3.1 Reynolds-number dependence

We first discuss the Re dependence of mean velocity and second-order moments in the MSU (i.e. ${L_{x}}^{+}\approx 400$ ). The distributions of the normalized mean velocity $\overline{U}^{+}$ and Reynolds shear stress $-\overline{u^{+}v^{+}}$ for $h^{+}=395$ and 1020 are compared in figures 1(a) and 1(d) with those for the LSD (i.e. $L_{x}=12.8h$ ). The experimental data of Wei & Willmarth (Reference Wei and Willmarth1989) and Monty et al. (Reference Monty, Stewart, Williams and Chong2007) and the DNS data of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for the VLSD are also included. Overall agreement with the LSD is satisfactory for $h^{+}=395$ , while there is an increased magnitude of $\overline{U}^{+}$ for $h^{+}=1020$ in the outer region ( $y^{+}>200$ , or equivalently, $y/h>0.2$ ) of the MSU than for the LSD/VLSD (see also figure 1 b of where the near parabolic profile is observed in the distribution of $\overline{U}^{+}$ for the MSU). There is also the well-established logarithmic mean velocity distribution for the MSU, which is intrinsically associated with the spanwise organization of large-scale structures, as discussed in § 3.3. Note that whilst the distribution of $-\overline{u^{+}v^{+}}$ for the MSU agrees reasonably well with that of the LSD/VLSD and of the experiment by Wei & Willmarth (Reference Wei and Willmarth1989) (figure 1 d), there is a discernible difference in the mean velocity gradient between the MSU and LSD in the outer region (see figure 1 c where the magnitude of $y^{+}(\unicode[STIX]{x2202}\overline{U}^{+}/\unicode[STIX]{x2202}y^{+})$ for the MSU is larger in the outer region than for the LSD). The turbulent eddy viscosity $\unicode[STIX]{x1D708}_{t}\equiv -\overline{uv}/(\unicode[STIX]{x2202}\overline{U}/\unicode[STIX]{x2202}y)$ is thus smaller in the outer region of the MSU than for the LSD and exhibits near constancy (the distribution is not shown here), which was also observed in the simulations with both streamwise and spanwise minimal units by Jiménez & Pinelli (Reference Jiménez and Pinelli1999) up to $h^{+}=633$ . The present results are most likely associated with the increasing two-dimensionality of the LSMs with $h^{+}$ , as discussed below.

Figure 1.

Distributions of $\overline{U}$ , $y(\unicode[STIX]{x2202}\overline{U}/\unicode[STIX]{x2202}y)$ and $-\overline{uv}$ normalized by wall units: (a,b) $\overline{U}^{+}$ ; (c) $y^{+}(\unicode[STIX]{x2202}\overline{U}^{+}/\unicode[STIX]{x2202}y^{+})$ ; (d) $-\overline{u^{+}v^{+}}$ . In (a,b), the distribution of $\overline{U}^{+}$ is shown in the half and whole channels, respectively. In (d), the distribution of the total shear stress, $\unicode[STIX]{x1D70F}_{total}^{+}=\unicode[STIX]{x2202}\overline{U}^{+}/\unicode[STIX]{x2202}y^{+}-\overline{u^{+}v^{+}}$ , is included. - - - - denotes the DNS data of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for $h^{+}=934$ with the VLSD ( $L_{x}=8\unicode[STIX]{x03C0}h$ ). ○ and ▫ refer to the experimental data of Wei & Willmarth (Reference Wei and Willmarth1989) for $h^{+}=1016$ and of Monty & Chong (Reference Monty and Chong2009) for $h^{+}=1040$ .

Figure 2.

Streamwise and spanwise two-point correlations of velocity fluctuations: (a) $R_{uu}(\unicode[STIX]{x0394}x)$ , $R_{vv}(\unicode[STIX]{x0394}x)$ and $R_{ww}(\unicode[STIX]{x0394}x)$ for the MSU for $h^{+}=395$ and 1020 at $y^{+}=5$ ; (b) $R_{uu}(\unicode[STIX]{x0394}x)$ , $R_{vv}(\unicode[STIX]{x0394}x)$ and $R_{ww}(\unicode[STIX]{x0394}x)$ for the MSU for $h^{+}=395$ and 1020 at $y/h=0.5$ ; (c) $R_{uu}(\unicode[STIX]{x0394}x)$ for the LSD for $h^{+}=1020$ ; (d) $R_{uu}(\unicode[STIX]{x0394}x)$ for the MSU for $h^{+}=1020$ ; (e) $R_{ww}(\unicode[STIX]{x0394}z)$ for the LSD for $h^{+}=1020$ ; (f) $R_{ww}(\unicode[STIX]{x0394}z)$ for the MSU for $h^{+}=1020$ . In (c–f), the $y$ locations of the plotted data are $y^{+}=5$ , 21, 61, 101 and $y/h=0.2$ , 0.5, 1.

Streamwise and spanwise two-point correlations are defined such that

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle R_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x0394}x)=\overline{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x0394}x)\,\unicode[STIX]{x1D6FC}(x+\unicode[STIX]{x0394}x)}/\overline{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}} & \displaystyle\end{eqnarray}$$

and

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle R_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x0394}z)=\overline{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x0394}z)\,\unicode[STIX]{x1D6FC}(z+\unicode[STIX]{x0394}z)}/\overline{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}\equiv u$ , $v$ or $w$ . Figures 2(a) and 2(b) show the distributions of $R_{uu}(\unicode[STIX]{x0394}x)$ , $R_{vv}(\unicode[STIX]{x0394}x)$ and $R_{ww}(\unicode[STIX]{x0394}x)$ in the inner and outer regions of the MSU for $h^{+}=395$ and 1020. Near the wall, the distributions remain invariably unchanged with $h^{+}$ . In the outer region ( $y/h=0.5$ ), $R_{uu}(\unicode[STIX]{x0394}x)$ for $h^{+}=1020$ shows a larger value ( ${\approx}0.9$ ) at the largest separation than that ( ${\approx}0.7$ ) for $h^{+}=395$ , whereas $R_{vv}(\unicode[STIX]{x0394}x)$ and $R_{ww}(\unicode[STIX]{x0394}x)$ for $h^{+}=1020$ show smaller values than those for $h^{+}=395$ . This indicates an increased two-dimensionality of $u$ in the outer region of the MSU at large $h^{+}$ . Panels 2(c) and 2(d) demonstrate how the magnitude of $R_{uu}(\unicode[STIX]{x0394}x)$ for $h^{+}=1020$ varies with increasing distance from the wall in the MSU and LSD. For the MSU, $R_{uu}(\unicode[STIX]{x0394}x)$ starts to show an increase in magnitude in the buffer region ( $y^{+}>20$ ) and exhibits a large value ( ${\geqslant}0.8$ ) in the outer region ( $y/h>0.2$ ) in which the distribution of $\overline{U}^{+}$ shows a larger value than that for the LSD. This highlights the close relationship between the two-dimensionality and the increased $\overline{U}^{+}$ in the outer region. Panels 2(e) and 2(f) show the distributions of $R_{ww}(\unicode[STIX]{x0394}z)$ for the MSU and LSD for $h^{+}=1020$ , which underline that the spanwise correlation of $w$ is reduced as a consequence of the two-dimensionality of large-scale structures. This behaviour is essentially associated with reduced spanwise meandering of the large-scale structures, as discussed in § 3.3.

Figure 3 shows distributions of the Reynolds normal stresses ( $\overline{u^{+}u^{+}}$ , $\overline{v^{+}v^{+}}$ , $\overline{w^{+}w^{+}}$ ) for both the MSU and LSD. Unlike the wall-normal ( $\overline{vv}$ ) and tangential ( $-\overline{uv}$ ) Reynolds stresses (figures 3 c,d and 1 d), the two wall-parallel components $\overline{uu}$ and $\overline{ww}$ for the MSU (figure 3 b,f) are somewhat different from those for the LSD (figure 3 a,e). Indeed, the magnitude of $\overline{uu}$ becomes more energetic as $h^{+}$ increases. In particular, there is almost a plateau instead of a large outer peak in $\overline{uu}$ when $h^{+}$ reaches 1020. The latter magnitude is comparable with that for the VLSD at $h^{+}=4000$ (Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014), which indicates the presence of energized LSMs in the MSU for a large $h^{+}$ (see § 3.3). On the other hand, the magnitude of $\overline{ww}$ for the MSU decreases significantly away from the wall. In high Re wall-bounded turbulent flows, the attached eddy model (Townsend Reference Townsend1976; Perry, Henbest & Chong Reference Perry, Henbest and Chong1986) predicts a logarithmic dependence for the two wall-parallel Reynolds stresses, viz.

