Heat transfer in rough-wall turbulent thermal convection in the ultimate regime

TL;DR

This study investigates heat transfer in turbulent thermal convection over rough walls, revealing that the ultimate regime with maximal heat transfer efficiency is unlikely to be achieved at finite Rayleigh numbers due to fundamental differences from momentum transfer.

Contribution

It provides a new theoretical analysis predicting heat transfer scaling in rough-wall turbulent convection, highlighting differences from momentum transfer and the unlikelihood of reaching the ultimate regime.

Findings

Heat transfer scales as Nu ~ Ra^{0.42} in rough turbulent convection.

Heat transfer does not reach the ultimate regime Nu ~ Ra^{1/2} at finite Ra.

The absence of pressure drag in temperature equations causes different scaling from momentum transfer.

Abstract

Heat and momentum transfer in wall-bounded turbulent flow, coupled with the effects of wall-roughness, is one of the outstanding questions in turbulence research. In the standard Rayleigh-B\'enard problem for natural thermal convection, it is notoriously difficult to reach the so-called ultimate regime in which the near-wall boundary layers are turbulent. Following the analyses proposed by Kraichnan [Phys. Fluids vol 5., pp. 1374-1389 (1962)] and Grossmann & Lohse [Phys. Fluids vol. 23, pp. 045108 (2011)], we instead utilize recent direct numerical simulations of forced convection over a rough wall in a minimal channel [MacDonald, Hutchins & Chung, J. Fluid Mech. vol. 861, pp. 138--162 (2019)] to directly study these turbulent boundary layers. We focus on the heat transport (in dimensionless form, the Nusselt number $Nu$) or equivalently the heat transfer coefficient (the Stanton number $C_h$). Extending the analyses of Kraichnan and Grossmann & Lohse, we assume logarithmic temperature profiles with a roughness-induced shift to predict an effective scaling of $Nu \sim Ra^{0.42}$, where $Ra$ is the dimensionless temperature difference, corresponding to $C_h \sim Re^{-0.16}$, where $Re$ is the centerline Reynolds number. This is pronouncedly different from the skin-friction coefficient $C_f$, which in the fully rough turbulent regime is independent of $Re$, due to the dominant pressure drag. In rough-wall turbulence the absence of the analog to pressure drag in the temperature advection equation is the origin for the very different scaling properties of the heat transfer as compared to the momentum transfer. This analysis suggests that, unlike momentum transfer, the asymptotic ultimate regime, where $Nu\sim Ra^{1/2}$, will never be reached for heat transfer at finite $Ra$.

Equations

$${\overline{u}}^{+}$$

$${\overline{\theta }}^{+}-{\overline{\theta }}_w^{+}$$

$$U^{+}\equiv {\overline{u}}^{+}\left(z=\frac{L}{2}\right)$$

$${\Theta }^{+}\equiv {\overline{\theta }}^{+}\left(z=\frac{L}{2}\right)-{\overline{\theta }}_w^{+}$$

