2 Experimental method

Figure 1 illustrates the experimental set-up and the coordinate system in the test section of the present study. As shown in figure 1(a), tap water is pumped up from a water tank equipped with a cooler (TRL-117GS2, THOMAS KAGAKU) to maintain a constant water temperature of 285 $\pm 1$ K, and its total flow rate is measured by a digital flow meter (FD-MH200A/500A, KEYENCE). The water flow is conditioned by a honeycomb-bundled nozzle, where its temperature is recorded by a digital thermometer (FD-T1, KEYENCE). Then the flow is fully developed in a driver section whose length is 3.0 m and enters the test section of length 1.0 m. Both the sections consist of solid smooth acrylic walls with a porous bottom layer. The sectional width $W$ and total height are, respectively, 0.305 m and 0.093 m, two-thirds of which are filled by a porous layer. For the porous layer, as in the previous reports (Suga et al. Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010, Reference Suga, Mori and Kaneda2011; Suga Reference Suga2016), three kinds of foamed ceramics (shown in figure 2) are used. They are named No. 20, No. 13 and No. 06, and correspond to low-, medium- and high-permeability porous media, respectively. All of them have an isotopic open-cell foam structure with the same porosity of $\unicode[STIX]{x1D711}=0.8$ , but the mean pore diameter, the permeability and the Forchheimer coefficient vary as $D_{p}=1.7{-}3.8~\text{mm}$ and $K=0.020{-}0.087~\text{mm}^{2}$ and $c_{F}=0.17{-}0.095$ , respectively, as listed in table 1. Here, in the measurement of Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) to obtain the permeability and the Forchheimer coefficient, the following Darcy–Forchheimer equation was applied between the pressure drop $\unicode[STIX]{x0394}P/\unicode[STIX]{x0394}x$ and the Darcy velocity $U_{d}$ :

(2.1) $$\begin{eqnarray}\displaystyle -\frac{\unicode[STIX]{x0394}P}{\unicode[STIX]{x0394}x}=\frac{\unicode[STIX]{x1D707}U_{d}}{K}+\frac{c_{F}}{K^{1/2}}\unicode[STIX]{x1D70C}U_{d}^{2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ are the fluid viscosity and density, respectively. Since the height of the clear fluid region is set to $H=0.03~\text{m}$ , the aspect ratio of the cross-section of the clear fluid region is approximately 10. To compare the results with those of solid smooth-walled channel flows, a different set of ducts (of height 0.03 m) are also used. By controlling the output of the pump with the power converter, inlet flow rates are adjusted. The measured range of the bulk Reynolds number $Re_{b}$ ( $U_{b}H/\unicode[STIX]{x1D708}$ ) is 1500–15 900, where $\unicode[STIX]{x1D708}$ is determined by the measured water temperature. To obtain the bulk mean velocity $U_{b}$ in the clear fluid region, the wall shear stress and the other parameters related to the mean velocity profile, additional corresponding measurements of $x$ – $y$ planes at the symmetry of the test section are also performed by the same method as reported in the previous study (Suga et al. Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010).

Figure 1. Experimental set-up: (a) channel flow facility; (b) schematic view of the test section.

Figure 2. Surface photographs of the foamed ceramics: (a) No. 20: low permeability, (b) No. 13: medium permeability, (c) No. 06: high permeability.

Table 1. Characteristics of the foamed ceramics; $\unicode[STIX]{x1D711},D_{p},K$ and $c_{F}$ are porosity, mean pore diameter, permeability and Forchheimer coefficient, respectively.

The employed planar PIV system consists of a double-pulse Nd-YAG laser (Dual Power 200-15, Litron) with $200~\text{mJ}~\text{pulse}^{-1}$ at a wavelength of 532 nm. The laser beam is formed into a sheet approximately 1.0 mm in thickness by several cylindrical lenses and illuminates $x$ – $z$ planes as shown in figure 1(b). Two CCD cameras (Flowsense 4M Mk2, DANTEC DYNAMICS) operating at 30 f.p.s with 85 mm $f/1.8$ lenses (AF Nikkor, Nikon) and extension tubes (Kenko Tokina 12 mm) are arranged in tandem. The recorded frame of each CCD camera covers a zone of $80(x)\times 80(z)~\text{mm}^{2}$ located in the middle of the test section with $2048\times 2048$ pixels, for which 20 % of the area overlaps with another camera’s frame.

