The Ultimate State of Turbulent Permeable-Channel Flow

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Fluid J. in publication for consideration Under e words: Key is wall permeable unity. the to set of property the and parameter fluid, the of density walls the on a velocity heat for mean bulk constant performed of flow been shear turbulent have heated internally simulations numerical Direct Motoki Shingo i.e. \ \ rdaeSho fEgneigSine sk University, Osaka Science, Engineering of School Graduate 1 1 sacneune h liaeha rnfri civd i achieved, is transfer heat ultimate the consequence, a As . ewe aall steml osi n emal wall permeable and no-slip isothermal, parallel, between , $ E ( D = D 1 1 ± \ ) 1 ota h alrdsiainlaw dissipation Taylor the that so , V? VD o equivalently (or anraporaet ril yewl perhr ntypes in here appear will type article to appropriate Banner / ubln etadmmnu rnfri emal hne f channel permeable in transfer momentum and heat Turbulent ubln iig iigehneet ublnesimulat Turbulence enhancement, Mixing mixing, Turbulent 1 d h eprtr is temperature The . 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Introduction One of the major interests among many researchers in engineering and geophysics is in the effect of wall surface properties on heat and momentum transfer in turbulent shear flows. In particular, turbulent flows over rough walls have been extensively investigated experimentally and numerically (see Jiménez 2004). In turbulent flows, surface roughness on a wall usually increases the drag thereon in comparison to a smooth wall. In the fully rough regime at high Reynolds numbers '4, the friction coefficient 2 5 can be independent of '4 as seen in the 0 Moody diagram (Moody 1944). The scaling 2 5 ∼ '4 corresponds to the Taylor dissipation law implying the inertial energy dissipation independent of the kinematic viscosity a. It is well known that in wall turbulence there exists the similarity between heat and momentum −2/3 transfer, empirically represented by (C ∼ %A 2 5 (Chilton & Colburn 1934) between the Stanton number (C (a dimensionless wall heat flux) and 2 5 , where %A is the Prandtl number. In rough-wall flows, however, (C decreases as '4 increases even in the fully rough regime 0 where 2 5 ∼ '4 (Dipprey & Sabersky 1963; Webb et al. 1971). This dissimilarity is a consequence of flow separation from roughness elements. In the fully rough regime at high '4 (for %A ∼ 1), the viscous sublayer separates from the roughnesselements to yield pressure drag on the rough wall, whereas the thin thermal conduction layer without any vortices is stuck to the rough surface (MacDonald et al. 2019a). The scaling (C ∼ '40 in forced convection means that a wall heat flux is independent of the thermal diffusivity ^. It relates to the well known ultimate scaling #D ∼ %A1/2'01/2 (also implying the ^-independent wall heat flux) suggested by Spiegel (1963); Kraichnan (1962) for turbulent thermal convection at extremely high '0, where #D is the Nusselt number, and '0 is the Rayleigh number. The ultimate scaling has been enthusiastically discussed in turbulent Rayleigh–Bénard convection (see Ahlers et al. 2009; Chillà & Schumacher 2012; Roche 2020). In thermal convection, it has been found that the roughness of heat transfer surface transiently yields the scaling #D ∼ %A1/2'01/2 in the limited range of '0 where the thermal conduction layer thickness is comparable to the size of roughness elements (Zhu et al. 2017, 2019; MacDonald et al. 2019b). However, it is still an open question whether or not the asymptotic ultimate scaling can be achieved in forced or thermal convection. Recently, Kawano et al. (2021) have found that the ultimate heat transfer #D ∼ %A1/2'01/2 can be achieved in turbulent thermal convection between permeable walls. In their study, the wall-normal transpiration velocity on the wall is assumed to be proportional to the local pressure fluctuations. This permeable boundary condition was originally introduced by Jiménez et al. (2001) to mimic a Darcy-type porous wall with a constant-pressure plenum chamber outside. They have investigated turbulent momentum transfer in permeable-channel flow, and found that the wall permeability leads to large-scale spanwise rolls over the permeable wall, significantly enhancing momentum transfer. By linear stability analyses, Jiménez et al. (2001) have clarified that the formation of the large-scale spanwise rolls originates from the Kelvin–Helmholtz type shear-layer instability over the permeable wall. Such large-scale turbulence structures have been observed numerically and experimentally in shear flows over porous media (see e.g. Suga et al. 2018; Nishiyama et al. 2020). In this study, we investigate the scaling properties of heat and momentum transfer in turbulent permeable-channel flow and report that the wall permeability can bring about 0 the ultimate state represented by the a-independent dissipation 2 5 ∼ '4 as well as the ^-independent heat flux (C ∼ '40.

