3.1.1 Length-scale self-similarity

The geometrical properties of the wall-attached $u$-structures ($PA_{u}$ and $NA_{u}$) in Re550 are examined by the joint p.d.f.s of $l_{x}^{+}$, $l_{z}^{+}$ with respect to $l_{y}^{+}$, as shown in figure 1. Hereafter, we mainly focus on the large-scale structures, i.e. those with $l_{y}^{+}>100$, whose wall-normal centres are approximately located at $y^{+}>50$. It can be seen that $l_{x}^{+}$ and $l_{z}^{+}$ increase with $l_{y}^{+}$ according to well-defined scaling laws. Nonlinear relationships are observed between $l_{x}^{+}$ and $l_{y}^{+}$, that is $l_{x}^{+}\propto (l_{y}^{+})^{0.70}$ for $PA_{u}$ and $l_{x}^{+}\propto (l_{y}^{+})^{0.88}$ for $NA_{u}$, and a linear relationship is well exhibited between $l_{z}^{+}$ and $l_{y}^{+}$. These sizes are approximately consistent with the observations in a turbulent boundary layer (Hwang & Sung Reference Hwang and Sung2018), suggesting that the geometrical self-similarity of spanwise length scales is also present in the wall turbulence even at much lower Reynolds numbers.

Figure 1.

(a,b) Joint p.d.f.s of the length ($l_{x}^{+}$) and width ($l_{z}^{+}$) of $PA_{u}$ in Re550 with respect to their height ($l_{y}^{+}$). The red dashed lines in (a,b) denote $l_{x}^{+}=17.41(l_{y}^{+})^{0.70}$ and $l_{z}^{+}=l_{y}^{+}$, respectively. (c,d) Joint p.d.f.s of the length ($l_{x}^{+}$) and width ($l_{z}^{+}$) of $NA_{u}$ in Re550 with respect to their height ($l_{y}^{+}$). The red dashed lines in (c,d) denote $l_{x}^{+}=9.88(l_{y}^{+})^{0.88}$ and $l_{z}^{+}=l_{y}^{+}$, respectively. All of these p.d.f.s are logarithmically distributed.

We next analyse the wall-normal decay of the population density of structures. Following Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012), we define the population density as $n_{d}(l_{y}^{+})=N_{i}(l_{y}^{+})/(2N_{F}A)$, where $N_{i}(l_{y}^{+})$ is the number of structures with a given $l_{y}^{+}$, $N_{F}$ is number of the flow fields and $A$ represents the area of the wall $L_{x}\times L_{z}$. Figure 2 displays $n_{d}$ as a function of $l_{y}^{+}$. The density of structures decays as $n_{d}\propto (l_{y}^{+})^{-1.9}$ for $NA_{u}$, while for $PA_{u}$ the density is well approximated by $n_{d}\propto (l_{y}^{+})^{-1.3}$. Since the expected theoretical estimation of $n_{d}$ obeys $n_{d}\propto (l_{y}^{+})^{-1}$ (Perry & Chong Reference Perry and Chong1982; Perry et al. Reference Perry, Henbest and Chong1986), the population density of $PA_{u}$ follows more closely the prediction of the attached-eddy theory than that of $NA_{u}$.

Figure 2.

The population density ($n_{d}$) of $PA_{u}$ and $NA_{u}$ as a function of $l_{y}^{+}$ in Re550. The red and black dashed lines represent $n_{d}\propto (l_{y}^{+})^{-1.9}$ and $n_{d}\propto (l_{y}^{+})^{-1.3}$, respectively.

