3.1 Effect of channel width, $L_{y}^{+}$ (set $A$ )

Figure 2. Mean (a) velocity and (b) temperature profiles for smooth-wall (solid) and rough-wall (dashed) channels with $k^{+}\approx 22$ (cross). Darker grey refers to increasing channel width (set $A$ , table 1). Dotted lines indicate unphysical region above the wall-normal critical height $z_{c}=0.4L_{y}$ (vertical mark). Insets show difference in smooth- and rough-wall profiles.

First, we will consider the effect of the channel width on mean velocity and temperature profiles, for both smooth- and rough-wall flows. This is done at a matched friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}\approx 395$ (set $A$ , table 1). The rough-wall minimal channel has been previously validated for momentum transfer, where it has been shown to be capable of reproducing the roughness function as well as the near-wall high-order statistics of a conventional full-span channel (Chung et al. Reference Chung, Chan, MacDonald, Hutchins and Ooi2015; MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016, Reference MacDonald, Chung, Hutchins, Chan, Ooi and García-Mayoral2017). In the present forced convection flow, temperature is a passive scalar and is simply advected by the velocity, suggesting that it will respond in the same manner as velocity to the minimal-span channel. However, Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) noted some differences between the passive scalar and velocity fields, especially in the outer (core) region of the flow. We will therefore compare mean velocity and temperature profiles for minimal and full-span channels. The mean streamwise velocity profile is shown in figure 2(a), where the dotted lines indicate the unphysical region above the wall-normal critical height, $z_{c}=0.4L_{y}$ , of the minimal channels. As expected, this scaling agrees well with the data, where we see that above the critical height (denoted by the vertical mark) the mean velocity increases relative to the full-span channel. Below this point we have what can be described as ‘healthy’ turbulence (Flores & Jiménez Reference Flores and Jiménez2010), as it is the same as in conventional (full-span) channel flow. Widening the channel by increasing $L_{y}^{+}$ extends the region of healthy turbulence further from the wall, as larger turbulent structures can fit inside the widened domain. The region above $z_{c}$ is unphysical due to the narrow constraints of the channel and is not relevant to the near-wall flow. The inset of figure 2(a) shows the difference in smooth- and rough-wall velocity profiles as a function of $z^{+}$ . This difference reaches a constant above $z^{+}\approx 40$ , which is the offset $\unicode[STIX]{x0394}U^{+}$ of the rough-wall flow relative to the smooth-wall flow. Since the difference is constant with $z^{+}$ , this indicates that the rough-wall flow has the same velocity profile as the smooth-wall flow and that the outer layers are similar. This is the case for both minimal-span and full-span channels and demonstrates that the minimal channel can accurately estimate the roughness function, as already discussed in our previous work (Chung et al. Reference Chung, Chan, MacDonald, Hutchins and Ooi2015; MacDonald et al. Reference MacDonald, Chung, Hutchins, Chan, Ooi and García-Mayoral2017).

Figure 2(b) shows the temperature profile in the same format as the velocity profile of figure 2(a). The critical height scaling obtained from the velocity profiles, $z_{c}=0.4L_{y}$ , is used here without alteration. There is excellent near-wall agreement between the minimal and conventional channels and it appears that the mean temperature does not increase as readily as the velocity when we are outside the healthy turbulence region (above $z_{c}$ , dotted lines). As with the velocity profiles, both the smooth- and rough-wall temperature profiles appear to scale with logarithmic distance from the wall above $z^{+}\gtrsim 50$ . The inset of figure 2(b) shows the difference in smooth- and rough-wall temperature profiles, which for the conventional full-span channel flow (black line) is tending towards a constant value above $z^{+}\approx 60$ . As with the velocity profile, this indicates that the smooth- and rough-wall temperature profiles are similar in the outer layer of the flow. This supports the rough-wall logarithmic temperature profile (1.3), and that we only need to estimate the temperature equivalent of the roughness function, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ , to describe the temperature profile. The difference in temperature profiles for the minimal-span channels (grey lines in inset of figure 2 b) shows more variation above $z_{c}^{+}$ than the corresponding velocity difference. The narrowest span channel with $z_{c}^{+}\approx 62$ (light grey line) shows the temperature difference tending towards a value of approximately 2.2 at the channel centreline, much lower than the conventional channel centreline value (black line) of 3.0. This difference in the minimal-span channel is possibly due to a lack of statistical convergence in the outer-layer region of the channel and would require a much longer run time to converge. However, the region above $z_{c}^{+}$ is inherently unphysical due to the minimal span and resulting lack of large-scale structures, which means obtaining statistical convergence in this region is not necessary or relevant to the near-wall healthy turbulence (MacDonald et al. Reference MacDonald, Chung, Hutchins, Chan, Ooi and García-Mayoral2017). The minimal channel simulations are therefore run to ensure converged statistics in the near-wall flow, up to the critical height, $z_{c}^{+}$ . As such, if we evaluate the temperature difference at the critical height $z_{c}^{+}$ to obtain $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ , we observe good agreement between all three channel widths, with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}\approx$ 3.0, 3.1 and 2.9 for $L_{y}^{+}=155$ , 310 and the full-span case, respectively. This result indicates that the minimal channel can be used for studying heat transfer, for both smooth- and rough-wall flows.

3.2 Effect of friction Reynolds number, $Re_{\unicode[STIX]{x1D70F}}$ (set $B$ )

Figure 3. Mean (a) velocity and (b) temperature profiles for smooth-wall (solid) and rough-wall (dashed) channels with $k^{+}\approx 22$ (cross), at $Re_{\unicode[STIX]{x1D70F}}\approx 395$ (grey) and $Re_{\unicode[STIX]{x1D70F}}\approx 590$ (black) (set $B$ , table 1). Dotted lines indicate unphysical region above the critical height $z_{c}=0.4L_{y}$ (vertical mark). Insets show difference in smooth- and rough-wall profiles.