2. Governing equations and numerical simulations Let us consider turbulentheatand momentumtransfer in internally heated shear ﬂow between parallel, isothermal, no-slip and permeable walls. The coordinates, G, H and I (or G1, G2 and 3

G3) are used for the representation of the streamwise, the wall-normal and the spanwise directions, respectively. The origin is on the midplane between the two walls positioned at H = ±ℎ. The corresponding components of the velocity u(x, C) are given by D,E and F (or D1,D2 and D3), respectively. The temperature \(x, C) is supposed to be a passive scalar. The governing equations are the Navier–Stokes equations for the divergence-free velocity and the energy equation for the temperature, ∇· u = 0, (2.1) mu 1 + (u · ∇)u = a∇2u + 5 e − ∇?, (2.2) mC G d m\ @ + (u · ∇)\ = ^∇2\ + , (2.3) mC d2 ? where ?(x, C) isthefluctuatingpressurewithrespecttothedrivingpressure %(G,C), d,a,^ and 2 ? are the mass density, the kinematic viscosity, the thermal diffusivity and the specific heat at constant pressure of the fluid, respectively. Here, vector eG is a unit vector in the streamwise direction, and 5 (C) (= −m%/mG/d > 0) and @(C) (> 0) are the spatially uniform driving force and internal heat source to maintain constant bulk mean velocity and temperature, D1 and \1, respectively. The velocity and temperature fields are supposed to be periodic in the G- and I-directions with the periods, !G and !I. On the permeable wall the wall-normal velocity E is assumed to be proportional to the local pressure fluctuation ? (Jiménez et al. 2001; Kawano et al. 2021). We impose the no-slip, permeable and isothermal conditions, ?(H = ±ℎ) D(H = ±ℎ) = F(H = ±ℎ) = 0, E(H = ±ℎ) = ±V ; \(H = ±ℎ) = 0, (2.4) d on the walls, where V (> 0) represents the permeability coefficient, and the impermeable conditions E(H = ±ℎ) = 0 are recovered for V = 0, while V → ∞ implies zero pressure fluctuations and an unconstrained wall-normal velocity. Note that the pressure fluctuation with zero mean instantaneously ensures a zero net mass flux through the permeable wall. The flow is characterised by the bulk Reynolds number '41 = 2ℎD1/a, the Prandtl number %A = a/^ and the dimensionless permeability VD1. The wall heat flux @F and the wall shear stress gF are quantified by the Stanton number (C and the friction coefficient 2 5 defined as 2@ D \ 2g D2 (C ≡ F = 2 g g , 2 ≡ F = 2 g , (2.5) d2 D \ D \ 5 2 2 ? 1 1 1 1 dD1 D1 = 1/2 = where Dg (∓adhDiGIC /dH|H=±ℎ) and \g ∓(^/Dg)dh\iGIC /dH|H=±ℎ are the friction velocity and the friction temperature, respectively. Hereafter, h·iGIC and h·iGHIC represent the plane-time (GIC-) average and the volume-time (GHIC-) average, respectively. We conduct direct numerical simulations (DNS) for turbulent heat and momentum transfer in shear flow between permeable walls. The present DNS code is based on the one developed for turbulent thermal convection between permeable walls (Kawano et al. 2021). The governing equations (2.1)–(2.3) are discretised employing the spectral Galerkin method based on the Fourier–Chebyshev expansions. Time advancement is performed with the aid of the implicit Euler scheme and the third-order Runge–Kutta scheme for the diffusion terms and the others, respectively. In this paper, we present the results obtained in the impermeable case VD1 = 0, the less-permeable case VD1 = 0.3 and the permeable case 3 4 VD1 = 0.5 for %A = 1. The simulations are carried out at '41 = 4 × 10 –4 × 10 for the periodicity (!G , !I) = (2cℎ,cℎ). The spatial grid spacings are less than 10 wall units in all the directions, and the data are accumulated for the duration of more than 30 wall units at VD1 = 0.5. 4 3. Heat flux, shear stress and energy budget In this section, we show the total heat flux, the total shear stress and the total energy budget in internally heated shear flow between permeable walls. We decompose the velocity u = u u′ and temperature into an GIC-average and a fluctuation about it as h iGIC + and = ′ \ h\iGIC + \ . Substituting them to (2.2) and (2.3), integrating their GIC-averages with respect to H, and supposing that the flow is statistically stationary, we obtain the total heat flux and the total shear stress, respectively, dh\i h@i GIC ′ ′ = C ^ −h\ E iGIC − (H + ℎ) + Dg \g, (3.1) dH d2 ? dhDi a GIC −hD′E′i = −h 5 i (H + ℎ) + D2 , (3.2) dH GIC C g ′ ′ where h·iC stands for the time average. Note that the turbulent heat flux h\ E iGIC and the ′ ′ = Reynolds shear stress −hD E iGIC have vanished on the walls (H ±ℎ) even in the permeable case due to the isothermal and no-slip conditions. The effects of the wall permeability appear in the total energy budget. By taking the GHIC-average of an inner product of the Navier– Stokes equation (2.2) with the velocity u and taking account of the boundary conditions (2.4), we obtain the total energy budget equation 2 D3 H=ℎ = 5 1 = 1 2 2 1 3 h 5 iC D1 n + E GIC + E GIC + E GIC , (3.3) 2 ℎ 2ℎV H=ℎ H=−ℎ 4ℎ h i H=−ℎ