3.1.2 Logarithmic wall-normal scaling of the statistical moments

To evaluate the scaling of the statistical moments and further examine the differences between $PA_{u}$ and $NA_{u}$, we define the conditional statistical moment of $u$ as

(3.1)$$\begin{eqnarray}\langle u_{a}^{2p+}\rangle ^{1/p}(y,l_{y}^{+})=\left\{\frac{1}{N_{u}(y,l_{y}^{+})}\mathop{\sum }_{i=1}^{N_{u}(y,l_{y}^{+})}\left[\frac{u_{i}(x,y,z)|_{l_{y}^{+}}}{u_{\unicode[STIX]{x1D70F}}}\right]^{2p}\right\}^{1/p},\end{eqnarray}$$

where $N_{u}(y,l_{y}^{+})$ denotes the number of grid points belonging to a structure with $l_{y}^{+}$ at a given $y$, and $u_{i}(x,y,z)|_{l_{y}^{+}}$ denotes the streamwise velocity fluctuation of the $i$th grid point of the structure. In this manner, $\langle u_{a}^{2p+}\rangle ^{1/p}$ represents the $2p$th root of the streamwise velocity moment carried by the structures with a given scale $l_{y}^{+}$.

Figure 3 shows the profiles of $\langle u_{a}^{2p+}\rangle ^{1/p}$, from $2p=2$ to $2p=8$, carried by structures with several selected $l_{y}^{+}$ in Re550. The figure reveals that, for increasing $l_{y}^{+}$, the profiles tend to decay logarithmically. To quantify the logarithmic decay in a systematic manner, we define the indicator function $\unicode[STIX]{x1D6EF}=y(\unicode[STIX]{x2202}\langle u_{a}^{2p+}\rangle ^{1/p}/\unicode[STIX]{x2202}y)$. Figure 4 shows the value of $\unicode[STIX]{x1D6EF}$ as a function of $y^{+}$, for the structures with $l_{y}^{+}=400$. Comparatively well-defined plateaus are observed for $PA_{u}$ regardless of the value of $p$. On the contrary, $\unicode[STIX]{x1D6EF}$ does not approach a constant value for $NA_{u}$. This phenomenon, together with the observations in § 3.1.1, suggests that $PA_{u}$ are more inclined to behave akin to ideal wall-attached eddies.

Figure 3.

Wall-normal variations of the $2p$th root of moments of the streamwise velocity fluctuations carried by $PA_{u}$ (a) and $NA_{u}$ (b) with several selected wall-normal heights in Re550. The profiles of the 4th, 6th and 8th roots of the moments are shifted upward by $\unicode[STIX]{x1D6E5}\langle u_{a}^{4+}\rangle ^{1/2}=10$, $\unicode[STIX]{x1D6E5}\langle u_{a}^{6+}\rangle ^{1/3}=20$ and $\unicode[STIX]{x1D6E5}\langle u_{a}^{8+}\rangle ^{1/4}=30$, respectively.

Figure 4.

Wall-normal variations of the corresponding indicator functions of the statistical moments from $2p=2$ to $2p=8$ for $PA_{u}$ (a) and $NA_{u}$ (b) with $l_{y}^{+}=400$ in Re550.

3.1.3 Inner peaks of the statistical moments of the streamwise velocity fluctuation

Another consequence of the attached-eddy model is the increase of the inner peaks of the second-order moment of $u$ as $\ln (Re_{\unicode[STIX]{x1D70F}})$, which has been validated in previous numerical and experimental studies (Marusic & Kunkel Reference Marusic and Kunkel2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Lee & Moser Reference Lee and Moser2015; Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017). By assuming that the inner peaks are the sole outcome of the wall-attached eddies residing in the logarithmic region, Meneveau & Marusic (Reference Meneveau and Marusic2013) derived a generalised formula for the inner peaks of the statistical moments of $u$,

(3.2)$$\begin{eqnarray}\langle u_{m}^{2p+}\rangle ^{1/p}\sim A_{p}\ln Re_{\unicode[STIX]{x1D70F}},\end{eqnarray}$$

where the coefficients $A_{p}$ are identical to those in (1.2), and represent the growth rates of the inner peaks.

