3.1. Effect of porosity under fixed grain spacing

First, let us consider deep porous (Pd) substrates with fixed $L^+$ but varying $\sqrt {K^+}$ by varying $g/L$ . Figure 4 shows instantaneous velocity fields on the $x$ – $y$ plane for six deep porous substrates with identical $L^+\approx 24$ but varying $g/L$ from $1/4$ to $3/4$ , corresponding to $\sqrt {K^+}$ from $0.4$ to $5.8$ . For increasing $g/L$ , the streamwise velocity $u^+$ of the subsurface flow generally increases, while regions of relatively low speed become more prevalent immediately above the interface. The changes in $u^+$ and $v^+$ suggest a gradually intensified penetration of the overlying flow into the substrate. There is strong impedance to the overlying turbulent eddies, which typically span multiple grains, penetrating into the substrate, and their footprint below the interface is much attenuated, and dispersed lengthscale-wise, by the presence of the individual grains.

Figure 4. Instantaneous fields of velocity components $u^+$ (a) and $v^+$ (b) on an $x$ - $y$ plane for deep-porous substrates with identical pitch $L^+\approx 24$ but different gap-to-pitch ratio $g/L=0.25\!$ – $0.75$ , from top to bottom substrates Pd-24-25/38/50/56/62/75. Colours from blue to red correspond for $u^+$ to $[0:5]$ and for $v^+$ to $[-0.8:0.8]$ .

Figure 5. (a,b) Mean velocity profile, ( $c$ ) Reynolds shear stress and ( $d$ – $f$ ) RMS velocity fluctuations for deep porous substrates with identical $L^+\!\approx \!24$ but different $g/L=0.25\!$ – $\!0.75$ . Colours from blue to red are for cases Pd-24-25/38/50/56/62/75, and dash-dotted lines for smooth-wall data. The dotted lines in panel (b) are Darcy–Brinkman analytical solutions (D3) for the mean velocity within the substrate, and the dashed lines mark the location of the free-flow/substrate interface.

The profiles of mean velocity $U^+$ , Reynolds shear stress $\overline {u^{+\prime }v^{+\prime }}$ and root-mean-square (RMS) velocity fluctuations $u^{+\prime }_{rms}$ , $v^{+\prime }_{rms}$ and $w^{+\prime }_{rms}$ for the above six substrates are portrayed in figure 5. For each case, the $U^+$ profile below the interface in figure 5(b) shows a near-interface region with strong shear, i.e. the Brinkman layer. Further below is a plateau where $U^+$ essentially results from the mean pressure gradient, i.e. the Darcy region. The Brinkman layer is thicker for larger $g/L$ , indicating a deepened penetration of shear of the overlying flow. This also results in a slight increase of the slip velocity, $U_s^+$ , but generally decreases $U^+$ above the interface, as shown in figure 5(a), which leads to an increase of the drag coefficient. Of the configurations discussed here, only those with smaller permeability, $\sqrt {K^+}\lesssim 1$ , exhibit a smooth-wall-like character. Those with greater permeability experience significant departures from smooth-wall turbulence, exhibiting the usual decrease of $u'$ and increase in $v'$ and $w'$ near the wall, with all three converging towards similar peak values, together with an increase in near-wall Reynolds shear stress and the corresponding increase in $\triangle U^+$ and drag. Given that similar intense departures from the smooth-wall-like regime occur across the whole set of configurations studied, the virtual-origin framework proposed in Ibrahim et al. (Reference Ibrahim, Gómez-de-Segura, Chung and García-Mayoral2021) will not be used here, as it only applies to smooth-wall-like turbulence.

The drag increase with increasing $g/L$ is directly associated with the changes in the $\overline {u^{+\prime }v^{+\prime }}$ profiles (see Gómez-de-Segura & García-Mayoral Reference Gómez-de-Segura and García-Mayoral2019, § 5.3), shown in figure 5(c) for $y^+\approx 0$ – $30$ . Their magnitude increases significantly relative to smooth-wall values for $g/L\gtrsim 0.4$ or $\sqrt {K^+}\gtrsim 1$ . These profiles of $\overline {u^{+\prime }v^{+\prime }}$ , together with $u^{+\prime }_{rms}$ , $v^{+\prime }_{rms}$ and $w^{+\prime }_{rms}$ in figure 5(d,e,f), illustrate the gradually enhanced penetration of turbulence into the substrates as $g/L$ increases. The penetration of $u^{+\prime }_{rms}$ is accompanied by a drop of its peak value above the interface, while such a drop is not observed for $v^{+\prime }_{rms}$ or $w^{+\prime }_{rms}$ . Similar trends have been observed not only for porous substrates (Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006) but also for rough surfaces (Ligrani & Moffat Reference Ligrani and Moffat1986; Abderrahaman-Elena et al. Reference Abderrahaman-Elena, Fairhall and García-Mayoral2019) and canopies (Sharma & García-Mayoral Reference Sharma and García-Mayoral2020b ). These trends have been interpreted by some authors (Jiménez Reference Jiménez2004; Flores & Jimenez Reference Flores and Jimenez2006) as the flow losing some of the anisotropic characters of the near-wall cycle. It is also notable that for all cases studied, $v^{+\prime }_{rms}$ decays with the depth into the substrate more slowly than $u^{+\prime }_{rms}$ and $w^{+\prime }_{rms}$ , a feature consistent with the study for dense canopies by Sharma & García-Mayoral (Reference Sharma and García-Mayoral2020b ). This implies that, for a finite-depth porous substrate, the wall-normal velocity fluctuations in the subsurface flow are more likely to perceive the presence of the substrate floor than the tangential (i.e. wall-parallel) fluctuations. This is further discussed in § 5.