= 2 where n (a/2)h mD8/mG 9 + mD 9/mG8 iGHIC is a total energy dissipation rate per unit mass. The leftmost equality is given by (2.5) and (3.2) for H = ℎ. The second term on the right- hand side of (3.3) denotes pressure power on the permeable walls. This term is strictly non-negative, being an energy sink. The third term represents outﬂow kinetic energy across the permeable walls. In the present DNS we have conﬁrmed that the second term is at most 1% of n whereas the third term is less than 0.01% of n. It turns out that the introduction of the wall permeability does not bring about any extra energy inputs.

4. Results and discussion Let us first consider the effects of the wall permeability on the Stanton number (C and the friction coefficient 2 5 . Figure 1 shows (C and 2 5 as a function of '41. In the impermeable case VD1 = 0 the present DNS data are in good agreement with the numerical result by Orlandi et al. (2015) for impermeable-channel flow in larger periodic domains (!G , !I) = 4 4 (12cℎ, 4cℎ) at '41 < 10 and (!G, !I) = (6cℎ, 2cℎ) at '41 > 10 .Asthewallpermeability increases from VD1 = 0 to VD1 = 0.5, not only the momentum transfer but the heat transfer are enhanced over the entire range of '41. In the less-permeable case VD1 = 0.3, (C and −1/4 2 5 can be seen to scale with '41 as in the impermeable case even at high Reynolds 4 numbers '41 & 10 , and they exhibit close similarity between heat and momentum transfer, i.e. (C ≈ 2 5 , as in impermeable-channel flow. In the permeable case VD1 = 0.5, on the other 0 0 hand, the ultimate state, (C ∼ '41 and 2 5 ∼ '41, can be observed specifically at the higher 4 Reynold number than the critical value '41 ∼ 10 , below which the classical similar scaling −1/4 (C ≈ 2 5 ∼ '41 appear as in the impermeable and less-permeable cases. Next, we present the remarkable difference in the mean temperature and velocity profiles in the less-permeable case VD1 = 0.3 and the permeablecase VD1 = 0.5. The mean temperature and velocity profiles normalised by the friction temperature \g and the friction velocity Dg as a function of the distance to the lower wall, (H + ℎ)/(a/Dg), are shown in figure 2. Focus on Fluids articles must not exceed this page length 5

Figure 1: Stanton number (C and friction coefficient 2 5 as a function of bulk Reynolds number '41. The filled red and open blue symbolds represent (C and 2 5 , respectively, in the permeable case VD1 = 0 (diamonds), less-permeable case VD1 = 0.3 (squares) and permeable case VD1 = 0.5 (circles) for Prandtl number %A = 1. The open black diamonds denote 2 5 in the DNS taken from Orlandi et al. (2015). The = −1 = −1/4 solid and dashed lines indicate 2 5 12'41 for laminar flow and the empirical formula 2 5 0.073'41 (Dean 1978) for turbulent flow.

