4.1 Mean velocity and Reynolds stresses

Owing to the spatial variation of wall structures, flows near porous media or rough walls do not always become homogeneous in a certain plane parallel to a wall. Hence, to statistically discuss such flows, spatial averaging has been introduced (Nikora et al. Reference Nikora, Goring, McEwan and Griffiths2001; Franca, Ferreira & Lemmin Reference Franca, Ferreira and Lemmin2008; Hurther & Lemmin Reference Hurther and Lemmin2008; Mignot, Hurther & Barthelemy Reference Mignot, Hurther and Barthelemy2009). The definition of the spatial (plane) averaging of $\unicode[STIX]{x1D719}(x,y,z)$ is introduced as

where $S$ and $A_{S_{f}}$ are the surface of the $x{-}z$ plane and the surface areas of the fluid phase contained within $S$ , respectively. To discuss turbulence statistics around the porous walls, double averaging, which consists of the plane and Reynolds averaging operations, is applied. In the following discussions, $\overline{\unicode[STIX]{x1D719}}$ denotes the Reynolds-averaged value and $\unicode[STIX]{x1D719}^{\prime }$ denotes the fluctuated value from $\overline{\unicode[STIX]{x1D719}}$ . For the Reynolds averaging, simulations are carried out over 30 ( $Re_{\unicode[STIX]{x1D70F}}=111$ ) or 10 ( $Re_{\unicode[STIX]{x1D70F}}=230$ ) cycle times after the flows reach the fully developed states.

Figure 2.

Plane-averaged streamwise mean velocity profiles normalized by $u_{\unicode[STIX]{x1D70F}}^{t}$ : (a) in porous and clear flow region; and (b) near the porous interface.

Figures 2 and 3 show the $x{-}z$ plane-averaged streamwise mean velocity $U^{t+}=[\overline{u}]^{f}/u_{\unicode[STIX]{x1D70F}}^{t}$ and Reynolds shear stress $R_{12}^{t+}=-[\overline{u^{\prime }v^{\prime }}]^{f}/(u_{\unicode[STIX]{x1D70F}}^{t})^{2}$ profiles normalized by the friction velocity of the smooth top wall. Therefore, in figure 3, the total stress reaches $-1$ at the top solid wall in all cases. Over the porous wall with spanwise and streamwise permeabilities (cases YZ, YZ and XYZ), $R_{12}^{t+}$ is significantly enhanced compared with that near the smooth top wall. In particular, the enhancement of $R_{12}^{t+}$ over the porous wall with streamwise permeability (cases XY and XYZ) is more pronounced than that in case YZ. Consequently, as shown in table 5, the enhancement of the wall shear stress of the porous wall $\unicode[STIX]{x1D70F}_{w}^{p}/\unicode[STIX]{x1D70F}_{w}^{t}$ in case XY is $1.28$ times larger than that in case YZ. As $R_{12}^{t+}$ increases, the friction Reynolds number of the porous-wall side $Re_{\unicode[STIX]{x1D70F}}^{p}$ increases, whereas the bulk mean Reynolds number $Re_{b}$ decreases; see table 5. In contrast, in case Y with pores only in the vertical direction, $U^{t+}$ rapidly converges to zero and the mean flow hardly penetrates into the bottom wall. Hence, even though the bottom wall is thick enough to neglect the influence of most of the bottom wall, the porous medium in case Y behaves as a rough wall. Furthermore, the pores of the bottom wall in case Y are so narrow that stable recirculation vortices that isolate the outer layer are sustained. Such flow is usually observed inside d-type roughness (Perry, Schofield & Joubert Reference Perry, Schofield and Joubert1969); thus it can be deduced that the bottom-wall characteristics hardly affect turbulence over the wall. Accordingly, the magnitudes of $R_{12}^{t+}$ in case Y over the porous and smooth top walls are almost the same and the profile of $U^{t+}$ in figure 2(a) is consequently almost symmetrical. Thus, as shown in table 5, the enhancement of the wall shear stress of the porous wall $\unicode[STIX]{x1D70F}_{w}^{p}/\unicode[STIX]{x1D70F}_{w}^{t}$ in case Y is $4\,\%$ at most. Figure 2(b) enlarges the plane-averaged mean velocity profiles beneath the porous interface. Although a slip velocity can be seen at the porous interface in all cases, that in case Y is smaller than those in cases XY, YZ and XYZ. Mean adverse flow can be seen just beneath the porous interface in cases YZ and Y; however, it eventually converges to zero inside the porous wall of $y/H<-0.2$ . The positive slip velocity and adverse flow just beneath the porous interface mean the presence of recirculation vortices just beneath the porous interface in cases Y and YZ. On the other hand, $U^{t+}$ in cases XY and XYZ does not converge to zero due to the streamwise permeability of the porous walls. The profiles in cases XY and XYZ are wavy corresponding to the porous structure, and clearly exhibit an inflection point ( $\text{d}^{2}U/\text{d}y^{2}=0$ ), which obeys a necessary condition for the inviscid instability such as the KH instability. The presence of the inflection point of the streamwise mean velocity was also reported by the DNS study for porous-walled channel flows (Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006; Kuwata & Suga Reference Kuwata and Suga2016b ,Reference Kuwata and Suga c ) and experiments for vegetated canopy flows (Finnigan Reference Finnigan2000; Ghisalberti & Nepf Reference Ghisalberti and Nepf2002; Poggi et al. Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004).