3.2. Effect of grain spacing under fixed porosity

Next, we consider the deep porous (Pd) substrates with the same gap-to-pitch ratio $g/L=1/2$ but different pitch $L^+$ . Figure 6 displays the changes in flow statistics as the pitch $L^+$ increases from $12$ to $48$ , which corresponds to $\sqrt {K^+}$ increasing from $0.9$ to $3.6$ . The changes in terms of drag, subsurface mean flow and turbulence penetration are qualitatively similar to those changes with $g/L$ increasing illustrated in figure 5. The dominant factor that underlies both the $g/L$ -induced changes and the $L^+$ -induced changes is discussed in § 4.2. In both instances, we note that the differences in the value of the mean velocity at the interface is small, while the differences far above the substrate are significant. This can be traced to the increase in the shear Reynolds stress profile, as discussed above. The latter shows a good correlation with the fluctuating transpiration $v^{+\prime }_{{\mathrm {rms}}}$ at the interface (Abderrahaman-Elena et al. Reference Abderrahaman-Elena, Fairhall and García-Mayoral2019), which, in turn, has been shown to correlate well with $\triangle U^+$ for rough surfaces (Orlandi et al. Reference Orlandi, Leonardi and Antonia2006; Orlandi & Leonardi 2006, Reference Orlandi and Leonardi2008). This suggests that the increase in drag is more closely connected to the transpiration than to the tangential velocity at the interface.

Figure 6. (a,b) Mean velocity profile, (c) Reynolds shear stress and (d–f) RMS velocity fluctuations for deep porous substrates with identical $g/L=0.50$ but different $L^+=12\!$ – $\!48$ . Colours from blue to red are for cases Pd-12/24/36/48-50, and dash-dotted lines for smooth-wall data. The dashed lines mark the location of the free-flow/substrate interface.

Figure 7. ( $a$ ) Shear-driven component of the mean velocity, ( $b$ ) Reynolds shear stress and ( $c$ – $e$ ) RMS velocity fluctuations for the flow within the substrate, normalised by the corresponding interfacial values and the thickness of one layer of cubes, $D$ , for the same cases of figure 6. Colours are as in figure 6.

In contrast to the cases with identical $L^+$ in figure 5, the four cases with identical $g/L=1/2$ in figure 6 share a similarity in substrate geometry, and thus exhibit some degree of similarity in the decay of subsurface flow properties. In figure 7, where the flow statistics are normalised by the corresponding interfacial values and the wall-normal coordinate is normalised by $D$ , the four cases have a similar decaying trend for $v^\prime _{rms}$ in most of the subsurface region with $y\lt 0$ , while for $u^\prime _{rms}$ and $w^\prime _{rms}$ the similarity occurs only for $y\lesssim -1D$ . This suggests that nonlinear inertial effects, which break the similarity of the subsurface flow, are largely limited to the near-interface region, and mainly influence tangential motions only. This would thus yield the difference in the decaying rate of near-interface $U$ among the four cases, shown in figure 7 $(b)$ .

3.3. Effect of substrate depth

Lastly, for a fixed pitch $L^+\approx 24$ and gap-to-pitch ratio $g/L=1/2$ , we compare four cases with varying depths $h/D=1,2,5$ , and $9$ , which are respectively labelled as Ro-24-50, Ps-24-50, Pd-24-50 and Pd-24-50-VD. As shown in figure 8, as the depth $h$ increases from $1D$ to $5D$ , we observe an increase in drag and a deeper penetration of turbulence, which are qualitatively similar to the changes with increasing $g/L$ in figure 5 or with increasing $L^+$ in figure 6. However, the present four cases demonstrate no substantial differences in $U^+$ , $u_{rms}^{+\prime }$ and $w_{rms}^{+\prime }$ in the range $y\approx -1$ – $0D$ , indicating that the change of $h$ has little influence on the penetration of tangential velocity components. Quantitative discussion on this phenomenon is presented in § 4.1.

Figure 8. ( $a$ , $b$ ) Mean velocity profile, ( $c$ ) Reynolds shear stress and ( $d$ – $f$ ) RMS velocity fluctuations for substrates with identical $L^+\approx 24$ and $g/L=0.50$ but different depth $h=1D$ - $9D$ . Colours from blue to red are for cases Ro-24-50 ( $h/D = 1$ ), Ps-24-50 ( $h/D = 2$ ), Pd-24-50 ( $h/D = 5$ ) and Pd-24-50-VD ( $h/D = 9$ ), and dash-dotted lines for smooth-wall data. The dashed lines mark the location of the free-flow/substrate interface.

The changes in flow statistics caused by the increase in $h$ diminish gradually. Eventually, for $h=5D$ and $9D$ , all the statistics become essentially indistinguishable except for $v^{+\prime }_{rms}$ deep inside the substrate, where the wall-normal fluctuations seem always able to penetrate to the floor, as discussed in Sharma & García-Mayoral (Reference Sharma and García-Mayoral2020b ). Nevertheless, $v_{rms}^{+\prime }$ becomes ultimately negligible below $y\approx -4D$ even for the substrate with depth $h=9D$ . Above $y\approx -4D$ , an asymptotic state is already reached for $h\geq 5D$ . This suggests that $h=5D$ is a depth sufficient for the overlying turbulence to essentially no longer perceive the floor. This concept of ‘sufficient depth’ is further investigated in § 5.

4.1. Slip and shear across the interface

The values of the mean slip velocity $U_s^+$ for all the cases in this study are portrayed versus different substrate parameters in figure 9. No apparent correlations are found between $U_s^+$ and any of the pitch $L^+$ , the gap size $g^+$ , the inclusion size $\ell ^+$ , the porosity $\varepsilon$ or the depth $h^+$ . The values of $U_s^+$ for deep porous (Pd) substrates, however, tend to correlate well with $\sqrt {K^+}$ , as shown in figure 9(f). This suggests that the slip velocity for deep substrates is essentially determined by their macroscale permeability, and is not directly associated with the microscale details of the individual grains. Similar results were also observed by Efstathiou & Luhar (Reference Efstathiou and Luhar2018).

