3.1 Mean flow

Figure 5(a) shows the mean velocity profile $\overline{u}$ as a function of the wall-normal coordinate. The black $+$ symbol is used for the reference case with rigid walls taken from Kim et al. (Reference Kim, Moin and Moser1987), and reproduced with the same code in Picano, Breugem & Brandt (Reference Picano, Breugem and Brandt2015); the solid lines indicate the results from the present simulations with blue, magenta, red and green used for cases with increasing elasticity (lower transverse elastic modulus $G$ ). This colour scheme will be used through the whole paper. The presence of one rigid and one elastic wall makes the mean velocity profile skewed, with its maximum $\overline{u}_{M}$ increasing and located closer to the bottom wall as the elasticity increases. Note that the mean velocity of the elastic wall is equal to zero.

Figure 5.

(a) Comparison of the streamwise mean velocity profile $U$ of a turbulent channel flow at $Re=2800$ over rigid walls ( $+$ symbols) and over different hyper-elastic walls (solid lines). The blue, magenta, red and green lines are used for the cases $G\downarrow$ , $G\downarrow \downarrow$ , $G\downarrow \downarrow \downarrow$ and $G\downarrow \downarrow \downarrow \downarrow$ , respectively, i.e. for increasing elasticity (lower transverse elastic modulus $G$ ). The results for the rigid case are taken from Kim et al. (Reference Kim, Moin and Moser1987). (b) Mean velocity profile $U$ versus the distance from the bottom rigid wall in wall units.

Figure 5(b) shows the mean velocity profiles versus the logarithm of the distance from the bottom rigid wall in wall units. For the turbulent channel flows over rigid impermeable walls (shown with the symbol $+$ ), we can usually identify three regions: firstly, the viscous sublayer for $y^{w+}<5$ where the variation of $\overline{u}^{w+}$ with $y^{w+}$ is approximately linear, i.e. 

(3.3) $$\begin{eqnarray}\overline{u}^{w+}=y^{w+}.\end{eqnarray}$$

In the so-called log-law region, $y^{w+}>30$ the variation of $\overline{u}^{w+}$ versus $y^{w+}$ is logarithmic, i.e. 

(3.4) $$\begin{eqnarray}\overline{u}^{w+}={\displaystyle \frac{1}{k}}\log y^{w+}+B,\end{eqnarray}$$

defined by the coefficients $k$ (the von Kármán constant) and $B$ . The values of $k$ and $B$ for smooth rigid walls are usually assumed to be $k=0.41$ and $B=5$ . However, for the Reynolds number that we are considering, a better fit with experimental and numerical simulations is obtained with $k=0.40$ and $B=5.5$ . Finally, the region between $5$ and $30$ wall units is called buffer layer and neither laws hold.

As shown in figure 5(b), the velocity profiles from the cases with an elastic wall still coincide, which indicates that the scaling of the mean flow near the bottom wall is not altered by the presence of the deformable top wall, the only differences being a reduced inertial range and the location of the maximum velocity at lower wall distances due to the skewness of the mean velocity profile.

Figure 6.

(a) Mean velocity profile $U$ versus the distance from the top rigid wall ${\tilde{y}}+d$ in wall units and (b) $({\tilde{y}}+d)^{+}\,\text{d}\overline{u}^{+}/\text{d}{\tilde{y}}^{+}$ as a function of ${\tilde{y}}^{+}$ for the case $G\downarrow \downarrow$ . The three lines in (b) correspond to three values of $d$ , i.e.  $0.00$ (solid line), $0.07$ (dashed line) and $0.14$ (dash-dotted line). The colour scheme is the same as in figure 5.