(a) (b)

Figure 2: Mean temperature and velocity proﬁles as a function of the distance to the lower wall (H+ℎ)/(a/Dg ) 3 4 in(a,c) the less-permeable case VD1 = 0.3 and (b,d) the permeable case VD1 = 0.5 at 4×10 6 '41 6 4×10 for %A = 1. The Reynolds number '41 increases in the direction of the arrows. The dashed lines denote 4 the DNS data (Pirozzoli et al. 2016) in impermeable-channel ﬂow at '41 = 3.96 × 10 . The solid lines = = represent the logarithmic law h\iGIC /\g hDiGIC /Dg (1/0.41) ln [(H + ℎ)/(a/Dg )] + 5.2. 6 (a) (b)

(e) ( f )

Figure 3: RMS temperature and velocity normalised by \1 and D1 as a fuction of the distance to the lower wall H/ℎ + 1 in (a,c,e) the less-permeable case VD1 = 0.3 and (b,d,f ) the permeable case VD1 = 0.5 at 3 4 4×10 6 '41 6 4×10 for %A = 1. The Reynolds number '41 increases in the direction of the arrows. The 4 dashed lines denote the DNS data (Pirozzoli et al. 2016) in impermeable-channel ﬂow at '41 = 3.96 × 10 .

The black dashed curves represent the mean profiles obtained by Pirozzoli et al. (2016) in impermeable-channel flow for %A = 1. In the less-permeable case (figure 2a,c), the mean temperature and velocity profiles scale with a, \g and Dg in the near-wall region, exhibiting the logarithmic law with the prefactor 1/0.41 and intercept 5.2 at (H + ℎ)/(a/Dg) & 30, commonly observed in wall turbulence. In the permeable case (figure 2b,d), on the other hand, the normalised mean temperature and velocity at (H + ℎ)/(a/Dg) & 1 decrease as 4 '41 increases at supercritical Reynolds numbers '41 & 10 . It has been confirmed that in the bulk region, the mean temperature and velocity profiles scale with ℎ, \1 and D1 in the permeable case (figure not shown), markedly differing from the scaling property in wall turbulence. Since the heat and momentum transfer on the permeable wall is dominated by thermal conduction and viscous diffusion due to the isothermal and no-slip boundary conditions, all the profiles in the less-permeable and permeable cases collapse onto a single line in the linear sublayer (H + ℎ)/(a/Dg) . 1. 7 (a) (b)

Figure 4: Instantaneous flow and thermal structures in the permeable VD1 = 0.5 for %A = 1 at (a) the 3 4 subcritical Reynolds '41 = 8 × 10 and (b) the supercritical Reynolds number '41 = 4 × 10 . The grey and dark grey objects represent the isosurfaces of the positive second invariant of the velocity gradient tensor 2 3 2 4 ′ (a) &/(Dg/ℎ) = 2 × 10 and (b) &/(Dg/ℎ) = 3 × 10 , and of the temperature fluctuation (a) \ /\g = 4 ′ and (b) \ /\g = 6, respectively. The colour represents the wall-normal velocity E on the walls H = ±ℎ. The = vectors indicate the spanwise-averaged velocity fluctuations (hDiI − hDiGIC , hEiI ) on I/ℎ 0.