Having validated that the minimal channel can accurately predict both the roughness function, $\unicode[STIX]{x0394}U^{+}$ , and the temperature difference, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ , we now assess the influence of the friction Reynolds number, $Re_{\unicode[STIX]{x1D70F}}$ . In Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015), it was shown that turbulent pipe-flow simulations at $Re_{\unicode[STIX]{x1D70F}}\approx 180$ lead to an overestimation of the roughness function compared to $Re_{\unicode[STIX]{x1D70F}}\gtrsim 360$ . This is primarily due to an upward shift of the logarithmic region in the smooth-wall case, caused by the pressure gradient effect inherent to low- $Re_{\unicode[STIX]{x1D70F}}$ turbulent flows. Here, we repeat this validation but also investigate the effect on heat transfer for two cases of $Re_{\unicode[STIX]{x1D70F}}=395$ and $Re_{\unicode[STIX]{x1D70F}}=590$ (set $B$ , table 1). The roughness size ( $k^{+}$ and $\unicode[STIX]{x1D706}^{+}$ ) and channel dimensions ( $L_{x}^{+}$ and $L_{y}^{+}$ ) are matched in viscous units. Figure 3 shows the mean velocity and temperature profiles for these two friction Reynolds numbers. As already demonstrated in Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015), the mean velocity profiles (figure 3 a) and velocity difference (inset) show that the Reynolds-number effect is negligible for $Re_{\unicode[STIX]{x1D70F}}\gtrsim 395$ . The roughness function is marginally overestimated for $Re_{\unicode[STIX]{x1D70F}}\approx 395$ by approximately 0.08 $U_{\unicode[STIX]{x1D70F}}$ , although this is similar to the level of uncertainty we would expect from the minimal-span channel (MacDonald et al. Reference MacDonald, Chung, Hutchins, Chan, Ooi and García-Mayoral2017).

The temperature profiles (figure 3 b) and difference (inset) shows a similar effect, with there being only a minor difference between $Re_{\unicode[STIX]{x1D70F}}\approx 395$ and $Re_{\unicode[STIX]{x1D70F}}\approx 590$ . Much of this difference is in the outer-layer region of the smooth-wall flow at $Re_{\unicode[STIX]{x1D70F}}\approx 395$ (solid grey line). However, the temperature difference (inset) is similar below the critical height $z_{c}^{+}$ , with the $Re_{\unicode[STIX]{x1D70F}}\approx 395$ case underestimating $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ by approximately $0.1\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70F}}$ . While this result does not give a lower bound on the friction Reynolds number for which Reynolds-number effects become negligible in heat transfer, we will assume that, as with velocity, $Re_{\unicode[STIX]{x1D70F}}\gtrsim 395$ is sufficient.

4.1 Mean profiles

Figure 4. Mean (a) velocity and (b) temperature profiles for smooth-wall (solid) and rough-wall (dashed) channels with increasing friction Reynolds numbers. Darker grey refers to increasing roughness height (set $C$ , table 1). The insets show the difference in smooth- and rough-wall profiles. Dotted lines indicate unphysical region above the critical height $z_{c}=0.4L_{y}$ (vertical mark). The blue dash-dotted line in (a) and (b) is the smooth-wall DNS data of Bernardini, Pirozzoli & Orlandi (Reference Bernardini, Pirozzoli and Orlandi2014) and Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016), respectively. (c) Rough-wall velocity profiles against wall-normal position normalised on roughness height, $z/k$ . The solid red line is the fully rough asymptote, $U^{+}\approx (1/0.4)\log (z/k)+5.1$ . (d) Rough-wall temperature profiles below the roughness crest. The insets in (c) and (d) show the velocity and temperature at the roughness crest, against $k^{+}$ . The dashed red line in the inset of (d) is the logarithmic temperature profile (1.3) evaluated at $z=k$ .

We now increase the roughness Reynolds number, $k^{+}$ , towards fully rough conditions (set $C$ , table 1). Figure 4(a) shows the mean velocity profiles for increasing friction Reynolds numbers, where the roughness height is fixed at $k=h/18$ (except for the smallest roughness case, where $k=h/36$ , to ensure $Re_{\unicode[STIX]{x1D70F}}\gtrsim 395$ ; see § 3.2). We see that increasing $Re_{\unicode[STIX]{x1D70F}}$ (and hence $L_{y}^{+}=\unicode[STIX]{x1D706}^{+}\approx 7.1k^{+}$ ) leads to the capturing of a larger region of the logarithmic layer for the smooth-wall flow (solid lines). However, the roughness increasingly reduces the near-wall velocity magnitude with Reynolds number (dashed lines), indicating that the roughness function is increasing (inset of figure 4 a). The rough-wall velocity profiles are plotted in figure 4(c) as a function of $z/k$ . In this scaling, we see that the cases with larger $k^{+}$ (darker grey lines) are collapsing onto the fully rough asymptote of $U^{+}\approx \unicode[STIX]{x1D705}_{m}^{-1}\log (z/k)+D$ (solid red line), where $\unicode[STIX]{x1D705}_{m}\approx 0.4$ and $D=A_{m}-C\approx 5.1$ . In a conventional flow we would also see the centreline velocity of the rough-wall flow tending towards a constant as the skin-friction coefficient becomes constant in the fully rough regime. This is not observed here as the channel width, $L_{y}=\unicode[STIX]{x1D706}$ , increases with Reynolds number, which affects the critical height, $z_{c}^{+}$ , and hence centreline velocity. However, the velocity at the roughness crest (inset of figure 4 c) is approximately constant, $U_{k}^{+}\equiv U^{+}(z=k)\approx 4.7$ , indicating that the drag coefficient defined on this velocity (e.g. Macdonald, Griffiths & Hall Reference Macdonald, Griffiths and Hall1998; Coceal & Belcher Reference Coceal and Belcher2004) is also approximately constant with Reynolds number, $C_{dk}\equiv \unicode[STIX]{x1D70F}_{w}/(1/2\unicode[STIX]{x1D70C}U_{k}^{2})=2/U_{k}^{+2}\approx 0.09$ .