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{uu}/U_{\unicode[STIX]{x1D70F}}^{2}=A_{1}-B_{1}\ln (y/h) & \displaystyle\end{eqnarray}$$

and

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}=A_{3}-B_{3}\ln (y/h), & \displaystyle\end{eqnarray}$$

where $A_{1}$ , $A_{3}$ , $B_{1}$ , $B_{3}$ are constants, while $\overline{vv}/U_{\unicode[STIX]{x1D70F}}^{2}\simeq \text{constant}$ and $-\overline{uv}/U_{\unicode[STIX]{x1D70F}}^{2}=1$ (see also figures 3 c,d and 1 d). Recently, it has become evident that $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ exhibits a logarithmic dependence even below $h^{+}=2000$ where, like $\overline{U}^{+}$ , $\overline{uu}/U_{\unicode[STIX]{x1D70F}}^{2}$ shows a departure from the logarithmic dependence due to very long and wide eddies in $u$ indicated by Jiménez & Hoyas (Reference Jiménez and Hoyas2008) (note that the profiles corresponding to (3.3) and (3.4) with $A_{1}=2.1$ , $A_{3}=0.85$ , $B_{1}=1.26$ , $B_{3}=0.45$ are plotted as ‘bold dashed lines’ in figures 3 a and 3 e). Figure 3 indicates that while the magnitude of $\overline{ww}$ for the MSU is attenuated, it exhibits a clearer logarithmic dependence than for the LSD (see figure 3 f where (3.4) with $A_{3}=0.15$ , $B_{3}=0.45$ is plotted and the agreement is convincing for the MSU when $h^{+}=1020$ ). This behaviour is most likely associated with reduced spanwise large scales in $w$ for the MSU (see figure 10 f) since small scales are likely to ‘lose’ the Re dependence more rapidly than large scales (see Abe & Antonia Reference Abe and Antonia2016, Reference Abe and Antonia2017 who noted that the overlap scaling for the energy and scalar dissipation rates is in fact established at a smaller value of $h^{+}$ than that at which the mean velocity and scalar log laws hold). Like $\overline{uu}/U_{\unicode[STIX]{x1D70F}}^{2}$ (see Jiménez & Hoyas Reference Jiménez and Hoyas2008), $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ is contaminated by the large-scale inactive motion. The attenuated spanwise large scales, associated with $w$ , essentially result from the fact that the LSMs have three-dimensional shear layers (see the x–y contours in figure 4 c and the y–z contours in figure 10 e,f), i.e. backs. In this context, Robinson (Reference Robinson1991) classified structures in a turbulent boundary layer into eight categories and noted that ‘backs’ are large-scale shear-layer interfaces between high (positive $u$ ) and low (negative $u$ ) speed fluid (i.e. sloping delta-scale discontinuities in the streamwise velocity) (see also figure 4 a). This will be further discussed in § 3.2, with a focus on the relationship with the pressure fluctuations (figure 4 b) and spanwise velocity fluctuations (figure 4 c).

Figure 3.

Distributions of normalized mean-square velocity fluctuations: (a,b) $\overline{uu}/U_{\unicode[STIX]{x1D70F}}^{2}$ ; (c,d) $\overline{vv}/U_{\unicode[STIX]{x1D70F}}^{2}$ ; (e,f) $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ . The data for the LSD are plotted in (a,c,e) together with those for the VLSDs in the literature, whereas those for the MSU are given in (b,d,f). In (a,c,e), – ⋅ – ⋅ –, $h^{+}=180$ ; $\cdots \cdots$ , $h^{+}=395$ ; - - - -, $h^{+}=640$ ; ——, $h^{+}=1020$ ; – – – –, $h^{+}=2003$ (Hoyas & Jiménez Reference Hoyas and Jiménez2008); — ⋅ ⋅ —, $h^{+}=4079$ (Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014); ○, Wei & Willmarth (Reference Wei and Willmarth1989) for $h^{+}=1016$ . The bold dashed-lines in (a) and (e) come from relations of (3.3) and (3.4), respectively, with $A_{1}=2.1$ , $A_{3}=0.85$ , $B_{1}=1.26$ , $B_{3}=0.45$ , while those in (f) are from (3.4) with $A_{3}=0.15$ , $B_{3}=0.45$ . In (b,d,f), – ⋅ – ⋅ –, $h^{+}=137$ (Toh & Itano Reference Toh and Itano2005);  $\cdots \cdots$ , $h^{+}=395$ ; ——, $h^{+}$ $=1020$ .

Figure 4.

Contours in the $x$ – $y$ plane of instantaneous $u$ , $p$ and $w$ for $h^{+}=1020$ : (a) $u$ ; (b)  $p$ ; (c) $w$ . In each panel, the top section shows the data for $L_{x}=12.8h$ (LSD), whereas the bottom section shows the data for $L_{x}=6.4h$ , $L_{x}=3.2h$ , $L_{x}=1.6h$ , $L_{x}=0.8h$ and $L_{x}=0.4h$ (MSU), the order being from left to right. In the top panel, the black solid lines have inclinations of 10 (left line) and 20 (right line) degrees.

3.2 Effect of varying $L_{x}$

In this subsection, we discuss how turbulence statistics and structures vary with decreasing $L_{x}$ for a sufficiently large Kármán number $h^{+}$ so that the inner and outer regions are sufficiently separated. Figure 4 shows contours of instantaneous streamwise velocity fluctuations $u$ in the $x$ – $y$ plane for $h^{+}=1020$ . Also included are contours of the instantaneous pressure fluctuations $p$ and spanwise velocity fluctuations $w$ . For the LSD ( $L_{x}=12.8h$ ), large-scale $u$ structures with a streamwise inclination of 10 to 20 degrees dominate in the outer region (see the two solid lines in figure 4 a), where positive and negative $p$ structures appear at the edge of negative $u$ structures (see the two solid lines in figure 4 b). This observation is consistent with the presence of transverse vortices along the edge of streamwise inclined negative $u$ structures (e.g. the PIV (particle image velocimetry) measurements of Christensen & Adrian Reference Christensen and Adrian2001 in connection with hairpin vortex packets and the DNS analysis of VLSMs by Lee & Sung Reference Lee and Sung2011). The streamwise inclination of the large-scale structures however becomes smaller with decreasing $L_{x}$ when $L_{x}<6.4h$ , which is intrinsically associated with the streamwise extent of the LSMs (i.e. approximately $2\sim 4h$ ) (see Balakumar & Adrian Reference Balakumar and Adrian2007). For the MSU (i.e. ${L_{x}}^{+}\approx 400$ ), the $u$ structures indeed exhibit a two-dimensional behaviour in the outer region, consistent with the result for the streamwise two-point correlations shown in figure 2(d).

Figure 5.

Effect of varying $L_{x}$ on premultiplied one-dimensional streamwise spectra of $u$ , $k_{x}\unicode[STIX]{x1D719}_{uu}(k_{x})$ , normalized by $U_{\unicode[STIX]{x1D70F}}^{2}$ for $h^{+}=1020$ : (a) $y/h=0.1$ ; (b) $y/h=0.3$ ; (c) $y/h=0.5$ ; (d) $y/h=0.7$ . – – – – denotes the DNS data of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for $h^{+}=934$ with the VLSD ( $L_{x}=8\unicode[STIX]{x03C0}h$ ). ○ and ▫ refer to the experimental data of Balakumar & Adrian (Reference Balakumar and Adrian2007) for $h^{+}=960$ and of Monty & Chong (Reference Monty and Chong2009) for $h^{+}=1040$ .

Figure 6.

Effect of varying $L_{x}$ on streamwise two-point correlations of $u$ , $R_{uu}(\unicode[STIX]{x0394}x)$ , for $h^{+}=1020$ : (a) $y/h=0.1$ ; (b) $y/h=0.3$ ; (c) $y/h=0.5$ ; (d) $y/h=0.7$ .

The streamwise variation of the large-scale $u$ structures may be quantified by the energy spectra. One-dimensional streamwise and spanwise spectra are defined such that

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\infty }\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}(k_{x})\,\text{d}k_{x}=\int _{0}^{\infty }k_{x}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}(k_{x})\,\text{d}(\log k_{x})=\overline{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}} & \displaystyle\end{eqnarray}$$