= ′2 1/2 Figure 3 shows the root mean square (RMS) temperature \rms h\ iGIC , the RMS = ′2 1/2 = ′2 1/2 streamwise velocity Drms hD iGIC and the RMS wall-normal velocity Erms hE iGIC normalised by the bulk mean temperature \1 and the bulk mean velocity D1 as a function of H/ℎ + 1. In the less-permeable case (figure 3a,c,e), the RMS temperature and the RMS 4 velocity exhibit almost the same behaviour, and the RMS velocities at '41 = 4 × 10 are in 4 good agreement with those at '41 = 3.96×10 in impermeable-channelflow (Pirozzoli et al. 2016). As '41 increases, the peaksof the profiles shift closer to the wall, and the RMS values in the bulk region decrease relatively with respect to \1 or D1, as commonly observed in wall turbulence. In the permeable case (figure 3b,d,f ), on the other hand, it can be seen that the temperature and velocity fluctuations are significantly enhanced at the supercritical 4 Reynolds number '41 & 10 . In the bulk region, the RMS temperature and velocity profiles are found to scale with ℎ, \1 and D1, being largely distinct from the scaling properties in wall turbulence. The intense wall-normal transpiration is induced on the permeable wall; however,it is suppresseddue to the presence of the wall, so that its intensity is approximately 4 1% of D1 even at '41 & 10 . Let us now draw attention to turbulence structures over the permeable wall. Instantaneous 3 flow and thermal structures at the subcritical Reynolds number '41 = 8 × 10 and at 4 the supercritical Reynolds number '41 = 4 × 10 in the permeable case VD1 = 0.5 are shown in figure 4. The grey objects represent the small-scale vortex structures identified in terms of the positive isosurfaces of the second invariant of the velocity gradient tensor & = ′ −mD8/mG 9 mD 9/mG8/2, and the dark grey objects show the high-temperature regions, \ > 0. 3 At '41 = 8 × 10 , there are almost the same turbulence structures as those observed in wall turbulence. The streamwise vortices and streaks near the walls appear roughly homogeneously 4 in the wall-parallel directions. At '41 = 4×10 for which the ultimate state has been observed, contrastingly, very different large-scale turbulence structures can be seen. The large-scale structures have been observed travelling downstream. The colour in the figures represents the wall-normal velocity on the permeable walls, exhibiting strong coherence in the spanwise direction. The small-scale vortex structures cluster around the blowing region, whereas high temperature concentrates in the suction region. The vectors on the plane I/ℎ = 0 show the spanwise-averaged velocity fluctuations, (hDiI − hDiGIC , hEiI), indicating large- scale spanwise rolls with the length scale comparable with the channel half width ℎ. This remarkable turbulence modulation originates from the Kelvin–Helmholtz type instability over the permeable wall (Jiménez et al. 2001). Such large-scale coherent structures have also 8 (a) (b)

Figure 5: Spanwise-averaged instantaneous temperature h\iI /\g and streamwise velocity hDiI /Dg near the lower wall at the same instant as in ﬁgure 4 in the permeable case VD1 = 0.5. The white lines indicate the = = = 3 isolines of h\iI /\g 0–4 and hDiI/Dg 0–4. (a,c) the subcritical Reynolds number '41 8 × 10 and 4 (b,d) the supercritical Reynolds number '41 = 4 × 10 .

been observed in shear flows in porous channel and duct (see e.g. Kuwata & Suga 2017; Kuwata et al. 2020). In the bulk region the large-scale rolls of the length scale ℎ, which undergo the velocity difference of $(D1), can induce the velocity fluctuations of $(D1), as 3 shown in figure 3 (b,d,f ). Accordingly, the Taylor dissipation law n ∼ D1/ℎ can hold, and 0 the total energy budget equation (3.3) provides us with 2 5 ∼ '41. In figure 5 are shown the spanwise-averaged instantaneous temperature and streamwise = velocity in the viscous sublayer, respectively. The white isolines, hDiI/Dg 0–4 and = h\iI/\g 0–4, indicate the thermal conduction layer and the (viscous) linear sublayer. Note that the null isolines have been observed to be always stuck to the wall surface. At the 4 supercritical Reynolds number '41 = 4 × 10 , the temperature and velocity distributions 3 near the wall differ greatly from those at the subcritical Reynolds number '41 = 8 × 10 , and significantly large-amplitude temperature and velocity fluctuations are induced even in 0 the close vicinity of the wall, (H + ℎ)/(a/Dg) ∼ 10 . The near-wall low-temperature and low-velocity fluids are blown up from the permeable wall, while the high-temperature and high-velocity fluids are sucked into the wall, highly inducing the turbulent heat flux and the Reynolds shear stress. In spite of such significant enchancement of heat and momentum transfer, there is no flow separation over the permeable wall unlike flows over a rough wall (see the stuck isolines of the null velocity). This is because of the relaxation of near-wall high pressure by the wall permeability, and implies the similarity between heat and momentum transfer. Heat transfer can be enhanced by the large-scale spanwise rolls comparably with momentum transfer, so that temperature fluctuations are of theorderof \1. As a consequence, 0 the wall-normal heat flux scales with D1\1, leading to the ultimate scaling (C ∼ '41. Finally, following the argument in Kawano et al. (2021), we would like to suggest the possibility of the ultimate state in practical applications. Let us consider a porous wall consisting of many fine through holes in the wall-normal direction with a constant-pressure plenum chamber outside. Supposing the flow through the holes to be the laminar Hagen– Poiseuille flow, the permeability coefficient V can be expressed rigorously as V = 32/(32a;), 9 and the dimensionless permeable parameter is given by 1 3 2 ℎ VD1= '41, (4.1) 32 ℎ ; where 3 and ; represent the diameter of the holes and the thickness of the wall, respectively. Taking into consideration that all the pressure power on the permeable wall in channel flow should be consumed to drive the viscous flow in the holes, the thickness of the porous wall would be estimated as ;/ℎ ∼ 1 (Kawano et al. 2021). Substitution of ;/ℎ ∼ 1 in 1/2 −1/2 (4.1) yields 3/ℎ ∼ (VD1) '41 . Thus, the porous wall with the geometry of ;/ℎ ∼ 1 −3 −2 0 and 10 . 3/ℎ . 10 could be characterised by the permeability parameter VD1 ∼ 10 at 4 6 10 . '41 . 10 ,wheretheultimatestateshouldbeobserved.Themeanvelocity in the holes −2 could be estimated to be E< ∼ 10 D1, since the RMS wall-normal velocity on the permeable 4 6 wall is approximately 1% of D1 at supercritical '41 for VD1 = 0.5. At 10 . '41 . 10 −2 −2 the Reynolds number of the flow in the holes, E<3/a ∼ 10 D13/a ∼ 10 '413/ℎ, is 0 1 in the range 10 . E<3/a . 10 , where the flow is laminar and fulfils the ‘Darcy law’. Therefore, we believe that the ultimate state can be achieved in the above realistic wall-flow configuration.

