3.1 Numerical scheme and computational domain

We perform a set of direct numerical simulations (DNS) of plane turbulent channel flows by solving the incompressible Navier–Stokes equations with a staggered, second-order, finite difference (Orlandi Reference Orlandi2000) and a fractional-step method (Kim & Moin Reference Kim and Moin1985) with a third-order Runge–Kutta time-advancing scheme (Wray Reference Wray1990). Periodic boundary conditions are imposed in the streamwise and spanwise directions, and no slip at the walls. The code has been validated in previous studies in turbulent channel flows (Lozano-Durán & Bae Reference Lozano-Durán and Bae2016; Bae et al. Reference Bae, Lozano-Durán, Bose and Moin2018a ,Reference Bae, Lozano-Durán, Bose and Moin b ) and flat-plate boundary layers (Lozano-Durán, Hack & Moin Reference Lozano-Durán, Hack and Moin2018).

Wall units are denoted by the superscript $^{+}$ and defined in terms of the kinematic viscosity $\unicode[STIX]{x1D708}$ and friction velocity at the wall  $u_{\unicode[STIX]{x1D70F}}$ . Accordingly, the friction Reynolds number is $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}$ . Velocities normalised by $u^{\ast }$ and $u^{\star }$ are denoted by the superscripts  $^{\ast }$ and  $^{\star }$ , respectively. Fluctuating quantities with respect to the mean are represented by $(\cdot )^{\prime }$ . The computational domain for the present simulations is $2\unicode[STIX]{x03C0}h\times 2h\times \unicode[STIX]{x03C0}h$ in the streamwise, wall-normal and spanwise directions, respectively. It has been shown that this domain size is large enough to accurately predict one-point statistics for $Re_{\unicode[STIX]{x1D70F}}$ up to 4200 (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014a ).

We compare our results with DNS data from del Álamo & Jiménez (Reference del Álamo and Jiménez2003) and Hoyas & Jiménez (Reference Hoyas and Jiménez2006) at $Re_{\unicode[STIX]{x1D70F}}\approx 550$ , 950 and 2000, which are labelled as NS550, NS950 and NS2000, respectively. The three cases have computational domains equal to $8\unicode[STIX]{x03C0}h\times 2h\times 3\unicode[STIX]{x03C0}h$ in the streamwise, wall-normal and spanwise directions, respectively.

3.2 Numerical experiments of channels driven by $x_{2}$ -dependent body force

We devise two sets of conceptual numerical experiments to unravel the characteristic scales of the outer-layer motions. The first set of experiments is a channel flow with no-slip walls driven by an $x_{2}$ -dependent body force per unit mass of the form

(3.1a-c ) $$\begin{eqnarray}f_{1}=\frac{u_{\unicode[STIX]{x1D70F}}^{2}}{h}[1+\unicode[STIX]{x1D716}(2x_{2}/h-x_{2}^{2}/h^{2})-2/3\unicode[STIX]{x1D716}],\quad f_{2}=0,\quad f_{3}=0,\end{eqnarray}$$

where $f_{i}$ is the component of the force in the $i$ th direction, and $\unicode[STIX]{x1D716}$ is a non-dimensional adjustable parameter. Equation (3.1) is such that $Re_{\unicode[STIX]{x1D70F}}$ remains unchanged with  $\unicode[STIX]{x1D716}$ . For $\unicode[STIX]{x1D716}=0$ , we recover the constant body force typically used to drive the channel, $f_{1}=u_{\unicode[STIX]{x1D70F}}^{2}/h$ . The goal of (3.1) is to alter the natural balance between eddies, which are forced to readjust their intensities to accommodate the new momentum flux. Two cases are considered at $Re_{\unicode[STIX]{x1D70F}}\approx 550$ : $\unicode[STIX]{x1D716}=4$ , labelled as NS550-p; and $\unicode[STIX]{x1D716}=-2$ , labelled as NS550-n, where NS denotes no-slip boundary condition at the walls. Note that case NS550 corresponds to $\unicode[STIX]{x1D716}=0$ .

The second set of experiments intends to clarify the characteristic length scales of the active energy-containing eddies. The change in $u^{\ast }$ and $t^{\ast }$ from NS550-p and NS550-n is not significant enough to assess conclusively the scaling proposed in (2.4). For that reason, two new simulations, NS550-s1 and NS550-s2, are considered by prescribing a synthetic mean velocity profile of the form

(3.2) $$\begin{eqnarray}\frac{\langle u_{1}\rangle }{u_{ref}}=\frac{\unicode[STIX]{x1D6FC}+2}{\unicode[STIX]{x1D6FC}+1}[1-(x_{2}/h-1)^{\unicode[STIX]{x1D6FC}+1}]+\frac{\unicode[STIX]{x1D6FD}+2}{\unicode[STIX]{x1D6FD}+1}[1-(x_{2}/h-1)^{\unicode[STIX]{x1D6FD}+1}],\end{eqnarray}$$

with $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})=(41,11)$ and $(3,3)$ for NS550-s1 and NS550-s2, respectively. The parameter $u_{ref}$ was adjusted to achieve $Re_{\unicode[STIX]{x1D70F}}\approx 550$ . The profiles from (3.2) are purposely tailored to create distinguishable $l^{\ast }$ with values equal to $0.06h$ , $0.03h$ and $0.04h$ at $x_{2}=0.10h$ , for cases NS550, NS550-s1 and NS550-s2, respectively. These last two simulations are similar to channel flows driven by $x_{2}$ -dependent body forces discussed above, in the sense that prescribing the mean velocity profile is equivalent to imposing an $x_{2}$ -dependent (and time-dependent) forcing as in (3.1). Simulations of turbulent channels with prescribed velocity profiles can also be found in Tuerke & Jiménez (Reference Tuerke and Jiménez2013). All the cases above are designed such that $u_{\unicode[STIX]{x1D70F}}$ and $x_{2}$ remain unchanged but do not coincide with the scaling proposed by $u^{\star }$ and  $l^{\ast }$ , in contrast with the traditional channel flow, where $u^{\star }\approx u_{\unicode[STIX]{x1D70F}}$ and $l^{\ast }\approx \unicode[STIX]{x1D705}x_{2}$ . This will allow us to assess the validity of each scaling. The list of cases is summarised in table 1. All the simulations were run for at least 10 eddy turnover times (defined as $h/u_{\unicode[STIX]{x1D70F}}$ ) after transients.