Figure 3.

Reynolds and total shear stress profiles normalized by $u_{\unicode[STIX]{x1D70F}}^{t}$ . The thin lines are the profiles of the total shear stress.

Figure 4.

Plane-averaged Reynolds normal stresses normalized by $u_{\unicode[STIX]{x1D70F}}^{t}$ : (a) streamwise component, (b) wall-normal component, and (c) spanwise component.

Figure 4 presents profiles of the $x{-}z$ plane-averaged Reynolds normal stresses $R_{ij}^{t+}=[\overline{u_{i}^{\prime }u_{j}^{\prime }}]^{f}/(u_{\unicode[STIX]{x1D70F}}^{t})^{2}$ normalized by the friction velocity of the smooth top wall. Note that, since $R_{ij}^{t+}$ is almost damped at the bottom of the porous wall, the porous-wall thickness is enough to neglect the effect of the most bottom boundary of the porous wall. The plane-averaged Reynolds normal stresses of $R_{ij}^{t+}$ in case Y near the porous and smooth top walls are almost the same whilst $R_{ij}^{t+}$ near the porous wall in cases YZ, XY and XYZ are enhanced. The enhancement of $R_{11}^{t+}$ in figure 4(a) is less prominent compared with that of $R_{22}^{t+}$ and $R_{33}^{t+}$ in figure 4(b,c). It is consistent with the previous experiments of Pokrajac & Manes (Reference Pokrajac and Manes2009), Manes et al. (Reference Manes, Pokrajac, McEwan and Nikora2009) and Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) and the DNS of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Kuwata & Suga (Reference Kuwata and Suga2016b ). The local maximum values of $R_{11}^{t+}$ in proximity to the porous interface are almost the same level in cases XY, YZ and XYZ; however, approaching the channel centre, $R_{11}^{t+}$ in case YZ decreases more rapidly than those in cases XY and XYZ. Since $R_{ij}^{t+}$ over the streamwise permeable wall in cases XY are more enhanced than those over the spanwise permeable wall in case YZ, it can be deduced that the streamwise permeability particularly contributes to the increase of $R_{ij}^{t+}$ over the porous walls. Moreover, it is interesting to note that, regardless of the difference of the porous structures between cases XY and XYZ, the profiles of $R_{ij}^{t+}$ over the porous wall are very similar. Inside the porous wall, $R_{ij}^{t+}$ in case Y is almost damped whereas $R_{ij}^{t+}$ in cases XY, YZ and XYZ still remains. However, in case YZ, since streamwise permeability $K_{xx}$ becomes zero, streamwise fluid motion is restricted and $R_{11}^{t+}$ consequently becomes zero. Likewise for $R_{11}^{t+}$ in case YZ, $R_{33}^{t+}$ in case XY converges to zero inside the porous wall. Since the significant pressure diffusion transports $R_{22}^{t+}$ deep inside the porous wall, as reported by Kuwata & Suga (Reference Kuwata and Suga2016c ), $R_{22}^{t+}$ in cases YZ, XY and XYZ decays slower than $R_{11}^{t+}$ and $R_{33}^{t+}$ .

Figure 5.

Comparisons of the mean values between the porous-wall and rough-wall results: (a) $U^{t+}$ and (b) $R_{12}^{t+}$ .