Figure 4(b) shows the temperature profile for the smooth- and rough-wall flows. The smooth-wall DNS data of Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) at $Re_{\unicode[STIX]{x1D70F}}\approx 2000$ are also shown by the blue dash-dotted line. We see excellent agreement with the present smooth-wall data below the critical height $z_{c}^{+}$ (vertical marks), further supporting the view made in § 3.1 that the minimal channel can be used to study the near-wall flow of forced convection heat transfer. Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) determined that the smooth-wall temperature profile in the logarithmic region follows $\unicode[STIX]{x1D6E9}^{+}=\unicode[STIX]{x1D705}_{h}^{-1}\log (z^{+})+A_{h}$ , with constants $\unicode[STIX]{x1D705}_{h}\approx 0.46$ and $A_{h}(Pr=0.7)\approx 3.2$ . As the roughness Reynolds number increases, the rough-wall temperature profiles (dashed lines) begin to collapse and follow a logarithmic trend with $z^{+}$ , the same as with the smooth-wall (solid lines) but offset. This agrees with the rough-wall logarithmic temperature profile introduced through (1.3). The temperature difference (inset of figure 4 b) begins to collapse as well for large $k^{+}$ , indicating that $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ is reaching a constant of approximately 4.4 that is independent of $k^{+}$ . While the temperature profile is collapsing in the logarithmic region for fully rough flow (figure 4 b), the temperature below the roughness crests continues to increase with $k^{+}$ , shown in figure 4(d). This is consistent with the logarithmic temperature profile (1.3), which becomes independent of $k^{+}$ in the fully rough regime, beginning almost immediately above the crest. This would then force the crest temperature to be $\unicode[STIX]{x1D6E9}_{k}^{+}\equiv \unicode[STIX]{x1D6E9}^{+}(z=k)\approx \unicode[STIX]{x1D705}_{h}^{-1}\log (k^{+})+A_{h}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ , with $A_{h}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}\approx 3.2-4.4\approx -1.2$ . Indeed, the inset of figure 4(d) shows that the temperature at the crest, $\unicode[STIX]{x1D6E9}_{k}^{+}$ , increases logarithmically with $k^{+}$ , in good agreement with the logarithmic temperature profile (1.3) evaluated at $z=k$ (dashed red line).

Figure 5. Roughness function, $\unicode[STIX]{x0394}U^{+}$ (●, blue), and temperature difference, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ (▪, red), for the present sinusoidal roughness, as a function of the equivalent sand-grain roughness, $k_{s}/k\approx 4.1$ . Other symbols are the roughness function for the sand-grain data of Nikuradse (Reference Nikuradse1933) (○) and for the same sinusoidal roughness geometry of Chung et al. (Reference Chung, Chan, MacDonald, Hutchins and Ooi2015) and Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) in a minimal channel (▴), a full channel (▾) and a pipe (♦).

Figure 5 shows the roughness function, $\unicode[STIX]{x0394}U^{+}$ , and the temperature difference, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ , for the present roughness simulations. These are shown with the roughness functions from Chung et al. (Reference Chung, Chan, MacDonald, Hutchins and Ooi2015) and Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) for the same sinusoidal roughness geometry, as well as for the sand-grain data of Nikuradse (Reference Nikuradse1933). The sinusoidal roughness data are plotted as a function of the equivalent sand-grain roughness, $k_{s}/k\approx 4.1$ , where this factor has been taken from Chung et al. (Reference Chung, Chan, MacDonald, Hutchins and Ooi2015). This scaling ensures the collapse of the present $\unicode[STIX]{x0394}U^{+}$ values with those of Nikuradse’s sand-grain roughness in the fully rough regime (here, $k_{s}^{+}\gtrsim 150$ ), where the roughness function scales as $\unicode[STIX]{x1D705}_{m}^{-1}\log (k_{s}^{+})+A_{m}-C_{N}\approx \unicode[STIX]{x1D705}_{m}^{-1}\log (k_{s}^{+})-3.5$ . Within the transitionally rough regime ( $k_{s}^{+}\lesssim 150$ ), the different roughness geometries have a unique behaviour and the roughness function is not guaranteed to collapse with that of Nikuradse (Jiménez Reference Jiménez2004). Despite having matched roughness geometries, there is a slight difference between the pipe (black diamonds) and present minimal channel (blue circles) data for the largest $k_{s}^{+}$ . This is likely to be due to the fundamental differences between the two domain geometries, as well as the difference in blockage ratios ( $k/h=1/6.75$ from Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) versus $k/h=1/18$ here).

From the temperature difference in figure 5, it appears that, for small $k_{s}^{+}$ , there may be a ‘thermodynamically smooth’ regime for heat transfer in which $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}\approx 0$ . This would be analogous to the hydrodynamically smooth regime for momentum transfer, in which $\unicode[STIX]{x0394}U^{+}\approx 0$ for $k_{s}^{+}\lesssim 4$ and the drag produced by the rough wall matches that of the smooth wall (Raupach, Antonia & Rajagopalan Reference Raupach, Antonia and Rajagopalan1991; Jiménez Reference Jiménez2004). A visual extrapolation of the present data to small $k_{s}^{+}$ suggests that this thermodynamically smooth regime would remain at larger $k_{s}^{+}$ values than the hydrodynamically smooth regime. This implies that for small $k_{s}^{+}$ the effects of roughness are first felt in momentum transfer before affecting heat transfer, presumably due to the present molecular Prandtl number being less than unity ( $Pr=0.7$ ). In this case, the thermal diffusive sublayer is slightly thicker than the viscous sublayer and would require a larger $k_{s}^{+}$ to overcome.

In the fully rough regime, the temperature difference appears to tend towards a constant value of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}\approx 4.4$ . This implies that further increases to the roughness Reynolds number result in the heat-transfer coefficient (Stanton number) decreasing, consistent with the experiments of Dipprey & Sabersky (Reference Dipprey and Sabersky1963). Presumably this constant $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ would be affected by the roughness geometry, and would depend on the solidity (a measure of the roughness density) among other roughness parameters. Note that when viewed in isolation, it may appear that $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ could have reached a maximum around $k_{s}^{+}\approx 250$ and could go on to decrease at larger Reynolds numbers, as opposed to remaining constant. However, the flow appears to have reached its asymptotic fully rough state, where $\unicode[STIX]{x0394}U^{+}$ scales as $(1/\unicode[STIX]{x1D705}_{m})\log (k_{s}^{+})$ and $U_{k}^{+}$ is constant (inset of figure 4 c). It is therefore difficult to conceive how the flow could undergo an additional change at even higher Reynolds numbers, well within the fully rough regime, that would lead to $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ decreasing. This decrease would imply that the temperature profile would eventually return to that of the smooth-wall flow while the smooth- and rough-wall velocity profiles continue to deviate. Moreover, the crest temperature, $\unicode[STIX]{x1D6E9}_{k}^{+}$ (inset of figure 4d), increases in a log-linear manner, which for increasing $k^{+}$ is only consistent if $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ is approaching a constant value. These observations therefore all suggest that $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ remains constant in the fully rough regime.

