uu\rangle _a^{{\ast}{+} } = 36.24 - 5.09\ln {y^ + }$ (slip) and $\langle uu\rangle _a^{{\ast}{+} } = 34.44 - 4.52\ln {y^ + }$ (no-slip). Two vertical dashed lines represent y ⁺ = 100 and y ⁺ = 220.

4.1. Structural features

The characteristics of wall-attached structures can be divided into buffer-layer $(l_y^ + < 70 \approx 3Re_\tau ^{0.5})$ and self-similar $(l_y^ + \ge 70)$ structures. As shown in figure 5, the upper part $({y^ + } \ge 70)$ of WASS $(l_y^ + \ge 70)$ is responsible for the logarithmic behaviour (figure 7b). The near-wall part (y ⁺ < 70) of WASS $(l_y^ + \ge 70)$ contributes to geometrical self-similarity (figure 5). The wall-attached structures can be decomposed into u_nws, u_uws and u_wb, which are defined as

(4.1a)\begin{gather}{u_{nws}}(\boldsymbol{x}) = \left\{ {\begin{array}{*{20}{l}} u&{\textrm{if}\;|u|\ge 1.6{u_{rms}},\;y_{min}^ +\approx 0,\;y_{max}^ + \ge 70,\;{y^ + } < 70,}\\ 0&{\textrm{otherwise,}} \end{array}} \right.\end{gather}

(4.1b)\begin{gather}{u_{uws}}(\boldsymbol{x}) = \left\{ {\begin{array}{*{20}{l}} u&{\textrm{if}\;|u|\ge 1.6{u_{rms}},\;y_{min}^ +\approx 0,\;y_{max}^ + \ge 70,\;{y^ + } \ge 70,}\\ 0&{\textrm{otherwise,}} \end{array}} \right.\end{gather}

(4.1c)\begin{gather}{u_{wb}}(\boldsymbol{x}) = \left\{ {\begin{array}{*{20}{l}} u&{\textrm{if}\;|u|\ge 1.6{u_{rms}},\;y_{min}^ +\approx 0,\;y_{max}^ + < 70,}\\ 0&{\textrm{otherwise}.} \end{array}} \right.\end{gather}

Here, the subscripts ‘nws’, ‘uws’ and ‘wb’ represent the near-wall part of WASS, upper part of WASS and wall-attached buffer-layer structures (WABS), respectively. From now on, we focus on the near-wall part of WASS, in which the viscosity effect is dominant.

Figure 8 illustrates 3-D iso-surfaces of u wall-attached structures for the no-slip case in an instantaneous flow field. Red and blue correspond to positive u and negative u, respectively. A schematic diagram in the inset of figure 9 helps to understand the decomposition of wall-attached structures: upper-part of self-similar, near-wall part of self-similar and buffer layer. Note that both u_nws and u_wb are located in the region of y ⁺ < 70. The near-wall part of WASS is more coherent than the WABS. In addition, the near-wall part of WASS (y ⁺ < 70) is easily observed in an instantaneous flow field, interpreted as the footprints of large-scale motions (Hutchins & Marusic Reference Hutchins and Marusic2007; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016).

Figure 8. 3-D iso-surfaces of u wall-attached structures: self-similar $(l_y^ + \ge 70 \approx 3Re_\tau ^{0.5})$; the upper part (u_uws) ($l_y^ + \ge 70$ and y ⁺ ≥ 70) and near-wall part (u_nws) ($l_y^ + \ge 70$ and y ⁺ < 70) and buffer-layer $(u_{wb}) (l_y^ + < 70)$. Red and blue correspond to positive u and negative u, respectively.

Figure 9. Two-point correlations of (a) u_nws and (b) u_wb. The reference wall-normal location is $y_{ref}^ + = 14.5$. Solid lines represent 5 % of the maximum of positive correlations, and dashed lines denote 50 % of the minimum of negative correlations. Cross symbols indicate spanwise centres of negative correlations. The blue line in panel (a) accords with R[u_wb, u_wb] for the slip case in panel (b).