Table 1. Tabulated list of cases for § 3. The numerical experiments are labelled following the convention NS[ $Re_{\unicode[STIX]{x1D70F}}$ ]-[specific case], where NS denotes a channel with no-slip walls. Here $\unicode[STIX]{x0394}x_{1}$ , $\unicode[STIX]{x0394}x_{2}$ and $\unicode[STIX]{x0394}x_{3}$ are the streamwise, wall-normal and spanwise grid resolutions, respectively. The last column shows the method employed to drive the channel flow. See text for details.

3.3 Assessment of characteristic velocity and length scales

We examine scaling (2.2) in turbulent channel flows driven by (3.1). The imposed $x_{2}$ -dependent body force breaks the global velocity scale with  $u_{\unicode[STIX]{x1D70F}}$ , and the new balance for the mean momentum flux requires that

(3.3) $$\begin{eqnarray}\langle u_{1}u_{2}\rangle \approx \int _{0}^{x_{2}}f_{1}(x_{2}^{\prime })\,\text{d}x_{2}^{\prime }.\end{eqnarray}$$

The total stresses consistent with (3.3) for cases NS550-p and NS550-n are shown in figure 2(a). Changes in the momentum flux propagate to the mean velocity profile as dictated by the integrated streamwise mean momentum equation, and the resulting profiles are shown in figure 2(b).

Figure 2. Mean (a) total tangential Reynolds stress and (b) streamwise velocity profile for NS550 (– – –), NS550-p (○) and NS550-n (●). Panels (c,d) contain the streamwise (circles and – – –), wall-normal (triangles and – ⋅ – ⋅ –) and spanwise (squares and $\cdots \cdots$ ) r.m.s. velocity fluctuations scaled with (c)  $u_{\unicode[STIX]{x1D70F}}$ and (d)  $u^{\star }$ . Symbols are for NS550-p (open) and NS550-n (closed). Lines without symbols are for NS550. For clarity, the profiles for the wall-normal and spanwise r.m.s. velocity fluctuations are shifted vertically by 2.0 and 3.4 wall units.

The three root-mean-squared (r.m.s.) fluctuating velocities for NS550, NS550-p and NS550-n are reported in figure 2(c). The pronounced lack of collapse among the three cases exposes the unsatisfactory scaling with  $u_{\unicode[STIX]{x1D70F}}$ . Conversely, when the r.m.s. fluctuating velocities are scaled with  $u^{\star }$ , which can be analytically evaluated for cases NS550-p and NS550-n, the agreement is excellent (figure 2 d). Appendix A shows results using  $u^{\ast }$ , which does not have any explicit functional dependence on  $x_{2}$ . Note that the argument above holds for Townsend’s active motions, i.e. those responsible for the mean momentum transfer, and that the inactive motions are expected to scale not with $u^{\star }$ (or  $u^{\ast }$ ) but with the bulk velocity or a mixed scale as suggested in previous works (Zagarola & Smits Reference Zagarola and Smits1998; De Graaff & Eaton Reference De Graaff and Eaton2000; del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004; Morrison et al. Reference Morrison, McKeon, Jiang and Smits2004).

Scaling (2.4) is investigated in cases NS550, NS550-s1 and NS550-s2 with mean velocity profiles shown in figure 3(a). Figure 3(b) contains the tangential Reynolds stress artificially generated to sustain $\langle u_{1}\rangle$ . The relevant length scale of the momentum-carrying eddies is examined in figure 4 by comparing the premultiplied, two-dimensional velocity spectra ( $\unicode[STIX]{x1D719}_{ii}$ , $i=1,2,3$ ) at $x_{2}=0.10h$ as a function of the streamwise and spanwise wavelengths ( $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ ) scaled by the distance to the wall (figure 4 a–c) and $l^{\ast }$ (figure 4 d–f). The spectra display a noticeable mismatch when the wavelengths are scaled by  $x_{2}$ , whereas the collapse is appreciably improved when $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ are normalised by  $l^{\ast }$ , especially for the most intense spectral cores. Therefore, $l^{\ast }$ stands as a more faithful characterisation of the eddy sizes compared to the distance to the wall.

Figure 3. Mean (a) streamwise velocity profile and (b) tangential Reynolds stress for NS550 (– – –), NS550-s1 (○) and NS550-s2 (●). The dash-dotted line is $x_{2}=0.1h$ .

Figure 4. Premultiplied streamwise $\unicode[STIX]{x1D719}_{11}$ (a,d), wall-normal $\unicode[STIX]{x1D719}_{22}$ (b,e) and spanwise $\unicode[STIX]{x1D719}_{33}$ (c,f) velocity two-dimensional spectra at $x_{2}=0.10h$ for NS550 (——), NS550-s1 (○) and NS550-s2 (●). The wavelengths are scaled by $x_{2}$ in (a–c) and by $l^{\ast }$ in (d–f). Contours are 0.1 and 0.6 of the maximum.

We further examine the performance of $l^{\ast }$ at different wall-normal heights using the traditional channel flow NS2000. Figure 5 shows the premultiplied velocity spectra for various wall-normal distances ranging from $x_{2}=0.1h$ to $x_{2}=0.4h$ . We test three different length scales to normalise the streamwise and spanwise wavelengths, namely, $h$ (figure 5 a–c), $x_{2}$ (figure 5 d–f) and $l^{\ast }$ (figure 5 g–i). Again, the best collapse is attained for  $l^{\ast }$ , although $x_{2}$ also provides a quantitative improvement with respect to  $h$ .