and

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\infty }\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}(k_{z})\,\text{d}k_{z}=\int _{0}^{\infty }k_{x}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}(k_{z})\,\text{d}(\log k_{z})=\overline{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}$ denotes the one-dimensional spectral density, $k_{x}$ and $k_{z}$ the streamwise and spanwise wavenumbers, $\unicode[STIX]{x1D706}_{x}(=2\unicode[STIX]{x03C0}/k_{x})$ and $\unicode[STIX]{x1D706}_{z}(=2\unicode[STIX]{x03C0}/k_{z})$ being the corresponding wavelengths and $\unicode[STIX]{x1D6FC}\equiv u$ , $v$ , $w$ or $p$ . Figure 5 shows the streamwise $u$ spectra $\unicode[STIX]{x1D719}_{uu}(k_{x})$ normalized by $U_{\unicode[STIX]{x1D70F}}^{2}$ at several $y$ stations in the outer region (viz. $y/h=0.1$ , 0.3, 0.5 and 0.7). Also included are the experimental data of Balakumar & Adrian (Reference Balakumar and Adrian2007) for $h^{+}=960$ and of Monty & Chong (Reference Monty and Chong2009) for $h^{+}=1040$ and the DNS data for the VLSD of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for $h^{+}=934$ . Whilst the streamwise $u$ spectra for the MSU agree reasonably well with those for the larger $L_{x}$ and also for the experiments up to their smallest non-zero wavenumbers, the most energetic outer spectral peak appearing at $k_{x}h=1.5{-}3$ (i.e. $\unicode[STIX]{x1D706}_{x}/h=2{-}4$ ) is not reproduced when $L_{x}<6.4h$ . The latter criterion of $L_{x}=6.4h$ is identical with the finding of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) that correct one-point statistics are obtained when $L_{x}$ is larger than $2\unicode[STIX]{x03C0}h$ . Also shown in figure 6 is the variation with $L_{x}$ of the streamwise two-point correlations of $u$ in the outer region (viz. $y/h=0.1$ , 0.3, 0.5 and 0.7). The correlation at the largest separation becomes smaller as $L_{x}$ increases. The present result is associated with the streamwise organization of the LSMs as observed in figure 4(a), which needs to be taken into account in order to reproduce correctly one-point statistics of velocity and pressure fluctuations (this will be discussed below). We also note a discernible difference in the streamwise $u$ spectra between the LSD and VLSD in the outer region where the low-wavenumber peak is resolved in the VLSD, but not in the LSD (see figure 5). This peak wavelength corresponds to the streamwise extent of the VLSMs (see Balakumar & Adrian Reference Balakumar and Adrian2007). The spanwise $u$ spectra however do not exhibit a noticeable difference between the LSD and VLSD (see figures 12 a,b and 16 a); mean and second-order statistics are also identical between the LSD and VLSD (see figures 1 a and 7).

Figure 7.

Effect of varying $L_{x}$ on the mean-square velocity and pressure fluctuations for $h^{+}=1020$ : (a) $\overline{uu}/U_{\unicode[STIX]{x1D70F}}^{2}$ ; (b) $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ ; (c) $\overline{vv}/U_{\unicode[STIX]{x1D70F}}^{2}$ ; (d) $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ . - - - - denotes the DNS data of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for $h^{+}=934$ with the VLSD ( $L_{x}=8\unicode[STIX]{x03C0}h$ ). ▫ refers to the experimental data of Monty & Chong (Reference Monty and Chong2009) for $h^{+}=1040$ . In (b,d), the bold dashed-lines are based on relations (3.4) with $A_{3}=0.85$ , $B_{3}=0.45$ and (3.7), respectively.

Figure 7 illustrates the $L_{x}$ dependence of the Reynolds normal stresses ( $\overline{uu}$ , $\overline{vv}$ , $\overline{ww}$ ) and mean-square pressure fluctuations ( $\overline{pp}$ ). Clearly, the magnitude of $\overline{uu}/U_{\unicode[STIX]{x1D70F}}^{2}$ increases with decreasing $L_{x}$ , whereas the magnitudes of $\overline{vv}/U_{\unicode[STIX]{x1D70F}}^{2}$ , $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ and $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ decrease with decreasing $L_{x}$ . This behaviour is essentially linked to the weakened energy redistribution (see figure 9 e). In particular, the rate of decrease in $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ and $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ is large. Figure 8 shows the premultiplied streamwise and spanwise spectra of $w$ and $p$ in the outer region ( $y/h=0.5$ ) for $h^{+}=1020$ . Both the streamwise and spanwise spectra are attenuated at large wavelengths when $L_{x}$ is below 6.4 $h$ . In the streamwise spectra, there is a close similarity between the $w$ and $p$ spectra, consistent with the good similarity in the instantaneous contours in the x–y plane of $w$ and $p$ (see figure 4 b,c where the sloping $w$ structures related to three-dimensional shear layers and the pressure structures associated with transverse vortices are observed). On the other hand, the spanwise $w$ spectra show more prominent peaks than the spanwise $p$ spectra. Indeed, there is a discernible difference in the peak wavelengths ( $\unicode[STIX]{x1D706}_{z,max}$ ) of the $w$ and $p$ spectra shown in figures 12(c) and 12(d) where the $p$ spectra for the LSD show a wider $y$ range of the linear increase of $\unicode[STIX]{x1D706}_{z,max}$ than the $w$ spectra. The latter corresponds to a wider $y$ region of the logarithmic dependence of $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ than that of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ (see figure 7 b,d). The instantaneous contours in the y–z plane of $w$ and $p$ for the LSD (see figures 10 e and 19 a) also show some differences: $w$ displays spanwise large-scale structures while $p$ exhibits smaller-scale structures than $w$ . This result corroborates the finding of Jiménez & Hoyas (Reference Jiménez and Hoyas2008) that because of the lack of very large scales, the pressure fluctuations away from the wall scale very well with $\unicode[STIX]{x1D70C}U_{\unicode[STIX]{x1D70F}}^{2}$ (or equivalently, $\unicode[STIX]{x1D70F}_{w}$ ). When $L_{x}<6.4h$ , the three-dimensional shear layers are not represented properly due to the less resolved streamwise behaviour of the LSMs where the pressure fluctuations exhibit localized structures (see § 3.4). Indeed, the peak wavelengths of both the spanwise $w$ and $p$ spectra for the MSU show a wide $y$ range of linear increase, consistent with the better logarithmic dependence of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ for the MSU. This underlines that spanwise large scales is responsible for the departure from the logarithmic dependence of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ .

Figure 8.

Effect of varying $L_{x}$ on premultiplied one-dimensional streamwise and spanwise spectra of $w$ and $p$ for $h^{+}=1020$ at $y/h=0.5$ : (a) $k_{x}\unicode[STIX]{x1D719}_{ww}(k_{x})/U_{\unicode[STIX]{x1D70F}}^{2}$ ; (b) $k_{z}\unicode[STIX]{x1D719}_{ww}(k_{z})/U_{\unicode[STIX]{x1D70F}}^{2}$ ; (c) $k_{x}\unicode[STIX]{x1D719}_{pp}(k_{x})/\unicode[STIX]{x1D70F}_{w}^{2}$ ; (d) $k_{z}\unicode[STIX]{x1D719}_{pp}(k_{z})/\unicode[STIX]{x1D70F}_{w}^{2}$ . In (a,b), – – – – denotes the DNS data of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for $h^{+}=934$ with the VLSD ( $L_{x}=8\unicode[STIX]{x03C0}h$ ).

Figure 9.

Effect of varying $L_{x}$ on budgets of the Reynolds normal stresses, energy dissipation rate, pressure strain and diffusion terms for $h^{+}=1020$ : (a) budget of $\overline{uu}$ ; (b) budget of $\overline{vv}$ ; (c) budget of $\overline{ww}$ ; (d) $\overline{\unicode[STIX]{x1D700}}h/U_{\unicode[STIX]{x1D70F}}^{3}$ ; (e) $\unicode[STIX]{x1D719}_{11}^{+}$ , $\unicode[STIX]{x1D719}_{22}^{+}$ and $\unicode[STIX]{x1D719}_{33}^{+}$ ; (f) $\overline{w^{+}w^{+}v^{+}}$ and $\overline{v^{+}p^{+}}$ . In (a–c), the normalization has been made by $U_{\unicode[STIX]{x1D70F}}^{4}/\unicode[STIX]{x1D708}$ . In (d), the bold dashed-line denotes the relation $\overline{\unicode[STIX]{x1D700}}h/U_{\unicode[STIX]{x1D70F}}^{3}=2.54/(y/h)-2.6$ , which is equivalent to relation (3.9) or (3.10).

Figure 10.

Contours in the $y$ – $z$ plane of instantaneous streamwise, wall-normal and spanwise velocity fluctuations, $u$ , $v$ , $w$ , for $h^{+}=1020$ : (a,b) $u$ ; (c,d) $v$ ; (e,f) $w$ . (a,c,e) Correspond to the LSD ( $L_{x}=12.8h$ ), while (b,d,f) are for the MSU ( ${L_{x}}^{+}\approx 400$ ). In (a,b), lines denote positive values of the second invariant of $Q^{+}$ (line contour levels are from $5\times 10^{-4}$ to $5\times 10^{-3}$ with an increment of $5\times 10^{-5}$ ).

The dependence of $\overline{pp}$ may be expressed as

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}=0.35-2.54\ln (y/h) & \displaystyle\end{eqnarray}$$

(note that the profile corresponding to (3.7) is plotted as a ‘bold dashed line’ in figure 7 d). Recently, Panton, Lee & Moser (Reference Panton, Lee and Moser2017) obtained an analogous relation to (3.7) by using matching argument. They found an interesting similarity between the slope of (3.7) and the magnitude of the von Kármán constant $\unicode[STIX]{x1D705}$ . They mentioned that ‘the physics governing the mean velocity and the pressure fluctuations is quite different’ but they did not point out that there is in fact a close similarity between (3.7) and what has been observed (Abe & Antonia Reference Abe and Antonia2016) for $\overline{\unicode[STIX]{x1D700}}$ . In this context, Poisson’s equation suggests a close relationship between the pressure derivative and energy dissipation rate (see Bradshaw & Koh Reference Bradshaw and Koh1981), although their characteristic $y$ scales are different (this difference was pointed out by one of the referees; see also the wall-normal two-point correlations for $p$ shown in figure 18 d). Abe & Antonia (Reference Abe and Antonia2016) found that the mean energy dissipation rate, i.e.