The mean velocity profiles in inner units based on the friction at the top moving walls are shown in figure 6(a). Differently from figure 5(b), the profiles are plotted versus ${\tilde{y}}+d=(2h-y)+d$ , where $d$ is a shift of the origin. This shift was introduced by Jackson (Reference Jackson1981) for rough walls and then used by Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) for permeable walls. Introducing $d$ , the log law is modified as

(3.5) $$\begin{eqnarray}\overline{u}^{+}={\displaystyle \frac{1}{\unicode[STIX]{x1D706}}}\log \frac{{\tilde{y}}+d}{y_{0}},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}$ is the slope of the inertial range, which may differ from $k$ , i.e.  $\unicode[STIX]{x1D706}=k+\unicode[STIX]{x0394}k$ , and $y_{0}$ is the so-called equivalent roughness height. Through simple manipulation, the previous relation can be rewritten as

(3.6) $$\begin{eqnarray}\overline{u}^{+}={\displaystyle \frac{1}{\unicode[STIX]{x1D706}}}\log ({\tilde{y}}+d)^{+}-\frac{1}{\unicode[STIX]{x1D706}}\log y_{0}^{+},\end{eqnarray}$$

and for a better comparison with (3.4) as

(3.7) $$\begin{eqnarray}\overline{u}^{+}={\displaystyle \frac{1}{k+\unicode[STIX]{x0394}k}}\log ({\tilde{y}}+d)^{+}+B-\unicode[STIX]{x0394}U^{+}.\end{eqnarray}$$

In order to identify the value of $d$ , we have first plot $({\tilde{y}}+d)^{+}\text{d}\overline{u}^{+}/\text{d}{\tilde{y}}^{+}$ as a function of ${\tilde{y}}^{+}$ for several values of $d$ . Inside the logarithmic layer this quantity must be constant and equal to the inverse of the slope in the inertial range, i.e.  $1/(k+\unicode[STIX]{x0394}k)$ . In this way, the correct value of the shift can be identified as the one providing a region with constant $({\tilde{y}}+d)^{+}d\overline{u}^{+}/d{\tilde{y}}^{+}$ (an equilibrium range). The procedure is exemplified in figure 6(b) where we report $({\tilde{y}}+d)^{+}d\overline{u}^{+}/d{\tilde{y}}^{+}$ for the case $G\downarrow \downarrow$ and three values of $d$ , i.e.  $0.00$ (solid line), $0.07$ (dashed line), and $0.14$ (dash-dotted line). Note that the slope in the inertial range is slightly positive for $d=0.0$ , while it becomes negative for $d=0.14$ ; only for the correct value $d=0.07$ the slope is null, providing $k+\unicode[STIX]{x0394}k=0.294$ . Once $d$ and $\unicode[STIX]{x0394}k$ are known, $\unicode[STIX]{x0394}U^{+}$ can be found by a simple fitting.

Table 2.

Summary of the log-law fitting parameters and wall deformations for the cases shown in table 1. The table provides the origin shift $d$ , the modified von Kármán constant $k+\unicode[STIX]{x0394}k$ , the shift of the velocity profile $\unicode[STIX]{x0394}U^{+}$ , the maximum and root mean square values of the wall deformation $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FF}^{\prime }$ .

This procedure has been repeated for all the cases, obtaining the values of the fitting parameters reported in table 2 and used for the scaling in figure 6(a); note that the dashed lines reported in the figure are computed using these values. All the cases with elastic walls under consideration show an increase in the slope of the logarithmic region, $\unicode[STIX]{x0394}k<0$ , when compared with the flow over rigid walls, as well as a downward shift $\unicode[STIX]{x0394}U^{+}$ . The change of slope in the logarithmic region, corresponding to an increase of drag, was also found in turbulent channel flow over porous walls by Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) for high values of permeabilities, while no change was found for low values of permeability in Rosti et al. (Reference Rosti, Cortelezzi and Quadrio2015) and for the turbulent channel flow over rough walls by Orlandi & Leonardi (Reference Orlandi and Leonardi2008); indeed the only effect of a rough wall was a downward shift of the mean profile, also associated with a drag increase.

Figure 7.

(a) Root-mean-square values of velocity fluctuations at the interface as function of the inverse of the elastic modulus ( $\unicode[STIX]{x1D70C}U_{b}^{2}/G$ ). The solid line is used for the streamwise component, the dashed line for the wall-normal component and the dash-dotted one for the spanwise component. (b) Root mean square of the wall-normal velocity component $v^{\prime }$ versus the velocity shift $\unicode[STIX]{x0394}U^{+}$ , see (3.9). The black circles are the data for the turbulent channel flow over porous wall investigated by Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006). The colour scheme is the same as in figure 5.