To investigate sizes of the near-wall part of WASS statistically, conditional two-point correlations of u_nws are examined at the reference wall-normal location $y_{ref}^ + = 14.5$, where the inner peak arises in $u_{rms}^ + $ (figure 2c). The two-point correlations of u_nws and u_wb are defined as

(4.2a)\begin{gather}R[{u_{nws}},{u_{nws}}]({r_x},y,{r_z},{y_{ref}}) = \frac{{\langle {u_{nws}}(x,{y_{ref}},z){u_{nws}}(x + {r_x},y,z + {r_z})\rangle }}{{{u_{nws,\,rms}}({y_{ref}}){u_{nws,\,rms}}(y)}},\end{gather}

(4.2b)\begin{gather}R[{u_{wb}},{u_{wb}}]({r_x},y,{r_z},{y_{ref}}) = \frac{{\langle {u_{wb}}(x,{y_{ref}},z){u_{wb}}(x + {r_x},y,z + {r_z})\rangle }}{{{u_{wb,\,rms}}({y_{ref}}){u_{wb,\,rms}}(y)}},\end{gather}

where u_nws,rms and u_wb,rms are the root-mean-square quantities of u_nws and u_wb, respectively. For comparison, the two-point correlation of u_wb is included.

Wall-parallel views of R[u_nws, u_nws] are shown in figure 9(a), where solid and dashed lines denote 5 % of the maximum of positive R[u_nws, u_nws] and 50 % of the minimum of negative R[u_nws, u_nws], respectively. The positive R[u_nws, u_nws] are extended to approximately 4.3δ in the streamwise direction, which is similar to streamwise lengths of the footprints of large-scale motions at the same location of $y_{ref}^ +\approx 14.5$ (Lee & Sung Reference Lee and Sung2013; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016; Yoon et al. Reference Yoon, Hwang, Lee, Sung and Kim2016b; Hwang & Sung Reference Hwang and Sung2017). In addition, the scaling of negative R[u_nws, u_nws] with δ indicates that the near-wall part of WASS is aligned side by side along the spanwise direction. The distance to the centre (cross symbol) of negative R[u_nws, u_nws] at r_x = 0 is 0.6δ, which is 20 % larger than that for the no-slip case. This result represents that the near-wall part of WASS is more sparsely distributed than those for the no-slip case (figure 4).

Figure 9(b) shows line contours of R[u_wb, u_wb] in the x–z plane. Compared to R[u_nws, u_nws], the negative R[u_wb, u_wb] is not observed, representing that the WABS are not influenced by adjacent other WABS. This is consistent with the observations in figure 8, where iso-surfaces of u_wb are more sparsely populated than those of u_nws. The line contour of R[u_wb, u_wb] is transversely away from the reference centre (r_x = 0 and r_z = 0) by 85v/u_τ, implying that the WABS are closely related to the near-wall streaks (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). The streamwise extension of positive R[u_wb, u_wb] is 500ν/u_τ at r_z = 0, which is 7.7 % longer than that for the no-slip case. The blue contour in figure 9(a) represents R[u_wb, u_wb] for the slip case, indicating that the WABS are much shorter in the streamwise direction than the near-wall part of WASS.

To further examine geometrical features of the near-wall part of WASS, two-point correlations of u_a (R[u_a, u_a]) at a given l_y are performed at $y_{ref}^ + = 14.5$. The conditional field (u_a) and two-point correlation of u_a are defined as

(4.3)\begin{equation}{u_a}(\boldsymbol{x},{l_y}) = \left\{ {\begin{array}{*{20}{l}} u&{\textrm{if}\;|u|\ge 1.6{u_{rms}},\;y_{min}^ +\approx 0,}\\ 0&{\textrm{otherwise},} \end{array}} \right.\end{equation}

(4.4)\begin{equation}R[{u_a},{u_a}]({r_x},y,{r_z},{y_{ref}},{l_y}) = \frac{{\langle {u_a}(x,{y_{ref}},z,{l_y}){u_a}(x + {r_x},y,z + {r_z},{l_y})\rangle }}{{{u_{a,\,rms}}({y_{ref}},{l_y}){u_{a,\,rms}}(y,{l_y})}},\end{equation}

where u_a,rms is the root-mean-square quantity of u_a. Figure 10(a) shows the x–z plane (at ${y^ + } = y_{ref}^ +$) and x–y plane (at $r_z^ + = 0$) views of R[u_a, u_a] for $l_y^ + = 300$. The wall-attached structures with the height of $l_y^ + = 300$ are more stretched in both upstream and downstream directions than those for the no-slip case. These observations support that weakened wall-shear stress due to the streamwise slip results in a long extension of wall-attached structures to the wall (Chung et al. Reference Chung, Monty and Ooi2014).