It should be noted that the above-discussed difference between the cases YZ and XY may be due to the difference of the wall roughness. To discuss the individual influence of the porous-wall roughness and permeability, figure 5 presents comparisons of the $x{-}z$ plane-averaged mean velocity $U^{t+}$ and Reynolds shear stress $R_{12}^{t+}$ profiles between the porous-wall cases (cases XY and YZ) and the rough-wall cases (cases XY-R and YZ-R). It is found that, although significant differences of $U^{t+}$ as shown in figure 5(a) can be observed between the cases XY and YZ, the mean velocity profiles over the different rough walls of cases XY-R and YZ-R almost overlap. Furthermore, in figure 5(b), enhancement of $R_{12}^{t+}$ by the porous wall is more remarkable than that by the rough walls. This implies that, as well as the wall roughness, the wall permeability itself significantly enhances turbulence. However, the Reynolds shear stress enhancement caused by the spanwise permeability in case YZ is less pronounced than that by the streamwise permeability in case XY. This suggests that the influence of the wall roughness in cases XY and YZ are almost the same whereas the remarkable enhancement of turbulence in case XY can be attributed to the streamwise permeability. It should be stressed that the impact of the anisotropic permeability on turbulence may be in contrast to that of the anisotropic rough walls. It was reported by the systematic DNS study of Orlandi, Leonardi & Antonia (Reference Orlandi, Leonardi and Antonia2006) that the roughness elements placed along the spanwise direction such as transverse ribs caused more drag increase than those placed along the streamwise direction. Moreover, it has been reported that drag reduction is even possible over the streamwise roughness elements, which are known as riblets (Bushnell Reference Bushnell1990; Goldstein, Handler & Sirovich Reference Goldstein, Handler and Sirovich1995).

4.2 Turbulence structures

Figure 6 displays isosurfaces of the second invariant of the velocity gradient tensor of $II^{t+}=0.025$ normalized by $u_{\unicode[STIX]{x1D70F}}^{t}$ , which are coloured based on the local values of the streamwise vorticity of $-0.25<\unicode[STIX]{x1D714}_{x}^{t+}<0.25$ . Although a few vortices are visualized in case Y, strong vortex motions are observed over the porous wall in cases XY and YZ. The vortex motion seems to be more active in case XY than that in case YZ, and the vortex generation in case XY seems to be of roughly the same level as that in case XYZ.

Figure 6.

Isosurfaces of the second invariant of the velocity gradient tensor $II^{t+}=0.025$ , coloured by the streamwise vorticity; blue–red colours correspond to $-0.25<\unicode[STIX]{x1D714}_{x}^{t+}<0.25$ .

Figure 7 shows snapshots of the wall-normal vorticity fluctuations normalized by $u_{\unicode[STIX]{x1D70F}}^{p}$ over the porous walls at $y^{p+}\approx 13$ . The streamwise elongated streaky structure, which is usually observed over a smooth solid wall, is also observed in case Y whilst the elongated structure looks shredded and twisted in case YZ. Moreover, the streaky structures completely vanish in cases XY and XYZ, which is consistent with the vortex structures over a highly permeable wall reported by the DNS of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Kuwata & Suga (Reference Kuwata and Suga2016b ). Although a strong mean velocity gradient is present in cases XY and XYZ as shown in figure 2, which is required to form high- and low-speed streaks, the streaky structure in cases XY and XYZ vanishes. It is considered that the active wall-normal velocity fluctuation as shown in figure 4(b) prevents the development of the elongated streaky structures. Furthermore, it is observed in cases XY and XYZ that the vortex structure is intermittently shrunken, as indicated by the region enclosed by black solid lines in figure 7.

Figure 7.

Snapshots of the instantaneous wall-normal vorticity fluctuations normalized by $u_{\unicode[STIX]{x1D70F}}^{p}$ at $y^{p+}\approx 13$ .

Figure 8.

Snapshots of the instantaneous ejection distributions normalized by $u_{\unicode[STIX]{x1D70F}}^{p}$ at $y^{p+}\approx 13$ .

To discuss the turbulence structures over the porous wall in detail, figures 8 and 9 present snapshots of the instantaneous ejections $Q_{2}^{\prime p+}=-(u^{\prime }v^{\prime })^{p+}$ ( $u^{\prime }<0$ and $v^{\prime }>0$ ) and sweeps $Q_{4}^{\prime p+}=-(u^{\prime }v^{\prime })^{p+}$ ( $u^{\prime }>0$ and $v^{\prime }<0$ ) over the porous walls at $y^{p+}\approx 13$ , respectively. Note that all visualizations in figures 6–9 are made by the same snapshots. The region enclosed by black solid lines indicates the same region as shown in figure 7. Likewise for the distribution of the wall-normal vorticity fluctuations in figure 7, both the ejection and sweep structures are elongated in case Y and they are both shredded in case YZ. In cases XY and XYZ, the structures are no longer elongated to the streamwise direction. On the contrary, they tend to be merged in the spanwise direction. The active $Q_{2}^{\prime p+}$ regions in cases XY and XYZ seem to periodically emerge and the streamwise pitch is roughly 500–800 porous-wall units. However, this feature cannot be observed in the sweep structures in cases XY and XYZ in figure 9. Since the active $Q_{2}^{\prime +}$ regions in cases XY and XYZ correspond to the regions where the vortices are intermittently and periodically shrunken as seen in figure 7, they are caused by the low-speed fluid pushed up from the porous media in cases XY and XYZ. Furthermore, it was reported by Kuwata & Suga (Reference Kuwata and Suga2016b ) that strong transverse wavy pressure fluctuations induced by the KH instability accelerated or decelerated fluid inside the porous wall, and this leads to large-scale velocity fluctuations. Since those large-scale velocity fluctuations are pushed up from the porous wall, the intermittent patched structures appear only in ejections, not sweeps.