As discussed by Orlandi & Leonardi (Reference Orlandi and Leonardi2008), previous studies employing DNS with different boundary conditions for the three velocity components at the wall have shown that the turbulent flow is strongly linked to the fluctuations of the wall-normal velocity component $v^{\prime }$ , with r.m.s. values denoted by a $^{\prime }$ in the following. This has been shown by Orlandi et al. (Reference Orlandi, Leonardi, Tuzi and Antonia2003) for the case of a rough surface made of two-dimensional square bars that can be mimicked by distributions of $v^{\prime }$ , by Flores & Jimenez (Reference Flores and Jimenez2006) who used synthetic velocity distributions, and by Jimenez et al. (Reference Jimenez, Uhlmann, Pinelli and Kawahara2001) who modelled a permeable wall adding a transpiration boundary condition proportional to the local pressure fluctuations $p^{\prime }$ , i.e.  $v=-\unicode[STIX]{x1D6FD}p^{\prime }$ , $\unicode[STIX]{x1D6FD}$ being a constant. Also, Cheng & Castro (Reference Cheng and Castro2002) experimentally showed that the roughness function correlates well with $v^{\prime }$ . Based on these previous data, Orlandi & Leonardi (Reference Orlandi and Leonardi2006, Reference Orlandi and Leonardi2008) proposed first a correlation between $\unicode[STIX]{x0394}U^{+}$ and $v^{\prime }$ , then improved by using $v^{\prime +}$ , finally showing that

(3.8) $$\begin{eqnarray}\unicode[STIX]{x0394}U^{+}\approx \frac{B}{k}v^{\prime +}.\end{eqnarray}$$

These correlation was validated against the results by Cheng & Castro (Reference Cheng and Castro2002) and Orlandi et al. (Reference Orlandi, Leonardi, Tuzi and Antonia2003), Flores & Jimenez (Reference Flores and Jimenez2006), Orlandi & Leonardi (Reference Orlandi and Leonardi2006, Reference Orlandi and Leonardi2008). Both numerical and experimental data fit well with relationship (3.8) for values of $v^{\prime +}$ up to $0.8$ ; above this values the linear correlation departs from the data, showing hence discrepancies at high $v^{\prime +}$ .

In order to test a similar correlation also for the present configuration, we display the r.m.s. of the velocity fluctuation components at the mean interface location, $y=2h$ , versus the elasticity $\unicode[STIX]{x1D70C}U_{b}^{2}/G$ in figure 7(a). In the picture, the solid, dashed and dash-dotted lines are used for the streamwise, wall-normal and spanwise velocity components. All the three components of the fluctuating velocity appear to vary almost linearly with the inverse of the elasticity, the wall-normal component of the velocity being the fastest and the spanwise the slowest. Figure 7(b) shows the log-law shift $\unicode[STIX]{x0394}U^{+}$ as a function of the ratio between the wall-normal velocity fluctuation $v^{\prime }$ and the change of the inertial range slope $\unicode[STIX]{x0394}k$ . For all the cases considered, we find that a linear relation well approximates the two quantities, i.e. 

(3.9) $$\begin{eqnarray}\unicode[STIX]{x0394}U^{+}\approx \frac{v^{\prime }}{\unicode[STIX]{x0394}k}.\end{eqnarray}$$

The proposed fit, given in (3.9), is here tested also for the case of a turbulent channel flow over porous wall by using the data of the simulations in Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006), shown in figure 7(b) as black circles. Interestingly, the agreement is quite good also in this case (please note, that the values of $\unicode[STIX]{x0394}U^{+}$ and $v^{\prime }$ are not reported but extracted from the figures).

3.2 Velocity fluctuations

Figure 8.

Wall-normal profiles of the different components of the Reynolds stress tensor, normalized with $U_{b}^{2}$ for a turbulent channel flow over hyper-elastic walls. The symbols indicate the flow between two rigid walls from Kim et al. (Reference Kim, Moin and Moser1987). Panels (a–c) show the diagonal components $u^{\prime }u^{\prime }$ , $v^{\prime }v^{\prime }$ and $w^{\prime }w^{\prime }$ , while (d) the cross-term $u^{\prime }v^{\prime }$ . The colour scheme is the same as in figure 5.