Figure 10. (a) Two-point correlations of u_a for $l_y^ + = 300$ in the x–y and x–z planes. The reference wall-normal location is $y_{ref}^ + = 14.5$. Line contours are R[u_a, u_a] = 0.05. (b) Streamwise and spanwise characteristic lengths ($R_x^ +$ and $R_z^ +$) of the near-wall part of WASS at $y_{ref}^ + = 14.5$ with respect to $l_y^ +$.

Figure 10(b) shows characteristic length scales of the near-wall part of WASS at $y_{ref}^ + = 14.5$. Their streamwise and spanwise lengths (R_x and R_z) can be defined in terms of the difference between the streamwise and spanwise displacements at $R[{u_a},{u_a}](r_x^ + ,{y^ + } = y_{ref}^ + ,r_z^ + = 0) = 0.05$ and $R[{u_a},{u_a}](r_x^ + = 0,{y^ + } = y_{ref}^ + ,r_z^ + ) = 0.05$, respectively. The magnitude of $R_x^ +$ and $R_z^ +$ gradually increases for both cases with the increase in $l_y^ +$. Interestingly, streamwise lengths of the near-wall part of WASS are longer than those for the no-slip case, and vice versa for the spanwise lengths up to $l_y^ + = 600$. The magnitude of $R_x^ +$ is larger than 1δ ⁺, and is thus related to the footprints of large-scale motions (Hutchins & Marusic Reference Hutchins and Marusic2007). The magnitude of $R_x^ +$ is approximately 1.3 times larger than that for the no-slip case up to $l_y^ + = 400$, over which the difference in $R_x^ +$ between two cases becomes larger. Although sizes of WASS are independent of the streamwise slip (figure 5), their roots are stretched and compressed in the streamwise and spanwise directions, respectively.

4.2. Conditional statistics

We investigate the turbulence statistics of the near-wall part of WASS along the wall-normal location. The turbulence statistics for the near-wall part of WASS can be conditionally averaged based on conditional fields, where velocity and vorticity fluctuations are obtained using equations similar to (4.1a). Conditionally averaged quantities are used to assess contributions of the near-wall part of WASS to the turbulence statistics in the region of y ⁺ < 70.

Figure 11 introduces wall-normal profiles of conditional averages for the near-wall part of WASS. Profiles of A_nws/A_xz (figure 11a) represent wall-parallel areas occupied by the near-wall part of WASS, which account for 5–7 % of the area of the x–z plane. Below y ⁺ = 30, the magnitude of A_nws/A_xz is larger than that for the no-slip case, in particular, 74.3 % larger near the wall. The streamwise slip induces long tails, observed from the conditional two-point correlations (figure 10a). Similar results were reported from the footprints of large-scale motions by a streamwise slip (Yoon et al. Reference Yoon, Hwang, Lee, Sung and Kim2016b). As shown in figure 11(b), the magnitude of $\langle uu\rangle _{nws}^ + $ is larger than that for the no-slip case below y ⁺ = 30, which indicates that the near-wall part of WASS is strengthened by the streamwise slip.

Figure 11. Conditionally averaged turbulence statistics of the near-wall part of WASS: wall-normal profiles of (a) A_nws/A_xz, (b) $\langle uu\rangle _{nws}^ + $, (c) $\langle - uv\rangle _{nws}^ + $ and (d) $\textrm{d}U_{nws}^ +{/}\textrm{d}{y^ + }$.