Figure 10. Mean slip length, $\ell _U^+$ , versus ( $a$ ) pitch $L^+$ , ( $b$ ) gap size $g^+$ , ( $c$ ) inclusion size $\ell ^+$ , ( $d$ ) porosity $\varepsilon$ , ( $e$ ) depth $h^+$ and ( $f$ ) permeability $\sqrt {K^+}$ . Symbols and colours are as in figure 9. The two dotted lines in $(f)$  are for $\ell _U^+=0.7\sqrt {K^+}$ and $\ell _U^+=1.0\sqrt {K^+}$ .

For $\sqrt {K^+}\lesssim 2$ , the values of $U_s^+$ for rough surfaces in figure 9 $(f)$ agree roughly with $U_s^+$ for their corresponding deep substrates. For $\sqrt {K^+}\gtrsim 2$ , however, the former are higher and more scattered than the latter. This is in spite of the slip length having a strong correlation with the permeability, as shown in figure 10, with the slip length defined as $\ell _U=U_s\,/\,\partial _{y}U|_0$ , where $\partial _{y}U|_0$ is the mean shear in the free flow at $y = 0$ . Figure 10 $(f)$ shows that $\ell _U^+$ is roughly proportional to $\sqrt {K^+}$ , with a constant of proportionality of order 0.7–1. This is consistent with the analysis (Abderrahaman-Elena & García-Mayoral Reference Abderrahaman-Elena and García-Mayoral2017; Gómez-de-Segura & García-Mayoral Reference Gómez-de-Segura and García-Mayoral2019) based on a homogenised model for the subsurface flow, which leads to $\ell _U^+\approx \sqrt {K^+}$ . The results in figure 10(f) indicate that under a fixed value of shear at the interface, the slip length is essentially independent of the substrate depth, even for depths as shallow as $h=1D=L/2$ . The different behaviours of the slip length $U_s^+$ and the slip velocity $\ell _U^+$ are caused by different non-zero Reynolds shear stresses $\overline {u^{+\prime }v^{+\prime }}$ at the interface plane. In their absence, the shear in viscous units would be $\partial _{y^+}U^+=1$ , and both quantities would have equal value. The comparison of figures 9 $(f)$ and 10 $(f)$ suggests that the substrate depth plays a key role in this difference.

Another feature of our porous and rough substrates in terms of tangential velocities is the discontinuity of shear $\partial _yU$ across the interface, shown in figure 11. A force balance in a thin volume containing the interface shows that the mean shear stress just above and below are different, as the shear stress above is partly balanced by the shear force between the fluid and the flat top surface of the substrate elements. This effect is in our case concentrated at the element tips, but can be expected to be more diffuse in substrates composed of rounder grains and with less even interfaces. Figure 12 portrays the ratio of inner to outer shear, $r_{sh}=\partial _yU|_{0^-}\,/\,\partial _yU|_0$ , where $\partial _yU|_{0^-}$ is the mean shear approaching $y=0$ from the substrate side, for all the cases. None of the length scales $L^+$ , $g^+$ , $L^+\!-\!g^+$ , $h^+$ and $\sqrt {K^+}$ scale the ratio $r_{sh}$ . Instead, $r_{sh}$ appears to correlate with the porosity $\varepsilon$ , suggesting that a denser substrate with lower $\varepsilon$ tends to have a stronger jump of shear, i.e. smaller $r_{sh}$ . This is consistent with the above observation that the discontinuity is caused by the shear absorbed at the exposed flat faces of the grains, as their surface area is a larger fraction of the interface plane for lower $\varepsilon$ . The ratio $r_{\textit {sh}}$ is roughly linear with $\varepsilon$ except for highly porous cases ( $\varepsilon \gtrsim 0.9$ ), of which $r_{\textit {sh}}$ adjusts to approach the no-jump asymptotic limit, $\lim _{\varepsilon \to 1}r_{\textit {sh}}=1$ .

Figure 11. Mean velocity profiles near the interface. $(a)$ , $(b)$ and $(c)$ are for the cases in figures 5, 6 and 8, respectively, with line styles as in the respective figure.

Figure 12. Ratio of inner to outer shear across the substrate interface, $r_{\textit {sh}}$ , versus ( $a$ ) pitch $L^+$ , ( $b$ ) gap size $g^+$ , ( $c$ ) inclusion size $\ell ^+$ , ( $d$ ) porosity $\varepsilon$ , ( $e$ ) depth $h^+$ and ( $f$ ) permeability $\sqrt {K^+}$ . Symbols and colours are as in figure 9. The dotted line in $(d)$ is for $r_{sh}=0.42\,\varepsilon +0.10$ .

The observations in this subsection suggest that, for both porous and rough substrates, the interfacial shear jump and the slip length are mainly influenced by the porosity and the permeability, respectively, with no significant direct influence of the substrate microscale details or depth. The scaling of slip length with permeability applies also to slip velocities only for small permeabilities, $\sqrt {K^+}\lesssim 2$ . For larger ones, the slip velocity also depends on he substrate depth, as the shear Reynolds stress at the interface becomes increasingly significant for the deeper substrates. In any event, we note that the values of $U_s^+$ are significantly smaller than those of $\triangle U^+$ , which implies that the slip plays only a small role in determining the drag. The effect of substrate granularity and depth are further investigated in §§ 4.2 and 5.

4.2. Drag increase and near-interface flow for deep substrates

Let us now focus on the drag increase of a substrate, given by $\triangle U^+$ as defined in § 2.3. Figure 13 portrays $\triangle U^+$ for all the cases simulated versus different substrate parameters. The values of $\triangle U^+$ do not correlate well with any of $L^+$ , $g^+$ , $L^+\!-\!g^+$ , $\varepsilon$ , and $h^+$ , individually. However, $\triangle U^+$ for deep porous substrates shows a good correlation with $\sqrt {K^+}$ , extending to shallow porous (Ps) and rough (Ro) substrates for small permeability, $\sqrt {K^+}\lesssim 1$ . Beyond this, $\triangle U^+$ for shallow and rough substrates is lower and exhibits more scatter than that for deep ones. In general, the discrepancies between deep and shallow substrates are considerably smaller than those between shallow and rough ones, implying an asymptotic behaviour of $\triangle U^+$ as the depth $h$ increases, similar to the observations in § 3.3. These results suggest that, in essence, permeability alone determines the drag for sufficiently deep porous substrates.