Figure 9.

Snapshots of the instantaneous sweep distributions normalized by $u_{\unicode[STIX]{x1D70F}}^{p}$ at $y^{p+}\approx 13$ .

4.3 Two-point correlation analysis

To investigate turbulence structures over the porous walls in more detail, streamwise and spanwise two-point correlations over the porous walls at $y^{p+}\approx 13$ are shown in figures 10 and 11. In figure 10(a,b), the streamwise correlations of $R_{uu,x}$ and $R_{vv,x}$ in cases YZ, XY and XYZ decay more rapidly compared with those in case Y. This substantiates the reduction of the streamwise elongated streaky structures as observed in figure 7. The correlations $R_{uu,x}$ and $R_{vv,x}$ in case YZ converge to zero, whilst, in cases XY and XYZ, both $R_{uu,x}$ and $R_{uu,z}$ show a wavy profile and a negative local minimum around $\unicode[STIX]{x0394}x^{p+}=200{-}300$ is observed. Such a negative local minimum can also be seen in the streamwise pressure correlation $R_{pp,x}$ as shown in figure 10(c). This is considered to be due to the streamwise large-scale perturbation induced by the KH instability as reported by the DNS of Jimenez et al. (Reference Jimenez, Uhlmann, Pinelli and Kawahara2001), Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Kuwata & Suga (Reference Kuwata and Suga2016b ) and the experiments of Finnigan (Reference Finnigan2000), Ghisalberti & Nepf (Reference Ghisalberti and Nepf2002) and Poggi et al. (Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004). From these results, it is found that the streamwise permeability allows development of the streamwise perturbation by the KH instability, which is triggered by the inflection of the streamwise mean velocity as seen in figure 2. The representative wavelength of this perturbation estimated by the location of the local minimum is approximately $600$ porous-wall units and this roughly corresponds to the pitch of the intermittent patched patterns of the ejections in figure 8. It is noticeable that, although pores in case XY are not interconnected in the spanwise direction, the negative local minimum value of the streamwise pressure correlation around $\unicode[STIX]{x0394}x^{p+}=200{-}300$ in case XY is slightly more significant than that in case XYZ whose pores are interconnected in the spanwise direction. This implies that flow communications in the spanwise direction do not enhance the coherence of the streamwise perturbation whereas those in the streamwise direction are essential to sustain the streamwise perturbation.

Figure 10.

Streamwise two-point correlations: (a) streamwise velocity fluctuations $R_{uu,x}$ , (b) wall-normal velocity fluctuations $R_{vv,x}$ , and (c) pressure fluctuations $R_{pp,x}$ .

Figure 11.

Spanwise two-point correlations: (a) streamwise velocity fluctuations $R_{uu,z}$ , (b) wall-normal velocity fluctuations $R_{vv,z}$ , and (c) pressure fluctuations $R_{pp,z}$ .

In figure 11, it is observed that the spanwise correlations $R_{uu,z}$ and $R_{vv,z}$ in cases Y and YZ exhibit a negative local minimum at $\unicode[STIX]{x1D6E5}_{z}^{p+}\approx 50$ , which is associated with the averaged spanwise distance of the high- and low-speed streaks. On the other hand, in cases XY and XYZ, such a negative local minimum cannot be observed in $R_{uu,z}$ and $R_{vv,z}$ . This suggests that the array of high- and low-speed streaks fades out in cases XY and XYZ, which is consistent with the observation in figure 7 and the previous DNS of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Kuwata & Suga (Reference Kuwata and Suga2016b ). In cases XY and XYZ, the spanwise correlation of $R_{pp,z}$ in figure 11(c) reaches a plateau, which implies the spanwise coherence across the full box. This result is consistent with the result of Jimenez et al. (Reference Jimenez, Uhlmann, Pinelli and Kawahara2001) and this spanwise coherence is considered to be due to the presence of the transverse pressure rolls by the KH instability as reported by Kuwata & Suga (Reference Kuwata and Suga2016b ).

