3.1 Mean velocity and secondary flows

Figure 3 shows contour maps of the time- and streamwise-averaged streamwise velocity:

(3.1)$$\begin{eqnarray}[\bar{u}]_{x}^{f}=\frac{\displaystyle \int _{0}^{4(D+d)}\bar{u}\,\text{d}x^{f}}{\displaystyle \int _{0}^{4(D+d)}\text{d}x^{f}},\end{eqnarray}$$

with the cross-sectional velocity vectors of case B at $Re=3400$ and 7700. (The vectors at $y/H=0$ are produced by interpolating the values at planes $y4$ and $y5$.) Here, the overbar denotes time averaging and $[\cdot ]_{x}^{f}$ denotes fluid-phase averaging in the $x$-direction. The contour maps are painted with the $x{-}z$ plane measurement data while the vectors are produced using the $x{-}y$ and $x{-}z$ plane measurement data. Irrespective of the Reynolds number, it is seen that the cross-sectional secondary flow pattern is very different from the well-known pattern in square duct flow. A large single recirculation is seen near the upper corner and relatively weak recirculation can be seen near the porous interface, while in a turbulent square duct flow (without a porous layer), the secondary flow consists of four pairs of counter-rotating vortices located at the duct corners. Accordingly, upward flows across the porous interface are observed in the middle region, though their magnitudes are rather small. Owing to the cross-sectional secondary flows, $[\bar{u}]_{x}^{f}$ shows a skewed distribution near the top wall at $y/H=1.0$. However, its distribution becomes much flatter near the porous interface at $y/H=0.0$.

These flow features verify the numerical simulation of Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) using the VANS equation for the porous region. Although Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) reported that the maximum magnitude of the secondary currents is approximately 8 % of the bulk velocity, which was four times as large as that of a regular duct of Vinuesa et al. (Reference Vinuesa, Noorani, Lozano-Durán, Khoury, Schlatter, Fischer and Nagib2014), the present results indicate approximately 6 % of the inlet velocity $U_{0}$. (The flow rates inside the porous layers are estimated as less than 5 % of the total flow rate by the mean velocity distribution discussed later. Hence, the difference between $U_{0}$ and the bulk velocity in the duct is in such an order.) It is considered that the magnitude difference of the secondary flows between the DNS and this study comes from the structural effects. Supporting this, our recent thermal field DNS (Kuwata, Tsuda & Suga Reference Kuwata, Tsuda and Suga2019) for the same flow geometry predicted the same order of the magnitude.

Figure 3.

Cross-sectional mean velocity and secondary flows of case B: (a) at $Re=3400$ and (b) at $Re=7700$.

For the kinetic energy of the secondary flow $K_{c}=(\bar{u}^{2}+\bar{v}^{2})/2$, Vinuesa, Schlatter & Nagib (Reference Vinuesa, Schlatter and Nagib2018) discussed spanwise variations of its wall-normal averaged values: $[K_{c}]_{y}=(1/H)\int _{0}^{H}K_{c}\,\text{d}y$, in several turbulent rectangular ducts. They reported that, in turbulent square duct flows, the minimum $[K_{c}]_{y}$ was located near the symmetry plane and $[K_{c}]_{y}$ increased towards the sidewall, having a couple of local maxima. However, figure 4 indicates that, for both the Reynolds numbers, $[K_{c}]_{y}$ tends to be larger towards the symmetry plane at $z/H=0$ by the strong downward motions. Owing to the centre of the large recirculation, which looks to be located near $z/H=0.2$ (figure 3), $[K_{c}]_{y}$ has a local minimum there. Towards the sidewall at $z/H=0.5$, $[K_{c}]_{y}$ tends to rise again due to the energetic vertical motion along the sidewall. For $Re=7700$, although a local kink is seen near $z/H=0.065$–0.13, its reason is unclear. Since the number of secondary flow data is $10\,(y)\times 7\,(z)$, it is not fine enough to capture local trends in detail, unfortunately. Since the magnitude of the present secondary flow velocity is significantly enhanced by the porous wall (three times larger than that in a square duct), the level of $[K_{c}]_{y}$ is approximately one order higher than those presented in Vinuesa et al. (Reference Vinuesa, Schlatter and Nagib2018). The magnitude level seems to be increased by the Reynolds number. Overall, in the porous duct flows, a higher concentration of energy appears near the symmetry plane as well as near the sidewall for both the Reynolds numbers.