Figure 5. Premultiplied streamwise $\unicode[STIX]{x1D719}_{11}$ (a,d,g), wall-normal $\unicode[STIX]{x1D719}_{22}$ (b,e,h) and spanwise $\unicode[STIX]{x1D719}_{33}$ (c,f,i) two-dimensional velocity spectra at $x_{2}/h=0.10$ , 0.15, 0.20, 0.30 and 0.40 (from black to red) for NS2000. The wavelengths are scaled by $h$ (a–c), $x_{2}$ (d–f) and $l^{\ast }$ (g–i). Contours are 0.1 and 0.6 of the maximum. The arrows in panels (a–c) indicate increasing distance from the wall.

The normalisation by $l^{\ast }$ is still far from perfect, although this may be expected considering that the new scaling is only applicable to the active motions responsible for carrying the tangential momentum flux. For that reason, both the lower spectral contours and the large-scale tails of $\unicode[STIX]{x1D719}_{11}$ and $\unicode[STIX]{x1D719}_{33}$ are not required to scale with  $l^{\ast }$ , the former due to contamination from small eddies decoupled from the mean shear, and the latter due to their lack of tangential stress despite their non-zero energy content.

In summary, we have shown that $u^{\ast }$ (or $u^{\star }$ ) and $l^{\ast }$ are tenable candidates to represent the characteristic scales of the momentum-carrying eddies in wall-bounded turbulence. Although we have focused on channel flows, we expect $u^{\ast }$ and $l^{\ast }$ to perform similarly in turbulent boundary layers. The proposed scales are still consistent with the classic scaling provided by $u_{\unicode[STIX]{x1D70F}}$ and $x_{2}$ for canonical flows, and can be considered as an extension for more general shear-dominated turbulence.

4.1 Numerical experiments of channels with Robin boundary conditions

We perform a set of DNS of turbulent channel flows using the same numerical scheme and computational domain as in § 3.1. The no-slip wall is replaced by a Robin boundary condition of the form

(4.1) $$\begin{eqnarray}u_{i}|_{w}=\left.l\,\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}n}\right|_{w},\quad i=1,2,3,\end{eqnarray}$$

where the subscript $w$ refers to quantities evaluated at the wall, and $n$ is the wall-normal (or boundary-normal) direction. We define $l$ to be the slip length, which, in general, may be a function of the spatial wall-parallel coordinates and time. The choice of $l$ must comply with the symmetries of the flow and, particularly for a channel flow configuration, (4.1) should satisfy

(4.2) $$\begin{eqnarray}\langle u_{i}|_{w}\rangle =\left\langle \left.l\,\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{2}}\right|_{w}\right\rangle =0,\quad i=2,3.\end{eqnarray}$$

In the present study, we consider a constant value for $l$ . This is consistent with (4.2) because $\langle u_{i}|_{w}\rangle =0$ and $\langle \unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{2}|_{w}\rangle =0$ for $i=2,3$ . The cases simulated are summarised in table 2. All the simulations were run for at least 10 eddy turnover times after transients. Throughout the text, we occasionally refer to cases with the Robin boundary condition as Robin-bounded and those with the no-slip condition as wall-bounded. Robin boundary conditions have been previously employed in the wall-parallel directions to model the flow over hydrophobic surfaces (Min & Kim Reference Min and Kim2004; Martell, Perot & Rothstein Reference Martell, Perot and Rothstein2009; Park, Park & Kim Reference Park, Park and Kim2013; Jelly, Jung & Zaki Reference Jelly, Jung and Zaki2014; Seo, García-Mayoral & Mani Reference Seo, García-Mayoral and Mani2015; Seo & Mani Reference Seo and Mani2016), but note that in the present study the boundary condition is also applied to the boundary-normal velocity, which is seldom done in the literature. The slip lengths used here are also larger than the typical values in hydrophobic works.

Table 2. Tabulated list of cases for § 4. The numerical experiments are labelled following the convention R[ $Re_{\unicode[STIX]{x1D70F}}$ ]-[specific case], where R denotes a channel with Robin boundary condition. Here $\unicode[STIX]{x0394}x_{1}$ , $\unicode[STIX]{x0394}x_{2}$ and $\unicode[STIX]{x0394}x_{3}$ are the streamwise, boundary-normal and spanwise grid resolutions, respectively. The slip length used in the Robin boundary condition is  $l$ . The last column shows the method employed to drive the channel flow. See text for details.

The motivation of using (4.1) is to provide a boundary for the flow that deviates from the behaviour of a regular wall. Indeed, for large values of  $l$ , (4.1) constitutes a significant modification of the classic no-slip boundary condition by suppressing the formation of near-wall viscous layers (Lozano-Durán & Bae Reference Lozano-Durán and Bae2016). The mean tangential Reynolds stress is shown in figure 6(a) for cases R550, R950 and R2000 with slip length $l=0.10h$ . For the three Reynolds numbers under consideration, $-\langle u_{1}u_{2}\rangle$ captures more than 90 % of the total stress, and this was the criterion used to select $l=0.10h$ as the reference slip length. As the Reynolds number increases, so does the contribution of $-\langle u_{1}u_{2}\rangle$ close to the wall at the expense of reducing the formation of near-wall viscous layers that appear prior to the proximity of the wall. Far from the boundaries, the tangential momentum flux is linear for the same reason as in wall-bounded channels, i.e. to balance the constant mean pressure gradient driving the flow. Cases R550-l1 and R550-l2 are for $l=0.25h$ and $l=0.50h$ , respectively, and are intended to test the effect of increasing slip lengths.