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D708}\overline{u_{i,j}(u_{i,j}+u_{j,i})}, & \displaystyle\end{eqnarray}$$

has an analogous overlap scaling,

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D700}}y/U_{\unicode[STIX]{x1D70F}}^{3}=1/\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D700}}-c(y^{+}/h^{+}) & \displaystyle\end{eqnarray}$$

and

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D700}}y/U_{\unicode[STIX]{x1D70F}}^{3}=1/\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D700}}-c(y/h), & \displaystyle\end{eqnarray}$$

with $1/\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D700}}=2.54$ and $c=2.6$ (see figure 9 d). The magnitude of $1/\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D700}}$ is identical to that of the slope in (3.7). In our previous studies (Abe & Antonia Reference Abe and Antonia2016, Reference Abe and Antonia2017) on the energy and passive-scalar dissipation rates, we noted that the overlap region for the dissipation is more readily established than that for the mean fields since the small scales (i.e. dissipation) are likely to ‘lose’ their dependence on the Reynolds number more rapidly than the large scales. Since small scales make up the source term in Poisson’s equation and the outer-layer similarity for both $\overline{pp}$ and $\overline{\unicode[STIX]{x1D700}}$ is established at much smaller $h^{+}$ than that for $\bar{U}$ , it seems reasonable to identify the scaling behaviour of $\overline{pp}$ with that of $\overline{\unicode[STIX]{x1D700}}$ . Note that unlike $\overline{pp}$ , the scaling for $\overline{\unicode[STIX]{x1D700}}$ is essentially unaffected by $L_{x}$ . This is because $\overline{pp}$ has wall-attached global nature (see the wall-normal two-point correlation of $p$ shown in figure 18 d), whilst $\overline{\unicode[STIX]{x1D700}}$ is essentially a small-scale quantity. When $h^{+}$ is sufficiently large ( ${\geqslant}5000$ ) for the velocity log law to be established over a region where $P_{k}\simeq \overline{\unicode[STIX]{x1D700}}$ and $-\overline{uv}\simeq \text{const.}$ (see figure 10 of Abe & Antonia Reference Abe and Antonia2016), the von Kármán constant $\unicode[STIX]{x1D705}$ can be identified with $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D700}}$ . Consistent with the similarity between $w$ and $p$ discussed above, the logarithmic dependence of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ is also established for a lower Kármán number than that at which the mean velocity log law is first observed. The lower $y$ bound of the logarithmic dependence of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ and $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ is indeed smaller than that of $\bar{U}$ , which indicates that $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ and $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ are less affected by viscous effects. An analogous trend was observed in a turbulent boundary layer by Spalart (Reference Spalart1988). In his simulations, the viscous correction obtained by Perry et al. (Reference Perry, Henbest and Chong1986) was successfully applied to $\overline{ww}$ rather than $\overline{uu}$ .

To gain further insight into the $L_{x}$ dependence of the Reynolds normal stresses, we examine the transport equation of $\overline{u_{i}u_{j}}$ defined such that

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle 0=P_{ij}+T_{ij}+\unicode[STIX]{x1D6F1}_{ij}+D_{ij}-\unicode[STIX]{x1D700}_{ij}, & \displaystyle\end{eqnarray}$$

where

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle P_{ij}^{+}=-\overline{u_{j}^{+}u_{k}^{+}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{U}_{i}^{+}}{\unicode[STIX]{x2202}x_{k}^{+}}}-\overline{u_{i}^{+}u_{k}^{+}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{U}_{j}^{+}}{\unicode[STIX]{x2202}x_{k}^{+}}}, & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle T_{ij}^{+}=-{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{k}^{+}}}\overline{u_{i}^{+}u_{j}^{+}u_{k}^{+}}, & \displaystyle\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F1}_{ij}^{+}=-\overline{u_{j}^{+}{\displaystyle \frac{\unicode[STIX]{x2202}p^{+}}{\unicode[STIX]{x2202}x_{i}^{+}}}}-\overline{u_{i}^{+}{\displaystyle \frac{\unicode[STIX]{x2202}p^{+}}{\unicode[STIX]{x2202}x_{j}^{+}}}}, & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle D_{ij}^{+}={\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x_{k}^{+2}}}\overline{u_{i}^{+}u_{j}^{+}}, & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D700}_{ij}}^{+}=2\overline{{\displaystyle \frac{\unicode[STIX]{x2202}u_{i}^{+}}{\unicode[STIX]{x2202}x_{k}^{+}}}{\displaystyle \frac{\unicode[STIX]{x2202}u_{j}^{+}}{\unicode[STIX]{x2202}x_{k}^{+}}}}. & \displaystyle\end{eqnarray}$$

$P_{ij}^{+}$ , $T_{ij}^{+}$ , $\unicode[STIX]{x1D6F1}_{ij}^{+}$ , $D_{ij}^{+}$ and $\overline{\unicode[STIX]{x1D700}_{ij}}^{+}$ denote the production, turbulent diffusion, velocity-pressure-gradient correlation (VPG), molecular diffusion, homogeneous dissipation, respectively. Figure 9(a–c) shows budgets of $\overline{uu}$ , $\overline{vv}$ , $\overline{ww}$ non-dimensionalized by $U_{\unicode[STIX]{x1D70F}}^{4}/\unicode[STIX]{x1D708}$ , which demonstrate the significant $L_{x}$ dependence of the spanwise Reynolds stress. For the $\overline{uu}$ budget, the production is essentially independent of $L_{x}$ , whereas the magnitudes of molecular diffusion and dissipation increase with decreasing $L_{x}$ close to the wall (figure 9 a). The $L_{x}$ independence of the production essentially comes from the forcing by the mean pressure gradient. As $L_{x}$ decreases, $\overline{\unicode[STIX]{x1D700}_{11}}$ increases whereas $\overline{\unicode[STIX]{x1D700}_{33}}$ decreases (figure 9 a,c) but the energy dissipation rate $\overline{\unicode[STIX]{x1D700}}$ ( $\equiv \overline{\unicode[STIX]{x1D700}_{ii}}/2$ ) is basically unchanged (figure 9 d). The total amount of energy (both produced and dissipated) is thus essentially unchanged. In the energy balance (i.e. integrals of the $k$ budget across the channel), the integral of the production is thus balanced by that of the dissipation since the integrals of convection and diffusion terms become zero. The latter relationship is satisfied independently of $L_{x}$ (see also section 2 of Abe & Antonia Reference Abe and Antonia2016). As for the turbulent flux, the turbulent diffusion terms ( $T_{22}^{+}$ , $T_{33}^{+}$ ) and the VPG ones ( $\unicode[STIX]{x1D6F1}_{11}^{+}$ , $\unicode[STIX]{x1D6F1}_{22}^{+}$ , $\unicode[STIX]{x1D6F1}_{33}^{+}$ ) decrease as $L_{x}$ decreases. The VPG term may be split into the pressure strain $\unicode[STIX]{x1D719}_{ij}^{+}$ and pressure diffusion $\unicode[STIX]{x1D711}_{ij}^{+}$ , respectively, i.e.

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F1}_{ij}^{+}=\underbrace{\overline{p^{+}\left({\displaystyle \frac{\unicode[STIX]{x2202}u_{i}^{+}}{\unicode[STIX]{x2202}x_{j}^{+}}}+{\displaystyle \frac{\unicode[STIX]{x2202}u_{j}^{+}}{\unicode[STIX]{x2202}x_{i}^{+}}}\right)}}_{\unicode[STIX]{x1D719}_{ij}^{+}}\underbrace{-\left({\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}^{+}}}\overline{u_{j}^{+}p^{+}}+{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}^{+}}}\overline{u_{i}^{+}p^{+}}\right)}_{\unicode[STIX]{x1D711}_{ij}^{+}}. & \displaystyle\end{eqnarray}$$

The pressure strain plays a role in redistributing the energy from $\overline{uu}$ to $\overline{vv}$ and $\overline{ww}$ . Note that $\unicode[STIX]{x1D719}_{11}^{+}$ and $\unicode[STIX]{x1D719}_{33}^{+}$ are equivalent to $\unicode[STIX]{x1D6F1}_{11}^{+}$ and $\unicode[STIX]{x1D6F1}_{33}^{+}$ , respectively, in the present flow. Figure 9(e) demonstrates that there is a weakened energy redistribution (see also Abe & Antonia Reference Abe and Antonia2016). For $L_{x}<6.4h$ , the magnitudes of $\unicode[STIX]{x1D719}_{11}^{+}$ and $\unicode[STIX]{x1D719}_{33}^{+}$ are attenuated significantly for $y^{+}<100$ due to the impaired global nature of the pressure (see also the wall-normal two-point correlations of $p$ shown in figure 18 d). Consistent with the behaviour of $T_{33}^{+}$ and $\unicode[STIX]{x1D6F1}_{22}^{+}$ , the energy fluxes of $\overline{w^{+}w^{+}v^{+}}$ and $\overline{p^{+}v^{+}}$ are also reduced noticeably across the channel when $L_{x}$ is small (see figure 9 f). This reflects the weakened turbulent diffusion resulting from the two-dimensionality in the MSU. This behaviour is essentially associated with more energized large-scale $u$ structures in the MSU than in the LSD (see figure 10 a,b), which is consistent with the increase in the mean streamwise velocity (see the near parabolic profile in figure 1 b). The magnitudes of $\overline{vv}$ and $\overline{ww}$ (figure 7 b,c) hence become smaller when $L_{x}$ decreases. The present observation for the MSU is consistent with the expectation of SSP at large Reynolds numbers (Waleffe & Kim Reference Waleffe, Kim and Panton1997). In particular, two wall-parallel components ( $\overline{uu}$ and $\overline{ww}$ ) are affected significantly by varying $L_{x}$ due to the absence of two properties of the large-scale structures, namely their streamwise inclination and spanwise meandering (the latter will be discussed further in the next subsection).