Figure 4.

Spanwise variation of the kinetic energy of the secondary flow averaged over the wall-normal direction in case B.

To see the effects of porous structures on the general flow fields, figure 5 compares the time- and streamwise-averaged vertical and streamwise velocity contour maps which are reconstructed using the $x{-}y$ plane measurement data. Although the structures of cases A and B are very different, irrespective of the Reynolds numbers, the mean velocity distributions in the clear duct region do not look very different from each other. For both the structures, strong downward velocity regions appear at $0.4<y/H<0.9$ near the symmetry planes of the clear ducts, while strong upward velocity regions appear towards the top corners. Those distributions are consistent with the secondary currents (figure 3) and the trend of $[K_{c}]_{y}$ (figure 4). It is then suggested that the present structural difference does not significantly change the secondary flow patterns in the clear duct regions. As for the flows under the porous interfaces, case B shows stronger upward flows in the central regions than case A due to the structural difference.

Figure 5.

Cross-sectional mean velocity contour maps: (a) case A at $Re=3300$, (b) case A at $Re=7400$, (c) case B at $Re=3400$ and (d) case B at $Re=7700$. The $z/H$ location for $[\bar{v}]_{x}^{f}$ is reversed for presentation.

3.2 Sectional flow characteristics

To show the examples of the detailed flow distributions inside the porous region, figures 6 and 7 show time-averaged streamwise velocity $\bar{u}$, Reynolds shear stress $-\overline{u^{\prime }v^{\prime }}$ and turbulent intensities (root-mean-square (r.m.s.) velocities) $u^{\prime },~v^{\prime }$ in the symmetry planes of $z/H=0$ at $Re=7400$ and 7700 for cases A and B, respectively. The position of $x/D=0$ corresponds to the symmetry plane of the transverse rods shown in figure 2(d). It is clear that, depending on the porous structure, the profiles of the turbulence quantities change significantly. Owing to the structures, the mean velocity shows apparent sinusoidal distribution profiles under the porous surfaces as seen in figures 6(a) and 7(a). However, when we focus on the penetration depth of the Reynolds shear stress shown in figures 6(b) and 7(b), which is assumed to be the length needed for the quantity to reach the asymptotic value under the interface, it is approximately equivalent to the rod height $d$. Indeed, the Reynolds shear stress is damped and almost vanishes up to $y=-d$. As for the r.m.s. velocities, figure 7(c,d) indicates that the penetration lengths are much longer, suggesting that turbulent fine eddies go deeply into the porous media depending on the porous structure. The budget term analysis for the turbulent flow over a permeable porous layer by Kuwata & Suga (Reference Kuwata and Suga2016b) found that the greater turbulence penetration towards the porous layer was due to the increased redistribution and pressure diffusion processes intensified significantly by the pressure fluctuations. Hence, it is considered that the enhanced turbulent intensities under the porous layer are primarily due to the enhanced pressure fluctuations.

Figure 6.

Time-averaged turbulence quantities in the symmetry plane (plane $z1$ at $z/H=0$) in case A at $Re=7400$: (a) streamwise velocity, (b) Reynolds shear stress, (c) streamwise r.m.s. velocity and (d) wall-normal r.m.s. velocity. Red broken lines show positions at $y=0$ and $y=-d$.

Figure 7.

Time-averaged turbulence quantities in the symmetry plane (plane $z1$ at $z/H=0$) in case B at $Re=7700$: (a) streamwise velocity, (b) Reynolds shear stress, (c) streamwise r.m.s. velocity and (d) wall-normal r.m.s. velocity. Red broken lines show positions at $y=0$ and $y=-d$.