Considering that $l_{y}^{+}$ can be interpreted as local $Re_{\unicode[STIX]{x1D70F}}$, by using the case of Re550, we further show in figure 5(a) the variations of $\langle u_{m}^{2p+}\rangle ^{1/p}$ as a function of $l_{y}^{+}$. For comparison, the wall-attached $u$-structures satisfying both (2.1) and (2.2), are also included and named $AA_{u}$, where $AA_{u}=PA_{u}+NA_{u}$. The inner peaks increase logarithmically with $l_{y}^{+}$ both for the high- and low-speed wall-attached $u$-structures, although the associated growth rates are different. In figure 5(b), the growth rates are compared with the high-Reynolds-number experimental results by Meneveau & Marusic (Reference Meneveau and Marusic2013). The growth rates of $PA_{u}$ are approximately in agreement with the experimental data, while the growth rates of $AA_{u}$ are approximately half of those of $PA_{u}$ due to the relatively smaller values of $NA_{u}$.

Figure 5.

(a) Variations of inner peaks ($\langle u_{m}^{2p+}\rangle ^{1/p}$) as a function of $l_{y}^{+}$ in Re550. The solid symbols denote $PA_{u}$, hollow symbols denote $AA_{u}$ and hollow symbols with ‘$+$’ denote $NA_{u}$. The symbols of $AA_{u}$ and $NA_{u}$ are shifted downward by $\unicode[STIX]{x1D6E5}\langle u_{m}^{2p+}\rangle ^{1/p}=5$ and $\unicode[STIX]{x1D6E5}\langle u_{m}^{2p+}\rangle ^{1/p}=12$, respectively. The solid lines are the fitting curves. ($b$) Variations of the growth rates of the inner peaks ($A_{m,p}$) as a function of the moment order $2p$ in Re550, compared with the $A_{p}$ obtained from experiments (Meneveau & Marusic Reference Meneveau and Marusic2013).

Recalling that both in experimental measurements and simulations the growth rates of the inner peaks are found to be lower than the theoretical predictions of $A_{p}$ in (3.2), an empirical factor $(1-\unicode[STIX]{x1D6FE})$ was introduced to correct $A_{p}$ with $\unicode[STIX]{x1D6FE}=0.5$ (Klewicki, Fife & Wei Reference Klewicki, Fife and Wei2009; Alfredsson, Segalini & Örlü Reference Alfredsson, Segalini and Örlü2011; Meneveau & Marusic Reference Meneveau and Marusic2013; Marusic et al. Reference Marusic, Baars and Hutchins2017). Note that, in these studies, no high- and low-speed scale separation has been conducted, similar to $AA_{u}$ in the present study. The results from figure 5 suggest that, although the features of the inner peaks of $AA_{u}$ are consistent with the experimental results for the empirical correction on $A_{p}$, the positive structures are again more in line with the theoretical predictions of the attached-eddy model.

3.1.4 An interpretation of the asymmetric behaviour between $PA_{u}$ and $NA_{u}$

The comparison of $PA_{u}$ and $NA_{u}$ has shown that the former follow more closely the theoretical predictions of the attached-eddy model framework. Hwang & Sung (Reference Hwang and Sung2019) observed a similar behaviour for the scale-based streamwise velocities in turbulent pipe flows up to $Re_{\unicode[STIX]{x1D70F}}=3008$. In their work, they attributed this phenomenon to the differences in the local $Re_{\unicode[STIX]{x1D70F}}$ conditioned to the presence of high- and low-speed streaks. In that respect, high-speed streaks are locally associated with higher Reynolds numbers, which was used to justify the emergence of logarithmic trends in quantities conditioned to the aforementioned structures. However, a similar analysis using our data (Re550) indicates that the friction Reynolds number conditioned to the presence of high-speed streaks is at most $Re_{\unicode[STIX]{x1D70F}}\approx 1000$, which is still too low to explain the appearance of the logarithmic trends in $PA_{u}$ discussed in previous subsections. Therefore, our results suggest that the differences in the local Reynolds number are probably not essential to understand the asymmetries between $NA_{u}$ and $PA_{u}$ reported above.