We continue our comparison between the turbulent channel flows with rigid and elastic walls by analysing the wall-normal distribution of the diagonal component of the Reynolds stress tensor; these are shown in 8(a–c) together with the data from Kim et al. (Reference Kim, Moin and Moser1987) for the rigid case. All the components are strongly affected by the moving wall, and the effect is not limited to the region close to the interface, but it extends also beyond the centreline. The peaks of the Reynolds stresses move farther from the walls as the elasticity is increased. As already showed in figure 7(a), all the components have non-zero values at $y=2h$ for the elastic cases, since the no-slip condition is now enforced on a wall which is moving, i.e.  $u_{i}^{f}=u_{i}^{s}$ . The Reynolds stress components decrease inside the solid layer, eventually becoming zero at the rigid top wall, located at $y=2.5h$ . Note that for the most elastic case considered ( $G\downarrow \downarrow \downarrow \downarrow$ ) all the stress components, especially the wall-normal one, do not clearly vanish until reaching the rigid top wall, thus indicating that the fluctuations propagate deeply inside the solid layer. In relative terms, the spanwise and wall-normal components are the most affected by the wall elasticity, which can be attributed to a weakening of the wall-blocking and wall-induced viscous effects (Perot & Moin Reference Perot and Moin1995a ,Reference Perot and Moin b ), and observed also in the experiments of Krogstad, Antonia & Browne (Reference Krogstad, Antonia and Browne1992) and Krogstadt & Antonia (Reference Krogstadt and Antonia1999) on boundary-layer flows over rough walls. The wall-normal component displays a secondary peak very close to the interface, which grows with the elasticity and eventually becomes its maximum value. This secondary peak is associated with the oscillatory movement of the wall. Indeed, the dashed line in figure 8(b) connects the maximum of the wall-normal oscillation amplitude $\unicode[STIX]{x1D6FF}/h$ reported in table 2 with the local value of the Reynolds stress, thus, clearly showing that the location of the peak coincides with the maximum oscillation of the wall. We also note that the peak of the streamwise component of the fluctuating velocity, $u^{\prime 2}$ , grows much less then that pertaining the other two components. This increase is even lower than the growth of the friction Reynolds number, corresponding to a reduction in wall units. The reduced amplitude of the streamwise fluctuations is usually associated with the absence or reduction of streaky structures above a wall (Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006; Rosti et al. Reference Rosti, Cortelezzi and Quadrio2015), which is true also in the present case, as shown in the next section.

Finally, figure 8(d) depicts the wall-normal profile of the off-diagonal component of the Reynolds stress tensor. Also this cross component is strongly affected by the presence of the moving wall. In particular, the maximum value increases and moves away from the wall as the elasticity increases. Consequently, also the location where the stress is zero $y_{uv}$ , i.e. zero turbulent production, moves closer to the bottom solid walls, from $y=1h$ for the rigid case to $y\approx 0.58h$ for the most elastic upper wall. Nevertheless the stress profiles vary linearly between the two peaks, of opposite sign, close to each wall, although with different slopes. At the interface the value of the stress is not null as in the rigid case, however, inside the elastic layer the stress vanishes quickly.

Figure 9.

Wall-normal profiles of the turbulent kinetic energy ${\mathcal{K}}=(u^{\prime 2}+v^{\prime 2}+w^{\prime 2})/2$ , normalized with the friction velocities of (a) the bottom rigid wall $u_{\unicode[STIX]{x1D70F}}^{w}$ and (b) of the top elastic wall $u_{\unicode[STIX]{x1D70F}}$ . Symbols indicate the flow between two rigid walls from the data in Kim et al. (Reference Kim, Moin and Moser1987). The colour scheme is the same as in figure 5.