Figure 12.

Premultiplied streamwise spectra: (a) streamwise velocity fluctuation $E_{uu,x}$ and (b) wall-normal velocity fluctuation $E_{vv,x}$ .

Figure 13.

Premultiplied spanwise spectra: (a) streamwise velocity fluctuation $E_{uu,z}$ and (b) wall-normal velocity fluctuation $E_{vv,z}$ .

4.4 Spectral analysis

Figures 12 and 13 give the premultiplied streamwise and spanwise energy spectra, respectively, over the porous walls at $y^{p+}\approx 13$ . At the wavelength of $\unicode[STIX]{x1D706}_{x}^{p+}=600$ , a pointed profile of $E_{uu,x}$ and $E_{vv,x}$ in cases XY and XYZ can be seen in figure 13, which roughly corresponds to the wavelength of the streamwise large-scale perturbation by the KH instability estimated by the two-point correlations in figure 10. Furthermore, $E_{uu,x}$ has a maximum value at $\unicode[STIX]{x1D706}_{x}^{p+}=600$ , which means that streamwise velocity fluctuations in cases XY and XYZ are dominated by the large-scale perturbation. Moreover, it is also found from figure 13 that, in cases XY and XYZ, the spanwise spectra of $E_{uu,z}$ and $E_{vv,z}$ in the relatively long-wavelength region of $\unicode[STIX]{x1D706}_{z}^{p+}>300$ become far larger than those in cases Y and YZ. In figure 13, it is confirmed that the streamwise velocity fluctuations in cases Y and YZ are dominated by the coherent streaks of $\unicode[STIX]{x1D706}_{z}^{p+}\approx 100$ . Since the streaky structures are less organized in case YZ as observed in figure 7, which is also confirmed by the spanwise correlation of the streamwise velocity fluctuation in figure 11, $E_{uu,z}$ in case YZ loses energy at $\unicode[STIX]{x1D706}_{z}^{p+}\approx 100$ compared with that in case Y whilst $E_{uu,z}$ of longer-wavelength regions $\unicode[STIX]{x1D706}_{z}^{p+}>100$ in case YZ increases. Because the streamwise elongated streaks are no longer developed in cases XY and XYZ as observed in figure 7, $E_{uu,z}$ at $\unicode[STIX]{x1D706}_{z}^{p+}\approx 100$ in cases XY and XYZ is far smaller than that in case Y. Moreover, in figure 12(a), $E_{uu,x}$ in cases XY and XYZ becomes smaller in the long-wavelength region of $\unicode[STIX]{x1D706}_{x}^{p+}\approx 1500\sim 2000$ compared with those in cases Y and YZ. From these results, it is summarized that the large-scale streamwise perturbation developed over the streamwise permeable walls in cases XY and XYZ substantially contributes to the increase of $R_{11}$ and $R_{22}$ . However, owing to the reduction of the streaks in cases XY and XYZ, the enhancement of $R_{11}$ is less prominent compared with that of $R_{22}$ , as shown in figure 4. Note that, since the wavelength of the small kinks of $E_{vv,x}$ and $E_{vv,z}$ at $\unicode[STIX]{x1D706}_{x}^{p+}\approx 45$ in figures 12(b) and 13(b) corresponds to the pitch of the holes $D_{2}^{p+}=45$ , it is considered to be associated with the printed pattern of the porous walls, which can also be observed in figure 7. However, since the kink is negligibly small, the printed pattern fluctuations in figure 7 is found to hardly contribute to the enhancement of $R_{22}$ . When the pore size is further reduced, the influence of the printed pattern fluctuations becomes negligible.