Figures 8 and 9 compare the sectional distributions of time- and streamwise-averaged quantities for cases A and B at $Re=$7400 and 7700, respectively. Among the seven planes, planes $z1$, $z4$ and $z6$ are plotted. For the streamwise mean velocity, it is clear that the distribution profiles are significantly skewed in the clear duct region of $y/H>0$ as seen in figures 8(a) and 9(a). In both cases, the profiles look narrower in the symmetry plane (plane $z1$) than in the other planes. This trend is considered to be from the secondary currents and was also seen in Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015). Owing to the sidewalls, the secondary flow motions towards the duct corners enhance the flow rate around the duct corners, leading to flatter velocity profiles near the sidewalls. Although the maximum velocities in cases A and B are $1.3U_{0}$, the locations of the maximum velocities are at $y/H=0.50$ and $y/H=0.52$, respectively, which are slightly different from $y/H=0.55$ predicted by Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015). Since the maximum values of the sinusoidal velocity profiles inside the porous layers are around $0.1U_{0}$, the flow rates inside the porous layers are estimated to be less than 5 % of the inlet flow rates. It is clear that the mean velocities in both cases are significantly damped just under the porous surfaces and this trend is consistent with that observed in figures 6(a) and 7(a).

Figure 8.

Distributions of streamwise-averaged turbulence quantities of case A at $Re=7400$: (a) streamwise velocity, (b) Reynolds shear stress, (c) streamwise r.m.s. velocity and (d) wall-normal r.m.s. velocity.

Figure 9.

Distributions of streamwise-averaged turbulence quantities of case B at $Re=7700$: (a) streamwise velocity, (b) Reynolds shear stress, (c) streamwise r.m.s. velocity and (d) wall-normal r.m.s. velocity.

Corresponding to the mean velocities, the streamwise-averaged Reynolds shear stresses of figures 8(b) and 9(b) show asymmetrical profiles in the clear duct region and are steeply damped just under the porous interfaces. As seen in figures 8(c,d) and 9(c,d), the streamwise-averaged r.m.s. velocity fluctuations show significantly different distribution profiles between cases A and B due to the structural difference. It is clear that, although the streamwise r.m.s. profiles drop steeply under the porous surfaces like the mean velocity and the shear stress profiles, the wall-normal r.m.s. profiles do not show such a trend in both cases.

3.3 Streamwise–spanwise plane-averaged flow characteristics

As discussed for figure 3, although the flows are not two-dimensional over the porous surfaces, to see the general characteristics of porous medium flows, $x$–$z$ plane averaging is applied by the trapezoidal rule using the $x$–$y$ plane measurement data of planes $z1$–$z5$ for $0\leqslant z/H\leqslant 0.26$. Here, the $x$–$z$ plane-averaged value is denoted as $[\cdot ]_{xz}$. By this procedure, the control area ($x$–$z$ plane) covers $4\times 1$ unit cells of the porous structure along the symmetry plane of the duct. From the time-averaged and $x$–$z$ plane-averaged (double-averaged) momentum equation (Whitaker Reference Whitaker1996), the total momentum flux across the pores on the porous surface can be written as

(3.2)$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{p}=\left(-\unicode[STIX]{x1D70C}[\overline{u^{\prime }v^{\prime }}]_{xz}^{f}-\unicode[STIX]{x1D70C}[\bar{u}]_{xz}^{f}[\bar{v}]_{xz}^{f}-\unicode[STIX]{x1D70C}[\tilde{\bar{u}}\,\tilde{\bar{v}}]_{xz}^{f}+\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}[\bar{u}]_{xz}^{f}}{\unicode[STIX]{x2202}y}\right)_{y=0},\end{eqnarray}$$

where $[\tilde{\bar{u}}\,\tilde{\bar{v}}]_{xz}^{f}$ is the dispersion stress in which the dispersion of $\bar{u}_{i}$ is defined as $\tilde{\bar{u}}_{i}=\bar{u}_{i}-[\bar{u}_{i}]_{xz}^{f}$. Hence, by using the measured quantities, the friction velocity $u_{\unicode[STIX]{x1D70F}}^{p}=\sqrt{\unicode[STIX]{x1D70F}_{p}/\unicode[STIX]{x1D70C}}$ on the porous surface can be obtained. Table 2 lists those friction velocities with the other parameters such as the permeability Reynolds numbers.