$$U_{FR}^{+}$$

$${\Theta }_{FR}^{+}$$

Full text

I Introduction

Heat transfer in wall-bounded turbulent flow is one of the outstanding problems in turbulence, both from a fundamental and an applied point of view. The canonical system to study it is Rayleigh–Bénard (RB) convection Ahlers et al. (2009a); Lohse and Xia (2010); Chilla and Schumacher (2012), i.e., the flow in a container heated from below and cooled from above. Here the key question is: how does the heat transfer (in dimensionless form, the Nusselt number $Nu$) scale with the temperature difference between the top and bottom walls (in dimensionless form, the Rayleigh number $Ra$)? And how does this scaling change with wall-roughness? For smooth walls in the so-called classical regime, in which the boundary layers (BLs) are of Prandtl–Blasius (i.e. basically laminar) type, the dependencies are reasonably understood along the unifying theory of thermal convection Grossmann and Lohse (2000, 2001); Ahlers et al. (2009a); Stevens et al. (2013). However, the situation is much less clear in the so-called ultimate regime, in which the BLs become turbulent Kraichnan (1962); Spiegel (1971); Grossmann and Lohse (2011, 2012) and the heat transfer is thus enhanced. In this regime Kraichnan Kraichnan (1962) predicted that $Nu\sim Ra^{1/2}[\log(Ra)]^{-3/2}$. Later, Grossmann & Lohse Grossmann and Lohse (2011, 2012) used logarithmic velocity and temperature profiles to quantify the logarithmic correction term. Beyond the transition, which for gases was predicted Grossmann and Lohse (2001) to occur around $Ra\sim 10^{14}$, both predictions imply an effective scaling $Nu\sim Ra^{\gamma_{eff}}$ with ${\gamma_{eff}}\approx 0.38$. As $Ra\rightarrow\infty$, the logarithmic correction terms become negligible and the flow approaches the so-called asymptotic ultimate regime where $Nu\sim Ra^{1/2}$. This asymptotic ultimate regime implies that viscosity and thermal diffusivity effects have a negligible impact on the flow. In contrast to Kraichnan (1962); Spiegel (1971); Grossmann and Lohse (2011), Owen and Thompson Owen and Thomson (1963) proposed that the asymptotic ultimate (and upper bound Howard (1963); Busse (1969); Doering and Constantin (1996)) scaling exponent 1/2 is never achieved.

Whether and when the transition to the ultimate regime indeed occurs, and to what turbulent state, has been hotly debated in the community. While Ahlers, Bodenschatz, and He experimentally found such a transition around $Ra\sim 10^{14}$ He et al. (2012a, b); Ahlers et al. (2014), Chavanne, Roche et al. Chavanne et al. (1997, 2001); Roche et al. (2010) observed it at lower $Ra\sim 10^{11}-10^{12}$ and others still do not find such a transition at all Niemela et al. (2000); Niemela and Sreenivasan (2010); Urban et al. (2014). To clarify this question, major numerical efforts are undertaken to solve the underlying Boussinesq equations in this large $Ra$ regime. While in 3D the required computational power is presently prohibitive, in 2D the onset of such a transition has recently been observed around $Ra\sim 10^{13}$ Zhu et al. (2018a); Krug et al. (2018), namely in the effective scaling of $Nu(Ra)$ and in the structure of the BLs, changing towards a logarithmic profile in the ultimate regime.

To trigger the onset of the ultimate regime, i.e. the transition from a laminar-type BL to a turbulent one, various wall-roughness elements have been employed, both in experiments Shen et al. (1996); Ciliberto and Laroche (1999); Du and Tong (2000); Roche et al. (2001); Qiu et al. (2005); Tisserand et al. (2011); Salort et al. (2014); Wei et al. (2014); Xie and Xia (2017); Tummers and Steunebrink (2019) and in numerical simulations Stringano and Verzicco (2006); Shishkina and Wagner (2011); Wagner and Shishkina (2015); Zhu et al. (2017). In general, these efforts have led to an enhanced $Nu$ versus $Ra$ scaling in some intermediate $Ra$ regime, and in limited $Ra$ regimes even an effective $Nu$ versus $Ra$ scaling exponent of 1/2 can be achieved. For large $Ra$ (but still below the onset of the ultimate regime) the effective scaling exponent settles back to a value close to the one in the classical regime Zhu et al. (2017), as then the thermal sublayer is thinner than the roughness elements, and starts to conform to the roughness topography. Only for even larger $Ra$ – hitherto not yet achieved in rough-wall RB flow – is the transition towards a turbulent BL throughout, and enhanced $Nu$ versus $Ra$, expected.