An overall view of the velocity fluctuations can be inferred considering the turbulent kinetic energy ${\mathcal{K}}=(u^{\prime 2}+v^{\prime 2}+w^{\prime 2})/2$ , shown in figure 9. We display in figure 9(a) the kinetic energy normalized by the friction velocity $u_{\unicode[STIX]{x1D70F}}^{w}$ at the bottom rigid wall as a function of the wall-normal distance $y/h$ . As usual, the symbols represent the profiles from the DNS of Kim et al. (Reference Kim, Moin and Moser1987) of turbulent flow between two solid rigid walls. Close to the bottom wall the profiles coincide, as expected, which indicates that the influence of the moving wall on the velocity fluctuations near the other wall is negligible. In the boundary layer close to the moving wall, instead, we observe a strong increase of the kinetic energy, with the maximum value becoming the double of the peak close to the bottom wall. Figure 9(b) reports the same turbulent kinetic energy profiles, but now normalized by the friction velocity $u_{\unicode[STIX]{x1D70F}}$ at the moving top wall. Except for the case $G\downarrow \downarrow \downarrow \downarrow$ , the profiles tend to coincide away from the wall (for $y$ in the range between $1h$ and $1.8h$ ), whereas the peak close to the moving wall is slightly lower for higher wall elasticities. The reduced peak is mainly due to the weak increase (decrease in wall units) of the streamwise component of the velocity fluctuations discussed above. The inset in figure 9(b) shows again the turbulent kinetic energy profile close to the moving wall versus the wall-normal distance ${\tilde{y}}$ divided by the coordinate ${\tilde{y}}_{uv}$ where the shear component of the Reynolds stress tensor crosses zero. This scaling compensates for the asymmetry in the geometry, similarly to what done by Leonardi, Orlandi & Antonia (Reference Leonardi, Orlandi and Antonia2007), and provides a better collapse of the profiles.

3.3 Solid deformation and flow structures

We now discuss in more details the deformation of the wall and its effect on the emerging flow coherent structures. Figure 10 shows one instantaneous configuration of the top elastic wall for the four cases considered here with growing elasticity from top to bottom, and with the flow from left to right. The wall surface is coloured by the wall-normal distance to the reference straight unstressed condition on the left column, with the colour scale ranging from $-0.13h$ (blue) to $0.13h$ (red), and by the pressure on the right column, with the colour scale ranging from $-1\unicode[STIX]{x1D70C}U_{b}^{2}$ (blue) to $1\unicode[STIX]{x1D70C}U_{b}^{2}$ (red). The wall surface deforms with a maximum amplitude of the wall deformation $\unicode[STIX]{x1D6FF}$ ranging approximately from 1 % of the channel half-height $h$ for the most rigid case ( $G\downarrow$ ) to 13 % in the most deformable one ( $G\downarrow \downarrow \downarrow \downarrow$ ). We recall that the values of $\unicode[STIX]{x1D6FF}$ are reported in table 2, together with the r.m.s. value of the wall deformation $\unicode[STIX]{x1D6FF}^{\prime }$ . It is interesting to note that the spanwise coherency of the wall deformation tends to increase as $G$ is decreased, i.e. the deformation increases.

Figure 10.

Instantaneous configuration of the top elastic wall, with transverse elastic modulus $G$ decreasing from top to bottom. The flow goes from left to right. The surfaces are coloured by the wall-normal distance in the left column, with the colour scale ranging from $-0.13h$ (blue) to $0.13h$ (red), and by the pressure on the right column, with the colour scale ranging from $-1\unicode[STIX]{x1D70C}U_{b}^{2}$ (blue) to $1\unicode[STIX]{x1D70C}U_{b}^{2}$ (red). The bottom row are zoomed views of the pictures in the right column.

Figure 11.

Profiles of the (a) streamwise $\unicode[STIX]{x1D70E}_{11}^{sh}$ and (b) cross $\unicode[STIX]{x1D70E}_{12}^{sh}$ hyper-elastic stress tensor components as a function of the wall-normal coordinate $y/h$ for the different values of the transverse elastic modulus $G$ under investigation. The colour scheme is the same as in figure 5.