In contrast to $\langle uu\rangle _{nws}^ + $, profiles of $\langle - uv\rangle _{nws}^ + $ for both cases collapse (figure 11c), similar to ${\langle - uv\rangle ^ + }$ (figure 2a). The streamwise slip results in the enhancement of negative u_nws and the attenuation of v_nws from the weighted JPDF of $u_{nws}^ + v_{nws}^ + $ (not shown here). Accordingly, the magnitudes $\langle - uv\rangle _{nws}^ + $ for both cases are similar. The near-wall part of WASS contributes to approximately 30 % of the total $\langle - uv\rangle $ despite of a small A_nws/A_xz (figure 11c), representing that they are the main energy-containing motions near the wall. Figure 11(d) shows the mean shear $(\textrm{d}U_{nws}^ + \textrm{/d}{y^ + })$ carried by the near-wall part of WASS. Below y ⁺ = 20, the magnitude of $\textrm{d}U_{nws}^ + \textrm{/d}{y^ + }$ is larger than that for the no-slip case, especially by 9 % in the vicinity of the wall. The magnitude of $\textrm{d}U_{nws}^ +{/}\textrm{d}{y^ + }$ at the wall can be interpreted as the ratio τ_w,nws/τ_w, where τ_w,nws is the wall shear stress carried by the near-wall part of WASS. Given that the wall shear stress is related to the skin friction coefficient, the high $\textrm{d}U_{nws}^ +{/}\textrm{d}{y^ + }$ at the wall is responsible for the frictional drag. The majority of discrepancy in $\textrm{d}U_{nws}^ +{/}\textrm{d}{y^ + }$ comes from the high A_nws/A_xz (figure 11a). The mean shear of the near-wall part of WASS for the slip case is restored above y ⁺ = 30. In contrast to the upper part of WASS, the near-wall part of WASS can be varied by the viscosity and wall conditions, but their contributions to the streamwise Reynolds stress and Reynolds shear stress are more significant.

Two velocity–vorticity correlations $({\langle v{\omega _z}\rangle _{nws}}$ and ${\langle - w{\omega _y}\rangle _{nws}})$ carried by the near-wall part of WASS are plotted in figure 12(a,c). The former is related to the advective vorticity transport, and the latter represents the vortex stretching (Tennekes & Lumley Reference Tennekes and Lumley1972). Here, ω_z and ω_y are the spanwise and wall-normal vorticity fluctuations, respectively. In addition, these velocity–vorticity correlations are important to near-wall turbulence, since they are directly related to the frictional drag (Yoon et al. Reference Yoon, Ahn, Hwang and Sung2016a; Hwang & Sung Reference Hwang and Sung2017; Yoon, Hwang & Sung Reference Yoon, Hwang and Sung2018). The magnitude of $\langle v{\omega _z}\rangle _{nws}^ + $ is significantly reduced near the positive peak at y ⁺ = 4.5 (figure 12a). Figure 12(b) shows the weighted JPDF of $v_{nws}^ + \omega _{z,nws}^ + $ at y ⁺ = 4.5, where $\langle v{\omega _z}\rangle _{nws}^ + $ has a positive peak (figure 12a). At the first quadrant $({v_{nws}} > 0\;\& \;{\omega _{z,nws}} > 0)$, the magnitude of mean $\omega _{z,nws}^ + $ is 28.2 % smaller than that for the no-slip case, and the magnitude of mean $v_{nws}^ + $ decreases by 18.4 %. The first quadrant of the weighted JPDF of vω_z at a positive peak of $\langle v{\omega _z}\rangle $ is interpreted as the vertical advection of sublayer streaks (Klewicki, Murray & Falco Reference Klewicki, Murray and Falco1994; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). Both the weakened near-wall part of WASS and upward motions lead to the reduction in positive $\langle v{\omega _z}\rangle _{nws}^ + $. In addition, a peak is evident in the negative $\langle v{\omega _z}\rangle _{nws}^ + $ at y ⁺ = 30, whereas a negative peak of ${\langle v{\omega _z}\rangle ^ + }$ is observed at y ⁺ = 18 (not shown here), corresponding to outward motions of hairpin vortex heads (Klewicki et al. Reference Klewicki, Murray and Falco1994; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). Given that vortical structures are in the form surrounding streaky structures (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Lee & Sung Reference Lee and Sung2009; Dennis & Nickels Reference Dennis and Nickels2011), the conditional field for the near-wall part of WASS does not fully form hairpin-like vortical structures.