For the tracer particles, polymer fluorescent particles containing Rhodamine B, whose mean diameter and specific gravity are respectively $10~\unicode[STIX]{x03BC}\text{m}$ and 1.50, are used. (By Stokes’ law, the settling velocity of the particles can be estimated as $2.2\times 10^{-5}~\text{m}~\text{s}^{-1}$ , which is negligibly small compared with the bulk mean velocity of $6.2\times 10^{-2}$ for the lowest Reynolds number case at $Re_{b}=1500$ .) Correspondingly, a long-pass filter with cutoff wavelength 560 nm is located in front of each camera lens. The seeding density is adjusted to obtain 10–15 particle-image pairs as in Adrian, Meinhart & Tomkins (Reference Adrian, Meinhart and Tomkins2000) in each interrogation window of size $32\times 32$ pixels. Thus the measurement sampling volume is $1.25(x)\times 1.0(y)\times 1.25(z)~\text{mm}^{3}$ . The image sampling rate is 4 Hz. The timing of the laser pulse interval is adjusted depending on the average particle displacement during the interval. In the present experiments, the average particle displacement is set to be approximately 25 % of the length (3–6 pixels) of the interrogation cell. To obtain the statistical data, in each zone, 4000 image pairs are processed in this study. The recorded data are processed by Dynamics Studio 3.1 (DANTEC DYNAMICS) software with the fast Fourier transform cross-correlation technique (Willert & Gharib Reference Willert and Gharib1991). Each image is processed to produce $127\times 127$ vectors from interrogation windows with a 50 % overlap in each direction. When the ratio of the first and the second correlation peaks in an interrogation window is smaller than 1.3, it is removed from the process as an error vector. Furthermore, the moving-average validation (Host-Madsen & McCluskey Reference Host-Madsen and McCluskey1994) is applied with an acceptance factor of 0.1. In the present study, the removed error vectors amount to less than 3 % of the total data processed.

By using image processing software, the average number of pixels for a particle image captured by the CCD camera in the present study is counted to be approximately 3–4 pixels. According to the discussions of Prasad et al. (Reference Prasad, Adrian, Landreth and Offutt1992), when particle images are well resolved so that the ratio of particle-image diameter to the size of a CCD pixel on the photograph is greater than 3–4, the uncertainty of the measurements is approximately $1/10$ – $1/20$ of the particle-image diameter. This indicates that the particle images are well resolved in the present experiments, and the uncertainty in the measured displacement can be expected to be approximately less than $1/10$ of the diameter of the particle image. Normalizing this uncertainty by the mean displacement length of particles indicates that the estimated error of the magnitude of the instantaneous velocity is less than 4 % of the maximum velocity near the channel centreline.

3 Results and discussions

Figure 3 shows examples indicating the two-dimensionality of the flow field and the reproducibility of the data between the $x$ – $y$ and $x$ – $z$ plane measurements. From figure 3(a), two-dimensionality can be reasonably attained in $z/W=0.1{-}0.9$ and the measured flow fields of $x$ – $z$ planes for $z/W=0.37{-}0.63$ are well assured to be two-dimensional. It is seen that the measured velocities for the $x$ – $z$ planes agree well with the data of the $x$ – $y$ plane measurement in figure 3(b). In the plots of the $x$ – $z$ planes, there is a small gap around $y/H=0.3$ . It suggests that the setting error for the $x$ – $z$ plane measurements was of such an order.

The $x$ – $z$ plane measurements are performed for solid smooth wall, No. 20, No. 13 and No. 06 porous wall cases, changing $Re_{b}$ from 1500 to 15 900 to achieve the conditions of $Re_{K}$ approximately at 1.0, 2.0, 3.0, 6.0 and 10.0. There are altogether 15 cases, as shown in table 2. For each case, several measurement planes in the wall normal direction are measured. In table 2, measured sections 5, 5 $^{\ast }$ , 14, 14 $^{\ast }$ , 15 and 18 correspond to sections of (1.0, 2.0, 3.0, 4.0, 5.0 mm), (0.5, 1.0, 1.5, 2.0, 2.5 mm), $(1.0,2.0,3.0,\ldots ,14.0,15.0~\text{mm})$ , $(1.0,2.0,3.0,\ldots ,13.0,14.0~\text{mm})$ , $(0.5,1.0,1.5,2.0,2.5,3.0,4.0,\ldots ,10.0,11.0~\text{mm})$ and $(0.5,1.0,1.5,2.0,2.5,3.0,4.0,\ldots ,14.0,15.0~\text{mm})$ from the bottom surface, respectively.

Figure 3. Two-dimensionality and reproducibility in the flow field measurements: (a) streamwise mean velocity distribution in the spanwise direction at $y/H=0.5$ and $Re_{b}=6200$ for No. 06 porous wall, (b) comparison of the streamwise mean velocity distribution between $x$ – $y$ and $x$ – $z$ plane measurements at $Re_{b}=3100$ for No. 13 porous wall.