Table 2.

Experimental conditions and measured parameters of the mean velocity fields. Here $Re$, $Re_{K_{x}}$, $Re_{K_{y}}$ and $Re_{K}^{\ast \ast }$ are the Reynolds number based on the inlet velocity $U_{0}$, the permeability Reynolds numbers based on $\sqrt{\unicode[STIX]{x1D612}_{xx}}$ and $\sqrt{\unicode[STIX]{x1D612}_{yy}}$ and the pore-scale Reynolds number defined by (3.5); $u_{\unicode[STIX]{x1D70F}}^{p}$ is the friction velocity over the porous wall calculated with the streamwise–spanwise plane-averaged values by (3.2); the boundary layer thickness $\unicode[STIX]{x1D6FF}_{w}$ and the slip velocity $U_{w}$ are from the plane-averaged mean velocity; $\unicode[STIX]{x1D705}$, $d_{0}$ and $h$ are the von Kármán coefficient, the zero plane displacement and the roughness scale, respectively; and $(\cdot )^{p+}$ corresponds to a value normalized by using $u_{\unicode[STIX]{x1D70F}}^{p}$.

Figure 10 compares the plane-averaged streamwise mean velocity, Reynolds stress and r.m.s. velocities at $Re\simeq 7500$. For the velocity profiles in the clear flow region shown in figure 10(a), although there are slight discrepancies between the cases A and B, these two cases show nearly the same profiles. The shown DNS profile of Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) is for the symmetry plane and the Reynolds number is $Re_{b}=5000$. Also their porous structure is different from those of the present cases. Even with such differences, the present results are in good accord with the DNS data. The slip velocity $U_{w}$ at the porous surface and $\unicode[STIX]{x1D6FF}_{w}$, which is the location where the mean velocity has the maximum value, also do not significantly change in the two cases as listed in table 2. They are $U_{w}=0.28U_{0}$ and $0.30U_{0}$ for cases A and B and $\unicode[STIX]{x1D6FF}_{w}=0.52H$ for both cases. From the velocity profiles under the surface, although the sinusoidal profiles of cases A and B are different underneath the surface due to the structural difference, the general trends are similar to each other. When we define the penetration depth as the location to the first local minimum, it is clear that the penetration depths of the mean velocity of both cases are less than $y/H=0.06$, which is the height $d$ of the square rod constructing the porous media. Corresponding to the mean velocities, the shear stresses of both cases damp quickly until $y/H=0.06$. Although some weak sinusoidal profiles are observed in the upper region of the porous layer, they eventually vanish deep inside the porous layer while the mean velocity does not show such a decay. Even though $\unicode[STIX]{x1D612}_{yy}$ is 8.3 times larger in case B, the penetration depths of the mean velocity and the Reynolds shear stress do not seem to increase.

Figure 10.

Comparison of plane-averaged turbulence quantities at $Re\simeq 7500$ (at $Re=7400$ for case A and $Re=7700$ for case B): (a) streamwise velocity, (b) Reynolds shear stress, (c) streamwise and wall-normal r.m.s. velocities and (d) dispersion stresses. The DNS data (Samanta et al. Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) are at $Re_{b}=5000$ in the symmetry plane.