Figure 12. Profiles of velocity–vorticity correlations carried by near-wall part of WASS: (a) $\langle v{\omega _z}\rangle _{nws}^ + $ and (c) $\langle - w{\omega _y}\rangle _{nws}^ + $. Weighted JPDF of (b) $v_{nws}^ + \omega _{z,nws}^ + $ at y ⁺ = 4.5 and (d) $w_{nws}^ + \omega _{y,nws}^ + $ at y ⁺ = 7.0. The solid and dashed lines correspond to positive and negative values, respectively. Contours are from −0.002 to +0.002 with increments of 0.001.

Figure 12(c) shows wall-normal profiles of $\langle - w{\omega _y}\rangle _{nws}^ + $, in which a positive peak is evident at y ⁺ = 7 and positive values are present up to y ⁺ = 50. The positive $\langle - w{\omega _y}\rangle $ are physically related to the simultaneous collapse of two adjacent hairpin vortex legs, leading to the stretching of hairpin vortices (Eyink Reference Eyink2008; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). The magnitude of $\langle - w{\omega _y}\rangle _{nws}^ +$ is similar to that for the no-slip case. Figure 12(d) represents the weighted JPDF of $w_{nws}^ + \omega _{y,nws}^ + $ at y ⁺ = 7, where two motions with negative correlations (w_nws > 0 and ω_y,nws < 0, and w_nws < 0 and ω_y,nws > 0) are dominant. Although the magnitudes of mean $w_{nws}^ + $ and mean $\omega _{y,nws}^ + $ at negative correlations are 25 % and 13 % smaller than those for the no-slip case, the magnitude of ${\langle - w{\omega _y}\rangle ^ + }$ at y ⁺ = 7 is similar to each other due to higher population density of negative correlations. The energy transfer via vortex stretching of the near-wall part of WASS is attenuated by the streamwise slip.

To determine the turbulence statistics reconstructed by the wall-attached structures at a given height, the streamwise velocity and its wall-normal gradient are conditionally averaged as a function of l_y,

(4.5)\begin{gather}{U_a}({l_y}) = \left\langle {{V_a}{{({l_y})}^{ - 1}}\int_{{V_a}} {\tilde{u}(\boldsymbol{x})\,\textrm{d}\boldsymbol{x}} } \right\rangle ,\end{gather}

(4.6)\begin{gather}\frac{{\textrm{d}{U_a}}}{{\textrm{d}y}}({l_y}) = \left\langle {{V_a}{{({l_y})}^{ - 1}}\int_{{V_a}} {\frac{{\textrm{d}\tilde{u}(\boldsymbol{x})}}{{\textrm{d}y}}\,\textrm{d}\boldsymbol{x}} } \right\rangle .\end{gather}

Figure 13(a) shows the variation in $U_a^ +$ with $l_y^ +$, i.e. the convection velocity of wall-attached structures at a given l_y. A large difference in $U_a^ +$ is observed at $l_y^ + < 30$, while the magnitude of $U_a^ +$ is similar in the region of $l_y^ + \ge 0.45{\delta ^ + }$. The discrepancy in $U_a^ +$ at lower $l_y^ +$ is mainly due to the slip velocity at the wall. Interestingly, the difference in $U_a^ +$ decreases as $l_y^ +$ increases. The wall-attached structures slide in the streamwise direction, leading to the increase in their volume close to the wall (figure 10a). As $l_y^ +$ increases, the convection velocity of wall-attached structures gets close to that for the no-slip case.

Figure 13. Profiles of (a) the convection velocity $(U_a^ + )$ and (b) the mean shear $(\textrm{d}U_a^ +{/}\textrm{d}{y^ + })$ carried by wall-attached structures.

The mean shear $(\textrm{d}U_a^ +{/}\textrm{d}{y^ + })$ carried by the wall-attached structures at a given l_y is shown in figure 13(b); it is suppressed in the region of $l_y^ + < 50$. Given that the mean shear is directly related to the formation of near-wall streaks (Waleffe Reference Waleffe1997), the weakened mean shear results in a lower population density of WABS for the slip case (figure 6a). However, profiles of $\textrm{d}U_a^ +{/}\textrm{d}{y^ + }$ for both cases collapse well over $l_y^ + < 70$, while the lower population of WASS is observed for the slip case, as shown in figure 6(a). This inconsistency reveals that the formation of WASS is affected by the population of WABS (not by the mean shear), supporting the hierarchical distributions of wall-attached structures (Perry & Chong Reference Perry and Chong1982).