To address the question of heat transport in smooth and rough-wall RB convection in the ultimate regime, in this paper we will assume the hypothesis proposed by Kraichnan Kraichnan (1962) and Grossmann & Lohse Grossmann and Lohse (2011, 2012) that the boundary layers are turbulent with logarithmic profiles. This allows us to employ our understanding of heat transfer in smooth and in particular rough-wall fully turbulent forced convection channel flow. The advantage of such flow is that the driving is mechanically supplied (namely by shear), which is much more efficient than the thermal driving in RB flow. Therefore, in numerical simulations the transition to turbulence in the boundary layers – manifesting itself in a logarithmic velocity profile – can easily be achieved Smits et al. (2011); Smits and Marusic (2013). Such turbulent boundary layers with a logarithmic velocity profile also exist in the shear-driven Taylor–Couette (TC) flow Grossmann et al. (2016), which is viewed as the “twin” of RB flow Busse (2012). For that flow, indeed the ultimate regime with the corresponding Nusselt number $Nu_{\omega}$ (the dimensionless angular velocity transport Eckhardt et al. (2007a)) scaling $Nu_{\omega}\propto Ta^{0.38}$ (where the Taylor number $Ta$ is the dimensionless mechanical driving strength) can be achieved both in experiments and in numerical simulations, see the review article Grossmann et al. (2016). In TC flow with a rough wall, even the asymptotic ultimate regime $Nu_{\omega}\propto Ta^{1/2}$ can be achieved, both experimentally Cadot et al. (1997); van den Berg et al. (2003); Zhu et al. (2018b) and numerically Zhu et al. (2018b). This regime corresponds to fully rough pipe or channel flow in which the friction factor becomes Reynolds number independent Nikuradse (1933); Hama (1954); Jiménez (2004); Zhu et al. (2018b). The reason is that in this regime the drag is determined by the pressure drag, and shear (viscous) drag hardly plays a role. However, this dominant pressure drag also implies that the analogy between heat transfer in RB and angular velocity transport in TC breaks down for roughness, due to the lack of a pressure-like component in the temperature advection equations Zhu et al. (2017). The quantitative effect of roughness on the heat transfer in RB flows, despite this qualitative understanding, is therefore not well understood, and we will address it here using forced convection channel flow. Note that, like RB and TC flows, numerical simulations of closed channel (and pipe) flows employing periodic boundary conditions in the flow direction also enjoy exact energy balances Eckhardt et al. (2007b).

In the present work, we will use the recent rough-wall turbulent forced convection results from MacDonald et al. (2019) as a model for the near-wall shear-dominated turbulent boundary layers observed in high $Ra$ natural convection flows, as envisioned by Kraichnan Kraichnan (1962). We will therefore seek to quantify and explain the effect of roughness on the scaling exponent of the Nusselt number in the ultimate regime. This involves extending the analysis of Ref. Grossmann and Lohse (2011) for smooth-wall ultimate RB flow with logarithmic velocity and temperature profiles, by quantifying the shift in the profiles induced by the roughness.