We provide an alternative interpretation of the differences between $PA_{u}$ and $NA_{u}$ based on the asymmetries of the fluid advected by sweeps and ejections. We measure $u$ and $v$ within the interior region of an identified structure to examine the averaged advection of the structure from the perspective of the quadrant decomposition (Wallace Reference Wallace2016). For the wall-attached $u$-structures ($PA_{u}$ and $NA_{u}$) with $l_{y}^{+}=400$, the joint p.d.f.s of $u^{+}$ and $v^{+}$ at $y/h=0.1$ and $y/h=0.2$ are displayed in figure 6. The results reveal that $PA_{u}$ with $l_{y}^{+}=400$ are associated with sweep events ($u^{+}>0$, $v^{+}<0$), while $NA_{u}$ with $l_{y}^{+}=400$ are strongly correlated with the ejection events ($u^{+}<0$, $v^{+}>0$). This observation holds for varying $l_{y}^{+}$.

Figure 6.

(a,b) The joint p.d.f.s of $u^{+}$ and $v^{+}$ at $y/h=0.1$ for $PA_{u}$ (a) and $NA_{u}$ (b) with $l_{y}^{+}=400$ in Re550. (c,d) The joint p.d.f.s of $u^{+}$ and $v^{+}$ at $y/h=0.2$ for $PA_{u}$ (c) and $NA_{u}$ (d) with $l_{y}^{+}=400$ in Re550. The red lines denote $v^{+}=0$.

Consequently, $PA_{u}$ are predominantly dominated by low shear flow from the upper region of the boundary layer, while $NA_{u}$ are strongly influenced by the higher shear and turbulence intensities of the near-wall region. The difference can be quantified more precisely using the Corrsin shear parameter (Corrsin Reference Corrsin1958) $S^{\ast }=SL_{\unicode[STIX]{x1D700}}/q$, which not only characterises the strength of the mean shear $S$, controlling the dynamics of eddies with integral length scale $L_{\unicode[STIX]{x1D700}}$ and turbulent kinetic energy $q^{2}$, but also stands for the ratio of the equilibration time $L_{\unicode[STIX]{x1D700}}/q$ (denoted as ET hereafter) and the life time $1/S$ (denoted as LT hereafter) of the eddies (Jiménez Reference Jiménez2018).

To shed light on the difference in the mean shear between $PA_{u}$ and $NA_{u}$, we define the conditional (scale-based) mean streamwise velocity, turbulent kinetic energy and dissipation as

(3.3)$$\begin{eqnarray}\displaystyle & \displaystyle \langle U_{a}\rangle (y,l_{y}^{+})=\left\{\frac{1}{N_{u}(y,l_{y}^{+})}\mathop{\sum }_{i=1}^{N_{u}(y,l_{y}^{+})}U_{i}(x,y,z)|_{l_{y}^{+}}\right\}, & \displaystyle\end{eqnarray}$$

(3.4)$$\begin{eqnarray}\displaystyle & \displaystyle \langle q_{a}^{2}\rangle (y,l_{y}^{+})=\left\{\frac{1}{N_{u}(y,l_{y}^{+})}\mathop{\sum }_{i=1}^{N_{u}(y,l_{y}^{+})}q_{i}^{2}(x,y,z)|_{l_{y}^{+}}\right\}, & \displaystyle\end{eqnarray}$$

(3.5)$$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D716}_{a}\rangle (y,l_{y}^{+})=\left\{\frac{1}{N_{u}(y,l_{y}^{+})}\mathop{\sum }_{i=1}^{N_{u}(y,l_{y}^{+})}\unicode[STIX]{x1D716}_{i}(x,y,z)|_{l_{y}^{+}}\right\}, & \displaystyle\end{eqnarray}$$

respectively, where $U_{i}(x,y,z)|_{l_{y}^{+}}$, $q_{i}^{2}(x,y,z)|_{l_{y}^{+}}$ and $\unicode[STIX]{x1D716}_{i}(x,y,z)|_{l_{y}^{+}}$ denote instantaneous streamwise velocity, turbulent kinetic energy and dissipation of the $i$th grid point of the structure with size $l_{y}^{+}$, respectively. The mean shear is calculated by $S_{a}(y)|_{l_{y}^{+}}=\unicode[STIX]{x2202}\langle U_{a}\rangle /\unicode[STIX]{x2202}y|_{l_{y}^{+}}$, and the conditional shear parameter is expressed as