Figure 11 shows the mean profile of the hyper-elastic stress tensor streamwise $\unicode[STIX]{x1D70E}_{11}^{sh}$ and shear $\unicode[STIX]{x1D70E}_{12}^{sh}$ components (see equation (2.8)). We note that the mean streamwise component is zero in the fully fluid region ( $y>2h-\unicode[STIX]{x1D6FF}$ ) and rapidly grows for $2h-\unicode[STIX]{x1D6FF}<y<2h+\unicode[STIX]{x1D6FF}$ , before reaching its positive maximum value at the solid wall, $y=2.5h$ . The shear component follows a similar trend, but has a maximum negative value at the wall, with the shear component being one or two orders of magnitude smaller then the streamwise one. While $\unicode[STIX]{x1D70E}_{11}^{sh}$ decreases as $G$ is reduced, i.e. for higher wall elasticity, on the contrary, $\unicode[STIX]{x1D70E}_{12}^{sh}$ increases (in absolute value). Note also that the values of the stress components approximately scale with $G$ .

Figure 12.

Instantaneous contour of the streamwise velocity fluctuations $u^{\prime }$ in the $x$ – $z$ plane at $({\tilde{y}}+d)^{+}=10$ for walls with transverse elastic modulus $G$ decreasing from top-left to bottom-right. The colour scale goes from $-0.35U_{b}$ (blue) to $0.35U_{b}$ (red). The black lines represent the levels $u^{\prime +}\pm 4$ .

Figure 13.

Instantaneous contour of the streamwise velocity fluctuations $u^{\prime }$ in the wall-parallel $x$ – $z$ plane at ${\tilde{y}}=0.15h$ for walls with transverse elastic modulus $G$ decreasing from top-left to bottom-right. The colour scale goes from $-0.35U_{b}$ (blue) to $0.35U_{b}$ (red). The black lines represent the levels $u^{\prime +}\pm 4$ .

The wall deformation affects the near-wall structure of the flow and this is visually confirmed by figure 12 that depicts the flow in the wall-parallel $x$ – $z$ plane at $({\tilde{y}}+d)^{+}=10$ . The figure reports instantaneous contours of the streamwise velocity fluctuations $u^{\prime }$ with colour scale from $-0.35U_{b}$ (blue) to $0.35U_{b}$ (red). In the same figure, we also show the levels $u^{\prime +}=\pm 4$ using black line contours. The goal of this figure is to identify the large-scale low- and high-speed near-wall streaks and show how they are affected by the wall elasticity. It is evident that the structures appear less elongated and more fragmented. Indeed, the flow is rich of small-scale features, consistent with a picture where the larger coherent structures are being broken into smaller pieces as a result of the wall movement. Figure 13 reports the same quantity but further from the wall, over the wall-parallel plane at ${\tilde{y}}=0.15h$ . As for the near-wall structures the low- and high-speed streaks become shorter the more elastic the walls. Another interesting flow feature can be inferred by this figure, i.e. as the structure become shorter they also increase their spanwise size, eventually extending almost over the entire spanwise extent of our domain (see figure 13, bottom right), consistently with our previous discussion about the wall deformation (see figure 10, fourth row). These visual observations will be now quantified statistically by analysing the autocorrelation functions.

Figure 14.

Line and colour contours of the one-dimensional autocorrelation of the (top) streamwise, (middle) wall-normal and (bottom) spanwise velocity fluctuations as a function of the streamwise spacing $x/h$ for different $y$ coordinates, shown across half-elastic-wall and half-channel with $G$ decreasing from left to right. The thick and thin lines correspond to positive and negative values, ranging from $-0.1$ to $0.9$ with a step of $0.2$ between two neighbouring lines. The colour scale ranges from $-0.1$ (white) to $1.0$ (black).