Figure 6. (a) Mean tangential Reynolds stress as a function of the wall-normal (or boundary-normal) coordinate for R550 (○), R950 (▾), R2000 (▫), NS550 (– – –), NS950 (– - – -) and NS2000 ( $\cdots \cdots$ ). (b) Premultiplied boundary-normal two-dimensional velocity spectra for Robin-bounded channels at $x_{2}/h=0$ . Contours are 0.1 and 0.6 of the maximum. The red dashed lines are $\unicode[STIX]{x1D706}_{1}/h=1$ and $\unicode[STIX]{x1D706}_{3}/h=1$ . Symbols are as in panel (a).

Another two important properties of the Robin boundary condition are that it allows for transpiration at all flow scales, and it does not encode any specific information regarding the linear wall-normal scaling of the log-layer eddies. The spectral density of the wall-normal velocity, $\unicode[STIX]{x1D719}_{22}$ , evaluated at $x_{2}/h=0$ for Robin-bounded cases is shown in figure 6(b) as a function of the streamwise and spanwise wavelengths, $\unicode[STIX]{x1D706}_{1}$ and  $\unicode[STIX]{x1D706}_{3}$ , respectively. The spectra are non-zero at the boundary with a non-negligible contribution from wavelengths up to $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ of  $O(h)$ . Moreover, the spectral energies obtained by integrating $\unicode[STIX]{x1D719}_{22}$ at $x_{2}/h=0$ are approximately  $u_{\unicode[STIX]{x1D70F}}^{2}$ , that is, of the same order as the values in the bulk flow. This implies that the Robin boundary should alter the behaviour of eddies with sizes up to $O(h)$ if they are controlled by the distance to the wall as commonly hypothesised (Townsend Reference Townsend1976).

One could still argue that $\langle u_{2}|_{w}\rangle$ is zero, analogously to the scenario encountered for impermeable walls. However, note that this is also the case for $\langle u_{2}\rangle$ at each wall-normal location for a no-slip channel, and that (4.2), and hence $\langle u_{2}|_{w}\rangle =0$ , is just the immediate consequence of the symmetries of the channel flow configuration rather than a result of the impermeability constraint. The only reminiscence of the wall in the Robin boundary condition comes from the fact that, by construction of (4.1), the velocities and their corresponding boundary-normal derivatives are expected to show similar length scales in the vicinity of $x_{2}=0$ akin to the production and dissipation in the buffer region of a smooth wall.

Instantaneous flow fields for the three velocity components at $Re_{\unicode[STIX]{x1D70F}}=2000$ for NS2000 and R2000 are compared in figure 7. The similarities between the two visualisations are striking, and their resemblance is confirmed by the quantitative analysis of the flow statistics presented in the following sections. Note that the Robin boundary condition is introduced in the present work just as a means to confine the flow within a boundary that differs from a no-slip wall, especially regarding transpiration. In this sense, the Robin boundary condition is just a tool and it does not aim to model any particular flow phenomena. Moreover, the focus of this paper lies on the outer region and we are less interested in the structure of the flow at the boundary. Nonetheless, a further characterisation of the flow at $x_{2}/h=0$ is provided in appendix B for completeness.

Figure 7. Instantaneous $x_{1}$ – $x_{2}$ planes of the streamwise (a,b), wall-normal (c,d) and spanwise (e,f) velocities for turbulent channels at $Re_{\unicode[STIX]{x1D70F}}=2000$ . (a,c,e) are for R2000, and (b,d,f) are for NS2000. The colour bars show velocity in wall units.

4.2 One-point statistics and spectra

The mean streamwise velocities for the Robin-bounded and wall-bounded channel flows are compared in figure 8. In figure 8(b), the mean profiles for Robin-bounded cases are vertically displaced to match the centreline velocity of the corresponding no-slip case. A first observation is that the shape of $\langle u_{1}\rangle$ remains roughly identical for $x_{2}\gtrsim 0.10h$ , and the Robin boundary condition is mainly responsible for a reduction of the total mass flux. The shifts required to match the Robin-bounded cases to their no-slip counterpart were positive and equal to 6.4, 8.0 and 8.9 plus units for R550, R950 and R2000, respectively. Nonetheless, we will not emphasise these values, as they can be trivially changed either by adding a constant uniform velocity to the right-hand side of the Robin boundary condition (4.1), or by a Galilean transformation of the velocity field.

The observations from figure 8 can be connected to the law of the wall (Prandtl Reference Prandtl1925; von Kármán Reference von Kármán1930; Millikan Reference Millikan, Hartog and Peters1938; Townsend Reference Townsend1976),

(4.3) $$\begin{eqnarray}\langle u_{1}^{+}\rangle =\frac{1}{\unicode[STIX]{x1D705}}\log (x_{2}^{+})+B+\unicode[STIX]{x1D6F1},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ and $B$ are the von Kármán and intercept constants for no-slip walls, respectively, and $\unicode[STIX]{x1D6F1}$ is an additional velocity displacement. The friction velocity for Robin-bounded cases is $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D708}\unicode[STIX]{x2202}\langle u_{1}\rangle /\unicode[STIX]{x2202}x_{2}-\langle u_{1}u_{2}\rangle }|_{x_{2}=0}$ . Hence, the Robin boundary condition introduces a non-zero tangential Reynolds stress at the wall, which acts as an effective drag with a major impact on $\unicode[STIX]{x1D6F1}$ (see Nikuradse Reference Nikuradse1933; Jiménez Reference Jiménez2004). On the other hand, we can hypothesise that the $x_{2}$ -dependent component, ${\sim}(1/\unicode[STIX]{x1D705})\log (x_{2})$ , is mainly controlled by the momentum flux for  $x_{2}>0$ .