To explore the characteristics of the near-wall part of WASS with respect to their height at the near-wall region, conditional averages of wall-parallel area, streamwise velocity, streamwise Reynolds stress and Reynolds shear stress at y ⁺ = 14.5 are examined, which are defined as

(4.7)\begin{gather}{S_{a,i}}({l_y}) = {S_a}(y,{l_y}){|_{{y^ + } = 14.5}}/(m{A_{xz}}),\end{gather}

(4.8)\begin{gather}{U_{a,i}}({l_y}) = U_a^\ast (y,{l_y}){|_{{y^ + } = 14.5}}/U(y){|_{{y^ + } = 14.5}},\end{gather}

(4.9)\begin{gather}{\langle uu\rangle _{a,i}}({l_y}) = \langle uu\rangle _a^\ast (y,{l_y}){|_{{y^ + } = 14.5}}/\langle uu\rangle (y){|_{{y^ + } = 14.5}},\end{gather}

(4.10)\begin{gather}{\langle - uv\rangle _{a,i}}({l_y}) = \langle - uv\rangle _a^\ast (y,{l_y}){|_{{y^ + } = 14.5}}/\langle - uv\rangle (y){|_{{y^ + } = 14.5}}.\end{gather}

Figure 14(a) shows profiles of the areas in the x–z plane (S_a,i) occupied by the near-wall part of WASS at y ⁺ = 14.5. The magnitude of S_a,i gradually decreases as $l_y^ +$ increases with a peak at $l_y^ + = 1.1{\delta ^ + }$. The quantities of U_a,i, ${\langle uu\rangle _{a,i}}$ and ${\langle - uv\rangle _{a,i}}$ describe the dependence on the height of the near-wall part of WASS in their contributions to the turbulence statistics (i.e. U, $\langle uu\rangle $ and $\langle - uv\rangle $) at y ⁺ = 14.5. Figure 14(b) represents profiles of U_a,i with respect to $l_y^ +$, defined as the convection velocity $(U_a^\ast )$ of the near-wall part of WASS normalized by U ⁺ at y ⁺ = 14.5. The region of $l_y^ +$, where the magnitude of U_a,i is larger than 1, indicates that positive u of the near-wall part of WASS at a given $l_y^ +$ is dominant near y ⁺ = 14.5, and vice versa. A peak of U_a,i is observed at $l_y^ + = 310$ (slip) and 370 (no-slip), and a concave peak of U_a,i is evident at $l_y^ + = 1.12{\delta ^ + }$. In particular, the profile of U_a,i is shifted downward from that for the no-slip case, especially in the region of $l_y^ + = 300 - {\delta ^ + }$. It shows that the population of wall-attached negative-u structures with $l_y^ +\approx {\delta ^ + }$ are more dominant than that for the no-slip case. These observations are consistent with the dominance of negative-u large-scale motions in the outer region (Yoon et al. Reference Yoon, Hwang, Lee, Sung and Kim2016b).

Figure 14. Conditional averages of the near-wall part of WASS at y ⁺ = 14.5: (a) S_a,i; (b) U_a,i; (c) ${\langle uu\rangle _{a,i}}$ and (d) ${\langle - uv\rangle _{a,i}}$. Dashed lines in panels (a) and (b) represent $l_y^ + = 1.1{\delta ^ + }$ and 1.12δ ⁺, respectively. A green line in panel (c) denotes the logarithmic variation: ${\langle uu\rangle _{a,i}} = 0.089\ln l_y^ +{+} 3.38$.