Table 2. Experimental conditions and measured parameters of the mean velocity fields; $u_{\unicode[STIX]{x1D70F}}^{t}$ and $u_{\unicode[STIX]{x1D70F}}^{p}$ are the friction velocities on the top solid and bottom porous walls; $\unicode[STIX]{x1D705},d$ and $h$ are the von Kármán coefficient, the zero-plane displacement and the roughness length scale, respectively; $(\cdot )^{p+}$ corresponds to a value normalized by using $u_{\unicode[STIX]{x1D70F}}^{p}$ .

3.1 Snapshots of streaks

Figure 4 shows snapshots of the instantaneous streamwise velocity fluctuation $u^{\prime }$ indicating low- and high-speed streaks and corresponding vorticity maps over porous walls at $y^{p+}\simeq 20$ , where $(\cdot )^{p+}$ is a value normalized by the friction velocity $u_{\unicode[STIX]{x1D70F}}^{p}$ on the porous wall which is estimated by extrapolating the Reynolds shear stress distribution obtained by the corresponding $x$ – $y$ plane measurement. Since in the previous study (Suga et al. Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) all the data for No. 20, No. 13 and No. 06 porous media were well correlated with $Re_{K}$ , figure 4 arranges the snapshots in the order of $Re_{K}$ irrespective of the porous media. (Indeed, the patterns of different porous medium cases are very similar to each other at similar $Re_{K}$ conditions.) Note that $y=0$ corresponds to the porous wall surface and all area sizes of the sub-figures are set to $L_{x}^{p+}\times L_{z}^{p+}=990\times 550$ to compare the scales of structural elements. The existence of the streaky structure is obvious in figure 4(a–d) at $Re_{K}=0.93{-}1.79$ although they are twisted. It is obvious that the patterns in the vorticity $\unicode[STIX]{x1D714}_{y}^{p+}$ maps are consistent with those in the maps of the high- and low-speed streaks. In figure 4(e,f) at $Re_{K}=2.94$ , the twisting degree significantly increases and the streaks tend to be torn into fragments. Such a tendency is obvious in the vorticity map of figure 4(f), and longitudinally elongated streaks can no longer be observed. Accordingly, it is confirmed that the quasi-streak-like structure is maintained near a wall at a lower $Re_{K}$ , while it tends to be destroyed as $Re_{K}$ increases.

Figure 4. Snapshots of streamwise velocity fluctuation indicating low- and high-speed streaks and instantaneous wall normal vorticity $\unicode[STIX]{x1D714}_{y}^{p+}=(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z-\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}x)\unicode[STIX]{x1D708}/(u_{\unicode[STIX]{x1D70F}}^{p+})^{2}$ in the window of $L_{x}^{p+}\times L_{z}^{p+}=990\times 550$ at $y^{p+}\simeq 20$ : (a,c,e,g,i) are fluctuation velocity contours and (b,d,f,h,j) are vorticity contours. (a,b) No. 20 porous wall at $Re_{K}=0.93$ and $Re_{b}=2500$ , (c,d) No. 13 porous wall at $Re_{K}=1.79$ and $Re_{b}=3100$ , (e,f) No. 06 porous wall at $Re_{K}=2.94$ and $Re_{b}=2700$ , (g,h) No. 20 porous wall at $Re_{K}=6.23$ and $Re_{b}=14800$ , (i,j) No. 06 porous wall at $Re_{K}=11.12$ and $Re_{b}=9900$ . The flow direction is left to right.

Figure 5. Snapshots of low- and high-speed fluid lumps in the measurement areas of $L_{x}/H\times L_{z}/H=4.8\times 2.7$ at $y^{p+}\simeq 20$ : (a) No. 20 porous wall at $Re_{K}=6.23$ and $Re_{b}=14800$ (corresponding to figure 4 g), (b) No. 06 porous wall at $Re_{K}=11.12$ and $Re_{b}=9900$ (corresponding to figure 4 i).

Figure 6. Two-point correlation of the streamwise velocity versus spanwise spacing: (a) solid wall at $Re_{b}=5600$ , (b) porous wall No. 20 at $Re_{K}=0.93$ and $Re_{b}=2500$ , (c) porous wall No. 13 at $Re_{K}=1.79$ and $Re_{b}=3100$ , (d) porous wall No. 06 at $Re_{K}=2.94$ and $Re_{b}=2700$ , (e) porous wall No. 20 at $Re_{K}=6.23$ and $Re_{b}=14\,800$ , (f) porous wall No. 06 at $Re_{K}=11.12$ and $Re_{b}=9900$ .