Figure 8. Mean streamwise velocity profiles as a function of $x_{2}$ in (a) linear scale and (b) logarithmic scale for R550 (○), R950 (▿), R2000 (▫), NS550 (– – –), NS950 (– - – -) and NS2000 ( $\cdots \cdots$ ). In panel (b), the velocity profiles of Robin-bounded cases are vertically shifted such that the centreline velocity coincides with their corresponding wall-bounded case. For clarity, profiles at $Re_{\unicode[STIX]{x1D70F}}\approx 950$ and $Re_{\unicode[STIX]{x1D70F}}\approx 2000$ are additionally shifted by 5 and 10 plus units, respectively.

The r.m.s. velocity fluctuations for the Robin-bounded and wall-bounded channels are shown in figure 9(a–c). The lack of a near-wall peak at $x_{2}^{+}\approx 15$ for the Robin-bounded streamwise velocity fluctuations is a consequence of the interruption of the classic near-wall cycle (Jiménez & Moin Reference Jiménez and Moin1991; Jiménez & Pinelli Reference Jiménez and Pinelli1999), which is also confirmed by visual inspection of the instantaneous near-wall velocities (see appendix B). The most remarkable observation from figure 9 is that the Robin-bounded fluctuating velocities match quantitatively their wall-bounded counterparts for $x_{2}\gtrsim 0.10h$ despite the lack of impermeable walls. The presence of a significant non-zero $u_{2}$ in the Robin-bounded cases is evidenced by the r.m.s. of $u_{2}$ at $x_{2}/h=0$ whose values are comparable to the r.m.s. in the bulk flow. The result is again an indication that the total contribution of the different eddies to the turbulence intensities is insensitive to the presence of impermeable walls. The pressure fluctuations are also accurately reproduced in figure 9(d) for the Robin-bounded cases. The results may appear surprising due to the global nature of the pressure, which is more prone to being affected by the changes in the boundary condition. However, the most important contribution of the pressure is to guarantee local continuity among eddies, and the satisfactory collapse of the r.m.s. velocity fluctuations seen in figure 9(a–c) also implies equally fair results for the pressure fluctuations. This is in accordance with Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2014), who concluded from the pressure correlations in boundary layers that $p$ is dominated by localised regions of strongly coupled small-scale structures.

Figure 9. (a) Streamwise, (b) wall-normal (or boundary-normal) and (c) spanwise r.m.s. velocity fluctuations, and (d) r.m.s. pressure fluctuations for R550 (○), R950 (▿), R2000 (▫), NS550 (– – –), NS950 (– - – -) and NS2000 ( $\cdots \cdots$ ). In all panels, the profiles for cases at $Re_{\unicode[STIX]{x1D70F}}\approx 950$ and $Re_{\unicode[STIX]{x1D70F}}\approx 2000$ are vertically shifted by 0.5 and 1 plus units, respectively, for clarity.

The spectral densities of the three velocity fluctuations, $\unicode[STIX]{x1D719}_{11}$ , $\unicode[STIX]{x1D719}_{22}$ and $\unicode[STIX]{x1D719}_{33}$ , are shown in figure 10 as a function of the streamwise and spanwise wavelengths. Several wall-normal heights are considered for R2000 and NS2000. The spectra for wall-bounded cases are zero at $x_{2}/h=0$ . Conversely, $\unicode[STIX]{x1D719}_{22}$ is non-zero at the boundary for the Robin-bounded case, and peaks at $\unicode[STIX]{x1D706}_{1}\approx 0.3h$ and $\unicode[STIX]{x1D706}_{3}\approx 0.10h$ , with a non-negligible contribution from wavelengths up to $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ of $O(h)$ . We can then estimate the flow scales that are expected to be affected by the transpiration of the boundary by assuming that the stress-carrying eddies follow $\unicode[STIX]{x1D706}_{1}\approx 1.5x_{2}$ and $\unicode[STIX]{x1D706}_{3}\approx x_{2}$ (as shown below in § 4.5). If the boundary is perceived as permeable for scales up to $O(h)$ , then attached motions below $x_{2}\approx 0.7h$ should adjust accordingly to accommodate transpiration effects, especially if they are organised by the wall. However, inspection of the spectra above $x_{2}\approx 0.10h$ shows that the agreement between wall-bounded and Robin-bounded channels is outstanding (figure 10 b,c,e,f,h,i). Consequently, the distance to the boundary (or non-existent wall) is not the relevant length scale controlling the size of the attached eddies, in contrast with the traditional argument by Townsend (Reference Townsend1976). Instead, the resemblance between wall-bounded and Robin-bounded cases presented above should be attributed to the common momentum transfer $u_{\unicode[STIX]{x1D70F}}^{2}$ characteristic of both cases as argued in § 2.

Figure 10. Wall-parallel premultiplied streamwise (a–c), wall-normal (or boundary-normal) (d–f) and spanwise (g–i) velocity spectra for R2000 (——) and NS2000 (– – –). The wall-normal (or boundary-normal) locations are $x_{2}^{+}=0$ (a,d,g), $x_{2}^{+}=1001$ (b,e,h) and $x_{2}^{+}=2003$ (c,f,i). Contours are 0.1 and 0.6 of the maximum. The red dotted lines in (d) are $\unicode[STIX]{x1D706}_{1}/h=1$ and $\unicode[STIX]{x1D706}_{3}/h=1$ .

It also worth noting that the excellent agreement of the one-point statistics and spectra occurs in spite of the absence of very-large-scale motions longer than ${\sim}6h$ (Guala et al. Reference Guala, Hommema and Adrian2006; Balakumar & Adrian Reference Balakumar and Adrian2007) that do not fit within the limited computational domain of the present simulations. The result is consistent with Flores & Jiménez (Reference Flores and Jiménez2010) and Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014a ), and scales larger than ${\sim}6h$ can be essentially modelled as infinitely long (due to the streamwise periodicity of the channel) while still interacting correctly with the smaller well-resolved eddies.