We now examine the streamwise Reynolds stress $({\langle uu\rangle _{a,i}})$ at y ⁺ = 14.5. Figure 14(c) shows the variation in ${\langle uu\rangle _{a,i}}$ with $l_y^ +$. The magnitude of ${\langle uu\rangle _{a,i}}$ gradually increases with increasing $l_y^ +$. The contributions of the near-wall part of WASS to the streamwise Reynolds stress are enhanced as $l_y^ +$ increases. In particular, the magnitude of ${\langle uu\rangle _{a,i}}$ is logarithmically proportional to $l_y^ +$ in the region of $100 < l_y^ + < 350\,({\langle uu\rangle _{a,i}}\sim \ln l_y^ + )$ (a green line in figure 15c). The inner-peak magnitude of streamwise Reynolds stress logarithmically increases with increasing Re_τ (Jiménez & Hoyas Reference Jiménez and Hoyas2008; Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017). Since l_y is related to hierarchical scales (figure 7a), the logarithmic variation in ${\langle uu\rangle _a}$ with respect to l_y reflects the hierarchical features of wall-attached structures (Hwang & Sung Reference Hwang and Sung2018); this logarithmic variation in ${\langle uu\rangle _a}$ is evident in the region of $200 < l_y^ + < 300$. A hierarchy of wall-attached structures is well established over the broader range of l_y.

Figure 15. Profiles of (a) pre-multiplied ${\zeta _{2a}}$ and (b) the density of skin friction $({\zeta ^{\prime}_{2a}})$ with respect to $l_y^ +$. Dashed lines represent $l_y^ + = 1.13{\delta ^ + }$. Open symbols in panel (b) represent ${\zeta ^{\prime}_{2a, - 34\,\%}}$.

Figure 14(d) shows profiles of ${\langle - uv\rangle _{a,i}}$, of which the value is larger than 3 in the region of $l_y^ + < {\delta ^ + }$. It means that the near-wall part of WASS carry three times more Reynolds shear stresses than $\langle - uv\rangle $ in the near-wall region (y ⁺ = 14.5). The magnitude of ${\langle - uv\rangle _{a,i}}$ is larger than that for the no-slip case below $l_y^ + < {\delta ^ + }$. These observations imply that the contributions of the near-wall part of WASS to the near-wall turbulence are dominant, since the Reynolds shear stress is directly related to turbulent contributions to the skin friction coefficient (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002).

5.1. Wall-attached structures

The turbulent term (C_{f ,2}) can be divided into contributions of wall-attached structures (C_{f ,2a}) and others (C_{f ,2others}), which captures contributions of wall-detached structures $(y_{min}^ + > 0)$ and weak turbulence $(-1.6{u_{rms}} < u < 1.6{u_{rms}})$ to C_{f ,2}. The contribution of wall-attached structures to C_{f ,2} can be obtained from the conditional fields:

(5.2)\begin{equation}{C_{f,2a}} = \int_{ - \infty }^\infty {{l_y}{\zeta _{2a}}({l_y})\,\textrm{d}\ln {l_y}} ,\quad \textrm{where}\;{\zeta _{2a}}({l_y}) = \frac{6}{{U_b^2}}\left\langle {\int_{{V_a}} { - (1 - y)u(\boldsymbol{x})v(\boldsymbol{x})\,\textrm{d}\boldsymbol{x}} } \right\rangle .\end{equation}

Here, ${l_y}{\zeta _{2a}}$ as a function of l_y is the integrand of C_{f ,2a}.

Profiles of $l_y^ + {\zeta _{2a}}$ are plotted on a semi-logarithmic scale in figure 15(a), where the area of profiles is proportional to C_{f ,2a}. Those profiles are biased towards higher $l_y^ +$, and peaks are evident at $l_y^ + = 1.13{\delta ^ + }$, representing that wall-attached δ-height structures make the dominant contributions to the frictional drag. The area of $l_y^ + {\zeta _{2a}}$ in the region of $l_y^ + \ge 70$ is the same as 97 % of the total area, showing that wall-attached structures with $l_y^ + \ge 70$ are responsible for almost all of the overall C_{f ,2a}.