Briefly we summarize the forced convection direct numerical simulations (DNSs) of Ref. MacDonald et al. (2019), in which buoyancy forces were neglected so that temperature was a passive scalar. Periodic boundary conditions were employed in the streamwise ($x$) and spanwise ($y$) directions and no-slip, impermeability and isothermal ($\theta_{w}=0$) conditions were applied to the top and bottom walls, with $z$ denoting the wall-normal (vertical) direction. A body forcing to the momentum equation was used to drive the flow at constant bulk velocity through the channel. An internal heating body force to the energy equation was used for temperature, representing a hot fluid being cooled by the walls. The Prandtl number was set to that of air at room temperature, $Pr\equiv\nu/\kappa=0.7$, where $\nu$ is the kinematic viscosity and $\kappa$ is the thermal diffusivity. Different friction Reynolds numbers, $Re_{\tau}=U_{\tau}h/\nu$ were simulated with $395\lesssim Re_{\tau}\lesssim 1680$, where $U_{\tau}$ is the friction velocity and $h$ is the channel half height, defined for the rough-wall flow to be distance between the channel center and the roughness mean height, corresponding to the hydraulic half height (Chan et al., 2015). Three-dimensional sinusoidal roughness with semi-amplitude of either $k=h/18$ or $k=h/36$ and wavelength $\lambda_{x}=\lambda_{y}=\lambda\approx 7.07k$ was applied to both the bottom and top walls. As the flow speed and friction Reynolds number increases, the roughness Reynolds number $k^{+}=kU_{\tau}/\nu$ increases towards the fully rough regime. Superscript $+$ indicates non-dimensionalization on $\nu$, $U_{\tau}\equiv\sqrt{\tau_{w}/\rho}$ and the friction temperature $\Theta_{\tau}\equiv[q_{w}/(\rho c_{p})]/U_{\tau}$, $\tau_{w}$ and $q_{w}$ being the temporally and spatially averaged momentum and heat fluxes at the wall, $\rho$ the fluid density and $c_{p}$ the specific heat at constant pressure. The minimal-span channel for rough wall flows was used (Chung et al., 2015; MacDonald et al., 2017), in which the spanwise domain width is purposely very narrow and only the near-wall turbulent flow is captured up to a critical height $z_{c}\approx 0.4L_{y}$, where $L_{y}$ is the channel span. Smooth-wall channel simulations with matched channel domain sizes were also conducted, to ensure that the differences between the smooth- and rough-wall flows were due to the roughness alone and not the channel span.

Fig. 1 shows instantaneous snapshots of the streamwise velocity and fluid temperature from the simulations of MacDonald et al. (2019). A white contour line for $u^{+}=4$ and $\theta^{+}-\theta_{w}^{+}=4$ has been selected to provide an indication of the viscous and thermal diffusive sublayers. The smooth-wall flow produces thin viscous and thermal sublayers close to the wall that appear similar, although the thermal sublayer is slightly larger due to the Prandtl number being less than unity. The roughness in the transitionally rough regime ($k^{+}\approx 21$) produces much thicker sublayers and, like the smooth wall, appear similar between the velocity and temperature fields. The final (right most) panel shows much larger roughness that is nominally fully rough ($k^{+}\approx 93$). The selected contour line of $u^{+}=4$ resides mostly above the roughness crests, indicating the fluid below the level of the roughness crests is nearly stationary due to the increased dominance of pressure drag. Conversely, the thermal diffusive sublayer is thin and closely follows the roughness geometry; it appears more like that of the smooth wall if the wall were contorted to match the roughness geometry.

II Turbulent Boundary Layers in the Rough-Wall Ultimate Regime

The turbulent boundary layers observed in high $Ra$ (ultimate regime) natural convection flows are characterized by local buoyancy forces that are much smaller than the shear forces. This leads to mean velocity and temperature profiles that are logarithmic functions of distance from the wall Grossmann and Lohse (2011), the same as in forced convection flows Kader (1981); Kawamura et al. (1999); Pirozzoli et al. (2016), given as

[TABLE]

where $\kappa_{m}\approx 0.4$ is the von Kármán constant, which is slightly larger for heat transfer with $\kappa_{h}\approx 0.46$ due to the turbulent Prandtl number (the ratio of momentum and heat transfer eddy diffusivities) being below unity Yaglom (1979); Pirozzoli et al. (2016). As in MacDonald et al. (2019) we take the smooth-wall offsets to be $A_{m}\approx 5.0$ and $A_{h}(Pr=0.7)\approx 3.2$. The enhanced skin friction and heat transfer of roughness lead to a downwards shift of these logarithmic profiles, represented by the roughness function, $\Delta U^{+}$ Hama (1954), and temperature difference, $\Delta\Theta^{+}$ Yaglom (1979); Miyake et al. (2001); Leonardi et al. (2007); MacDonald et al. (2019). These two quantities are a flow property of a given rough surface and vary with the roughness Reynolds number. Evaluating Eqs. (1–2) at the middle of an RB cell, $z=L/2$, yields

[TABLE]

where the Reynolds number $Re=UL/\nu$. Eqs. (3) and (4) thus describe $U^{+}$ and $\Theta^{+}$ for a given flow state governed by $Re$ and $Pr$ and by the relative roughness $k/L$, provided $\Delta U^{+}$ and $\Delta\Theta^{+}$ are known.