The trend that the flow variables look insensitive to the wall-normal permeability supports our previous conclusion in Suga et al. (Reference Suga, Okazaki, Ho and Kuwata2018), which suggested that, although turbulence generation over porous media was enhanced by the permeable surface, it is rather insensitive to the wall-normal permeability $\unicode[STIX]{x1D612}_{yy}$ compared with $\unicode[STIX]{x1D612}_{xx}$ when $R_{y/x}\geqslant 1.0$. We then suggested that flow was reasonably correlated to the permeability Reynolds number $Re_{K_{x}}$ based on the streamwise permeability. Indeed, although $\unicode[STIX]{x1D612}_{yy}$ of case B is 8.3 times larger than that of case A, the discrepancy between the two cases cannot be considered to reflect such a large difference. Instead of the insensitivity to $\unicode[STIX]{x1D612}_{yy}$, Suga et al. (Reference Suga, Okazaki, Ho and Kuwata2018) showed that a 25 % increase of $\unicode[STIX]{x1D612}_{xx}$ produced a certain amount of turbulence enhancement. While the fluids penetrating into the porous layer move towards the streamwise or spanwise direction, due to the large turbulent surface shear, most of the penetrating fluids move towards the streamwise direction. Since the penetration depth is not large, it is considered that the moving distances of the penetrating fluids in the streamwise direction are longer than those in the wall-normal direction. Hence, it is considered that among the wall-normal and streamwise permeabilities, the streamwise permeability affects more the turbulence near the porous surface. For the present data including the case at $R_{y/x}=0.8$ (case A), as seen in figure 10(b), case A whose $\unicode[STIX]{x1D612}_{xx}$ is 19 % larger than that of case B shows 9.8 % larger shear stress near the interface. Also, the profiles of the streamwise r.m.s. velocities shown in figure 10(c) indicate that case A at $Re_{K_{x}}=6.37$ is more turbulent than case B at $Re_{K_{x}}=5.80$. Hence, the level of surface turbulence follows the order of $Re_{K_{x}}$ even including the case of $R_{y/x}<1.0$

As for the r.m.s. velocities in figure 10(c), although the plots of the DNS by Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) are for the symmetry plane and applied the averaged friction velocity over the whole porous surface, comparison is made to confirm the turbulence level. The corresponding streamwise permeability Reynolds numbers are $Re_{K_{x}}=8.90$, 6.37 and 5.80 for the DNS, cases A and B, respectively. Here, for the abscissa $\unicode[STIX]{x1D6FF}_{w}$ is the location of the maximum plane-averaged velocity. Although the levels of the present r.m.s. results are in the order of $Re_{K_{x}}$, the level of the DNS is somewhat lower than the present experiments even with the higher $Re_{K_{x}}$. Indeed, it is seen that, although the levels of the r.m.s. velocities by the DNS are similar to those of the present data, the peak value of the streamwise component and the levels of the r.m.s. velocities in the porous layer are lower than the present experiments. Following Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006), Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) assumed the permeability and the Forchheimer coefficient as $\unicode[STIX]{x1D612}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}=d_{p}^{2}\unicode[STIX]{x1D711}^{3}/[180(1-\unicode[STIX]{x1D711})^{2}]$ and $\unicode[STIX]{x1D60A}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}^{F}=\unicode[STIX]{x1D711}d_{p}/[100(1-\unicode[STIX]{x1D711})]$, where $d_{p}$ is the particle size for the loosely packed beds. In the DNS, $\unicode[STIX]{x1D711}=0.95$ and $d_{p}/H=0.01$ were applied and these values produced the Forchheimer coefficient as $\unicode[STIX]{x1D60A}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}^{F}/H=1.9\times 10^{-3}$, while in the present cases $\unicode[STIX]{x1D60A}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}^{F}/H$ is in the range of $(1.5{-}6.6)\times 10^{-4}$, which is one order smaller than that of the DNS. Since the effect of a smaller $\unicode[STIX]{x1D60A}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}^{F}$ is comparable to that of a larger $\unicode[STIX]{x1D612}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}}$, it is considered that, when the larger permeability enhances turbulence, the smaller Forchheimer coefficient also enhances turbulence. Note that in connection with (2.1), the drag force term in the momentum equation can be written as $f_{i}=\unicode[STIX]{x1D711}\unicode[STIX]{x1D707}\unicode[STIX]{x1D612}_{ij}^{-1}\langle u_{j}\rangle ^{f}+\unicode[STIX]{x1D711}\unicode[STIX]{x1D707}\unicode[STIX]{x1D612}_{ik}^{-1}\unicode[STIX]{x1D60D}_{kj}\langle u_{j}\rangle ^{f}$. Hence, the Forchheimer tensor works similarly to the inverse of the permeability tensor in the double-averaged equation system. Also the larger Forchheimer coefficient and thus larger form drag caused the steep damping for the r.m.s. velocities inside the porous layer for the DNS.