Figure 6. Turbulent Prandtl number against wall-normal distance, $z^{+}$ , for (a) increasing channel width with $Re_{\unicode[STIX]{x1D70F}}=395$ (set $A$ , table 1) and (b) increasing Reynolds number (set $C$ , table 1), for smooth-wall (solid) and rough-wall (dashed) flows. Data are only shown from crest to critical height, $z_{c}^{+}$ , and in (b) are staggered by $-0.2$ for decreasing $Re_{\unicode[STIX]{x1D70F}}$ .

An important quantity in heat-transfer models is the turbulent Prandtl number, defined as the ratio of momentum and heat-transfer eddy diffusivities,

(4.1) $$\begin{eqnarray}Pr_{t}=\frac{\unicode[STIX]{x1D708}_{t}}{\unicode[STIX]{x1D6FC}_{t}}=\frac{\langle \overline{u^{\prime }w^{\prime }}\rangle }{\langle \overline{w^{\prime }\unicode[STIX]{x1D703}^{\prime }}\rangle }\frac{\text{d}\unicode[STIX]{x1D6E9}/\text{d}z}{\text{d}U/\text{d}z},\end{eqnarray}$$

where $\langle \overline{u^{\prime }w^{\prime }}\rangle$ is the Reynolds shear stress and $\langle \overline{w^{\prime }\unicode[STIX]{x1D703}^{\prime }}\rangle$ is the turbulent heat flux. Under sufficiently high Reynolds and Péclet numbers, dimensional arguments predict that $Pr_{t}$ should be constant in the logarithmic layer (Cebeci & Bradshaw Reference Cebeci and Bradshaw1984). Indeed, Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) showed that the smooth-wall turbulent Prandtl number is close to unity at the wall, before reducing to become approximately constant in the logarithmic layer, with $Pr_{t}\approx 0.85$ over $z^{+}\gtrsim 100$ and $z/h\lesssim 0.5$ . Studies of temporally developing boundary layers (Kozul, Chung & Monty Reference Kozul, Chung and Monty2016) and statistically stationary homogeneous shear flows (which can be treated as a model for the logarithmic layer – see Chung & Matheou (Reference Chung and Matheou2012) and Sekimoto, Dong & Jiménez (Reference Sekimoto, Dong and Jiménez2016)) also suggest $Pr_{t}\approx 1.0$ . In the wake region, $Pr_{t}$ reduces further towards values of 0.5, in line with free-shear flows, or the wake behind a bluff body (Cebeci & Bradshaw Reference Cebeci and Bradshaw1984).

In figure 6(a), the turbulent Prandtl number is shown for increasing channel widths at fixed $Re_{\unicode[STIX]{x1D70F}}=395$ (set $A$ , table 1), where the data are only shown for $z<z_{c}$ . The full-span channel (black line) shows $Pr_{t}$ tending towards values of 0.85–0.9 within the logarithmic layer (around $z^{+}\approx 60$ ), although the relatively low Reynolds numbers and short logarithmic layer do not enable the constant- $Pr_{t}$ region to be obtained. Above the logarithmic layer, in the wake, the magnitude of the full-span $Pr_{t}$ continues to decrease, as discussed above. The narrowest channel with $L_{y}^{+}\approx 153$ (light grey solid line) has a turbulent Prandtl number that is slightly less than the wider span cases, even at the wall. If we consider the turbulent momentum and heat fluxes as integrals of their respective one-dimensional spanwise energy spectra, $\langle \overline{u^{\prime }w^{\prime }}\rangle =\int _{0}^{\infty }E_{uw}\,\text{d}k_{y}$ and $\langle \overline{w^{\prime }\unicode[STIX]{x1D703}^{\prime }}\rangle =\int _{0}^{\infty }E_{w\unicode[STIX]{x1D703}}\,\text{d}k_{y}$ , we see that some length scales (namely $k_{y}<2\unicode[STIX]{x03C0}/L_{y}$ ) will be missing due to the use of the minimal channel, even below $z<z_{c}$ . However, MacDonald et al. (Reference MacDonald, Chung, Hutchins, Chan, Ooi and García-Mayoral2017) showed that many of the dynamically relevant scales are still captured in these energy spectra, and we see that the difference in $Pr_{t}$ between the different channel spans close to the wall in figure 6(a) is fairly small. Moreover, $L_{y}^{+}$ increases with $Re_{\unicode[STIX]{x1D70F}}$ , so this effect is less significant for our larger-Reynolds-number cases. More noticeable is the reduction in $Pr_{t}$ at higher $z^{+}$ values (while still below $z_{c}^{+}$ ) for the minimal channels, relative to the full-span channel. This is likely to be due to the wake region, which now starts at $z_{c}^{+}$ , encroaching into the immature logarithmic layer, leading to the early reduction in $Pr_{t}$ . Note that $Pr_{t}$ is a highly sensitive measure of the relative slopes of the mean velocity and temperature profiles, and requires very high Reynolds and Péclet numbers to yield meaningful results. The differences in these mean profiles, $\unicode[STIX]{x0394}U^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ , meanwhile, are much less sensitive to the slopes and do not suffer as severely from this wake-encroachment issue. They are also the primary source of uncertainty in estimating the skin-friction and heat-transfer coefficients and, as shown above, are accurately measured with the minimal channel.