To clarify the contribution of the large-scale streamwise perturbation to the plane-averaged Reynolds shear stress, the premultiplied cospectrum of $E_{uv,x}$ and $E_{uv,z}$ normalized by $R_{12}$ is given in figure 14. From the streamwise cospectrum of $E_{uv,x}$ in figure 14(a), the contribution of the fluctuations in long-wavelength regions of $\unicode[STIX]{x1D706}_{x}^{p+}>800$ to $R_{12}$ in cases YZ, XY and XYZ becomes smaller compared with that in case Y whereas the contribution increases in the wavelength of $\unicode[STIX]{x1D706}_{x}^{p+}<500$ . Moreover, the profile of $E_{uv,x}$ in cases XYZ and XY exhibits a pointed peak at $\unicode[STIX]{x1D706}_{x}^{p+}=600$ . This means that, over the porous wall with the streamwise permeability, the large-scale streamwise perturbation contributes to $R_{12}$ , leading to the increase of friction. From the spanwise cospectrum of $E_{uv,z}$ in figure 14(b), it is found that $R_{12}$ in cases Y and YZ are dominated by fluctuations associated with streaks near $\unicode[STIX]{x1D706}_{z}^{p+}\approx 100$ whereas the locations of the global maximum in cases XY and XYZ shift to the longer-wavelength regions of $\unicode[STIX]{x1D706}_{z}^{p+}\approx 200$ . Although there are small kinks of $E_{uv,z}$ in cases XY and XYZ at wavelength of $\unicode[STIX]{x1D706}_{z}^{p+}\approx 45$ , which are associated with the printed pattern fluctuations in figure 7, the contribution to $R_{12}$ is found to be marginal.

Figure 14.

Premultiplied cospectra normalized by $R_{12}$ : (a) streamwise cospectra $E_{uv,x}$ and (b) spanwise cospectra $E_{uv,z}$

4.5 Reynolds-number dependence

In order to discuss the characteristics of the streamwise mean velocity profiles in the semi-logarithmic scale of figure 15, the logarithmic velocity profile is considered as

where parameters $\unicode[STIX]{x1D705}$ , $d$ and $h_{r}$ are, respectively, the Kármán constant, the zero-plane displacement and the equivalent roughness height, which are listed in table 6. (See Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) or Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) for the procedure to obtain the parameters.) Distinct logarithmic profiles cannot be observed in case Y; however, since the mean velocity profile in case Y is almost symmetrical, as shown in figure 2, the value of $\unicode[STIX]{x1D705}$ is assumed to be the standard value of $\unicode[STIX]{x1D705}=0.41$ , and the values of $d$ and $h_{r}$ are assumed to be $d=0$ and $h_{r}=0$ . As the wall shear stress of the porous wall $\unicode[STIX]{x1D70F}_{w}^{p}/\unicode[STIX]{x1D70F}_{w}^{t}$ increases as seen in table 5, the profile shifts downwards from the standard log-law line. Furthermore, both the zero-plane displacement $d$ and the equivalent roughness height $h_{r}$ increase whilst the Kármán constant decreases as $\unicode[STIX]{x1D70F}_{w}^{p}/\unicode[STIX]{x1D70F}_{w}^{t}$ increases. Through the comparison between the different Reynolds-number cases XY and XY-HR or cases YZ and YZ-HR, it is found that, as the Reynolds number increases, the profile shifts rightwards and both $d$ and $h_{r}$ increase whilst $\unicode[STIX]{x1D705}$ decreases.

Figure 15.

Streamwise mean velocity profiles with porous-wall scaling.

Figure 16.

Zero-plane displacement versus the roughness height.

Table 6.

Characteristic parameters of the streamwise mean velocity: Kármán constant, zero-plane displacement and roughness height.

To compare those values with the previous experimental data of Manes et al. (Reference Manes, Pokrajac, McEwan and Nikora2009) and Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) and the DNS data of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Kuwata & Suga (Reference Kuwata and Suga2016b ), figure 16 plots $d^{p+}$ versus $h_{r}^{p+}$ . From figure 16, it is understood that, although the presently focused porous media have strong anisotropy and the structures are totally different from those considered in the experimental and DNS studies, there is a clear correlation between $d^{p+}$ and $h_{r}^{p+}$ . This result supports the discussions by Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) and Manes et al. (Reference Manes, Poggi and Ridolfi2011) that the increase of $d^{p+}$ and $h_{r}^{p+}$ can both be attributed to relaxation of wall blocking. It was reported by Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) and Manes et al. (Reference Manes, Poggi and Ridolfi2011) that the trend of $d^{p+}$ and $h_{r}^{p+}$ could be well characterized by the permeability Reynolds number $Re_{K}$ . However, since the streamwise permeability Reynolds number $Re_{Kx}$ in table 5 becomes zero in cases YZ and YZ-HR despite the fact that $d^{p+}\neq 0$ and $h_{r}^{p+}\neq 0$ , the streamwise permeability Reynolds number cannot be used as $Re_{K}$ straightforwardly. The wall-normal and spanwise permeabilities also need considering to represent the relaxation of the wall-blocking effect.