5. Summary and outlook We have investigated turbulent heat and momentum transfer numerically in internally heated permeable-channel flow of the constant bulk mean velocity and temperature, D1 and \1, for %A = 1. On the permeable walls at H = ±ℎ the wall-normal velocity is assumed to be proportional to the local pressure fluctuations, i.e. E(H = ±ℎ) = ±V?/d. In the permeable channel VD1 = 0.5, we have found the critical transition of the scaling of the Stanton number (C and the friction coefficient 2 5 from the Blasius empirical law (C ≈ −1/4 0 0 2 5 ∼ '41 to the ultimate state of (C ∼ '41 and 2 5 ∼ '41 at the bulk Reynolds number 4 4 '41 ∼ 10 . At the subcritical Reynolds number '41 . 10 , there are no significant changes in turbulence statistics or structures from the impermeable case VD1 = 0. The ultimate state found at the supercritical Reynolds number '4 & 104 is attributed to the appearance of large-scale spanwise rolls stemming from the Kelvin–Helmholtz type shear-layer instability over the permeable walls. On the permeable wall surface the blowing and suction are excited by the Kelvin–Helmholtz wave which is roughly uniform in the spanwise direction. Near-wall low-temperature and low-velocity fluids are blown up from the permeable wall, while the high-temperature and high-velocity fluids are sucked into the wall, largely producing the turbulent heat flux and the Reynolds shear stress. Such remarkable turbulence modulation 0 extends to the close vicinity of the wall, |H ± ℎ|/(a/Dg ) ∼ 10 . Unlike rough walls, there is no flow separation, so that heat transfer is enhanced comparably with momentum transfer. The key to the achievement of the ultimate state in supercritical permeable-channel flow is the significant heat and momentum transfer enhancement without flow separation by large-scale spanwise rolls of the length scale of $(ℎ). The large-scale rolls can induce the large-amplitude velocity fluctuations of $(D1) and similarly the temperature fluctuations of 3 0 $(\1), leading to the Taylor dissipation law n ∼ D1/ℎ (or equivalently 2 5 ∼ '41) and the 0 ultimate scaling @F /(d2 ?) ∼ D1\1 (or equivalently (C ∼ '41). In this study the ultimate state has been achieved in internally heated permeable-channel flow for the permeability VD1 = 0.5 and the streamwise period !G = 2cℎ. If we consider a different thermal configuration, e.g. constant temperature difference Δ\ between the permeable walls, the same large-scale rolls appear to induce large-amplitude temperature fluctuations of $(Δ\), so that the ultimate scaling @F /(d2 ?) ∼ D1Δ\ should be achieved 10 as well. Concerning the dependence of the ultimate state on VD1 and !G, our preliminary study has shown that slight reduction to VD1 = 0.45 delays the onset of the ultimate state 4 until '41 ∼ 2 × 10 and that the longer !G = 4cℎ can occasionally accommodate the larger streamwise wavelength of the spanwise rolls, yielding the lower onset '41 and greater values of the prefactor of the ultimate scaling. A detailed examination is left for a future study.