At the higher- $Re_{K}$ cases of $Re_{K}=6.23$ and 11.12, in figure 4(g–j), although low- and high-speed fluid lumps exist, obvious streak-like vorticity patterns cannot be observed in the windows of $L_{x}^{p+}\times L_{z}^{p+}=990\times 550$ . To see those fluid lumps on a wider scale, figure 5 shows patterns of low- and high-speed fluid lumps for the cases of figure 4(g,i). The window size of these sub-figures is $L_{x}/H\times L_{z}/H=4.8\times 2.7$ , which corresponds to $L_{x}^{p+}\times L_{z}^{p+}=6670\times 3670$ and $5660\times 3100$ for figures 5(a) and 5(b), respectively. Obviously, the pattens shown in figure 5 are very different from the longitudinally elongated streak structure, and large-scale spanwise patterns seem to be developing as in figure 5(b). Indeed, in the streamwise direction, black and white block areas can be seen in turn. This is considered to be footprints of the spanwise transverse rolls induced by the Kelvin–Helmholtz (K–H) instability as reported numerically by Jiménez et al. (Reference Jiménez, Uhlmann, Pinelli and Kawahara2001) and Kuwata & Suga (Reference Kuwata and Suga2016a ).

Figure 7. Distribution of spanwise spacing of streaks in the wall normal direction: (a) solid wall, (b) porous wall No. 20, (c) porous wall No. 13, (d) porous wall No. 06; hatching zone is the zone where the plots start to deviate largely from the broken line; broken line indicates the fitting line of the solid-wall cases.

3.2 Scales of spanwise structure

To discuss scales of the coherent structure, the normalized two-point correlation function:

(3.1) $$\begin{eqnarray}\displaystyle \widehat{R_{ij}}(\unicode[STIX]{x0394}z)=R_{ij}(\unicode[STIX]{x0394}z)/R_{ij}(0)=\overline{u_{i}^{\prime }(z)u_{j}^{\prime }(z+\unicode[STIX]{x0394}z)}/\overline{u_{i}^{\prime }(z)u_{j}^{\prime }(z)}, & & \displaystyle\end{eqnarray}$$

in the spanwise direction is discussed. Figure 6 shows $\widehat{R_{11}}(\unicode[STIX]{x0394}z)$ distributions which correspond to the solid-wall case: figure 6(a), and the five $Re_{K}$ cases shown in figure 4. For all the cases local minima can be seen, while their depths (magnitudes) and positions change depending on the wall normal distance. This trend is consistent with the results of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006). Moreover, generally the depth tends to decrease, and the oscillations observed after the local minima vanish as $Re_{K}$ increases. Since the local minimum corresponds to the average spanwise distance between low-speed and neighbouring high-speed regions, the decrease of its magnitude and vanishing oscillations indicate that the streak structure tends to vanish as $Re_{K}$ increases.

From the locations of the local minima, the distributions of the spanwise spacing of streaks $\unicode[STIX]{x1D706}_{z}^{p+}$ (or $\unicode[STIX]{x1D706}_{z}^{+}$ for solid walls) against the wall normal distance (plus the zero-plane displacement) are obtained as in figure 7. Figure 7(a) compares the solid smooth wall results and published data in the literature (Smith & Metzler Reference Smith and Metzler1983; Iritani, Kasagi & Hirata Reference Iritani, Kasagi and Hirata1985; Kim, Moin & Moser Reference Kim, Moin and Moser1987; Tomkins & Adrian Reference Tomkins and Adrian2003). It is seen that all the data align well on a single curve, indicating that the streaks over the solid smooth walls are well organized and their spacing is approximately 100 wall units under $y^{+}=20$ , while it gradually increases as the wall normal distance increases.

In figure 7(b–d), it is found that the points of $\unicode[STIX]{x1D706}_{z}^{p+}$ collapse around the curve under $(y+d)^{p+}\simeq 100$ reasonably well when the zero-plane displacement $d$ is combined with the wall normal distance. The zero-plane displacement is a parameter in the log-law form for the mean velocity over porous walls. (See appendix A for comments on the parameters of log-law velocities listed in table 2.) Indeed, in figure 7(b–d) plots for No. 20, No. 13 and No. 06 cases are in reasonable agreement with the line representing the correlation of the solid-wall case under $(y+d)^{p+}\simeq 70$ , 90 and 80, respectively. It is considered that quasi-coherent streaks over a porous wall generated by the wall shear are pushed into the porous wall by downward motions of the K–H waves. Since the zero-plane displacement is considered to be a length scale associated with the penetration, the agreement between the plots and lines in figure 7(b–d) suggests that the structure retains the characteristics of streaks even if their bottom parts penetrate into the porous walls. Hence, $\unicode[STIX]{x1D706}_{z}^{p+}$ shows the correlation with $(y+d)^{p+}$ . However, at positions of certain distances from the wall surfaces, in the regions which are marked by hatching zones, the plotted values start to deviate largely from the correlation line. Although the ranges of the hatching zones are different in the porous layer cases, their locations are around $(y+d)^{p+}\simeq 100$ . This region where the trend changes is considered to be a structural transition region. It is considered that the quasi-streaky structure is smeared or may no longer exist over such regions. Indeed, the snapshots shown in figure 5 are the cases inside and over the hatching zones.