As far as the profiles inside the porous layer are concerned, both cases A and B show very similar levels of the streamwise and wall-normal r.m.s. velocities while some minor structural effects are seen. Unlike the shear stress profiles, both r.m.s. velocities maintain certain levels and continue decaying inside the porous layer. Obviously the decaying rate is more gentle than those of the mean velocity and the shear stress. It is considered that, although the shear generation is very weak, turbulent vortices penetrate against the upward secondary flow motion and dissipate at a deeper position inside the porous media. Particularly for the wall-normal direction, it is supposed that flow fluctuations are induced by the pressure fluctuations over the surface, which propagate more deeply inside the porous media. These trends confirm the results reported by Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) who applied the VANS equation for the porous medium region. Figure 10(d) compares the dispersion stresses $[\tilde{\bar{u}}_{i}\,\tilde{\bar{u}}_{j}]_{xz}^{f}$. It is obvious that the dispersion stresses are generally very small compared with the turbulent stresses. However, the streamwise component $[\tilde{\bar{u}}\,\tilde{\bar{u}}]_{xz}^{f}$ becomes comparable to the turbulent stress only in the region across the porous interface. This trend is consistent with that shown in Kuwata & Suga (Reference Kuwata and Suga2016b).

To see the trend of the r.m.s. velocity in more detail, to smooth the sinusoidal profiles, the square-root values of the volume-averaged $\overline{u_{i}^{\prime \,2}}$,

(3.3)$$\begin{eqnarray}\langle \overline{u_{i}^{\prime \,2}}\rangle ^{f}=\frac{1}{4V_{REV}^{f}}\int _{0}^{D+d}\int _{y}^{y+D_{p_{y}}}\int _{0}^{4(D+d)}\overline{u_{i}^{\prime \,2}}\,\text{d}v^{f},\end{eqnarray}$$

are plotted in figure 11, where $V_{REV}^{f}$ is a representative elementary volume (REV) of the porous media defined as the fluid-phase volume of $(D+d)\times D_{p_{y}}\times (D+d)$ in the homogeneous structure region. To the REV size in the $y$-direction, $D_{p_{y}}=4d$ is applied. Although case B has the structural interval of $2d$, the structural interval of case A is used for comparison. (Since for both cases the structures are homogeneous at $y<-d$, figure 11 shows the values under $y/D_{p_{y}}=-0.75$, which is the uppermost location of the REV centre.) For both Reynolds numbers, although the streamwise component is dominant over the surfaces, the wall-normal component surpasses it below the surfaces and becomes dominant until $y/D_{p_{y}}\simeq -1.25$ and $y/D_{p_{y}}\simeq -1.5$ for cases A and B, respectively. Then, the streamwise component becomes dominant again and maintains the dominance deep inside the porous layer. This trend confirms that the wall-normal fluctuations penetrate more deeply than the streamwise ones while their decaying rate is higher, suggesting that turbulence anisotropy is maintained inside the porous layer. The effects of the wall-normal permeability are not significant enough to change the trend since the difference between the locations where the streamwise component surpasses the wall-normal component is rather small. The present trend of turbulence anisotropy at $\unicode[STIX]{x1D711}\simeq 0.8$ and in the range of $Re_{K_{x}}=2.49{-}6.37$ is thus consistent with those observed in our previous DNS studies (Kuwata & Suga Reference Kuwata and Suga2016b, Reference Kuwata and Suga2017), which are for $\unicode[STIX]{x1D711}=0.7{-}0.84$ at $Re_{K}=3.8{-}6.1$.

Figure 11.

R.m.s. velocities by the REV averaging: (a) case A and (b) case B, for $D_{p_{y}}=4d$.