Acknowledgements This work was supported by the Japanese Society for Promotion of Science (JSPS) KAKENHI Grant Numbers 19K14889 and 18H01370.

REFERENCES Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh– Bénard convection. Rev. Mod. Phys. 81, 503–537. Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35 (58). Chilton, T. H. & Colburn, A. P. 1934 Mass transfer (absorption) coefficients prediction from data on heat transfer and fluid friction. Ind. Eng. Chem. 26, 1183–1187. Dean, R. B. 1978 Reynolds number dependence of skin friction and other bulk flow variables in two- dimensional rectangular duct flow. J. Fluids Engng. 100, 215–223. Dipprey, D. F. & Sabersky, R. H. 1963 Heat and momentum transfer in smooth and rough tubes at various Prandtl numbers. Int. J. Heat and Mass Transfer 6, 329–353. Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173–196. Jiménez, J., Uhlmann, M., Pinelli, A. & Kawahara, G. 2001 Turbulent shear flow over active and passive porous surfaces. J. Fluid Mech. 442, 89–117. Kawano, K., Motoki, S., Shimizu, M. & Kawahara, G. 2021 Ultimate heat transfer in ‘wall-bounded’ convective turbulence. J. Fluid Mech. 914, A13. Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5 (1374). Kuwata, Y. & Suga, K. 2017 Direct numerical simulation of turbulence over anisotropic porous media. J. Fluid Mech. 831, 41–71. Kuwata, Y., Tsuda, K. & Suga, K. 2020 Direct numerical simulation of turbulent conjugate heat transfer in a porous-walled duct flow. J. Fluid Mech. 904, A9. MacDonald, M., Hutchins, N. & Chung, D. 2019a Roughness effects in turbulent forced convection. J. Fluid Mech. 861, 138–162. MacDonald, M., Hutchins, N., Lohse, D. & Chung, D. 2019b Heat transfer in rough-wall turbulent thermal convection in the ultimate regime. Phys. Rev. Fluids 4 (071501(R)). Moody, L. F. 1944 Friction factors for pipe flow. Trans. ASME 66, 671–684. Nishiyama, Y., Kuwata, Y. & Suga, K. 2020 Direct numerical simulation of turbulent heat transfer over fully resolved anisotropic porous structures. Int. J. Heat and Fluid Flow 81 (108515). Orlandi, P., Bernardini, M. & Pirozzoli, S. 2015 Poiseuille and Couette flows in the transitional and fully turbulent regime. J. Fluid Mech. 770, 424–441. Pirozzoli, S., Bernardini, M. & Orlandi, P. 2016 Passive scalars in turbulent channel flow at high Reynolds number. J. Fluid Mech. 788, 614–639. Roche, P.-E. 2020 The ultimate state of convection: a unifying picture of very high Rayleigh numbers experiments. New J. Phys. 22 (073056). Spiegel, E. A. 1963 A generalization of the mixing-length theory of thermal convection. Astrophys. J. 138, 216–225. Suga, K., Okazaki, Y., Ho, U. & Kuwata, Y. 2018 Anisotropic wall permeability effects on turbulent channel flows. J. Fluid Mech. 855, 983–1016. Webb, R. L., Eckert, E. R. G. & Goldstein, R. J. 1971 Heat transfer and friction in tubes with repeated-rib roughness. Int. J. Heat and Mass Transfer 14, 601–617. Zhu, X., Stevens, R. J. A. M., Shishkina, O., Verzicco, R. & Lohse, D. 2019 Roughness effects in turbulent forced convection. J. Fluid Mech. 869, R4. Zhu, X., Stevens, R. J. A. M., Verzicco,R. & Lohse, D. 2017 Roughness-facilitated local 1/2 scaling does not imply the onset of the ultimate regime of thermal convection. Phys. Rev. Lett. 119 (154501).

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