3.3 Spanwise organization of velocity fluctuations

Here, we discuss the spanwise organization of velocity fluctuations for the MSU and LSD (i.e. $L_{x}=0.4h$ and $12.8h$ ) for $h^{+}=1020$ . Contours of instantaneous streamwise, wall-normal and spanwise velocity fluctuations ( $u$ , $v$ , $w$ ) in the $y$ – $z$ plane for $h^{+}=1020$ are shown in figure 10. This figure underlines the presence of hierarchical $u$ structures for both the MSU and LSD. Clearly, the spanwise organization of streamwise velocity fluctuations $u$ is essentially identical between the two cases, whilst wall-normal and spanwise velocity fluctuations $v$ and $w$ away from the wall are weakened as $L_{x}$ decreases. These behaviours are supported statistically by figure 11 where the premultiplied spanwise spectra of $u$ , $v$ , $w$ are shown at several $y$ locations. Indeed, the MSU reproduces the spanwise organization of large-scale structures in the log region properly even at sufficiently large $h^{+}$ , consistent with the well-established logarithmic mean velocity distribution (see figure 1 a). One discernible difference from the LSD is that large-scale structures in the outer region ( $y/h>0.2$ ) are more energized for the MSU due to the reduced turbulent diffusion resulting from the approach of the large-scale structures towards the two-dimensionality (see figure 9 f); consequently, a large majority of the turbulent kinetic energy (approximately 80 %) is carried by the large-scale $u$ structures in the MSU. The negative $u$ structures (see figure 10 a,b where the positive values of the second invariant of the velocity gradient tensor $Q(\equiv -u_{i,j}u_{j,i}/2)$ are also depicted as a vortex indicator and preferably associated with the negative $u$ structures in the outer region) are also closely associated with negative uv regions as for the LSD and active in the context of the momentum transport (the instantaneous product of $u$ and $v$ is not shown here). The latter underlines that the infinitely long $u$ structures observed for the MSU are responsible for the increase in the mean velocity distribution in the outer region (see the near parabolic profile in figure 1 b).

Figure 11.

One-dimensional premultiplied spanwise spectra of $u$ , $v$ and $w$ for $h^{+}=1020$ : (a,b) $k_{z}\unicode[STIX]{x1D719}_{uu}(k_{z})/\overline{uu}$ ; (c,d) $k_{z}\unicode[STIX]{x1D719}_{vv}(k_{z})/\overline{vv}$ ; (e,f) $k_{z}\unicode[STIX]{x1D719}_{ww}(k_{z})/\overline{ww}$ . (a,c,e) Correspond to the LSD ( $L_{x}=12.8h$ ), while (b,d,f) are for the MSU ( ${L_{x}}^{+}\approx 400$ ). In each figure, the $y$ locations of the plotted data are $y^{+}=5$ , 21, 61, 101 and $y/h=0.2$ , 0.5, 1.

Figure 12.

Most energetic spanwise scales, $\unicode[STIX]{x1D706}_{z,max}$ , obtained from peaks of premultiplied spanwise spectra of $u$ , $w$ and $p$ for the MSU and LSD $h^{+}=1020$ : (a), $\unicode[STIX]{x1D706}_{z,max}/h$ for $u$ (b) $\unicode[STIX]{x1D706}_{z,max}^{+}$ for $u$ ; (c) $\unicode[STIX]{x1D706}_{z,max}/h$ for $w$ ; (d) $\unicode[STIX]{x1D706}_{z,max}/h$ for $p$ . In (a,b), ▫ denotes the DNS data of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for $h^{+}=934$ with the VLSD ( $L_{x}=8\unicode[STIX]{x03C0}h$ ). In (a); the dashed, one-dotted and two-dotted lines denote $\unicode[STIX]{x1D706}_{z,max}/h=0.8$ , 1.3 and 1.6, respectively; the ‘diamonds’ refer to the data of Tomkins & Adrian (Reference Tomkins and Adrian2005) in a zero-pressure-gradient turbulent boundary layer for $\unicode[STIX]{x1D6FF}^{+}=426$ . In (b–d), the dashed lines are linear fits to the data.

We now examine the most energetic spanwise scales of the hierarchical $u$ structures, $\unicode[STIX]{x1D706}_{z,max}$ , by plotting the peak wavelengths of the premultiplied spectra $k_{z}\unicode[STIX]{x1D719}_{uu}(k_{z})$ shown in figure 11(a,b). Note that the peak wavelength of the premultiplied spectrum corresponds to the most energetic scale contributing to the mean square value. Figure 12(a) demonstrates that there are three energetic spanwise scales for both the MSU and LSD, which are essentially identical to those for the VLSD of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004). Near the wall, a large peak appears at ${\unicode[STIX]{x1D706}_{z}}^{+}\approx 100$ , which corresponds to mean spanwise spacing of near-wall streaks (Smith & Metzler Reference Smith and Metzler1983). Away from the wall, the most energetic spanwise scales tend to increase linearly with distance from the wall in the range $40\leqslant y^{+}\leqslant 80$ (see figure 12 b), which is consistent with the approximate self-similarity of the energetic spanwise scales and agrees well with the experimental results (Nakagawa & Nezu Reference Nakagawa and Nezu1977; Smith & Metzler Reference Smith and Metzler1983; Tomkins & Adrian Reference Tomkins and Adrian2003, Reference Tomkins and Adrian2005). The linear increase in $\unicode[STIX]{x1D706}_{z,max}$ is better observed for the $w$ and $p$  spectra (see figure 12 c,d). Here, $\unicode[STIX]{x1D706}_{z,max}$ associated with $w$ shows a linear increase over a wider range of $y$ for the MSU than for the LSD. The $y$ range of the linear increase corresponds well to that for the logarithmic dependence of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ . This implies that the departure from the logarithmic behaviour of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ is essentially associated with the large-scale spanwise meandering observed in the LSD/VLSD, which corroborates Jiménez & Hoyas’s (Reference Jiménez and Hoyas2008) finding that most of the departure from the logarithmic behaviour of $\overline{uu}/U_{\unicode[STIX]{x1D70F}}^{2}$ is due to the very long and wide eddies in $u$ . Given the notion of Nickels et al. (Reference Nickels, Marusic, Hafez, Hutchins and Chong2007) that any eddy with a size that scales with its distance from the wall may be considered to be attached to the wall, the well-established logarithmic dependence of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ in the MSU is intrinsically associated with the attached eddy hypothesis of Townsend (Reference Townsend1976). On the other hand, the $\unicode[STIX]{x1D706}_{z}/h=1.3\sim 1.6$ peak dominates in the outer region ( $y/h=0.2{-}1$ ) (see figure 12 a), which corresponds to the outer global mode (del Álamo & Jiménez Reference del Álamo and Jiménez2003; Abe et al. Reference Abe, Kawamura and Choi2004b ; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014). The present result indicates that while the outer global modes are observed in the longest and widest modes of the $u$ spectra (i.e. $\unicode[STIX]{x1D706}_{x}/h>5\sim 6h$ and $\unicode[STIX]{x1D706}_{z}/h>1\sim 2h$ ) for the VLSD (Jiménez & Hoyas Reference Jiménez and Hoyas2008), the MSU and LSD also capture the spanwise global mode in the spectra. This underlines that the spanwise organization of the VLSMs can be represented approximately as infinitely long structures.

Figure 13.

One-dimensional premultiplied spanwise spectra of $u$ for $h^{+}=1020$ for the LSD (the black line) and MSU (the red line) at $y/h=0.2$ . The spectra are compared with those of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) and Hoyas & Jiménez (Reference Hoyas and Jiménez2008) for $h^{+}=934$ and 2003 at $y/h=0.16$ (the green line) and $y/h=0.2$ (the blue line), respectively, for the VLSD ( $L_{x}=8\unicode[STIX]{x03C0}h$ ) in the present flow. Also included are the spectra of Tomkins & Adrian (Reference Tomkins and Adrian2005) in a zero-pressure-gradient turbulent boundary layer for $\unicode[STIX]{x1D6FF}^{+}=2216$ at $y/\unicode[STIX]{x1D6FF}=0.2$ (circles) and those of Ahn et al. (Reference Ahn, Lee, Lee, Kang and Sung2015) in a turbulent pipe for $R^{+}=3008$ at $y/R=0.1$ (triangles).