4.3 Existence of a virtual wall versus adaptation layer from the boundary

It could still be argued that the flow is influenced by an effective virtual wall whose origin is vertically displaced with respect to the boundary. In that case, different values of the slip length $l$ would be accompanied by changes in the origin of the virtual wall. The virtual-wall argument, often invoked in roughness studies (Raupach, Antonia & Rajagopalan Reference Raupach, Antonia and Rajagopalan1991; Jiménez Reference Jiménez2004), proved not to be necessary in § 4.2, where it was shown that the flow recovers above some wall-normal distance, $l_{a}$ , without requiring a shift in  $x_{2}$ . Nevertheless, the position of the hypothetical virtual wall, $\unicode[STIX]{x1D6FF}x_{2}$ , can be estimated by a least-squares fit of the mean velocity profile to the logarithmic law (Raupach et al. Reference Raupach, Antonia and Rajagopalan1991),

(4.4) $$\begin{eqnarray}\langle u_{1}^{+}\rangle =\frac{1}{\unicode[STIX]{x1D705}}\log (x_{2}^{+}-\unicode[STIX]{x1D6FF}x_{2}^{+})+B-\unicode[STIX]{x1D6E5}\langle u_{1}^{+}\rangle ,\end{eqnarray}$$

with $\unicode[STIX]{x1D705}=0.392$ and $B=4.48$ (Luchini Reference Luchini2017), within the range $x_{2}\in [0.1h,0.2h]$ . For this purpose, we will consider two additional cases with $l=0.25h$ and $l=0.50h$ at $Re_{\unicode[STIX]{x1D70F}}\approx 550$ , labelled respectively as R550-l1 and R550-l2 in table 2. The spectral density $\unicode[STIX]{x1D719}_{22}$ for the three different slip lengths is reported in figure 11(a) at $x_{2}/h=0$ . As $l$ increases, the spectrum moves towards higher wavelengths along the ridge $\unicode[STIX]{x1D706}_{1}\sim \unicode[STIX]{x1D706}_{3}$ , and transpiration is allowed for larger scales, which in principle should change the location of the virtual wall. However, the virtual wall offset computed with (4.4) is $\unicode[STIX]{x1D6FF}x_{2}^{+}\approx O(10)$ for all the Robin-bounded cases in table 2, which is not significant enough to alter in a meaningful manner the energy-containing eddies populating the log layer. Furthermore, when the $x_{2}$ direction was remapped into $\tilde{x}_{2}=x_{2}+\unicode[STIX]{x1D6FF}x_{2}(1-x_{2}/h)$ (Flores & Jiménez Reference Flores and Jiménez2006), the collapse of the r.m.s. velocity fluctuations worsened despite the minor improvements in $\langle u_{1}\rangle$ . Consequently, the results do not comply with the existence of a virtual wall. Note that the same conclusion does not apply to the viscous eddies close to the wall with sizes comparable to $\unicode[STIX]{x1D6FF}x_{2}$ , where changes in the permeability of the boundary are expected to have a considerable impact on the near-wall flow dynamics as typically described in flow control strategies (Choi, Moin & Kim Reference Choi, Moin and Kim1994; Abderrahaman-Elena & García-Mayoral Reference Abderrahaman-Elena and García-Mayoral2017).

Instead of a virtual wall, we analyse the results assuming the existence of an adaptation layer from the boundary. The vertical distance from the boundary above which the flow recovers to the nominal no-slip flow statistics, $l_{a}$ , is plotted in figure 11(b) and is referred to as the adaptation length. More specifically, $l_{a}$ is defined as the maximum boundary-normal distance from which

(4.5) $$\begin{eqnarray}{\mathcal{E}}^{+}(x_{2})=|\unicode[STIX]{x1D713}^{+}(x_{2})-\unicode[STIX]{x1D713}_{NS}^{+}(x_{2})|<\unicode[STIX]{x1D6FC},\end{eqnarray}$$

where the subscript NS denotes variables for no-slip cases, and $\unicode[STIX]{x1D713}$ takes the value of $\langle u_{1}\rangle$ or $\langle {u_{i}^{\prime }}^{2}\rangle ^{1/2}$ with $i=1,2,3$ , depending on whether the adaptation length refers to the recovery of the mean velocity profile, or the streamwise, boundary-normal or spanwise r.m.s. velocity fluctuations, respectively. The thresholding error $\unicode[STIX]{x1D6FC}$ is set to be equal to 0.5 plus units for the mean profile and 0.1 for the r.m.s. velocity fluctuations. The results show that the flow statistics collapse to the corresponding no-slip channel above $l_{a}\sim l$ for the quantities assessed and various Reynolds numbers. The selected values of $\unicode[STIX]{x1D6FC}$ are admittedly arbitrary and other choices may be preferred without any major consequences other than a vertical shift of the adaptation length in figure 11(b).

Figure 11. (a) Wall-parallel premultiplied boundary-normal velocity spectra at $x_{2}^{+}=0$ for R550 (——), R550-l1 (○) and R550-l2 (●). Contours are 0.1 and 0.6 of the maximum. The red dashed line is $\unicode[STIX]{x1D706}_{3}=0.7\unicode[STIX]{x1D706}_{1}$ . (b) Adaptation length $l_{a}$ for the mean velocity profile (♢), and r.m.s. streamwise (○), boundary-normal (▫) and spanwise (▿) velocity fluctuations as a function of the slip length  $l$ . Colours are black for R550, blue for R950 and red for R2000.