Suga et al. (Reference Suga, Mori and Kaneda2011) observed another transitional change in the distributions of turbulence fluctuation velocities over porous walls. The shape of the joint probability density function $p(u^{\prime },v^{\prime })$ of the fluctuation velocity at $y^{p+}\simeq 15$ changed in the condition under $Re_{K}\simeq 3$ . (The turbulence anisotropy tended to rapidly weaken as $Re_{K}$ increased up to $Re_{K}\simeq 3$ .) The location at $y^{p+}\simeq 15$ under $Re_{K}\simeq 3$ corresponds to the region under $(y+d)^{p+}\simeq 47$ if the correlation $d^{p+}\simeq 15.1Re_{K}-13.5$ , which was seen in Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010), is valid. (In the present cases of $Re_{K}=2.90{-}3.00$ , $y^{p+}=15$ corresponds to $(y+d)^{p+}=36-60$ .) This indicates that the transitional change of the spanwise turbulence scale occurs after the transitional change of the near-wall turbulence anisotropy.

To see the vortex length scales, figure 8 shows the distribution of the spanwise integral length $\unicode[STIX]{x1D6EC}_{11}$ , which is defined as

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}_{ij}=\frac{1}{2\overline{u_{i}^{\prime }(z)u_{j}^{\prime }(z)}}\int _{-\infty }^{\infty }R_{ij}(r)\,\text{d}r. & & \displaystyle\end{eqnarray}$$

It is seen that a cluster involving the solid-wall case extends up to $(y+d)^{p+}\simeq 300$ and the length scale size ranges as $\unicode[STIX]{x1D6EC}_{11}^{p+}=5{-}25$ . However, it is indicated that at $(y+d)^{p+}>100$ , flow motions with much larger scales coexist and start to be dominant. This suggests that fragments of the torn longitudinal vortex tubes are entrained by the large-scale fluid lumps moving with travelling transverse waves.

Figure 8. Integral length variations in the wall normal direction.

4 Concluding summary

At a lower $Re_{K}$ , low- and high-speed streaks, which are similar to those of solid-wall turbulence, are observed near the permeable walls. In case of a large $Re_{K}$ , although some large-scale patterns are observed in the velocity field, they are confirmed to be very different from those of the solid-wall turbulence structure. It is found that the spanwise spacing of the streaks $\unicode[STIX]{x1D706}_{z}$ and the spanwise integral length $\unicode[STIX]{x1D6EC}_{11}$ can be reasonably correlated with the wall normal distance including the zero-plane displacement $d$ of the log-law mean velocity profile. Since there is a correlation between $d^{p+}$ and $Re_{K}$ (Suga et al. Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010), these phenomena have a correlation with $Re_{K}$ via $d^{p+}$ . It is considered that quasi-coherent streaks over permeable walls generated by the wall shear are pushed into the porous walls by the downward motions of the K–H waves. Since the zero-plane displacement $d$ is considered to be a length scale associated with the penetration, the distribution profiles of those scales ( $\unicode[STIX]{x1D706}_{z}$ and $\unicode[STIX]{x1D6EC}_{11}$ ) indicate that the structure keeps the characteristics of streaks even if their bottom parts penetrate into the porous walls. Accordingly, the surviving elements of the streaks exist and their scales maintain correlations with the wall normal distance under the transitional range starting from $(y+d)^{p+}\simeq 100$ , while those elements tend to be disturbed by the K–H instability as the wall normal distance increases. From the region at $(y+d)^{p+}>100$ , flow motions with much larger spanwise length scales start to be dominant. Such flow motions are considered to be transverse roll cells which are generated by the K–H instability and destroy the longitudinal vortex trails. It is considered that fragments of the torn longitudinal vortex tubes surf over the transverse travelling waves. When $Re_{K}$ is large enough, $d^{p+}$ becomes close to or even surpasses 100, resulting in the loss of the quasi-streak structure near the surfaces.