Note that near the upper limit ( $y/h\approx 0.2$ ) of the logarithmic region, two large peaks are observed for the spanwise $u$ spectra at large wavelengths, $\unicode[STIX]{x1D706}_{z}/h=0.8\sim 0.9$ and $1.3\sim 1.6$ (see figure 11 a,b). The former peak appears in the region $y^{+}=100$ to $y/h=0.2$ , whereas the latter one in the region $y/h=0.2{-}1$ , implying the existence of two different large-scale modes (see also Kawamura, Abe & Matsuo Reference Kawamura, Abe and Matsuo2004). The peak at $\unicode[STIX]{x1D706}_{\text{z}}/h=0.8\sim 0.9$ is not prominent at low Reynolds numbers (del Álamo & Jiménez Reference del Álamo and Jiménez2003; Abe et al. Reference Abe, Kawamura and Choi2004b ) but appears clearly at $h^{+}=1020$ . In this context, the recent DNS work of Ahn et al. (Reference Ahn, Lee, Lee, Kang and Sung2015) in a pipe flow for $R^{+}=3008$ ( $R$ denotes the pipe radius) reported the double peaks (i.e. $\unicode[STIX]{x1D706}_{z}/R=0.7$ and 1.3) of the premultiplied spanwise $u$ spectra in the logarithmic region. The present spectra have been compared with those of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) and Hoyas & Jiménez (Reference Hoyas and Jiménez2008) at $h^{+}=934$ and 2003, respectively, for the VLSD in the present flow, those of Ahn et al. (Reference Ahn, Lee, Lee, Kang and Sung2015) in a pipe flow for $R^{+}=3008$ and of Tomkins & Adrian (Reference Tomkins and Adrian2005) in a laboratory boundary layer for $\unicode[STIX]{x1D6FF}^{+}=2216$ , and are shown in figure 13. Indeed, the double peaks are evident in the logarithmic region when $h^{+}$ becomes large. Also, while the outer flow is completely different between the internal and external flows (e.g. the presence of the $\unicode[STIX]{x1D706}_{z}/h=1.3\sim 1.6$ or $\unicode[STIX]{x1D706}_{z}/R=1.3$ mode in the internal flows), there seems to be some similarity in the spectral peak at a lower wavelength ( $\unicode[STIX]{x1D706}_{z}/h=0.8\sim 0.9$ (channel), $\unicode[STIX]{x1D706}_{z}/R=0.7$ (pipe), $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}=0.8$ (boundary layer)) in the logarithmic region, which corroborates the fact that the magnitude of the von Kármán constant $\unicode[STIX]{x1D705}$ does not differ significantly between the three flows and is approximately 0.39 at extremely large Reynolds number (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). We note that the spanwise outer global mode with $\unicode[STIX]{x1D706}_{z}/h=1.3\sim 1.6$ (del Álamo & Jiménez Reference del Álamo and Jiménez2003; Abe et al. Reference Abe, Kawamura and Choi2004b ; Jiménez & Hoyas Reference Jiménez and Hoyas2008; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014) is quite dominant in a channel flow, which masks the peak in the logarithmic region in the LSD/VLSD. In the MSU, the latter peak however becomes prominent due to the tight coupling of the neighbouring structures, as discussed below. These results also highlight a close relationship between the spanwise organization of large-scale structures and the logarithmic mean velocity distribution.

Figure 14.

Time variation of premultiplied spanwise spectra of $u$ at $y^{+}=101$ for $h^{+}=1020$ : (a) LSD ( $L_{x}=12.8h$ ); (b) MSU ( ${L_{x}}^{+}\approx 400$ ). The spectra have been normalized by $U_{\unicode[STIX]{x1D70F}}^{2}$ . In (a), —— and – ⋅ – ⋅ – correspond to the spectral peaks at $\unicode[STIX]{x1D706}_{z}/h=0.8$ and 1.3, respectively, for the LSD. In (b), – ⋅ – ⋅ – and — ⋅ ⋅ — correspond to the spectral peaks at $\unicode[STIX]{x1D706}_{z}/h=0.9$ and 1.6, respectively, for the MSU.

Figure 14 shows the time variation of the spanwise $u$ spectra at $y^{+}=101$ , which highlights the persistence of two outer peaks, i.e. the $\unicode[STIX]{x1D706}_{z}/h=0.8\sim 0.9$ and $1.3\sim 1.6$ peaks (see also the two lines of $\unicode[STIX]{x1D706}_{z}/h=0.8$ and 1.3 for the LSD and $\unicode[STIX]{x1D706}_{z}/h\approx 0.9$ and 1.6 for the MSU shown in figure 14). These peaks are observed more clearly in the MSU than in the LSD due to the limited number of organization in the former domain so that a longer sampling time period is required for the MSU for obtaining the statistics than for the LSD. Note that the time period shown in figure 14 (i.e. $t^{+}=10^{4}$ ) corresponds to approximately 3 % and 73 % of the total sampling time period for the MSU and LSD, respectively. Figure 14(a) demonstrates that the time scales for the $\unicode[STIX]{x1D706}_{z}/h=0.8\sim 0.9$ and $1.3\sim 1.6$ peaks for the LSD are approximately $t^{+}=500\sim 1000$ and $2000\sim 3000$ , respectively. The present time scales agree reasonably well with the relation of $t=6y/U_{\unicode[STIX]{x1D70F}}$ obtained by Flores & Jiménez (Reference Flores and Jiménez2010) given that the $\unicode[STIX]{x1D706}_{z}/h=0.8\sim 0.9$ and $1.3\sim 1.6$ peaks dominate in the logarithmic ( $y^{+}=100$ to $y/h=0.2$ ) and outer ( $y/h=0.2{-}1$ ) regions, respectively. Note however that the time scale for the $\unicode[STIX]{x1D706}_{z}/h=1.3\sim 1.6$ peak for the MSU is longer than for the LSD and quite persistent in time (see figure 14 b). This is intrinsically associated with the approach of the large-scale structures towards two-dimensionality in the present MSU at $h^{+}=1020$ , which yields a tight coupling between neighbouring structures; as a consequence, the large-scale structures do not move significantly in the $y$ and $z$ directions, but are present in the domain for a long period of time (see figure 17). There is thus a strong interaction between neighbouring structures. A tight coupling between large-scale structures and near-wall streaks was first observed by Toh & Itano (Reference Toh and Itano2005) for the MSU at $h^{+}=137$ and 349. To examine this behaviour for $h^{+}=1020$ in detail, we introduce $u^{2D}$ defined such that

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle u^{2D}(y,z,t)={\displaystyle \frac{1}{L_{x}}}\int _{0}^{L_{x}}u(x,y,z,t)\,\text{d}x, & \displaystyle\end{eqnarray}$$

given that the MSU allows for only one near-wall and one large-scale structure with respect to the $x$ direction (Toh & Itano Reference Toh and Itano2005). In figure 15, the t–z contours of $u^{2D}$ for the MSU are compared with the x–z contours of $u$ for the LSD. Two wall-normal locations ( $y^{+}=15$ and $y/h=0.5$ ) are examined. The similarity between $u$ and $u^{2D}$ is convincing in the inner region. In this figure, we also see the good correspondence in locations between the dense clustering patterns of near-wall streaks and the LSMs. The former are thus footprints of the LSMs (see Abe et al. Reference Abe, Kawamura and Choi2004b ; Hutchins & Marusic Reference Hutchins and Marusic2007a ,Reference Hutchins and Marusic b ). We stress that the dense clustering patterns become more energetic for the MSU than for the LSD. Figure 16(a) shows the spanwise spectra of $u$ , $v$ , and $w$ normalized by $U_{\unicode[STIX]{x1D70F}}^{2}$ and $\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}$ for $h^{+}=1020$ at $y^{+}=15$ for the MSU and LSD/VLSD. The high-wavenumber behaviour is nearly identical between the MSU and LSD/VLSD, while the energy at low and intermediate wavenumbers in the $u$ and $w$ spectra are increased and decreased, respectively, for the MSU. In particular, the energy at low and intermediate wavenumbers (i.e. $k_{z}h=5\sim 50$ ) in the $u$ spectra is approximately 30 % larger for the MSU than for the LSD (see figure 16 b). Also, the energy in the $u$ spectra for the MSU is more enhanced at very low wavenumbers (i.e. $k_{z}h<5$ ) than at low and intermediate wavenumbers (i.e. $k_{z}h=5\sim 50$ ), where the peak is invariably observed at $k_{z}h\approx 2$ , or equivalently, $\unicode[STIX]{x1D706}_{z}/h\approx 3$ across the channel (figure 16 d). In the LSD, the latter spectral peak is exclusively observed in the core region (figure 16 c). This indicates that the superimposition of the LSMs as a low-wavenumber mean shift onto the near-wall region (see Hutchins & Marusic Reference Hutchins and Marusic2007b ) is significantly enhanced for the MSU. Note that the large-scale spanwise meandering (Hutchins & Marusic Reference Hutchins and Marusic2007a ,Reference Hutchins and Marusic b ) and waviness of the large-scale structures (Flores & Jiménez Reference Flores and Jiménez2010) in the logarithmic region are not observed in the MSU (see figure 10 f). The latter behaviour is essentially associated with the decreased logarithmic dependence of $\overline{ww}$ shown in figure 3(f).