Then, results from figure 11(b) can be interpreted without the need of a virtual wall if the boundary condition is understood as a flow distortion stirred by the turbulent background for a depth $l_{a}^{2}\sim \unicode[STIX]{x1D708}_{t}t$ during a time period of $t\sim l_{a}/\langle u_{2}^{2}\rangle ^{1/2}$ , where $\unicode[STIX]{x1D708}_{t}$ is the eddy viscosity. Assuming that $\unicode[STIX]{x1D708}_{t}\sim \langle u_{1}u_{2}\rangle /\langle \unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{2}\rangle$ , and that the flow follows (4.1) within the adaptation layer defined by $x_{2}\lesssim l_{a}$ , then

(4.6) $$\begin{eqnarray}l_{a}\sim \frac{\langle u_{1}u_{2}\rangle }{\langle u_{2}^{2}\rangle ^{1/2}\left\langle {\displaystyle \frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{2}}}\right\rangle }\approx l\frac{\left\langle {\displaystyle \frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}x_{2}}}\right\rangle }{\left\langle \left({\displaystyle \frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}x_{2}}}\right)^{2}\right\rangle ^{1/2}\left\langle {\displaystyle \frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{2}}}\right\rangle },\end{eqnarray}$$

from where it is reasonable to assume that $l_{a}\sim l$ in first-order approximation, consistent with the results reported in figure 11(b). The existence of this adaptation layer due to the imposition of an unphysical Robin boundary condition together with the observations from previous sections suggest that both the Robin-bounded and wall-bounded channels share identical flow motions for $x_{2}>l_{a}$ once the disturbance by the boundary vanishes.

4.4 Logarithmic layer without inner–outer scale separation

The Robin boundary condition from (4.1) imposes a new length scale to the eddies in the near-wall region. The characteristic flow length scales of the no-slip and Robin-bounded channels are plotted in figure 12(a) as a function of  $x_{2}$ . The small and large scales are represented by the Kolmogorov length scale $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D700})^{1/4}$ and the integral length scale $L_{\unicode[STIX]{x1D700}}=(2k/3)^{3/2}/\unicode[STIX]{x1D700}$ , respectively, where $\unicode[STIX]{x1D700}$ is the rate of kinetic energy dissipation and $k$ is the turbulence kinetic energy. Note that $L_{\unicode[STIX]{x1D700}}$ drops rapidly to zero as $x_{2}$ approaches the wall for the no-slip channel, whereas it remains roughly constant in the Robin-bounded cases. Moreover, the comparison of $L_{\unicode[STIX]{x1D700}}$ at three different $Re_{\unicode[STIX]{x1D70F}}$ for the Robin-bounded channels shows that the integral length scale collapses in outer units across the entire boundary-layer thickness, including the region close to $x_{2}/h=0$ . These results can be read as the disruption of the classic viscous scaling of the active energy-containing eddies at the wall, i.e. their sizes are a fixed fraction of $h$ and do not decrease with  $Re_{\unicode[STIX]{x1D70F}}$ .

Figure 12. (a) Kolmogorov length scale $\unicode[STIX]{x1D702}$ for NS550 (– – –) and R550 (○), and integral length scale $L_{\unicode[STIX]{x1D700}}$ for NS550 (——), R550 (▫), R950 (♢) and R2000 (▵). (b)  ${\mathcal{E}}_{l}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ . Closed and open symbols are for wall-bounded and Robin-bounded cases, respectively. The dashed line is ${\mathcal{E}}_{l}^{+}\sim Re_{\unicode[STIX]{x1D70F}}^{-1}$ .

In spite of the lack of inner–outer layer scale separation with increasing  $Re_{\unicode[STIX]{x1D70F}}$ , the mean velocity profile for Robin-bounded cases (figure 8 a) tends towards a log layer as in wall-bounded channels. This is quantified in figure 12(b), which contains the error function

(4.7) $$\begin{eqnarray}{\mathcal{E}}_{l}^{+}=\left[\frac{1}{0.1}\int _{0.1}^{0.2}\left(x_{2}^{+}\frac{\unicode[STIX]{x2202}\langle u_{1}^{+}\rangle }{\unicode[STIX]{x2202}x_{2}^{+}}-\frac{1}{\unicode[STIX]{x1D705}}-\frac{x_{2}}{h}\right)^{2}\text{d}(x_{2}/h)\right]^{1/2},\end{eqnarray}$$

with $\unicode[STIX]{x1D705}=0.384$ (Lee & Moser Reference Lee and Moser2015) as a function of $Re_{\unicode[STIX]{x1D70F}}$ . Equation (4.7) measures the deviation of $x_{2}^{+}\unicode[STIX]{x2202}\langle u_{1}^{+}\rangle /\unicode[STIX]{x2202}x_{2}^{+}$ with respect to its asymptotic value for a well-developed log layer, $1/\unicode[STIX]{x1D705}-x_{2}/h$ , within the range $x_{2}/h\in [0.1,0.2]$ . The asymptotic value of $x_{2}^{+}\unicode[STIX]{x2202}\langle u_{1}^{+}\rangle /\unicode[STIX]{x2202}x_{2}^{+}$ has been derived by several authors using matched asymptotic expansions (Mellor Reference Mellor1972; Afzal & Yajnik Reference Afzal and Yajnik1973; Afzal Reference Afzal1976; Phillips Reference Phillips1987; Jiménez & Moser Reference Jiménez and Moser2007),