We now focus on the two-point correlation, defined as follows

(3.10) $$\begin{eqnarray}R_{ii}(\boldsymbol{x},\boldsymbol{r})=\frac{\overline{u_{i}^{\prime }(\boldsymbol{x})u_{i}^{\prime }(\boldsymbol{x}+\boldsymbol{r})}}{\overline{u_{i}^{\prime 2}(\boldsymbol{x})}},\end{eqnarray}$$

where the bar denotes average over time and homogeneous directions, and the prime the velocity fluctuation. Figure 14 illustrates the structure of the correlation functions along the streamwise direction $x$ . These are one-dimensional correlations using the velocity difference along $x$ , displayed for different $y$ coordinates across half of the elastic wall and the channel, i.e. for $y\in [1h,2.25h]$ . For the most rigid wall, the correlations, and in particular that for the streamwise component $u$ , present in the fluid region structures of relatively large length especially close to the walls. These are associated with the presence of elongated streaky structures that characterize the near-wall region. For a completely rigid wall, the streaks have typically a length of the order of $1000$ wall units, which would correspond to a correlation distance of approximately $5h$ . As already seen in the visualisation in figure 12, the typical length scale is strongly reduced when increasing the wall elasticity, and indeed the correlation distance decreases significantly close to the wall, which is consistent with the reduction/absence of low- and high-speed streaks.

It is also interesting to note that the correlations change rapidly but continuously across the interface: the typical correlation length becomes much shorter for the $u$ component inside the elastic material as this cannot support streaky structures, while it remains comparable to the rigid channel length scales for the other two velocity components. Inside the elastic wall, the correlations present an alternating sequence of progressively weaker negative and positive local peaks, a so-called cell-like pattern, associated with large-scale pressure fluctuations within the elastic material and just above the interface, as suggested by Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) for the flow over a porous wall. The typical wavelength of this oscillation for the most deformable case is approximately $1h$ , thus corresponding to around $300$ plus units, a result consistent with the undulations shown in figure 12(bottom right).

Figure 15.

Line and colour contours of the one-dimensional autocorrelation of the (top) streamwise, (middle) wall-normal and (bottom) spanwise velocity fluctuations as a function of the spanwise spacing $z/h$ for different $y$ coordinates, shown across half-elastic-wall and half-channel with $G$ decreasing from left to right. The thick and thin lines correspond to positive and negative values, ranging from $-0.1$ to $0.9$ , with a step of $0.2$ between two neighbouring lines. The colour scale ranges from $-0.1$ (white) to $1.0$ (black).

Finally, we consider the spanwise correlation functions for all three velocity components, see figure 15. On a rigid wall (Kim et al. Reference Kim, Moin and Moser1987), the spanwise autocorrelation of the streamwise velocity component $u$ (figure 15, top row) exhibits a local minimum at $z^{w+}\approx 50$ in the region close to the wall, usually associated with the average spanwise distance between a low-speed and a neighbouring high-speed streak. Note that the periodicity of the streaks is deduced from the oscillations of the autocorrelation for larger spanwise spacings. The flow over elastic walls, conversely, display a larger correlation distance. Considering the wall-normal component of the velocity (figure 15, middle row), we observe a local minimum at approximately $y^{w+}\approx 25$ close to the rigid wall, which is consistent with the presence of quasi-streamwise vortices. The value of this local minimum decreases ( $G\downarrow$ and $G\downarrow \downarrow$ ), and eventually disappear ( $G\downarrow \downarrow \downarrow$ and $G\downarrow \downarrow \downarrow \downarrow$ ) in the case of moving elastic walls, suggesting that the quasi-streamwise vortices are strongly reduced. All the three velocity components present relatively large spanwise scales, and we thus find a statistical signature of the spanwise vortical structures or rolls similarly to those that have been identified in the turbulent flow over porous walls (Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006; Rosti et al. Reference Rosti, Cortelezzi and Quadrio2015; Samanta et al. Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015), rough surfaces (Jimenez et al. Reference Jimenez, Uhlmann, Pinelli and Kawahara2001) and plant canopies (Finnigan Reference Finnigan2000). These structures are also believed to be the main cause of the performance loss of drag-reducing riblets when the riblet size is increased above the optimum (Garcia-Mayoral & Jimenez Reference Garcia-Mayoral and Jimenez2011).

3.4 Effect of viscosity ratio

Figure 16.