Figure 6(b) shows the turbulent Prandtl number for increasing friction Reynolds numbers. We see that the rough-wall turbulent Prandtl number (dashed lines) is initially unity at the roughness crest (within the roughness sublayer) and therefore larger than that of the smooth wall at matched $z^{+}$ . For small $k^{+}$ values, the difference in the smooth- and rough-wall turbulent Prandtl numbers is minor and a reasonable collapse is observed. For the two largest $k^{+}$ values which are nominally fully rough (dark grey dashed lines), the rough-wall $Pr_{t}$ at the crest is noticeably larger than the smooth-wall value at matched $z^{+}$ . It then rapidly reduces with wall-normal distance until it is slightly less than that of the smooth wall at $z_{c}^{+}$ . From Townsend’s outer-layer similarity hypothesis (Townsend Reference Townsend1976), we would expect the smooth- and rough-wall turbulent Prandtl numbers to eventually collapse in the logarithmic layer (with constant $Pr_{t}$ ), assuming sufficiently large $Re_{\unicode[STIX]{x1D70F}}$ and wall-normal extent of the logarithmic layer.

4.2 Skin-friction and heat-transfer coefficients

Figure 7. (a) Skin-friction coefficient $C_{f}$ , (b) heat-transfer coefficient (Stanton number) $C_{h}$ and (c) ratio $2C_{h}/C_{f}$ , as functions of bulk Reynolds number $Re_{b}=2U_{bf}h/\unicode[STIX]{x1D708}$ . (d) Ratio of momentum and heat-transfer roughness lengths $z_{0m}/z_{0h}$ against $z_{0m}^{+}$ . Symbols: ▪, smooth-wall data; ●, rough-wall data. These are estimated by fitting full-span composite profiles to the minimal channel velocity and temperature profiles for $z>z_{c}$ . Line styles: — - —, smooth-wall power-law correlations (Dean Reference Dean1978; Kays, Crawford & Weigand Reference Kays, Crawford and Weigand2005); ——, smooth-wall log-law estimate using (4.3) and (4.4); - - -, fully rough heat-transfer model (4.5) of Dipprey & Sabersky (Reference Dipprey and Sabersky1963); — —, fully rough log-law estimate (4.6)–(4.10).

The skin-friction and heat-transfer coefficients, $C_{f}=2/U_{bf}^{+2}$ and $C_{h}=1/(U_{bf}^{+}\unicode[STIX]{x1D6E9}_{mf}^{+})$ , are given in figures 7(a) and 7(b) for the present smooth-wall and rough-wall data with $k/h=1/18$ (symbols). Here, the expected full-span velocity ( $U_{f}$ ) and temperature ( $\unicode[STIX]{x1D6E9}_{f}$ ) profiles are computed by fitting the composite profile of Nagib & Chauhan (Reference Nagib and Chauhan2008) for full-span channel flow to the minimal channel data for $z>z_{c}$ . We use slope coefficients of $\unicode[STIX]{x1D705}_{m}=0.4$ and $\unicode[STIX]{x1D705}_{h}=0.46$ (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2016) and the same empirical wake function, $\unicode[STIX]{x1D6F9}$ , of Nagib & Chauhan (Reference Nagib and Chauhan2008) for both velocity and temperature profiles. However, the wake parameter, $\unicode[STIX]{x1D6F1}$ , is set to 0.08 for velocity and 0.03 for temperature, where these constants comes from fitting the outer-layer composite profile to the data of Bernardini et al. (Reference Bernardini, Pirozzoli and Orlandi2014) and Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) for velocity and temperature, respectively. The offsets, $A_{m}$ and $A_{h}$ , are set such that the profile is continuous at $z_{c}$ .

Empirical correlations are also given in figure 7(a,b). The power law of Dean (Reference Dean1978) for the smooth-wall skin-friction coefficient, $C_{fs}=0.073Re_{b}^{-1/4}$ , is provided (dash-dotted line in figure 7 a), as well as the Prandtl–von Kármán logarithmic skin-friction law (solid line). The latter comes from integrating the logarithmic smooth-wall mean velocity profile (as in (1.1) with $\unicode[STIX]{x0394}U^{+}=0$ ) across the entire channel to obtain an implicit equation,

(4.2) $$\begin{eqnarray}U_{bs}^{+}=\frac{1}{\unicode[STIX]{x1D705}_{m}}\log \left(\frac{\frac{1}{2}Re_{b}}{U_{bs}^{+}}\right)-\frac{1}{\unicode[STIX]{x1D705}_{m}}+A_{m}.\end{eqnarray}$$

This can be solved for $C_{fs}$ in terms of $Re_{b}$ using the product logarithm (or Lambert’s ${\mathcal{W}}$ -function), resulting in

(4.3) $$\begin{eqnarray}U_{bs}^{+}=\sqrt{\frac{2}{C_{fs}}}=\frac{1}{\unicode[STIX]{x1D705}_{m}}{\mathcal{W}}\left(\frac{1}{2}Re_{b}\unicode[STIX]{x1D705}_{m}\text{e}^{(A_{m}\unicode[STIX]{x1D705}_{m}-1)}\right).\end{eqnarray}$$

While both the power-series and log-law skin-friction coefficient models agree well with the present smooth-wall data at moderate Reynolds number (figure 7 a), recent studies of smooth-wall channel flow have shown that the log-law equation agrees better with high-Reynolds-number data than the smooth-wall power-series correlations (Schultz & Flack Reference Schultz and Flack2013; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014).

Similarly, the smooth-wall power-series heat-transfer coefficient of Kays et al. (Reference Kays, Crawford and Weigand2005) is given in figure 7(b), $C_{hs}=0.021Re_{b}^{-0.2}Pr^{-0.5}$ . As with the velocity profile, the smooth-wall logarithmic temperature profile (as in (1.3) with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}=0$ ) can be integrated across the entire channel to obtain a similar expression to (4.2), with

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{as}^{+}=\frac{1}{\unicode[STIX]{x1D705}_{h}}\log \left(\frac{\frac{1}{2}Re_{b}}{U_{bs}^{+}}\right)-\frac{1}{\unicode[STIX]{x1D705}_{h}}+A_{h},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{h}\approx 0.46$ and $A_{h}\approx 3.2$ for the present $Pr=0.7$ flow (figure 4 b). Hence, the smooth-wall Stanton number $C_{hs}=1/(U_{bs}^{+}\unicode[STIX]{x1D6E9}_{ms}^{+})$ can be estimated using (4.3) and (4.4), where (1.4) is used to get the mixed mean temperature, $\unicode[STIX]{x1D6E9}_{m}^{+}$ , in terms of the arithmetic mean temperature, $\unicode[STIX]{x1D6E9}_{a}^{+}$ . At moderate Reynolds numbers, both the power-series correlation of Kays et al. (Reference Kays, Crawford and Weigand2005) and the log-law formulae show good agreement with the present data for the smooth-wall heat-transfer coefficient (squares in figure 7 b). At higher Reynolds numbers, however, we might expect the log-law equations to perform better like they do with the skin-friction coefficient.