Figure 13. Velocity deficit $\triangle U^+$ versus ( $a$ ) pitch $L^+$ , ( $b$ ) gap size $g^+$ , ( $c$ ) inclusion size $\ell ^+$ , ( $d$ ) porosity $\varepsilon$ , ( $e$ ) depth $h^+$ and ( $f$ ) permeability $\sqrt {K^+}$ . Symbols and colours are as in figure 9.

Figure 14. ( $a$ , $b$ ) Mean velocity profile, ( $c$ ) Reynolds shear stress and ( $d$ – $f$ ) RMS velocity fluctuations for deep porous substrates. Blue to red dotted lines are for substrates with different $L^+$ and $\varepsilon$ but similar $\sqrt {K^+}\approx 1$ , cases Pd-12-50, Pd-24-38, Pd-36-33 and Pd-48-28; dashed for similar $\sqrt {K^+}\approx 2.5$ , cases Pd-24-56, Pd-36-50 and Pd-48-44; and solid for similar $\sqrt {K^+}\approx 6$ , Pd-24-75, Pd-36-67 and Pd-48-61. The dash-dotted lines are for smooth-wall data, and the vertical dashed lines mark the location of the free-flow/substrate interface.

Focusing for now on deep substrates, we observe that the turbulent statistics in general also depend essentially on permeability alone. Figure 14 shows that the substrates with different $L^+$ and $\varepsilon$ but similar $\sqrt {K^+}$ have fairly similar mean velocity profiles, turbulent shear stress and RMS fluctuations in the overlying flow. Differences in the turbulent stress and RMS occur below the interface, where cases with larger $L^+$ or lower $\varepsilon$ tend to have larger magnitudes. These differences are likely attributable to the different substrate geometries causing different dispersive or grain-coherent stresses, although this would require more in-depth analysis.

Figure 15. Instantaneous fields of ( $a$ – $j$ ) $u^\prime$ and ( $k$ – $t$ ) $v^\prime$ at $y^+\!\approx \!3$ for the same deep porous substrates of figure 14. Columns from left to right correspond to substrates with $L^+\approx 12$ , $24$ , $36$ and $48$ , respectively. (a,b,c,d) and (k,l,m,n) substrates with $\sqrt {K^+}\!\approx \!1$ ; (e,f,g) and (o,p,q) substrates with $\sqrt {K^+}\!\approx \!2.5$ ; (h,i,j) and (r,s,t) substrates with $\sqrt {K^+}\!\approx \!6$ . Colours from dark to clear are for the value range $[-2:2]$ relative to the RMS value of the variable at that plane.

Some more details of the structure of turbulence near the interface can be illustrated by instantaneous flow fields and energy spectral densities. The flow fields portrayed in figure 15 exhibit a signature of the grain-coherent flow with a characteristic length scale $L^+$ . Superimposed with this signature, we can observe the grain-incoherent features of the background turbulence. Just as the flow statistics in figure 14, the background turbulence is visually similar for cases with different $L^+$ and $\varepsilon$ but similar $\sqrt {K^+}$ , while the grain-coherent flow varies greatly with $L^+$ . The four cases with $\sqrt {K^+}\approx 1$ exhibit the typical features of smooth-wall turbulence in the streamwise elongated shapes in $u^\prime$ , shown in figure 15( $a$ – $d$ ), and $v^\prime$ , shown in figure 15( $k$ – $n$ ). The streamwise elongation of these structures is disrupted for $\sqrt {K^+}\approx 2.5$ , as shown in figure 15( $e$ – $g,o$ – $q$ ), and entirely lost for $\sqrt {K^+}\approx 6$ , figure 15( $h$ – $j$ , $r$ – $t$ ), for which eddies have an $x$ – $z$ aspect ratio closer to unity.

Figure 16. Pre-multiplied spectra $\alpha _x\alpha _z\Phi _{**}$ at (a,c,e) $y^+\!\approx \!3$ and (b,d,f) $y^+\!\approx \!11$ for the same deep porous substrates of figure 14. (a,b) Substrates with $\sqrt {K^+}\!\approx \!1$ ; (c,d) substrates with $\sqrt {K^+}\!\approx \!2.5$ ; (e,f) substrates with $\sqrt {K^+}\!\approx \!6$ . Dashed lines are for $L^+\!\approx \!12$ , shaded contours for $L^+\!\approx \!24$ , solid lines for $L^+\!\approx \!36$ and dotted lines for $L^+\!\approx \!48$ . The contours mark values [0.044:0.044:0.264] relative to the corresponding variance or covariance.

The above discussion is also consistent with the statistical information displayed in the spectral density maps of figure 16. At $y^+\approx 3$ , these maps present two distinct features: a main spectral region attributable to the background turbulence, and smaller lobes centred about the grain-spacing wavelengths, caused by the grain-coherent flow. These lobes extend beyond the mere harmonics of the texture because of the amplitude modulation of the grain-coherent flow by the background turbulence (Abderrahaman-Elena et al. Reference Abderrahaman-Elena, Fairhall and García-Mayoral2019; Khorasani et al. Reference Khorasani, Morteza, Luhar and Bagheri2024). The cases with similar $\sqrt {K^+}$ show good agreement for the background turbulence, but differ in the regions produced by the grain-coherent flow due to their different $L^+$ . The grain-coherent flow quickly decays away from the interface, as evidenced in the maps at $y^+\approx 11$ . For the background turbulence, as $\sqrt {K^+}$ increases, the spectral densities become lower in $x$ -elongated wavelengths but higher in wider $z$ -wavelengths.