Figure 12.

Distributions of the mean velocities and log-law parameters against the pore-scale Reynolds number: (a) streamwise mean velocity distributions in semi-log scale, (b) zero-plane displacement, (c) von Kármán coefficient and (d) roughness scale. The red lines in panel (a) are fitting lines. The data of Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006), Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010, Reference Suga, Nakagawa and Kaneda2017), Manes et al. (Reference Manes, Poggi and Ridol2011) and Kuwata & Suga (Reference Kuwata and Suga2016a) are for isotropic porous media, while the data of Kuwata & Suga (Reference Kuwata and Suga2017) and Suga et al. (Reference Suga, Okazaki, Ho and Kuwata2018) are for anisotropic porous media.

To confirm whether the mean velocity profiles near the symmetry plane are similar to those over two-dimensional porous-wall flows, figure 12(a) shows the mean velocities over the porous surfaces in semi-logarithmic charts with fitting lines of the log law:

(3.4)$$\begin{eqnarray}[\bar{u}]_{xz}^{p+}=\frac{1}{\unicode[STIX]{x1D705}}\ln \left(\frac{y+d_{0}}{h}\right).\end{eqnarray}$$

This log-law form is usually applied to flows over porous media and canopies (Best Reference Best1935; Nikora et al. Reference Nikora, Koll, McLean, Dittrich and Aberle2002; Nepf & Ghisalberti Reference Nepf and Ghisalberti2008). Our previous report (Suga et al. Reference Suga, Okazaki, Ho and Kuwata2018) showed that, irrespective of anisotropy of the permeability, the von Kármán coefficient $\unicode[STIX]{x1D705}$, the zero-plane displacement $d_{0}$ and the roughness scale $h$ were generally well correlated with the pore-scale Reynolds number $Re_{K}^{\ast \ast }$. The pore-scale Reynolds number was defined as

(3.5)$$\begin{eqnarray}Re_{K}^{\ast \ast }=\frac{u_{\unicode[STIX]{x1D70F}}^{p}D_{p_{x}}}{3.8\unicode[STIX]{x1D708}},\end{eqnarray}$$

where $D_{p_{x}}$ is the streamwise length of the pore. (Note that, since this Reynolds number has a strong correlation with a permeability Reynolds number based on the porosity and the streamwise permeability, Suga et al. (Reference Suga, Okazaki, Ho and Kuwata2018) called it the ‘surrogate’ permeability Reynolds number.) Table 2 lists the values of $\unicode[STIX]{x1D705}$, $d_{0}^{p+}~(=u_{\unicode[STIX]{x1D70F}}^{p}d_{0}/\unicode[STIX]{x1D708})$ and $h^{p+}~(=u_{\unicode[STIX]{x1D70F}}^{p}h/\unicode[STIX]{x1D708})$ that are obtained by the fitting procedure described in Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010, Reference Suga, Nakagawa and Kaneda2017, Reference Suga, Okazaki, Ho and Kuwata2018). The corresponding figure 12(b–d) shows that the present parameters are well in the clusters of the plots of a wide range of porous-wall flows. This confirms that the turbulent flow over the porous surfaces near the symmetry plane of the square duct maintains the general trend of the mean velocity in the two-dimensional porous-wall turbulence even though the secondary flows are enhanced. Note that for the von Kármán coefficient, the correlation with the pore-scale Reynolds number is not strong compared with the other parameters. Since the zero-plane displacement and the roughness scale are geometrical parameters related to the porous structure, the pore scale is a reasonable parameter for the characteristic length scale. However, $\unicode[STIX]{x1D705}y$ corresponds to the mixing length for near-wall turbulence and thus it is considered that the correlation between $\unicode[STIX]{x1D705}$ and the pore scale is not very strong.