The present rough-wall skin-friction coefficient (circles in figure 7 a) is seen to initially increase in value in the transitionally rough regime. However, for sufficiently large bulk Reynolds number, it is tending towards a constant value of $C_{f_{FR}}\approx 0.021$ (dashed line). This indicates that the flow is approaching the asymptotic fully rough state. The rough-wall heat-transfer coefficient (circles in figure 7 b), meanwhile, increases to a maximum in the transitionally rough regime, before monotonically reducing in the fully rough regime. The dotted line here shows the heat-transfer model of Dipprey & Sabersky (Reference Dipprey and Sabersky1963, equation (28)), in which the fully rough (subscript $FR$ ) heat-transfer coefficient for a pipe was given as

(4.5) $$\begin{eqnarray}C_{h_{FR}}=\frac{\displaystyle \frac{C_{f_{FR}}}{2}}{1+\sqrt{\displaystyle \frac{C_{f_{FR}}}{2}}\left[k_{f}\left(Re_{b}\sqrt{\displaystyle \frac{C_{f_{FR}}}{2}}\displaystyle \frac{k_{s}}{2h}\right)^{0.2}Pr^{0.44}-8.48\right]},\end{eqnarray}$$

where $k_{s}/(2h)=\exp [\unicode[STIX]{x1D705}_{m}(3.0-\sqrt{2/C_{f_{FR}}})]$ is the blockage in terms of the equivalent sand-grain roughness. The constant $k_{f}$ is roughness dependent, where Dipprey & Sabersky (Reference Dipprey and Sabersky1963) suggested $k_{f}=5.19$ for granular roughness. Here, we use $k_{f}=5.6$ , obtained from a least-squares fit to the fully rough data ( $k^{+}\gtrsim 66$ ). While (4.5) was developed for pipe flow, we see that it correctly predicts the trend of $C_{h}$ reducing with Reynolds number for the present channel flow cases.

Alternatively, we can use the assumed logarithmic velocity and temperature profiles to obtain an expression for the fully rough heat-transfer coefficient (Stanton number). Starting from the definition of the Stanton number and (1.4) we get

(4.6) $$\begin{eqnarray}C_{h_{FR}}=\frac{1}{U_{b_{FR}}^{+}\unicode[STIX]{x1D6E9}_{m_{FR}}^{+}}=\frac{\unicode[STIX]{x1D705}_{m}\unicode[STIX]{x1D705}_{h}}{1+\unicode[STIX]{x1D705}_{m}\unicode[STIX]{x1D705}_{h}U_{b_{FR}}^{+}\unicode[STIX]{x1D6E9}_{a_{FR}}^{+}},\end{eqnarray}$$

where $U_{b_{FR}}^{+}$ is the fully rough bulk velocity, which can be obtained by integrating the logarithmic velocity profile (1.1) with $\unicode[STIX]{x0394}U^{+}=\unicode[STIX]{x1D705}_{m}^{-1}\log (k^{+})+C$ , to yield a constant,

(4.7) $$\begin{eqnarray}\displaystyle \displaystyle U_{b_{FR}}^{+} & = & \displaystyle A_{m}-C-\frac{1}{\unicode[STIX]{x1D705}_{m}}\left(1+\log \left(\frac{k}{h}\right)\right)\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle \displaystyle & = & \displaystyle C_{N}-\frac{1}{\unicode[STIX]{x1D705}_{m}}\left(1+\log \left(\frac{k_{s}}{h}\right)\right),\end{eqnarray}$$

and $\unicode[STIX]{x1D6E9}_{a_{FR}}^{+}$ can be obtained by integrating the logarithmic rough-wall temperature profile, as in (1.3) with constant temperature difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ , to yield

(4.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{a_{FR}}^{+}=\frac{1}{\unicode[STIX]{x1D705}_{h}}\log \left(\frac{\frac{1}{2}Re_{b}}{U_{b_{FR}}^{+}}\right)-\frac{1}{\unicode[STIX]{x1D705}_{h}}+A_{h}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}. & & \displaystyle\end{eqnarray}$$

Here, the log-law constants, $A_{m}\approx 5.0$ , $\unicode[STIX]{x1D705}_{m}\approx 0.4$ , $\unicode[STIX]{x1D705}_{h}\approx 0.46$ and $A_{h}(Pr=0.7)\approx 3.2$ , are all known and $C_{N}=8.5$ is Nikuradse’s constant. This means that we only need the roughness function offset, $C$ , from $\unicode[STIX]{x0394}U^{+}=\unicode[STIX]{x1D705}_{m}^{-1}\log (k^{+})+C$ in (4.7), or alternatively the equivalent sand-grain roughness, $k_{s}$ , in the form given by (4.8), as well as the constant $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ to predict the Stanton number at any Reynolds number in the fully rough regime. These dynamical parameters ( $k_{s}/k$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}$ ) are geometry dependent and must be measured in a dynamic procedure. Here, the unknowns can be determined using the minimal channel technique; for the present sinusoidal roughness, these values are $k_{s}/k\approx 4.1$ and $\unicode[STIX]{x0394}{\unicode[STIX]{x1D6E9}_{FR}}^{+}\approx 4.4$ (figure 5).