The discussion in this subsection suggests that the effect of deep porous substrates on the overlying turbulence is essentially governed by the permeability, a characteristic not directly associated with the geometric microscale detail of individual grains in a porous medium. The grain-coherent flow near the interface, in turn, manifests the effect of the granularity, but decays quickly away from the substrate, at least for the grain pitches here considered, $L^+\lesssim 50$ and $g^+\lesssim 30$ .

5.1. An equivalent permeability incorporating the effect of depth

In §§ 3.3 and 4.1, we have discussed the relatively small influence of depth on the tangential velocity and interfacial shear and slip, i.e. a porous substrate and a typical rough surface with identical grain geometry have fairly similar subsurface decay of the tangential mean and fluctuating velocities, as shown in figure 8(b,d,f), and also similar interfacial slip and shear properties, as shown in figures 10 $(f)$ and 12 $(d)$ . Therefore, we can infer that the apparent differences in drag increase between porous and rough surfaces in figure 13 $(f)$ principally originate from their differences in interfacial transpiration. This is further supported by figures 17( $a$ ) and 17( $b$ ), which show that, for all the substrates studied, the drag increase is highly correlated with the intensity of the interfacial wall-normal velocity fluctuation, rather than with the tangential one, as is the case also for rough surfaces (Orlandi et al. Reference Orlandi, Leonardi and Antonia2006; Orlandi & Leonardi Reference Orlandi and Leonardi2006, Reference Orlandi and Leonardi2008).

Figure 17. Velocity deficit $\triangle U^+$ for all the cases studied as a function of: ( $a$ ) the RMS of the interfacial  $u^\prime$ ; ( $b$ ) the RMS of the interfacial $v^\prime$ ; ( $c$ ) the slip-based equivalent permeability $K_{eq}^{s+}$ ; and ( $d$ ) the transpiration-based equivalent permeability $K_{eq}^{t+}$ (right). Lines and symbols are as in figure 9. In ( $d$ ), $K_{eq}^{t+}$ has been calculated for a characteristic near-wall pressure lengthscale $\lambda _p^+=200$ , and the error bars represent the range $\lambda _p^+=150$ – $250$ . The values of $\triangle U^+$ versus $\sqrt {K^+}$ from figure 13 $(f)$ are displayed in grey for comparison.

The effect of substrate depth on the interfacial slip and transpiration can be observed in the analytical solution of the homogenised Darcy–Brinkman model for the flow within the substrate (Gómez-de-Segura et al. Reference Gómez-de-Segura, Sharma and García-Mayoral2018a ; Gómez-de-Segura & García-Mayoral Reference Gómez-de-Segura and García-Mayoral2019). Appendix D presents this model for the case of isotropic substrates and discusses the effect of depth as deduced from its solution. The model ultimately results in an admittance, linear relationship (D10), which gives the interfacial slip and transpiration velocities in response to the interfacial shear and normal stresses. This relationship involves five coefficients, $\mathcal {L}_{slip}$ , $\mathcal {K}_{slip}$ , $\mathcal {N}_{trsp}$ , $\mathcal {K}_{trsp}$ and $\mathcal {L}_{slip}^{\bot }$ , which characterise the effect of the substrate on the overlying free flow. Each coefficient can be written as a product of two parts: a component that is a function of permeability $K$ and a dimensionless attenuating function $f_{*}(\alpha ,h)$ that depends on the wavenumber $\alpha$ of the exciting stress and the substrate depth $h$ .

The function $f_{*}(\alpha ,h)$ tends to unity for deep substrates and long exciting waves, and to vanish for shallow substrates or for small waves, making the substrates effectively impermeable and smooth in the latter cases. The first component can be thus identified as the admittance coefficient relating interfacial velocity and stress for large-scale flows over sufficiently deep substrates. One exception to this, though, is the coefficient $\mathcal {K}_{trsp}=K f_{\mathcal {K}t}(\alpha ,h)$ relating the transpiration velocity $v^\prime$ and the pressure fluctuation $p^\prime$ , which also vanishes for very long waves, and is maximum for an intermediate wavelength comparable to the depth. The discussion in (D3) highlights that, as $h$ increases, $\mathcal {K}_{trsp}$ reaches its asymptotic value significantly more slowly than the other four coefficients for the typical wavelengths in wall turbulence. This indicates that the principal effect of a finite depth is to decrease $\mathcal {K}_{trsp}$ , i.e. to suppress the pressure-excited transpiration at the interface. Given the strong correlation of this transpiration with $\triangle U^+$ discussed above, we would expect this suppression effect to play a leading role in determining the effect of substrate depth on drag.

The above discussion is based on Darcy–Brinkman solutions, which are after all a mere model for the homogenised subsurface flow: they fail to capture, for instance, the discontinuity in macroscopic velocity shear across the interface, or the direct effect of granularity, which grows with the texture size $L^+$ . We thus simply use these solutions to guide us in proposing an empirical “equivalent permeability” $K_{eq}^t$ to incorporate the effect of substrate depth on transpiration,

(5.1) \begin{equation} K_{eq}^t \equiv K \, f_{\mathcal {K}t}(\tilde {\alpha },\tilde {h}), \end{equation}

where the attenuating function $f_{\mathcal {K}t}(\tilde {\alpha },\tilde {h})$ is calculated from (D12 $d$ ), and where the dimensionless wavenumber $\tilde {\alpha }$ and depth $\tilde {h}$ are defined by (D13) and (D4). The value of $\alpha$ in (5.1) should be chosen to represent the characteristic scale of the typical near-wall pressure fluctuations that excite transpiration at the interface. Informed by the spectra in figure 16, we assume a characteristic wavelength $\lambda _p=150$ – $250\,\nu /u_\tau$ and, thus, $\alpha =\alpha _p=2\pi /\lambda _p$ . In addition to $K_{eq}^t$ defined by (5.1), for comparison we also define another ‘equivalent permeability’ that incorporates the effect of substrate depth on slip,