Figure 17 plots $\unicode[STIX]{x1D705}$ versus $d/\unicode[STIX]{x1D6FF}_{p}$ . Although Kármán constants in the range of smaller $d/\unicode[STIX]{x1D6FF}_{p}$ cases ( $d/\unicode[STIX]{x1D6FF}_{p}<0.025$ ) by Manes et al. (Reference Manes, Pokrajac, McEwan and Nikora2009) exhibit lower values than those reported for smooth wall and lower-permeability walls (Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006; Suga et al. Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010), the consistent trend can be confirmed among the data of present DNS and Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006), Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) and Kuwata & Suga (Reference Kuwata and Suga2016b ). In the range of $d/\unicode[STIX]{x1D6FF}_{p}<0.15$ , the Kármán constant proportionally decreases with increasing $d/\unicode[STIX]{x1D6FF}_{p}$ whereas the decreasing rate becomes smaller in the range of $d/\unicode[STIX]{x1D6FF}_{p}>0.15$ . The clear correlations between $\unicode[STIX]{x1D705}$ and $d/\unicode[STIX]{x1D6FF}_{p}$ observed in figure 17 support the discussions of Manes et al. (Reference Manes, Pokrajac, McEwan and Nikora2009) that the Kármán constant can be scaled by the ratio of the boundary-layer thickness and the zero-plane displacement, meaning the ratio of the inner and outer length scales. This was also supported by the DNS of impermeable rough-wall turbulence studied by Leonardi & Castro (Reference Leonardi and Castro2010).

Figure 17.

Kármán constant distribution versus the ratio of the zero-plane displacement and the boundary-layer thickness of the porous side.

Figure 18 compares the plane-averaged Reynolds stresses normalized by $u_{\unicode[STIX]{x1D70F}}^{t}$ at different Reynolds numbers of $Re_{\unicode[STIX]{x1D70F}}=111$ and $230$ . As the Reynolds number increases, the plane-averaged Reynolds stresses of $R_{22}^{t+}$ , $R_{33}^{t+}$ and $R_{12}^{t+}$ increase substantially over the porous walls whilst $R_{11}^{t+}$ does not increase. It is found that the enhancement of $R_{12}$ due to the increase of $Re_{\unicode[STIX]{x1D70F}}$ in case XY is more remarkable than that in case YZ. Accordingly, as $Re_{\unicode[STIX]{x1D70F}}$ increases from $111$ to $230$ , the porous-wall shear stress in table 5 increases 5 % in case YZ and increases 21 % in case XY. This confirms that the Reynolds-number dependence is more remarkable in case XY than in case YZ. Inside the porous wall of $y/H<0$ , the plane-averaged Reynolds normal stresses are increased by the rise of $Re_{\unicode[STIX]{x1D70F}}$ ; however, increase of the plane-averaged Reynolds shear stress is marginal. It rapidly converges to zero inside the porous wall even in the case of $Re_{\unicode[STIX]{x1D70F}}=230$ .

Figure 18.

Plane-averaged Reynolds stresses at $Re_{\unicode[STIX]{x1D70F}}=111$ and $Re_{\unicode[STIX]{x1D70F}}=230$ normalized by $u_{\unicode[STIX]{x1D70F}}^{t}$ : (a) streamwise component, (b) wall-normal component, (c) spanwise component, and (d) off-diagonal component.

To investigate the Reynolds-number dependence of the large-scale streamwise perturbation, figure 19 compares the streamwise and spanwise two-point correlations of streamwise velocity fluctuations. As shown in figure 19(a), the streamwise correlation of $R_{uu,x}$ in case YZ-HR rapidly converges to zero whereas that in case XY-HR does not converge to zero and it shows a wavy profile. This indicates that, even for higher-Reynolds-number flows of $Re_{\unicode[STIX]{x1D70F}}=230$ , the streamwise perturbation does not develop over the spanwise permeable wall but develops over the streamwise permeable wall. The location of the local minimum shifts from $\unicode[STIX]{x1D6E5}_{x}^{p+}=300$ in case XY to $\unicode[STIX]{x1D6E5}_{x}^{p+}=700$ corresponding to the wavelength of $\unicode[STIX]{x1D706}_{x}^{p+}=1400$ . The magnitude of the local minimum value in case XY-HR at $\unicode[STIX]{x1D6E5}_{x}^{p+}=700$ is slightly larger than that at $\unicode[STIX]{x1D6E5}_{x}^{p+}=300$ in case XY. Moreover, it is also clear from figure 19(b) that, since the convergence of $R_{uu,z}$ in case XY-HR becomes slower compared with that in case XY, the stronger spanwise coherence is present across the full box. These results mean that, over the streamwise permeable wall, as the friction Reynolds number increases from $111$ to $230$ , the wavelength of the streamwise perturbation increases from $\unicode[STIX]{x1D706}_{x}^{p+}=600$ to $\unicode[STIX]{x1D706}_{x}^{p+}=1400$ and the perturbation becomes more organized.