(a) Comparison of the streamwise mean velocity profile $U$ of a turbulent channel flow at $Re=2800$ over rigid walls ( $+$ symbols) and over different hyper-elastic walls (red lines). The dotted, solid and dashed lines are used for the cases $\unicode[STIX]{x1D707}^{s}/\unicode[STIX]{x1D707}^{f}=0.1$ , $1.0$ and $10.0$ at a fixed elastic modulus $G=1.0$ . The results for the rigid case are taken from Kim et al. (Reference Kim, Moin and Moser1987). (b) Mean velocity profile $U$ versus the distance from the top deformable wall in wall units.

We now consider the effect of the viscosity of the hyper-elastic material on the turbulence statistics for the case $G\downarrow \downarrow \downarrow$ . Three viscosity ratios are examined, i.e.  $\unicode[STIX]{x1D707}^{s}/\unicode[STIX]{x1D707}^{f}=0.1$ , $1.0$ and $10.0$ , see table 1 for a direct comparison of the values of the main global statistics.

When the elastic layer is less viscous than the fluid ( $\unicode[STIX]{x1D707}^{s}\downarrow$ ), the friction Reynolds number is equal to $240$ , same value obtained when the two viscosities are the same, while when the layer is more viscous than the fluid ( $\unicode[STIX]{x1D707}^{s}\uparrow$ ), the Reynolds number decreases by 6 % to $226\,\%$ . This trend is shown by the mean velocity profiles, reported in figure 16(a), where the dotted, solid, and dashed lines correspond to the flow with $\unicode[STIX]{x1D707}^{s}/\unicode[STIX]{x1D707}^{f}=0.1$ , $1.0$ and $10.0$ , respectively.

The mean velocity profile is less skewed in the high viscosity case ( $\unicode[STIX]{x1D707}^{s}\uparrow$ ), with its maximum velocity $u^{M}$ decreasing and located closer to the centreline. The same velocity profiles are displayed versus the logarithm of the distance from the top wall in wall units in figure 16(b). Again, the cases with $\unicode[STIX]{x1D707}^{s}<\unicode[STIX]{x1D707}^{f}$ and $\unicode[STIX]{x1D707}^{s}=\unicode[STIX]{x1D707}^{f}$ are almost undistinguishable, while the downward shift of the profile is reduced when $\unicode[STIX]{x1D707}^{s}<\unicode[STIX]{x1D707}^{f}$ , moving the logarithmic part of the velocity profile towards that for a rigid wall. A similar picture emerges after examining other quantities, such the wall-normal profile of the Reynolds stress tensor components reported in figure 17.

Figure 17.

Components of the Reynolds stress tensor as a function of the wall-normal coordinate $y$ , normalized with the bulk velocity $U_{b}$ for a turbulent channel flow over a hyper-elastic wall. Symbols are for the rigid case from Kim et al. (Reference Kim, Moin and Moser1987) while the dotted, solid, and dashed lines are used for the cases $\unicode[STIX]{x1D707}^{s}/\unicode[STIX]{x1D707}^{f}=0.1$ , $1.0$ , and $10.0$ at a fixed elastic modulus $G=1.0$ . The first three panels (a–c) show the diagonal components $u^{\prime }u^{\prime }$ , $v^{\prime }v^{\prime }$ and $w^{\prime }w^{\prime }$ , whereas the cross-term $u^{\prime }v^{\prime }$ is depicted in (d).

Overall, the results suggest that the behaviour of the flow in the limit of large and small values of $\unicode[STIX]{x1D707}^{s}$ and $G$ is different. Indeed, as $G$ or $\unicode[STIX]{x1D707}^{s}/\unicode[STIX]{x1D707}^{f}$ tend to infinity, the wall deformation decreases and the flow approaches the one over a rigid wall. On the contrary, while $G$ tending to $0$ increases the wall deformation, thus enhancing the turbulence, the flow appears to be almost unaffected by a decrease of $\unicode[STIX]{x1D707}^{s}/\unicode[STIX]{x1D707}^{f}$ below unity. Indeed, this behaviour suggests that when the viscosity ratio is low, the deformation is determined by the fluid fluctuations, while when the ratio $\unicode[STIX]{x1D707}^{s}/\unicode[STIX]{x1D707}^{f}$ is high the deformation is limited by the properties of the elastic layer.