(5.2) \begin{equation} K_{eq}^s \equiv K \, f_{\mathcal {L}s}(\tilde {\alpha },\tilde {h}), \end{equation}

where the attenuating function $f_{\mathcal {L}s}(\tilde {\alpha },\tilde {h})$ is calculated from (D12 $a$ ). This $K_{eq}^s$ gives a measure of the response of the substrate to an overlying shear, and is therefore an inherently interfacial quantity, while $K_{eq}^t$ gives a measure of the penetration of wall-normal flow due to pressure fluctuations, and therefore accounts for the properties of the substrate not just near the interface but deeper within. An alternative set of admittance relationships to (D10) can be derived from homogenisation (Bottaro & Naqvi Reference Bottaro and Naqvi2020; Naqvi & Bottaro Reference Naqvi and Bottaro2021). This gives a set of upscaled coefficients that play the same role of $K_{eq}^t$ , $K_{eq}^s$ and the other coefficients derived from a Darcy–Brinkman model. The scaling of the roughness function with those upscaled coefficients is portrayed in Appendix E, and the collapse is generally not as good as the one discussed below with $\sqrt {K_{eq}^{t+}}$ . In the authors’ view, the reason is likely the lack of separation of scales between turbulence and surface texture, which is a central assumption in homogenisation.

The values of $\triangle U^+$ against the newly defined $\sqrt {K_{eq}^{s+}}$ and $\sqrt {K_{eq}^{t+}}$ are portrayed in figure 17 for all porous and rough cases. Figure 17 $(c)$ shows that the values of $\sqrt {K_{eq}^{s+}}$ are close to those of $\sqrt {K^+}$ . This is consistent with the observations in § 4.1 and the analysis in Appendix D.3, which suggests that the effect of depth on interfacial slip is small. As a result, $\sqrt {K_{eq}^{s+}}$ and $\sqrt {K^+}$ produce a similarly poor collapse for $\triangle U^+$ across all the substrates studied. Meanwhile, the values of $\triangle U^+$ across all substrates collapse well with $\sqrt {K_{eq}^{t+}}$ , as shown in figure 17 $(d)$ . For rough surfaces, $\sqrt {K_{eq}^{t+}}$ is significantly smaller than the original $\sqrt {K^+}$ , indicating that their small depths suppress significantly the interfacial transpiration. For deep porous substrates, such suppression tends to vanish and the differences between $\sqrt {K_{eq}^{t+}}$ and $\sqrt {K^+}$ are small.

Figure 18. ( $a$ , $b$ ) Mean velocity profile, ( $c$ ) Reynolds shear stress and ( $d$ – $f$ ) RMS velocity fluctuations for cases with similar $\sqrt {K_{eq}^{t+}}$ . Blue, yellow and red are for deep porous (Pd) substrates ( $h/D\geq 5$ ), shallow porous (Ps) substrates ( $h/D=2$ ) and rough surfaces ( $h/D=1$ ), respectively. Dotted lines are for the cases with different $L^+$ , $\varepsilon$ , and $h/D$ but similar $\sqrt {K_{eq}^{t+}}\approx 1$ : Pd-48-28 ( $\sqrt {K^+} = 0.98$ , $\sqrt {K_{eq}^{t+}} = 0.97$ ), Ps-36-33 ( $\sqrt {K^+} = 1.10$ , $\sqrt {K_{eq}^{t+}} = 0.96$ ) and Ro-24-50 ( $\sqrt {K^+} = 1.82$ , $\sqrt {K_{eq}^{t+}} = 0.94$ ). Dashed lines are for the cases with similar $\sqrt {K_{eq}^{t+}}\approx 1.7$ : Pd-24-50 ( $\sqrt {K^+} = 1.82$ , $\sqrt {K_{eq}^{t+}} = 1.75$ ), Ps-48-38 ( $\sqrt {K^+} = 1.88$ , $\sqrt {K_{eq}^{t+}} = 1.66$ ) and Ro-48-44 ( $\sqrt {K^+} = 2.66$ , $\sqrt {K_{eq}^{t+}} = 1.70$ ). Solid lines are for the cases with similar $\sqrt {K_{eq}^{t+}}\approx 3.5$ : Pd-48-50 ( $\sqrt {K^+} = 3.64$ , $\sqrt {K_{eq}^{t+}} = 3.44$ ), Ps-24-75 ( $\sqrt {K^+} = 5.80$ , $\sqrt {K_{eq}^{t+}} = 3.68$ ) and Ro-48-61 ( $\sqrt {K^+} = 6.28$ , $\sqrt {K_{eq}^{t+}} = 3.55$ ). The dash-dotted lines are for smooth-wall data, and the vertical dashed lines mark the location of the free-flow/substrate interface.

Figure 19. Instantaneous fields of ( $a$ – $i$ ) $u^\prime$ and ( $j$ – $r$ ) $v^\prime$ at $y^+\!\approx \!3$ for the same substrates of figure 18. Columns from left to right correspond to deep porous (Pd), shallow porous (Ps) and rough (Ro) substrates, respectively. (a,b,c) and (j,k,l) Substrates with $\sqrt {K_{eq}^{t+}}\!\approx \!1$ ; (d,e,f) and (m,n,o) substrates with $\sqrt {K_{eq}^{t+}}\!\approx \!1.7$ ; (g,h,i) and (p,q,r) substrates with $\sqrt {K_{eq}^{t+}}\!\approx \!3.5$ . Colours from dark to clear are for the value range $[-2:2]$ relative to the RMS value of the variable at that plane.

Figure 20. Premultiplied spectra $\alpha _x\alpha _z\Phi _{**}$ at (a,c,e) $y^+\!\approx \!3$ and (b,d,f) $y^+\!\approx \!11$ for the same deep porous substrates of figure 18. (a,b) Substrates with $\sqrt {K_{eq}^{t+}}\!\approx \!1$ ; (c,d) substrates with $\sqrt {K_{eq}^{t+}}\!\approx \!1.7$ ; (e,f) substrates with $\sqrt {K_{eq}^{t+}}\!\approx \!3.5$ . Shaded contours are for deep porous (Pd) substrates, yellow dotted lines for shallow porous (Ps) substrates and red solid lines for rough (Ro) surfaces. The contours mark values [0.044:0.044:0.264] relative to the corresponding variance or covariance.