We define the skin-friction coefficient as $C_{f}\equiv\tau_{w}/(\tfrac{1}{2}\rho U^{2})=2/U^{+2}$ and heat-transfer coefficient (Stanton number) as $C_{h}\equiv q_{w}/(\rho c_{p}U)=1/(U^{+}\Theta^{+})$. The temperature profiles from the top and bottom walls must match at the centerline so that $2\Theta=\Delta$, where $\Delta$ is the driving temperature difference in RB domains and thus we define the Nusselt number as $Nu=\mbox{{Pr}\hskip 0.56917pt}Re/(U^{+}\Delta^{+})=\mbox{{Pr}\hskip 0.56917pt}ReC_{h}/2$. In order to describe the rough-wall $C_{f}$, $C_{h}$ and $Nu$ as a function of Reynolds number, we therefore require knowledge of $\Delta U^{+}$ and $\Delta\Theta^{+}$. Note that for the smooth wall case ($\Delta U^{+}=\Delta\Theta^{+}=k/L=0$), Eq. (3) yields the implicit Prandtl–von Kármán logarithmic skin-friction law, which can be solved using Lambert’s $\mathcal{W}$-function with $U^{+}=(1/\kappa_{m})\mathcal{W}\left(Re\hskip 1.13791pt\kappa_{m}\exp\left(A_{m}\kappa_{m}\right)/2\right)$. Grossmann & Lohse Grossmann and Lohse (2011) obtained the smooth-wall ultimate-regime Nusselt number scaling exponent of ${\gamma_{eff}}\approx 0.38$ using this result.

Fig. 2(a) shows a sketch of a sinusoidal rough-wall RB domain. We can obtain the roughness function $\Delta U^{+}$ and temperature difference $\Delta\Theta^{+}$ from the turbulent forced convection system of MacDonald et al. (2019). These two quantities are shown in Fig. 2(b), as a function of the equivalent sand-grain roughness Reynolds number, $k_{s}^{+}\approx 4.1k^{+}$. This scaling guarantees a collapse of the roughness function with that of Nikuradse’s sand-grain data in the fully rough regime (here $k_{s}^{+}\gtrsim 150$), where $\Delta U_{FR}^{+}=(1/\kappa_{m})\log(k_{s}^{+})+A_{m}-C_{n}$ (blue dashed line), with $C_{n}\approx 8.5$ being Nikuradse’s constant Nikuradse (1933); Schlichting (1936). Within the asymptotic fully rough regime, viscous effects are negligible and the pressure (or form) drag is dominant (Schultz and Flack, 2009; Busse et al., 2017; Zhu et al., 2018b). Note that $k_{s}$ must be determined dynamically for a given rough surface and is not a simple geometric length scale of the roughness. The temperature difference, meanwhile, is tending towards a constant value of $\Delta\Theta_{FR}^{+}\approx 4.4$ in the fully rough regime (red dashed line). Like $k_{s}$, the exact value of $\Delta\Theta_{FR}^{+}$ is a dynamic parameter that is likely to be roughness dependent. Ultimately however, with this information, in the fully rough regime Eqs. (3) and (4) become

[TABLE]

That is, the friction-normalized centerline velocity is constant and only depends on the relative roughness $k_{s}/L$, while the centerline friction-normalized temperature remains dependent on the Reynolds number.