(3.6)$$\begin{eqnarray}S_{a}^{\ast }(y)=\frac{S_{a}\langle q_{a}^{2}\rangle }{\langle \unicode[STIX]{x1D716}_{a}\rangle }.\end{eqnarray}$$

Figure 7(a) shows the variations of $S_{a}^{\ast }$ as a function of $y/h$ for $PA_{u}$ and $NA_{u}$ with $l_{y}^{+}=400$ in Re550. The shear parameter conditioned to $NA_{u}$ is visibly larger than that of $PA_{u}$ for $y/h>0.02$. This suggests that $NA_{u}$ originate under the strong shear conditions of the near-wall region, while $PA_{u}$ are controlled by the low shear flow in the upper domain of the boundary layer. Moreover, the wall-normal variations of ET and LT of $PA_{u}$ and $NA_{u}$ with $l_{y}^{+}=400$ are displayed in figure 7(b). For $PA_{u}$, the values of ET and LT are closer to each other than those of $NA_{u}$ when $y/h>0.03$. Consequently, it can be hypothesised that the flow associated with $NA_{u}$ is less likely to equilibrate during their life time to obey the logarithmic trends than that of $PA_{u}$, which is consistent with the asymmetrical features observed between $PA_{u}$ and $NA_{u}$.

Figure 7.

(a) Wall-normal variations of the conditional shear parameter ($S_{a}^{\ast }$) carried by $PA_{u}$ and $NA_{u}$ with $l_{y}^{+}=400$ in Re550. (b) Wall-normal variations of the life time and equilibration time of $PA_{u}$ and $NA_{u}$ with $l_{y}^{+}=400$ in Re550.

Finally, the trends for the inner peaks shown in figure 5 can also be interpreted from the sweep/ejection viewpoint. The inner peaks of $PA_{u}$ arise from the top-to-bottom wall interactions of sweep events. In this context, it is reasonable to assume that the $\ln (Re_{\unicode[STIX]{x1D70F}})$ growth rate of the inner peak is mainly a consequence of the outer flow transported by $PA_{u}$.

3.3.1 The wall-attached $u$-structures

We first focus on the statistical properties of the wall-attached $u$-structures. As for the geometrical features, the relationships between $l_{x}^{+}$, $l_{z}^{+}$ and $l_{y}^{+}$ can be expressed as $l_{x}^{+}\propto (l_{y}^{+})^{\unicode[STIX]{x1D6FE}_{1}}$ and $l_{z}^{+}\propto (l_{y}^{+})^{\unicode[STIX]{x1D6FE}_{2}}$, respectively. Figure 12(a) shows the variations of $\unicode[STIX]{x1D6FE}_{1}$ and $\unicode[STIX]{x1D6FE}_{2}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$. Note that only the structures with $l_{y}^{+}>100$ are taken into account. As $Re_{\unicode[STIX]{x1D70F}}$ increases, $\unicode[STIX]{x1D6FE}_{1}$ seems to converge to a constant value; whereas remarkable asymmetries are always observed between $PA_{u}$ and $NA_{u}$, regardless of $Re_{\unicode[STIX]{x1D70F}}$. The value of $\unicode[STIX]{x1D6FE}_{2}$ is in close proximity that when $Re_{\unicode[STIX]{x1D70F}}\geqslant 547$, indicating the self-similarity of the $u$-structures. The population density of the structures is represented by $n_{d}\propto (l_{y}^{+})^{-\unicode[STIX]{x1D6FE}_{3}}$, and the variations of the index $\unicode[STIX]{x1D6FE}_{3}$ are shown in figure 12(b) as a function of $Re_{\unicode[STIX]{x1D70F}}$. As seen, the values of $\unicode[STIX]{x1D6FE}_{3}$ for $PA_{u}$ are invariably close to the predictions of the attached-eddy model.