The transpiration-based equivalent permeability, $\sqrt {K_{eq}^{t+}}$ , also characterises well the turbulence for substrates with different depths. In figures 18, 19 and 20, we consider three groups of substrates, each including a deep porous (Pd), a shallow porous (Ps) and a rough (Ro) case with different depth $h/D$ but similar $\sqrt {K_{eq}^{t+}}$ . Figure 18 shows that substrates with similar $\sqrt {K_{eq}^{t+}}$ also share similarity in their free-flow mean velocity profiles, RMS velocity fluctuations and Reynolds stress, in spite of their differences in subsurface flow, to be expected given the different depths and granularities. This similarity can also be observed in instantaneous realisations for the flow just above the interface, as illustrated in figure 19. The flows with similar $\sqrt {K_{eq}^{t+}}$ exhibit similar features for the background turbulence, but differences in the grain-coherent flow, directly attributable to the different grain topologies. The same effects can be observed in the spectral density maps of figure 20, which quantify statistically the similarities in the fluctuations at different lengthscales for the background turbulence, and also display the different intensities and lengthscales of the grain-coherent flow for different topologies.

The collapse observed for $\sqrt {K_{eq}^{t+}}$ , and the comparatively lack of collapse observed for $\sqrt {K_{eq}^{s+}}$ , suggest that effect of the substrate on the overlying turbulence is mainly governed by the wall-normal transpiration triggered by overlying pressure fluctuations. The effect of the interfacial slip, which is characterised by $\sqrt {K_{eq}^{s+}}$ , plays in contrast a minor role. The good collapse with $\sqrt {K_{eq}^{t+}}$ may also suggest that the Darcy–Brinkman model captures reasonably well the fluctuating transpiration induced by overlying pressure waves, even if it fails to capture other features of the flow such as the tangential shear and velocity near the interface (Sanchez-Palencia Reference Sanchez-Palencia1982). It is somewhat surprising that this is the case even for substrates with one or two layers of grains, for which the lack of sufficient depth would in principle preclude the type of homogenisation in $y$ implicit in Darcy–Brinkman. An answer to this is, however, out of the scope of the present paper, where we limit ourselves, as stated above, to using the Darcy–Brinkman solutions to guide us in proposing an otherwise empirical coefficient to collapse the DNS results. We note that the present analysis extends to substrates that exhibit a rough interface due to their intrinsic granularity, and leaves out surfaces that are rough but would not have an obvious permeable substrate counterpart, e.g. wavy walls or smoothly curved surfaces. For those surfaces, we could expect the slip dynamics to play a more dominant role, as occurs also for drag-reducing textures (Gómez-de-Segura et al. Reference Gómez-de-Segura, Sharma and García-Mayoral2018b ; Ibrahim et al. Reference Ibrahim, Gómez-de-Segura, Chung and García-Mayoral2021). For porous substrates such as the present ones, the interfacial roughness can potentially be eliminated by making the surface artificially flat, which would partially suppress the resulting increase in drag (see Kim et al. (Reference Kim, Blois, Best and Christensen2020), or figure 1 and the corresponding discussion in Rosti et al. (Reference Rosti, Cortelezzi and Quadrio2015)). It is also possible to generate an interfacial roughness completely unrelated to the substrate topology and decouple their effects, as in Wangsawijaya et al. (Reference Wangsawijaya, Jaiswal and Ganapathisubramani2023). Both cases are beyond the scope of this study. Likewise, we have also only considered isotropic substrates. Anisotropic substrates such as those of Khorasani et al. (Reference Khorasani, Morteza, Luhar and Bagheri2024) would be beyond the present scope. In the latter, the permeability is tensorial, and the expression in (5.1) for an effective $\sqrt {K_{eq}^{t+}}$ would not be valid. It is possible, however, to find an analogous solution for the anisotropic Darcy–Brinkman equations. We have done so in the past for the case of principal directions aligned with $x$ , $y$ and $z$ (Sharma et al. Reference Sharma, Gómez-de Segura and García-Mayoral2017; Gómez-de-Segura et al. Reference Gómez-de-Segura, Sharma and García-Mayoral2018a ; Gómez-de-Segura & García-Mayoral Reference Gómez-de-Segura and García-Mayoral2019); the solution presented in Appendix D is actually a particularisation of such solutions. A simple explicit expression is not possible in that case, but from our previous work we would expect that the role of $K$ in (5.1) was played by $K_y$ for streamwise preferential substrates (Gómez-de-Segura & García-Mayoral Reference Gómez-de-Segura and García-Mayoral2019), and by $\sqrt {K_x K_y}$ for wall normal preferential ones (Sharma et al. Reference Sharma, Gómez-de Segura and García-Mayoral2017). Here, $K_x$ and $K_y$ are the streamwise and the wall-normal permeability, respectively.

The reader may wonder if the present findings apply only to the regular staggered-cube and mesh topologies for which DNSs have been conducted in this paper, with relatively small grain pitch $L^+\lesssim 50$ . In Appendix F we compare our results with data compiled for different substrate layouts and from previous studies, namely the randomly packed spheres of Zippe & Graf (Reference Zippe and Graf1983) and Karra et al. (Reference Karra, Apte, He and Scheibe2023), the staggered cubes of Kuwata & Suga (Reference Kuwata and Suga2016a ), the collocated spheres of Kim et al. (Reference Kim, Blois, Best and Christensen2020), the reticulated foams of Esteban et al. (Reference Esteban, Rodríguez-López, Ferreira and Ganapathisubramani2022), and the isotropic mesh lattices of Habibi Khorasani et al. (Reference Khorasani, Morteza, Luhar and Bagheri2024). The substrates in those studies have generally larger grain and, therefore, larger $K^+$ , and data for the roughness function was sometimes obtained indirectly, especially for experimental works, but the agreement is generally good. We note that some of those studies were for irregular substrate topologies. We expect our conclusions to hold beyond the homogeneous substrates of our DNSs, so long as the heterogeneity occurs on a scale small enough not to interact with individual turbulent eddies. If heterogeneities were present over lengthscales of ${\sim}100$ viscous units or larger, they would need to be accounted for and would probably require a unique ad hoc analysis for each instance.