The empirical heat-transfer model of Dipprey & Sabersky (Reference Dipprey and Sabersky1963) in (4.5) as well as the integrated log-law equations in (4.8) and (4.9) are shown in figure 7(b), along with the present data (circles). Both models show good agreement with the present data at moderate Reynolds number, with the heat-transfer coefficient reducing with Reynolds number. However, the advantage of the log-law equations is that the only modelling assumption made is that the temperature profile follows a logarithmic profile across the entire channel. This is phenomenologically consistent with our understanding of forced convection wall turbulence and does not require any ad hoc terms like $k_{f}$ in (4.5). The integrated log-law equations can also be extended to include the effect of the wake through additive constants $L_{m}=\int _{0}^{h}2\unicode[STIX]{x1D6F1}_{m}/(\unicode[STIX]{x1D705}_{m})\unicode[STIX]{x1D6F9}\,\text{d}z$ and $L_{h}=\int _{0}^{h}2\unicode[STIX]{x1D6F1}_{h}/(\unicode[STIX]{x1D705}_{h})\unicode[STIX]{x1D6F9}\,\text{d}z$ . These are small for channel flows so do not significantly alter the results here, but are larger for pipes and boundary layers.

Figure 7(c) shows the ratio of the heat-transfer and skin-friction coefficients, $2C_{h}/C_{f}$ , for the present data as well as the log-law equations. We see that the smooth-wall ratio is constant with Reynolds number, in support of the Reynolds analogy in which momentum transfer is proportional to heat transfer. In contrast, the rough-wall ratio reduces with Reynolds number, indicating that the Reynolds analogy for rough-wall flow is breaking down for these bulk measures of momentum and heat transfer. Note that this is only in regard to the bulk quantities; the analogy may still hold within the flow when looking at quantities such as the turbulent diffusivity for momentum and heat at a given wall-normal location. In an engineering sense, this ratio of coefficients can be regarded as the heat-transfer rate per unit pumping power (Dipprey & Sabersky Reference Dipprey and Sabersky1963), or that the roughness is advantageous for heat-transfer applications when $C_{hr}/C_{fr}>C_{hs}/C_{fs}$ . Evidently, the present roughness and flow conditions are not as efficient as the smooth-wall flow. However, Dipprey & Sabersky (Reference Dipprey and Sabersky1963) noted that increasing the Prandtl number above $Pr\gtrsim 3$ may enable advantageous heat transfer for the rough-wall flow to be obtained. This favourable condition typically only occurs when the flow is in the transitionally rough regime and the rough-wall heat-transfer coefficient has reached a maximum. Note that there are also alternative measures for the performance of heat-transfer systems depending on the engineering design constraints present (Webb Reference Webb1981).

Finally, in meteorology, the logarithmic velocity profile (1.1) is often given as $U^{+}=(1/\unicode[STIX]{x1D705}_{m})\log (z/z_{0m})$ , where $z_{0m}$ is the roughness length for momentum transfer. This is related to the equivalent sand-grain roughness by a constant, $k_{s}=\exp [\unicode[STIX]{x1D705}_{m}C_{N}]z_{0m}\approx 30z_{0m}$ (Jiménez Reference Jiménez2004). In the fully rough regime, $k_{s}/k$ and $z_{0m}/k$ are constants that depend only on the roughness geometry. In a similar manner, the logarithmic temperature profile is defined as $\unicode[STIX]{x1D6E9}^{+}=(1/\unicode[STIX]{x1D705}_{h})\log (z/z_{0h})$ , where $z_{0h}$ is the roughness length for heat transfer (e.g. Owen & Thomson Reference Owen and Thomson1963; Chamberlain Reference Chamberlain1966; Wood & Mason Reference Wood and Mason1991). Relating this to (1.3) in the fully rough regime, we obtain

(4.10) $$\begin{eqnarray}z_{0h}^{+}=\exp [-\unicode[STIX]{x1D705}_{h}(A_{h}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+})],\end{eqnarray}$$

where for the present roughness and flow conditions, $z_{0h}^{+}\approx 1.7$ . Importantly, we see that this inner-normalised roughness length for heat transfer (or any passive scalar) is constant and does not depend on the roughness height, $k$ , like its momentum counterpart. This is the same form as suggested by Garratt & Hicks (Reference Garratt and Hicks1973), although there the authors used a slightly different roughness-independent constant, $z_{0h}^{+}\approx 1/(\unicode[STIX]{x1D705}Pr)\approx 3.5$ , where the heat-transfer slope constant was assumed to be equal to the momentum constant with $\unicode[STIX]{x1D705}=0.41$ . In any case, this constant value is larger than the smooth-wall constant, which would be $z_{0hs}^{+}=\exp (-\unicode[STIX]{x1D705}_{h}A_{h})\approx 0.23$ . Garratt & Hicks (Reference Garratt and Hicks1973) presented data from a collection of experimental studies which for moderate Reynolds numbers provided strong support for the rough-wall constant- $z_{0h}^{+}$ form above. However, at higher Reynolds numbers of $Re_{\unicode[STIX]{x1D70F}}\gtrsim 10^{4}$ , some of the experimental data for the ratio $z_{0m}/z_{0h}$ increased faster than was predicted by using a constant $z_{0h}^{+}$ (i.e. $z_{0m}/z_{0h}\approx \unicode[STIX]{x1D705}Prz_{0m}^{+}$ ). In the present formulation using (4.10), this ratio would take the form $z_{0m}/z_{0h}\approx \exp [\unicode[STIX]{x1D705}_{h}(A_{h}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+})]z_{0m}^{+}=Bz_{0m}^{+}$ where the constant $B\approx 0.58$ for the present sinusoidal roughness. This is shown in figure 7(d) and agrees well with the data for the present moderate Reynolds numbers. A more rapid increase in the ratio for very large Reynolds numbers would correspond to $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ decreasing. More complex models for $z_{0m}/z_{0h}$ involving nonlinear expressions of $z_{0m}^{+}$ have been suggested to account for this increase (e.g. Owen & Thomson Reference Owen and Thomson1963; Brutsaert Reference Brutsaert1975; Andreas Reference Andreas1987). However, there is large scatter in the experimental data for high-Reynolds-number flows, with some data even suggesting that $z_{0m}/z_{0h}$ eventually reduces (Garratt & Hicks Reference Garratt and Hicks1973), making it difficult to assess the true behaviour of $z_{0m}/z_{0h}$ (or $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ ). For completeness, while an equivalent sand-grain roughness for heat transfer, $k_{sh}$ , does not appear to be used in the literature, it would take the form $k_{sh}=\exp [\unicode[STIX]{x1D705}_{h}C_{Nh}]z_{0h}$ , where $C_{Nh}$ would be the heat-transfer constant for Nikuradse’s sand-grain roughness, which is currently unknown.