3.4 Kelvin–Helmholtz instability waves

Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) observed that there were short spanwise rollers in the middle region and their spanwise size was approximately $0.4H$. Following the discussion by Jiménez et al. (Reference Jiménez, Uhlmann, Pinelli and Kawahara2001), those spanwise rollers are considered to be induced by the KH instability since the mean velocity has an inflection point under the porous surface as seen in figure 10. Although Samanta et al. (Reference Samanta, Vinuesa, Lashgari, Schlatter and Brandt2015) did not analyse the characteristics of those rollers, we try to detect and analyse the wave motions over porous media. Figure 13 shows the one-dimensional spectrum of streamwise velocity fluctuations $E_{11,x}$ at $y/H=0.1$, $0.0$ and $-0.02$ in the symmetry planes. Those locations correspond to over, at and under the surface, respectively. For $Re\simeq 3500$, $y/H=0.1$ and $-0.02$ correspond to $y^{p+}\simeq 30$ and $-6$, respectively, while for $Re\simeq 7500$, they correspond to $y^{p+}\simeq 75$ and $-15$, respectively. Both cases A and B at the surfaces ($y/H=0$) show clear peaks. At $Re\simeq 3500$ shown in figure 13(a,b), the peaks of $y/H=0.0$ are located at $f=0.25~\text{Hz}$, which corresponds to the wavelength of $\unicode[STIX]{x1D706}_{w}=U_{w}/f=1.4H$ due to the slippage velocity of $U_{w}=0.28U_{0}$ as listed in table 2. At $Re\simeq 7500$ of case A, the peak of $y/H=0.0$ is located at $f=0.5~\text{Hz}$, which corresponds to the wavelength of $\unicode[STIX]{x1D706}_{w}=1.8H$, while for case B, it is located at $f=0.6~\text{Hz}$, which corresponds to the wavelength of $\unicode[STIX]{x1D706}_{w}=1.6H$ since $U_{w}$ increases to $0.3U_{0}$. Accordingly, clear effects of the wall-normal permeability on the wavelengths are not observed also. Although the locations of the peaks of $y/H=-0.02$ shift slightly from those of $y/H=0.0$, the corresponding frequencies are very similar to those of $y/H=0.0$. This is because the energy of the waves generated at the surfaces propagates to both over- and under-surface regions.

Figure 13.

One-dimensional streamwise spectrum of $u^{\prime }$ in the symmetry plane: (a) case A at $Re=3300$, (b) case B at $Re=3400$, (c) case A at $Re=7400$ and (d) case B at $Re=7700$. Black broken vertical lines correspond to the frequencies of peak locations at the surface: $y/H=0.0$.

In the fully developed mixing layers, the wavelength of the KH-type coherent eddies normalized by the vorticity thickness, $C_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}_{x}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}$, is known to be $3.5\leqslant C_{\unicode[STIX]{x1D706}}\leqslant 5$ (Dimotakis & Brown Reference Dimotakis and Brown1976; Rogers & Moser Reference Rogers and Moser1994). For flows over porous walls, the DNS by Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Kuwata & Suga (Reference Kuwata and Suga2017) and the experiments by Suga et al. (Reference Suga, Okazaki, Ho and Kuwata2018) indicated $3.4\leqslant C_{\unicode[STIX]{x1D706}}\leqslant 5.5$ when the boundary layer thickness $\unicode[STIX]{x1D6FF}_{w}$ is considered to be equivalent to the vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FA}}$. The wavelength $\unicode[STIX]{x1D706}_{w}$ of cases A and B at $Re\simeq 3500$ produces $C_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}_{w}/\unicode[STIX]{x1D6FF}_{w}=3.0$ and 2.9, respectively. At $Re\simeq 7500$, for cases A and B they are $C_{\unicode[STIX]{x1D706}}=3.5$ and 3.1, respectively. Although these values are close to the range of the reported values for KH waves, they are somewhat smaller than the reported values of turbulence over porous media. Although there are many unknown things regarding the interaction between the KH instability and the sidewall turbulent boundary layers, we believe that such interaction influences the instability, resulting in shorter wavelengths than those in two-dimensional boundary layer type of flows. This sidewall confinement effect is stronger at lower Reynolds numbers since the sidewall boundary layer is thicker at lower Reynolds number as seen in figure 3.