Figure 12.

(a) The variations of $\unicode[STIX]{x1D6FE}_{1}$ and $\unicode[STIX]{x1D6FE}_{2}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ for $u$-structures. (b) The variations of $\unicode[STIX]{x1D6FE}_{3}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ for $u$-structures.

Figure 13.

Wall-normal variations of the corresponding indicator function of the second-order moment for $PA_{u}$ (a) and $NA_{u}$ (b) with selected $l_{y}^{+}$ for all cases.

We next examine Reynolds-number effects on the statistical moments of the $u$-structures. Figure 13 shows the corresponding indicator function $\unicode[STIX]{x1D6EF}$ of the second-order moment of the $u$-structures for all cases. It can be seen that comparatively well-defined plateaus are observed for $PA_{u}$. The plateaus start from $y^{+}\approx 80$, which corresponds to the lower limit of the logarithmic region (Jiménez Reference Jiménez2018; Baars & Marusic Reference Baars and Marusic2020). For $NA_{u}$, however, $\unicode[STIX]{x1D6EF}$ does not approach a constant value. Note that the wrinkles in figure 13 may be caused by a lack of DNS samples. To the authors knowledge, this is the first time the logarithmic decay of the streamwise turbulence intensity at such low Reynolds numbers ($Re_{\unicode[STIX]{x1D70F}}\approx 180{-}550$) has been shown.

Figure 14.

Variations of the growth rates of the inner peaks ($A_{m,p}$) associated with $PA_{u}$ at different Reynolds numbers as a function of the moment order $2p$, compared with the $A_{p}$ obtained from experiments (Meneveau & Marusic Reference Meneveau and Marusic2013).

Finally, the growth rates of the inner peaks associated with $PA_{u}$ at different Reynolds number are examined and displayed in figure 14. The growth rates of the inner peaks are increased with the Reynolds number and tend to be consistent with the experimental results when $Re_{\unicode[STIX]{x1D70F}}\geqslant 547$. The mismatches in Re180 and Re350 are probably related to the viscous effect, which plays a non-negligible role in outer flow transport.

3.3.2 The wall-attached $w$-structures

We examine the Reynolds-number effects on statistical properties of the wall-attached $w$-structures. Due to the lack of asymmetry between $PA_{w}$ and $NA_{w}$, we only take $AA_{w}$ ($AA_{w}=PA_{w}+NA_{w}$) into account. Figure 15 shows the variations of $\unicode[STIX]{x1D6FE}_{1}$, $\unicode[STIX]{x1D6FE}_{2}$ and $\unicode[STIX]{x1D6FE}_{3}$ as functions of $Re_{\unicode[STIX]{x1D70F}}$ for $AA_{w}$. As $Re_{\unicode[STIX]{x1D70F}}$ increases, the values of $\unicode[STIX]{x1D6FE}_{1}$ and $\unicode[STIX]{x1D6FE}_{2}$ tend to be one, indicating the formation of the self-similarity. For $\unicode[STIX]{x1D6FE}_{3}$, its value is not far from the theoretical prediction ($\unicode[STIX]{x1D6FE}_{3}=1$), by and large.

To assess the logarithmic scaling of the spanwise turbulence intensity, figure 16 shows the corresponding indicator function $\unicode[STIX]{x1D6EF}$ of the second-order moment of the $w$-structures. It is readily noted that the plateau is observed in the range $80<y^{+}<250$ in Re950. It would be hypothesised that the logarithmic scaling of spanwise turbulence intensity is more significant at higher Reynolds numbers.

Figure 15.

The variations of $\unicode[STIX]{x1D6FE}_{1}$ , $\unicode[STIX]{x1D6FE}_{2}$ and $\unicode[STIX]{x1D6FE}_{3}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ for $AA_{w}$.

Figure 16.

Wall-normal variations of the corresponding indicator function of the second-order moment for $AA_{w}$ with selected $l_{y}^{+}$ for all cases.