Figure 15.

Comparison between x–z contours of $u$ for the LSD ( $L_{x}=12.8h$ ) (a,b) and t–z contours of $u^{2D}$ for the MSU (c,d) for $h^{+}=1020$ . The data at $y^{+}=15$ are plotted in (a,c), whereas those at $y/h=0.5$ are shown in (b,d). In (a,b), the instantaneous $u$ data in the LSD have been plotted at the two $y$ locations in the x–z plane. In (c,d), the time evolution data of $u^{2D}$ , defined in (3.18), in the MSU have been plotted at the two $y$ locations as a function of the spanwise location.

Figure 16.

Normalized one-dimensional spanwise spectra of velocity fluctuations: (a) $\unicode[STIX]{x1D719}_{uu}(k_{z})/U_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D708}$ , $\unicode[STIX]{x1D719}_{vv}(k_{z})/U_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D708}$ , $\unicode[STIX]{x1D719}_{ww}(k_{z})/U_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D708}$ at $y^{+}=15$ ; (b) $\unicode[STIX]{x1D719}_{uu}(k_{z})/U_{\unicode[STIX]{x1D70F}}^{2}h$ at $y^{+}=15$ and $y/h=0.5$ ; (c) $\unicode[STIX]{x1D719}_{uu}(k_{z})/U_{\unicode[STIX]{x1D70F}}^{2}h$ for the LSD at $y^{+}=5$ , 61 and $y/h=0.2$ , 0.5, 0.8; (d) $\unicode[STIX]{x1D719}_{uu}(k_{z})/U_{\unicode[STIX]{x1D70F}}^{2}h$ for the MSU at $y^{+}=5$ , 61 and $y/h=0.2$ , 0.5, 0.8. In (a,b), the spectra for the MSU have been compared with those for the LSD. In (c,d), the spectra have been plotted in inner scales to highlight the low-wavenumber behaviour. In (a), – – – – denotes the DNS data of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) for $h^{+}=934$ with the VLSD ( $L_{x}=8\unicode[STIX]{x03C0}h$ ).

Figure 17.

Time variation of the large-scale structures in the MSU for $h^{+}=1020$ : (a) $t^{+}=0$ ; (b) $t^{+}=600$ ; (c) $t^{+}=1200$ ; (d) $t^{+}=1800$ : red, $u^{+}>1.5$ ; blue, $u^{+}<-1.5$ . Spanwise scales below 0.5 $h$ are filtered out with the use of a spectral sharp cutoff filter. The visualization domain size shown here is the whole computational domain for the MSU (i.e. $0.4h\times 2h\times 6.4h$ in the $x$ , $y$ and $z$ directions, respectively).

Figure 17 demonstrates how the large-scale structures in the MSU for $h^{+}=1020$ evolve in time, where structures with a spanwise scale larger than 0.5 $h$ are illustrated with the use of a sharp cut filter. Indeed, large-scale structures become increasingly larger with increasing distance from the wall. The positive and negative large-scale $u$ structures with a spanwise spacing of approximately 0.8 $h$ are observed below the log region ( $y/h\leqslant 0.2$ ), whereas those with a spanwise spacing of 1.6 $h$ dominate in the outer region ( $y/h>0.2$ ), consistent with the peak wavelengths of the spanwise premultiplied $u$ spectra (figure 12 a). Note that we also see the large-scale structures spanning the channel half-width with a spanwise spacing approximately 3 $h$ for the MSU, as discussed below. Near the centreline, large-scale structures originating from one wall interact with those from the other wall, which is reminiscent of the large flow recirculation observed by Toh & Itano (Reference Toh and Itano2005) at low Reynolds numbers. There is however a discernible difference between the present MSU and that of Toh & Itano (Reference Toh and Itano2005) due to the closer approach towards two-dimensionality in the present MSU (see figure 2 d). Consequently, the large-scale structures in the present MSU are present for a long period of time, as opposed to those in Toh & Itano (Reference Toh and Itano2005) (see also the persistent outer spectral peak for the MSU shown in figure 14 b). This is most likely due to the lack of nonlinear evolution of the large-scale structures in the present MSU. Figure 17 underlines that antisymmetric pairs (with respect to centreline) of large-scale structures, with a spanwise spacing of approximately 3 $h$ , dominate in the MSU. The latter spanwise spacing corresponds to the peak wavelength observed in the spanwise $u$ spectra across the channel (figure 16 d). Because of the two-dimensionality of the large-scale structures (in the MSU), which carry a large amount of the turbulent kinetic energy, we have associated this peak wavelength with that in the transient growth simulation. Indeed, the peak wavelength in the present MSU (approximately 3 $h$ ) is very close to the optimal spanwise wavelength of 3 $h$ reported by del Álamo & Jiménez (Reference del Álamo and Jiménez2006). Pujals et al. (Reference Pujals, Manuel Garcia-Villalba, Cossu and Depardon2009) also reported an optimal spanwise spacing of 4 $h$ for the optimal transient growth of perturbations in a turbulent channel flow. Also, to provide statistical evidence regarding the antisymmetric pairs of large-scale structures, wall-normal two-point correlations, defined such that

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle R_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}(y_{r},y)={\displaystyle \frac{\overline{\unicode[STIX]{x1D6FC}(y_{r})\unicode[STIX]{x1D6FC}(y)}}{\overline{\unicode[STIX]{x1D6FC}(y_{r})\unicode[STIX]{x1D6FC}(y_{r})}}}, & \displaystyle\end{eqnarray}$$

where $y_{r}$ is a reference $y$ location and $\unicode[STIX]{x1D6FC}\equiv u$ , $v$ , $w$ or $p$ , have been examined. Figure 18 exhibits the anomalous behaviour for the $u$ component of the MSU, which supports the antisymmetric pattern of large-scale structures observed in figure 17. Note that for this quantity, the averaging is based on a time period $t^{+}\approx 1300$ and 1800, for the LSD and MSU, respectively. The sampling time period for the MSU corresponds to the time period, shown in figure 17, related to the evolution of the large-scale structures. The present sampling time period is approximately 6–8 times larger than that used by Kim & Hussain (Reference Kim and Hussain1993) in the analysis of the space–time correlations for $h^{+}=180$ ; the longer sampling yields a better convergence for correlations at large separations. While the $u$ correlations for the LSD tend to approach zero at large separations, those for the MSU indeed exhibit asymmetric patterns with respect to the centreline (see figure 18). The antisymmetric pattern is also identical with the most amplified outer mode of a transient growth of perturbations (see del Álamo & Jiménez Reference del Álamo and Jiménez2006; Pujals et al. Reference Pujals, Manuel Garcia-Villalba, Cossu and Depardon2009). In this context, Butler & Farrell (Reference Butler and Farrell1992) noted in their analysis of laminar transient growth that the parabolic shape of the mean flow leads to the growth of $u$ in antisymmetric pairs. Indeed, when $h^{+}$ reaches 1020 and the two-dimensionality is established approximately for the MSU, there is an increase in $\overline{U}^{+}$ (figure 1 a) and the distribution of $\overline{U}^{+}$ is closer to the parabolic profile (figure 1 b). The present results underline that the most energetic mode in the transient growth of perturbations is most likely the origin of the global $u$ mode in a turbulent channel flow.

Figure 18.

Wall-normal two-point correlations for $h^{+}=1020$ : (a) $R_{uu}(y_{r},y)$ ; (b) $R_{vv}(y_{r},y)$ ; (c) $R_{ww}(y_{r},y)$ ; (d) $R_{pp}(y_{r},y)$ . In this figure, the averaging is based on a time period $t^{+}\approx 1300$ and 1800, for the LSD and MSU, respectively. The locations of $y_{r}$ are $y/h=0$ , 0.2, 0.5 and 1. The data at $y_{r}=0$ have been obtained at the first grid point.

Figure 19.

Contours in the $y$ – $z$ plane of instantaneous total, slow rapid pressure for $h^{+}=1020$ : (a,b) $p$ ; (c,d) $p_{s}$ ; (e,f) $p_{r}$ . (a,c,e) correspond to the LSD ( $L_{x}=12.8h$ ), while (b,d,f) are for the MSU ( ${L_{x}}^{+}\approx 400$ ). The visualized instantaneous field is the same as in figure 10.