5.2. A conceptual regime diagram for finite-depth substrates

The above results and discussion suggest that, for any given substrate topology, its effect on the overlying turbulence can be essentially characterised by $K_{eq}^{t+}$ alone. This coefficient is proportional to the substrate permeability $K^+$ , which encodes the relevant information on the grain topology, but also depends on the substrate depth through $f_{\mathcal {K}t}$ . Based on this, we propose a regime diagram to conceptually illustrate the relationship between porous and rough surfaces for a given grain topology. As an example, in figure 21 we consider our staggered-cube topology with gap-to-pitch ratio $g/L=1/2$ , corresponding to a constant $\sqrt {K}\approx 0.076 L$ . For other topologies, the regime diagram would be qualitatively similar.

Figure 21. Substrate regime diagram for a staggered-cube topology with $g/L=1/2$ . The shaded contours are for the transpiration-based equivalent permeability $\sqrt {K_{eq}^{t+}}$ , taken as a surrogate for the drag increase. The dashed and solid contour lines are for $f_{\mathcal {L}s}$ and $f_{\mathcal {K}t}$ , respectively, for values [0.1 : 0.1 : 0.9] from red to blue. The results have been obtained for a characteristic wavelength for the overlying turbulence $\lambda _p^+=200$ . The dotted straight line represents impermeable rough surfaces with $h=L/2$ . The dashed lines represent boundaries between regimes. The markers represent the present twelve DNSs with this topology, $g/L = 1/2$ .

The diagram portrays, as a function of $\sqrt {K^+}$ and $h^+$ , the equivalent permeability $K_{eq}^{t+}$ , taken as a surrogate for the roughness function $\triangle U^+$ . To illustrate the effect of substrate depth on slip and transpiration, we also portray the corresponding values of the attenuation coefficients $f_{\mathcal {L}s}$ and $f_{\mathcal {K}t}$ . To construct the diagram, these have been calculated for a characteristic exciting wavelength $\lambda _p^+\approx 200$ , as discussed in § 5.1. Note that this would be valid so long as the signature of the overlying pressure fluctuations at the interface plane retained roughly its characteristic size for smooth-wall flows. This is the case for the texture sizes studied here, $L^+\lesssim 50$ , as shown in figures 16 and 20, but would eventually break down for larger sizes. Therefore, in the following, we only consider substrates with $\sqrt {K^+}\lesssim 10$ or $L^+\lesssim 100$ , which are the scope of the present study and for which we would expect the diagram to be representative.

In the diagram, a dotted line marks rough surfaces with cubic roughness elements of pitch $L$ and height $h=L/2$ , which includes our four rough DNSs with the topology under consideration, $g/L=1/2$ . Following this line for increasing $L^+$ is equivalent to increasing the Reynolds number for a fixed surface, thus increasing its dimensions in viscous units, and its roughness function accordingly, as if following a Moody chart (Moody Reference Moody1944). Substrates to the right of this line would also be rough, featuring roughness elements with the same pitch and element width and gap but smaller depth, $h\lt L/2$ . The isocontours for $f_{\mathcal {L}s}$ indicate that, in terms of depth, the slip properties of the surfaces with $h=L/2$ would already be maxed out, i.e. beyond the line labelled ‘slip saturated’: no further increases would take place when increasing $h$ further. Meanwhile, the shallower rough substrates with $h\lt L/2$ would experience comparatively reduced slip.

As discussed in § 5.1, however, the effect of slip through $K_{eq}^{s+}$ is secondary, and the drag is mainly governed by $K_{eq}^{t+}$ , for which the relevant attenuating function is the transpiration one, $f_{\mathcal {K}t}$ . The latter is a continuous, smooth function of the substrate depth, which tends to zero for vanishing depth and to unity for deep substrates. Its isocontours show that, while $f_{\mathcal {L}s}$ saturates at much shallower depths $h$ , i.e. already for simply rough, impermeable surfaces, $f_{\mathcal {K}t}$ saturates only for deeper substrates. Its isocontours indicate that, for the present textures with $\sqrt {K^+}\lesssim 10$ , the saturation depth is essentially independent of $K^+$ or $L^+$ , and is roughly $h^+\approx 50$ . This can be traced to the fact that $h$ is scaled by $\lambda _p$ in the expression for $f_{\mathcal {K}t}$ , and that $\lambda _p^+\approx 200$ independently of the substrate, as discussed above. Whether a substrate is shallow or sufficiently deep is governed by whether pressure fluctuations of this wavelength would penetrate deeper in the absence of an impermeable floor at a finite depth $y^+=-h^+$ . Perhaps counterintuitively, this depends on the depth in viscous units, rather than in permeability lengths or number of inclusion layers.

From this, we can distinguish a series of regions in the diagram which are characterised by different flow regimes. For substrates deeper than $h^+\approx 50$ , the flow would perceive the substrate as deep enough to exhibit its permeable character fully. We refer to these as ‘sufficiently deep porous substrates’, for which any further increase in depth would not result in any significant changes to the flow. For substrates shallower than this but deeper than the rough surfaces with cubic inclusions, $h=L/2$ , the substrate would exhibit an intermediate character, where the transpiration is partially suppressed compared to a deeper substrate with the same $K^+$ . We refer to these as ‘shallow porous substrates’. Substrates with $h=L/2$ would be so shallow that they are conventional rough, impermeable surfaces. Substrates even shallower than $h=L/2$ would likewise be rough, but with shorter-height roughness elements, further suppression of transpiration and, in addition, suppression of slip.