Figure 19.

Two-point correlations of the streamwise velocity fluctuations at $Re_{\unicode[STIX]{x1D70F}}=111$ and $Re_{\unicode[STIX]{x1D70F}}=230$ : (a) streamwise correlation and (b) spanwise correlation.

This kind of large-scale coherent eddy motion is also observed near vegetated canopy flows. Raupach et al. (Reference Raupach, Finnigan and Brunei1996) and Finnigan (Reference Finnigan2000) summarized that the wavelength of this large-scale eddy motion $\unicode[STIX]{x1D706}_{x}$ was proportional to the vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ as $\unicode[STIX]{x1D706}_{x}=C_{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ . Here, the constant value of $C_{\unicode[STIX]{x1D706}}=4.05$ was the best fitted value in a wide range of canopy types. Since this value was in the range of the spacing of KH-type coherent eddies in the fully developed mixing layer, $3.5\leqq \unicode[STIX]{x1D706}_{x}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}\leqq 5$ (Dimotakis & Brown Reference Dimotakis and Brown1976; Rogers & Moser Reference Rogers and Moser1994), it was evident that turbulent flows over a canopy behaved as a plane mixing layer rather than a boundary layer. To discuss the length scale of the streamwise perturbation over porous media, table 7 summarizes the wavelength of the perturbation in cases XY and XY-HR and other data in the literature of flows over porous walls (Jimenez et al. Reference Jimenez, Uhlmann, Pinelli and Kawahara2001; Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006; Kuwata & Suga Reference Kuwata and Suga2016b ). In terms of porous-wall turbulence, it is clear that no meaningful tendency can be observed by the normalization of the inner scale whereas the wavelength normalized by the outer layer scale such as the boundary-layer thickness $\unicode[STIX]{x1D706}_{x}/\unicode[STIX]{x1D6FF}_{p}$ coincides with $\unicode[STIX]{x1D706}_{x}/\unicode[STIX]{x1D6FF}_{p}\approx 3.4$ , despite the fact that the flow conditions and the porous-medium characteristics (such as porosity, permeability and Forchheimer tensors) focused in those studies are entirely different. Supposing that the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{p}$ represents the vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ over the porous walls, the constant value of the relation $\unicode[STIX]{x1D706}_{x}=C_{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ becomes $C_{\unicode[STIX]{x1D706}}=3.4$ , which is close to the value obtained in canopy flows ( $C_{\unicode[STIX]{x1D706}}=4.05$ ) and the fully developed mixing layer ( $3.5\leqq C_{\unicode[STIX]{x1D706}}\leqq 5$ ). This result implies that the mechanism of the generation of the streamwise perturbation is the same as that in the fully developed mixing layer. Moreover, it is interesting to stress that the wavelength of the streamwise perturbation does not depend on the porous-medium characteristics.

Table 7.

Streamwise wavelength of the large-scale streamwise perturbation.

To examine the coherence of the streamwise perturbation, the magnitude of the local minimum value of R $_{uu,x}$ is examined. It is expected that, as the streamwise wall permeability decreases (the viscous drag increases), large-scale turbulence motion penetrated into the porous walls tend to be blocked and dissipated. Consequently, the streamwise perturbation loses coherence. However, even though the streamwise permeability Reynolds number is almost the same in case XY-HR ( $Re_{Kx}=10$ ) and case E95 ( $Re_{Kx}=9.35$ ) of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006), the magnitude of the local minimum value of $R_{uu,x}$ in case XY-HR shows $0.21$ whilst that in case E95 is less than 0.1. This discrepancy may be attributed to inertial effects. When the inertial effect, which is expressed as the Forchheimer drag term (second term on the right-hand side of (3.1)), becomes more significant, it is expected that turbulence penetration tends to be blocked and disturbance inside porous media tends to be rapidly decayed. Consequently, the large-scale perturbation loses coherence due to the inertial effects. Since the pore in case XY goes through along the streamwise direction, the inertial effect (streamwise Forchheimer drag) is less significant than for the porous medium of E95 in Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006). Indeed, the coherence is stronger over the porous media in case XY than that in case E95 of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006). Hence, it is further expected that, over a porous wall with lower permeability and with higher Forchheimer number, such as geological rocks, the streamwise perturbation loses coherence and finally disappears.