To determine the contributions of wall-attached structures at a given l_y to the frictional drag, the density of skin friction $({\zeta ^{\prime}_{2a}})$ is defined as

(5.3)\begin{equation}{\zeta ^{\prime}_{2a}}({l_y}) = {\zeta _{2a}}({l_y})\frac{{m{A_{xz}}\delta }}{{{n_a}{V_a}}}.\end{equation}

Figure 15(b) shows the profile of ${\zeta ^{\prime}_{2a}}$ with respect to $l_y^ +$. A smooth peak in ${\zeta ^{\prime}_{2a}}$ is evident at $l_y^ + = 100$, which indicates that the wall-attached structures with the height of $l_y^ +\approx 100$ contain the highest density of skin friction. The magnitude of ${\zeta ^{\prime}_{2a}}$ is more than four times larger than C_f beyond $l_y^ + = 30$, in particular, nine times larger near $l_y^ + = 100$. This means that each wall-attached structure is responsible for the generation of turbulent frictional drag (de Giovanetti, Hwang & Choi Reference de Giovanetti, Hwang and Choi2016). Open symbols in figure 15(b) represent ${\zeta ^{\prime}_{2a, - 34\,\%}}$. The profile of ${\zeta ^{\prime}_{\textrm{2}a, - 34\,\%}}$ is shifted downward by as much as 34 % from ${\zeta ^{\prime}_{2a}}$. The magnitude of ${\zeta ^{\prime}_{2a}}$ at $l_y^ + = 1.13{\delta ^ + }$ (red dashed line) is similar to that of ${\zeta ^{\prime}_{\textrm{2}a, - 34\,\%}}$ (grey dashed line), representing that the density of skin friction of wall-attached structures with $l_y^ + = 1.13{\delta ^ + }$ is similar for both cases. The large difference in $l_y^ + {\zeta _{2a}}$ at $l_y^ + = 1.13{\delta ^ + }$ (figure 15a) is caused by the volume of wall-attached δ-height structures (figure 6b). The magnitude of ${\zeta ^{\prime}_{2a}}$ is 34 % smaller than that of ${\zeta ^{\prime}_{2a}}$ for the no-slip case, especially 44 % in the region of $250 < l_y^ + < 400$.

5.2. Near-wall part of wall-attached self-similar structures

To scrutinize contributions of the near-wall part of WASS to the skin friction in detail, the turbulent component of C_f is conditionally averaged as functions of y and l_y,

(5.4)\begin{gather}C_{f,2a}^\ast (y,{l_y}) = \frac{6}{{U_b^2}}\left\langle {\int_{{S_a}} { - (1 - y)u(\boldsymbol{x})v(\boldsymbol{x})\,\textrm{d}\kern0.06em x\,\textrm{d}z} } \right\rangle ,\end{gather}

(5.5)\begin{gather}{C_{f,i}}({l_y}) = C_{f,2a}^\ast (y,{l_y}){|_{{y^ + } = 14.5}}/C_{f,2}^\ast (y){|_{{y^ + } = 14.5}},\end{gather}

where $C_{f,2}^\ast $ is the integrand of C_{f ,2} in (5.2) and the double integration of $C_{f,2a}^\ast $ with respect to y and l_y is the same as C_{f ,2a}.

Figure 16 shows C_{f ,i} with respect to $l_y^ +$, where the area of C_{f ,i} is proportional to the contribution of the near-wall part of WASS to $C_{f,2}^\ast $ at y ⁺ = 14.5. The magnitude of C_{f ,i} exponentially decreases with the increase in $l_y^ +$. A smooth peak is observed at $l_y^ + = 1.1{\delta ^ + }$, where S_a,i exhibits a large peak (figure 14a). Dominant contributions of the near-wall part of wall-attached δ-height structures to C_{f ,2} result from their large area. The contribution of the near-wall part to C_{f ,2} in the region of $l_y^ + < 400$ is greater than for the no-slip case, although S_a,i is similar below $l_y^ + = 400$ (figure 14a). This is due to the enhanced streamwise Reynolds stress and Reynolds shear stress in the vicinity of the wall (figure 14c,d). The near-wall part of WASS is responsible for 19.1 % of $C_{f,2}^\ast {\textrm{|}_{{y^ + } = 14.5}}$, which is larger than the no-slip case (17.3 %). Although the near-wall part of WASS occupy only a small area, i.e. approximately 6.3 % of the x–z plane area at y ⁺ = 14.5 (figure 11a), its contributions to the skin frictional drag for turbulence are much larger than others.

Figure 16. Profiles of pre-multiplied C_{f ,i} versus $l_y^ +$. Dashed lines denote $l_y^ + = 1.1{\delta ^ + }$.