(4.8) $$\begin{eqnarray}x_{2}^{+}\frac{\unicode[STIX]{x2202}\langle u_{1}^{+}\rangle }{\unicode[STIX]{x2202}x_{2}^{+}}\approx \frac{1}{\unicode[STIX]{x1D705}}+\frac{x_{2}}{h}+\frac{\tilde{\unicode[STIX]{x1D6FD}}}{Re_{\unicode[STIX]{x1D70F}}},\end{eqnarray}$$

with $\tilde{\unicode[STIX]{x1D6FD}}$ a Reynolds-number-independent constant. The results, plotted in figure 12(b), show that ${\mathcal{E}}_{l}$ is larger for Robin-bounded cases than for wall-bounded cases, but both set-ups converge to the expected value at a rate close to $Re_{\unicode[STIX]{x1D70F}}^{-1}$ as predicted by (4.8). The fact that Robin-bounded cases approach a logarithmic profile for increasing $Re_{\unicode[STIX]{x1D70F}}$ without the inner–outer scale separation challenges the log-layer formulations derived from Millikan’s argument (see, among others, Millikan Reference Millikan, Hartog and Peters1938; Wosnik et al. Reference Wosnik, Castillo and George2000; Oberlack Reference Oberlack2001; Buschmann & Gad-el Hak Reference Buschmann and Gad-el Hak2003). Nonetheless, the Reynolds numbers in the present work are too low to attain a well-developed log layer, and therefore the results are indicative but not conclusive of the convergence of Robin-bounded cases to an actual wall-bounded log layer. As we are concerned with the outer layer, the analysis above was performed for a range of wall-normal distances fixed in outer units, $x_{2}/h\in [0.1,0.2]$ , and the slip length set to a given fraction of  $h$ . The effect of keeping the slip length constant in wall units as $Re_{\unicode[STIX]{x1D70F}}$ increases is discussed in appendix C, where we show the development of the log layer in the wall-normal direction.

4.5 Wall-attached eddies without walls

A more detailed analysis of the size of the eddies is provided by the investigation of three-dimensional regions of the flow where a quantity of interest is particularly intense. We focus on the regions of high momentum transfer from Lozano-Durán, Flores & Jiménez (Reference Lozano-Durán, Flores and Jiménez2012) (see also Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b ; Lozano-Durán & Borrell Reference Lozano-Durán and Borrell2016), defined as spatially connected points in the flow satisfying

(4.9) $$\begin{eqnarray}-u_{1}^{\prime }u_{2}^{\prime }>H\langle {u_{1}^{\prime }}^{2}\rangle ^{1/2}\langle {u_{2}^{\prime }}^{2}\rangle ^{1/2},\end{eqnarray}$$

where $u_{1}^{\prime }u_{2}^{\prime }$ is the instantaneous pointwise fluctuating tangential Reynolds stress, and $H$ is a thresholding parameter equal to 1.75 obtained from a percolation analysis (Moisy & Jiménez Reference Moisy and Jiménez2004). Three-dimensional structures are then constructed by connecting neighbouring grid points fulfilling relation (4.9) and using the six-connectivity criterion (Rosenfeld & Kak Reference Rosenfeld and Kak1982). The cases under investigation are NS2000 and R2000, and the total number of structures identified is of the order of $10^{6}$ . Since we are interested in the outer-layer eddies, the region $x_{2}^{+}<100$ was excluded from case NS2000 in order to avoid spurious contributions from the viscous layer that are not present in R2000. This is consistent with Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017), who showed that objects identified by (4.9) are artificially elongated in the streamwise direction when the region below $x_{2}^{+}=100$ is included.

Figure 13 shows two examples of actual objects extracted from NS2000 and R2000, and underlines the complex geometries that may arise. Although visual impressions should not substitute statistical analysis, the comparison between the two examples reinforces the idea that the structures in Robin-bounded and wall-bounded channels are comparatively similar.

Figure 13. Example of instantaneous three-dimensional momentum transfer structures extracted from R2000 (a) and NS2000 (b). The mean flow is from bottom left to top right. Axes are normalised in plus units. The colour gradient indicates distance to the wall/boundary.

Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014b ) showed that objects defined by (4.9) are responsible for most of the momentum transfer in the log layer, justifying the choice of intense regions of $u_{1}^{\prime }u_{2}^{\prime }$ as representative structures of the flow. Moreover, these regions are geometrically and temporally self-similar and can be considered as sensible contenders for the active wall-attached eddies envisioned by Townsend (Reference Townsend1976). Therefore, it is interesting to compare the geometry of these structures in wall-bounded and Robin-bounded channel flows. The sizes of the objects are measured by circumscribing each structure within a box aligned to the Cartesian axes, whose streamwise, wall-normal (or boundary-normal) and spanwise sizes are denoted by $\unicode[STIX]{x1D6E5}_{1}$ , $\unicode[STIX]{x1D6E5}_{2}$ and  $\unicode[STIX]{x1D6E5}_{3}$ , respectively. Figure 14 shows the joint probability density functions (p.d.f.s) of the logarithms of $\unicode[STIX]{x1D6E5}_{i}$ , $p(\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2})$ and $p(\unicode[STIX]{x1D6E5}_{3},\unicode[STIX]{x1D6E5}_{2})$ , for NS2000 and R2000. Both cases collapse remarkably well except for small objects close to the boundary, and follow fairly well-defined linear laws

(4.10a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{1}\approx 1.5\unicode[STIX]{x1D6E5}_{2}\quad \text{and}\quad \unicode[STIX]{x1D6E5}_{3}\approx \unicode[STIX]{x1D6E5}_{2},\end{eqnarray}$$

consistent with the good agreement obtained for the spectra in § 4.2. The similarity laws in (4.10) were used in § 4.2 to estimate the sizes of the eddies at a given $x_{2}$ location. The quantitative resemblance of the momentum eddies in the two flow configurations highlights once again that the impermeability of the wall (and hence the distance to the wall) is not a fundamental feature of the stress-carrying motions in the outer layer of wall-bounded flows.

Figure 14. Joint p.d.f.s of the logarithms of the box sizes of structures, (a)  $p(\unicode[STIX]{x1D6E5}_{1}^{+},\unicode[STIX]{x1D6E5}_{2}^{+})$ and (b)  $p(\unicode[STIX]{x1D6E5}_{3}^{+},\unicode[STIX]{x1D6E5}_{2}^{+})$ , for NS2000 (——) and R2000 (– – –). Contours contain 50 % and 99.8 % of the p.d.f. The dashed straight lines are $\unicode[STIX]{x1D6E5}_{1}=1.5\unicode[STIX]{x1D6E5}_{2}$ and $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{3}$ , respectively.