The dotted lines in Fig. 2(b) are curve fits to the DNS data. For the roughness function, we use the fit from Ref. Monty et al. (2016) with $\Delta U_{fit}^{+}\approx(1/\kappa_{m})\log(k_{s}^{+})+A_{m}-C_{n}-(a/k_{s}^{+})^{b}$, where $a\approx 89.3$ and $b\approx 1.12$. While the fitting constants are different to Monty et al. (2016) due to the different roughness geometries, this function correctly tends towards the fully rough asymptote for large $k_{s}^{+}$. However, the function quickly reaches zero at $k_{s}^{+}\approx 25$, much more rapidly than for sinusoidal roughness or sand-grain roughness (see figure 5 of MacDonald et al. (2019)). We therefore only use $\Delta U_{fit}^{+}$ for $k_{s}^{+}\gtrsim 35$. For the temperature difference, a sigmoid function across the entire range of $k_{s}^{+}$ is used, with $\Delta\Theta_{fit}^{+}\approx B+K/[1+\exp(c+dk_{s}^{+})]$ where $B\approx-1.66$, $K\approx 6.11$, $c\approx 0.97$ and $d\approx-0.0239$. This function correctly goes to zero for small $k_{s}^{+}$ (i.e. a smooth wall) and tends towards $\Delta\Theta_{FR}^{+}\approx 4.4$ in the limit of $k_{s}^{+}\rightarrow\infty$ (i.e. the fully rough regime). We can therefore obtain $C_{f}$, $C_{h}$ and $Nu$ numerically using $\Delta U_{fit}^{+}$ and $\Delta\Theta_{fit}^{+}$ with Eqs. (3–4), although the functional form of $\Delta U_{fit}^{+}$ precludes an analytical solution for $U^{+}$.

III Effective heat-transfer scaling in the ultimate regime

Fig. 3(a) and (b) shows the skin-friction and heat-transfer coefficients as a function of Reynolds number. Here, the solid lines show the smooth-wall $C_{f}$ and $C_{h}$ calculated using the logarithmic velocity and temperature profiles (Eqs. (3–4) with $\Delta U^{+}=\Delta\Theta^{+}=0$). The dotted lines correspond to the rough-wall $C_{f}$ and $C_{h}$ calculated using Eqs. (3–4) with the curve fits $\Delta U_{fit}^{+}$ and $\Delta\Theta_{fit}^{+}$ from Fig. 2(b), while the dashed lines show the asymptotic fully rough regime (Eqs. 5–6). The relative roughness ratio $k_{s}/L$ is varied, given by the different colors, where it is assumed $\Delta U^{+}$ and $\Delta\Theta^{+}$ are independent of $k_{s}/L$. The symbols are the DNS data MacDonald et al. (2019), assuming that the channel centerline $h$ is equal to the middle of the RB cell, $L/2$. For the smooth-wall flow, both of these coefficients monotonically reduce with Reynolds number at the same rate. Roughness enhances momentum and thermal transfer, leading to an increase in these coefficients relative to the smooth wall. In the fully rough regime ($Re\gtrsim 2\times 10^{4}$ for $k_{s}/L=1.14\times 10^{-1}$, black dashed line) the skin-friction coefficient becomes constant with Reynolds number, with $C_{f}\approx 0.013$. This indicates that the viscous effects are negligible and the momentum transfer has attained an asymptotic state Zhu et al. (2017). Conversely, the heat-transfer coefficient reduces with Reynolds number in the fully rough regime, similar to the smooth wall, and does not appear to reach any asymptotic state. At the present Reynolds numbers, Eqs. (3–6) predict $C_{h}\propto Re^{-0.2}$ for the smooth wall (e.g. Kays et al., 2005), while for the rough wall with $k_{s}/L=1.14\times 10^{-1}$ it scales as $C_{h}\propto Re^{-0.16}$.