3.4 Pressure fluctuations

In this subsection, we examine the reason why the wall-attached pressure fluctuations are attenuated in the MSU. As in Jiménez & Hoyas (Reference Jiménez and Hoyas2008), we adopt the pressure splitting method, i.e. the slow and rapid pressure ( $p_{s}$ , $p_{r}$ ) obtained by solving Poisson’s equation, viz.

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}p_{s}^{+}=-{\displaystyle \frac{\unicode[STIX]{x2202}u_{i}^{+}}{\unicode[STIX]{x2202}x_{j}^{\#}}}{\displaystyle \frac{\unicode[STIX]{x2202}u_{j}^{+}}{\unicode[STIX]{x2202}x_{i}^{\#}}} & \displaystyle\end{eqnarray}$$

and

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}p_{r}^{+}=-2{\displaystyle \frac{\unicode[STIX]{x2202}v^{+}}{\unicode[STIX]{x2202}x^{\#}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{U}^{+}}{\unicode[STIX]{x2202}y^{\#}}} & \displaystyle\end{eqnarray}$$

(see also Abe, Matsuo & Kawamura Reference Abe, Matsuo, Kawamura, Humphrey, Gatski, Eaton, Friedrich, Kasagi and Leschziner2005). Note that the superscript # denotes normalization by the channel half-width. The boundary conditions at the wall are $\unicode[STIX]{x2202}p_{s}^{+}/\unicode[STIX]{x2202}y^{\#}=(1/h^{+})(\unicode[STIX]{x2202}^{2}v^{+}/\unicode[STIX]{x2202}y^{\#2})$ and $\unicode[STIX]{x2202}p_{r}^{+}/\unicode[STIX]{x2202}y^{\#}=0$ ; the Stokes pressure is included in the slow pressure. The rapid and slow pressure are sometimes referred to as the linear and nonlinear pressure, respectively, (see Kim Reference Kim1989) because the Poisson equations for the slow and rapid pressure have nonlinear–nonlinear (the term on the right-hand side of (3.20)) and linear–nonlinear (the term on the right-hand side of (3.21)) interactions, respectively. Here, we use 50 and 600 time consecutive datasets for the LSD and MSU, respectively, with a time interval of $\unicode[STIX]{x0394}t^{+}\approx 3$ for $h^{+}=1020$ . In the following statistics, the time averaging has been performed with these time consecutive datasets, viz. $t^{+}\approx 150$ and 1800 for the LSD and MSU, respectively.

Figure 19 shows the instantaneous contours in the y–z plane of the total, rapid and slow pressure for the LSD and MSU for $h^{+}=1020$ . In contrast to the x–y plane, the instantaneous contours in the y–z plane show a noticeable difference between $w$ and $p$ (see figures 10 e and 19 a). This is essentially associated with a difference between the three-dimensional shear layers and the wall-attached structures of the pressure (see also the different behaviours in the wall-normal two-point correlations of $w$ and $p$ shown in figure 18 c,d). The pressure splitting decomposes the shear-layer structures and tall ones. The slow pressure $p_{s}$ for the LSD (figure 19 c) shows rounded structures in the y–z plane, which are the cross-sections of streamwise sloping structures (see figure 4 c). On the other hand, the rapid pressure $p_{r}$ for the LSD (figure 19 e) shows tall global structures, which is most likely a consequence of the behaviour that high momentum large-scale structures impact on low momentum large-scale ones (see figure 4 a). When the streamwise domain $L_{x}$ is reduced to the size of MSU, these structures become indeed smaller (see figure 19) and exhibit localized structures. In particular, there is a significantly reduced magnitude of the rapid pressure, which is essentially associated with the weakened linear–nonlinear interaction in the Poisson equation (viz. the term on the right-hand side of (3.21)).

Figure 20.

Distributions of mean-square pressure fluctuations ( $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ , $\overline{p_{s}p_{s}}/\unicode[STIX]{x1D70F}_{w}^{2}$ , $\overline{p_{r}p_{r}}/\unicode[STIX]{x1D70F}_{w}^{2}$ ) for $h^{+}=1020$ : (a) LSD ( $L_{x}=12.8h$ ); (b) MSU ( ${L_{x}}^{+}\approx 400$ ). In each figure, the relation of (3.8) is plotted as the two-dotted line.

To provide the corresponding statistical evidence, figure 20 shows the mean square values for both the MSU and LSD. Indeed, the magnitudes of $p_{s}$ and $p_{r}$ are reduced in the MSU where $p_{s}$ is the sole contributor to the pressure fluctuations away from the wall, consistent with the result in the instantaneous fields (figure 19). Figure 21(a,b) shows the streamwise and spanwise power spectra of the total, slow rapid pressure for $h^{+}=1020$ in the outer region ( $y/h=0.5$ ) of the LSD and MSU. The streamwise spectra for the MSU agree reasonably well with those for the LSD up to their smallest non-zero wavenumbers, whilst the spanwise spectra for the MSU show a smaller magnitude at low wavenumbers than for the LSD. Indeed, there is a better spectral similarity between the total and slow pressure for the MSU in the outer region than for the LSD (see figure 21 a,b), consistent with the result in the mean-square values (see figure 20). This result indicates that the spectral similarity between $w$ and $p$ in the outer region observed in figure 8 is intrinsically associated with the slow pressure. We also note that there is a significant decrease in power at low wavenumbers in the spanwise $p_{r}$ spectra of the MSU (see figure 21 b). Also, a local peak appearing in the streamwise $p_{r}$ spectra (not in the streamwise $p_{s}$ spectra) at $k_{x}h=2\sim 3$ (i.e. $\unicode[STIX]{x1D706}_{x}/h=2\sim 3$ ) across the channel for the LSD is not reproduced for the MSU (see figure 21 c,d). These behaviours are consistent with the weakened structures for the MSU observed in figures 4(b) and 19(b) and essentially associated with the significantly decreased logarithmic dependence of $\overline{pp}/\unicode[STIX]{x1D70F}_{w}^{2}$ (figure 7 d). We have also examined the pressure strain terms for the total $\unicode[STIX]{x1D719}_{ij}^{t}$ , slow $\unicode[STIX]{x1D719}_{ij}^{s}$ , rapid $\unicode[STIX]{x1D719}_{ij}^{r}$ pressure (figure 22) to gain further insight into the weakened redistribution in the MSU (see figure 9 e). Indeed, the magnitudes of $\unicode[STIX]{x1D719}_{11}^{r}$ and $\unicode[STIX]{x1D719}_{33}^{r}$ are attenuated significantly in the MSU, consistent with the weakened structures (figure 10 f). Given that $\unicode[STIX]{x1D719}_{33}^{r}$ is the major contributor to $\unicode[STIX]{x1D719}_{33}^{t}$ away from the wall, the reduced $\unicode[STIX]{x1D719}_{33}^{r}$ in the MSU is essentially associated with the reduced magnitude of $\overline{ww}/U_{\unicode[STIX]{x1D70F}}^{2}$ (figure 7 b). The reason for the reduced energy redistribution in the MSU is therefore essentially associated with the impaired global nature of the pressure largely resulting from the reduced linear–nonlinear interaction in the Poisson equation (i.e. the rapid pressure).

Figure 21.

Streamwise and spanwise power spectra of the total, slow and rapid pressure normalized by $\unicode[STIX]{x1D70F}_{w}^{2}$ and $h$ , $\unicode[STIX]{x1D719}_{pp}(k_{x})/\unicode[STIX]{x1D70F}_{w}^{2}h$ and $\unicode[STIX]{x1D719}_{pp}(k_{z})/\unicode[STIX]{x1D70F}_{w}^{2}h$ , for $h^{+}=1020$ : (a) $\unicode[STIX]{x1D719}_{pp}(k_{x})/\unicode[STIX]{x1D70F}_{w}^{2}h$ at $y/h=0.5$ ; (b) $\unicode[STIX]{x1D719}_{pp}(k_{z})/\unicode[STIX]{x1D70F}_{w}^{2}h$ at $y/h=0.5$ ; (c) $\unicode[STIX]{x1D719}_{pp}(k_{x})/\unicode[STIX]{x1D70F}_{w}^{2}h$ for the slow pressure; (d) $\unicode[STIX]{x1D719}_{pp}(k_{x})/\unicode[STIX]{x1D70F}_{w}^{2}h$ for the rapid pressure. In (c,d), the $y$ stations of the spectra plotted are $y/h=0$ , 0.2, 0.5, 0.7 and 1.

Figure 22.

Distributions of the pressure strain terms for the total, slow and rapid pressure normalized by wall units for $h^{+}=1020$ : (a) $\unicode[STIX]{x1D719}_{11}^{t}$ , $\unicode[STIX]{x1D719}_{11}^{s}$ , $\unicode[STIX]{x1D719}_{11}^{r}$ ; (b) $\unicode[STIX]{x1D719}_{22}^{t}$ , $\unicode[STIX]{x1D719}_{22}^{s}$ , $\unicode[STIX]{x1D719}_{22}^{r}$ ; (c) $\unicode[STIX]{x1D719}_{33}^{t}$ , $\unicode[STIX]{x1D719}_{33}^{s}$ , $\unicode[STIX]{x1D719}_{33}^{r}$ .