Ultimately we would like to know the dependency of the Nusselt number on the Rayleigh number, $Nu\sim Ra^{{\gamma_{eff}}}$. However, the log equations (Eqs. 3–6) as well as the forced convection DNS data are functions of the Reynolds number, requiring an assumption of the form $Re=A\cdot Ra^{\beta}$ in order to determine the Rayleigh number dependency. Grossmann & Lohse Grossmann and Lohse (2011) made the same assumption in their work, where they used $Re=0.346\cdot Ra^{0.443}$, corresponding to boundary layers that were not yet turbulent. As we are explicitly dealing with turbulent (logarithmic) boundary layers, we use the ultimate-regime scaling of $\beta=1/2$ Grossmann and Lohse (2011). More recently, Shishkina et al. Shishkina et al. (2017) derived this relation with $\beta=1/2$ from the Prandtl BL equations with very weak assumptions. We take the coefficient $A=0.7$, which results in $Nu(Ra=10^{14})\approx 2500$ for the smooth wall, in agreement the laboratory experiments of Ref. He et al. (2012b). The exact choice of $A$ and $\beta$ will alter the absolute values of $Nu$ and ${\gamma_{eff}}$ for a given $Ra$, however we emphasize that the assumption made here does not alter the main conclusions of this paper.

Figure 4(a) shows the Nusselt number as a function of the Rayleigh number. The inset highlights just the DNS data, where the smooth-wall forced convection data (circles) have an effective scaling exponent of ${\gamma_{eff}}\approx 0.38$ (solid black line), matching that observed in ultimate RB convection Grossmann and Lohse (2011), as expected. Note that the mechanically supplied shear of forced convection corresponds to a very strong wind in RB convection, explaining the relatively low $Ra$ values in the inset of Fig. 4(a). A realistic RB flow in the ultimate regime typically has higher $Ra$ values due to a weaker wind (i.e., a lower Reynolds number, or coefficient $A$ in the assumed $Re-Ra$ relationship above). The transitionally rough regime, meanwhile, has an enhanced exponent of ${\gamma_{eff}}\approx 0.55$ shown by the dotted lines (similar to the rough-wall RB experiments by Xie and Xia (2017)), however in the asymptotic fully rough regime the scaling reduces towards ${\gamma_{eff}}\approx 0.42$. Fig. 4(b) shows the compensated form, $Nu/Ra^{1/2}$, demonstrating that while the transitionally rough regime has a scaling near ${\gamma_{eff}}\approx 0.55$, the fully rough regime clearly has a reduced scaling exponent.

Figure 4(c) shows the effective scaling exponent, $\mathrm{d}\log_{10}(Nu)/\mathrm{d}\log_{10}(Ra)$, computed using the log-law formulas for smooth-wall and fully rough flows (Eqs. 3–6). This is also done using the curve fits of $\Delta U_{fit}^{+}$ and $\Delta\Theta_{fit}^{+}$ (Fig. 2b) in Eqs. (3–4), shown by the dotted lines. This figure is similar to figure 2a of Ref. Grossmann and Lohse (2011), where for fully turbulent smooth-wall convection (solid line, Fig. 4c) we see the scaling exponent is in the range $0.35\lesssim{\gamma_{eff}}\lesssim 0.42$. The transitionally rough regime has a much larger exponent, with $0.45\lesssim{\gamma_{eff}}\lesssim 0.58$ for varying $k_{s}/L$ ratios and $Ra$. While the behavior of this exponent with Rayleigh number is dependent on the functional forms of $\Delta U_{fit}^{+}$ and $\Delta\Theta_{fit}^{+}$, the important aspect is that the use of these fits shows an exponent close to 0.55. As the flow enters the fully rough regime and $\Delta\Theta^{+}$ becomes constant ($Ra\approx 10^{9}$ for $k_{s}/L=1.14\times 10^{-1}$, black dashed line), the scaling exponent reduces to approximately $0.42\lesssim{\gamma_{eff}}\lesssim 0.45$, consistent with Fig. 4(a), although still larger than that of the smooth-wall flow. Crucially, from Eqs. (5–6) we see that the rough-wall exponent must depend on the equivalent sand-grain roughness, $k_{s}/L$, as well as the temperature difference, $\Delta\Theta_{FR}^{+}$, making it distinct from the smooth-wall scaling exponent.

IV Discussion and conclusions

While we have used simple curve fits to obtain ${\gamma_{eff}}\approx 0.55$ in the transitionally rough regime, they show how large scaling exponents in the transitionally rough regime can be obtained (Xie and Xia, 2017; Zhu et al., 2017; Tummers and Steunebrink, 2019). Echoing Ref. Zhu et al. (2017), these large exponents do not indicate that the asymptotic ultimate regime has been obtained, as both viscous and thermal diffusivity effects are still important in the transitionally rough regime (referred to as Regime I in Zhu et al. (2017)). It is only once the flow enters the fully rough regime (Regime II in Zhu et al. (2017)), when the skin-friction coefficient is constant, that viscous actions become negligible and the scaling exponent reduces in value. Critically, however, thermal diffusivity effects will always remain important, as they do in the smooth wall. From Eq. 6, we see the fully rough centerline temperature and hence Nusselt number scales with the logarithm of $Re$, indicating that only for asymptotically large $Ra$ does the heat transfer of rough-wall flows approach the asymptotic ultimate regime.

The origin for this major difference between momentum transfer and heat transfer in rough-wall shear flow lies in the pressure drag, which dominates the momentum transfer, but whose analog is absent for the heat transfer Owen and Thomson (1963); Cebeci and Bradshaw (1984); MacDonald et al. (2019). It is this absence which leads to an effective scaling $Nu\sim Ra^{0.42}$ in the fully rough ultimate regime, rather than the upper bound exponent 1/2. Our $Nu$ vs $Ra$ scaling prediction also seems to be consistent with recent rough-plate RB experiments in the Göttingen U-Boat facility, which in the ultimate regime yield an exponent of 0.43 for $Pr=0.8$ and $Ra\approx 10^{13}$ (E. Bodenschatz, private communication).

We finally note that due to the limitations in fabrication every surface is rough to some degree ($k_{s}/L>0$). For example, the Göttingen U-Boat system with $L=2.24$ m uses either aluminum (HPCF-I) or copper (HPCF-II) top and bottom plates with average roughness heights of $R_{a}\approx 1.6$ $\mu$m and 0.2 $\mu$m, respectively Ahlers et al. (2009b). At what Rayleigh number will this roughness become visible in the $Nu(Ra)$ relation? Unfortunately, such an estimate is very difficult as it strongly depends on the prefactors and exact values of the scaling exponents. With the assumption of our above analysis, these surfaces could only be considered as fully smooth until $Ra\approx 1\times 10^{15}$ and $9\times 10^{16}$ for the aluminum and copper plates, respectively, before becoming transitionally rough with an enhanced scaling exponent. These numbers should be taken with utmost care, as, as mentioned above, these estimates strongly depend on the coefficients in the $Re-Ra$ relationship assumed in Section III, with the present coefficients ($Re=0.7Ra^{0.5}$) assuming fully turbulent boundary layers. If we use $Re=0.346Ra^{0.443}$ from Grossmann–Lohse theory Grossmann and Lohse (2011), then the corresponding critical Rayleigh numbers are $4\times 10^{17}$ and $6\times 10^{19}$, respectively. To obtain either set of values, the roughness is assumed to be hydrodynamically and thermodynamically smooth ($\Delta U^{+}=\Delta\Theta^{+}=0$) until $k_{s}^{+}=4$, and that the plate surface equivalent sand-grain roughness is that of sinusoidal roughness, $k_{s}\approx 4.1k$, where the sinusoidal semi-amplitude is related to the mean roughness height by $k=2.46R_{a}$ Chan et al. (2015). While there are uncertainties in the exact $Ra$ values above, they provide some indication of the level of surface smoothness required in laboratory experiments to ensure the smooth-wall scaling exponent in the ultimate regime is observed. Regardless, as shown in Figs. 3 and 4, where the different line colors correspond to varying $k_{s}/L$, for sufficiently large $Ra$ the surfaces will inevitably cease to be dynamically smooth.

Acknowledgements.

The authors would like to gratefully acknowledge the financial support of the Australian Research Council through a Discovery Project (DP170102595).