4.3 Wall fluxes

Figure 8. (a) Ratio of pressure to total drag force, $F_{p}/F_{T}$ , against equivalent sand-grain roughness, $k_{s}^{+}=4.1k^{+}$ . Symbols: ●, blue, present data; ♦, pipe flow with the same sinusoidal roughness geometry (Chan et al. Reference Chan, MacDonald, Chung, Hutchins and Ooi2015). (b) Momentum and heat fluxes non-dimensionalised on crest velocity and temperature, $U_{k}$ and $\unicode[STIX]{x1D6E9}_{k}$ . Line styles: — —, pressure drag contribution $\unicode[STIX]{x1D70F}_{p}/((1/2)\unicode[STIX]{x1D70C}U_{k}^{2})$ ; - - -, viscous drag contribution $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D708}}/((1/2)\unicode[STIX]{x1D70C}U_{k}^{2})$ ; ——, wall heat flux $q_{w}/(\unicode[STIX]{x1D70C}c_{p}U_{k}\unicode[STIX]{x1D6E9}_{k})$ .

The ratio of pressure drag to total drag (pressure plus viscous drag) is shown in figure 8(a). Equivalently, the total stress can be decomposed into pressure and viscous stress contributions, $\unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D70F}_{p}+\unicode[STIX]{x1D70F}_{v}$ , so that figure 8(a) also represents the inner-normalised pressure stress, $\unicode[STIX]{x1D70F}_{p}^{+}$ . Also shown in figure 8(a) are the pipe-flow data of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) for the same sinusoidal roughness geometry, with good agreement observed between the two datasets. From the roughness function values shown in figure 5, it appears that the two roughness cases where $k_{s}^{+}\gtrsim 275$ are tending towards the fully rough asymptote of $\unicode[STIX]{x1D705}_{m}^{-1}\log (k_{s}^{+})+A_{m}-8.5$ , where $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ is approximately constant. In figure 8, this then corresponds to the pressure drag being larger than approximately 75 % of the total drag force in the fully rough regime. Comparing figures 5 and 8(a) shows that, while the temperature difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ has become approximately constant for $k_{s}^{+}\gtrsim 275$ , the pressure drag continues to increase in a log-linear manner with $k_{s}^{+}$ . There is no distinct change in this trend between the transitionally and fully rough regimes and the viscous drag is still somewhat significant at 25 % of the total drag force.

Figure 8(b) shows the momentum and heat fluxes non-dimensionalised with the crest velocity and temperature, $U_{k}$ and $\unicode[STIX]{x1D6E9}_{k}$ . The momentum flux has been decomposed into its pressure and viscous drag contributions (dashed and dotted lines, respectively), where the sum of these two contributions gives the drag coefficient, $C_{dk}=2/U_{k}^{+2}\approx 0.09$ , which was seen to be approximately constant in the inset of figure 4(c). Note that the reduction in the pressure drag component for $k_{s}^{+}\approx 90$ is due to the slight increase in $U_{k}^{+}$ for this $k^{+}$ ; the ratio of pressure to total drag force (figure 8 a) increases with $k_{s}^{+}$ as expected. Alongside the momentum fluxes shown in figure 8(b) is the wall heat flux, $q_{w}/(\unicode[STIX]{x1D70C}c_{p}U_{k}\unicode[STIX]{x1D6E9}_{k})=1/(U_{k}^{+}\unicode[STIX]{x1D6E9}_{k}^{+})$ , which follows a similar trend to the viscous contribution. This suggests that in rough-wall forced convection, the Reynolds analogy can only be used to relate the heat flux to the viscous drag contribution of the momentum flux. The analogy does not hold for the overall momentum flux due to the pressure-drag contribution, and explains why the ratio $2C_{h}/C_{f}$ (figure 7 c) reduces with Reynolds number for the rough-wall case.

Figure 9. (a,c) Viscous stress and (b,d) wall heat flux for roughness with (a,b) $k^{+}\approx 33$ (transitionally rough) and (c,d) $k^{+}\approx 93$ (fully rough), averaged over time and for each repeating roughness element. The dashed contour shows the zero-stress (recirculation) region; the dotted contour in (a) shows negative stress with value $-0.2U_{\unicode[STIX]{x1D70F}}^{2}$ . Contours are equally spaced with intervals of (a,c) $0.2U_{\unicode[STIX]{x1D70F}}^{2}$ and (b,d) $0.2\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70F}}U_{\unicode[STIX]{x1D70F}}$ . Flow is from lower left to upper right.

The average viscous stress and heat flux on the rough wall are shown in figure 9 for (a,b) transitionally rough flow ( $k^{+}\approx 33$ ) and (c,d) nominally fully rough flow ( $k^{+}\approx 93$ ). These have been temporally averaged from data outputted approximately every $35\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}^{2}$ as well as averaged over each repeating roughness element in the domain. The viscous stress (figure 9 a,c) is strongest at the roughness crests, as there is high-speed fluid passing over the crests, resulting in substantial shear. This region of strong viscous stress extends down towards the saddles between neighbouring crests. A region of negative stress forms on the lee side of the roughness due to recirculation behind the roughness crests. The negative stress is stronger for the transitionally rough flow; however, the area covered by this negative stress (the dashed contour line) is approximately constant, at 40 % of the roughness plan area for both the transitionally (figure 9 a) and nominally fully rough (figure 9 c) flows. The heat flux (figure 9 b,d) appears similar to the viscous stress contours, with strong heat flux localised to the roughness crests and along the saddles. This occurs because these regions are exposed to faster flow and higher fluid temperature, leading to enhanced viscous stress and heat flux.

Figure 10. Instantaneous streamwise velocity (left) and temperature (right) in the streamwise–vertical plane, where $\unicode[STIX]{x1D703}_{w}=0$ is the isothermal wall temperature. The horizontal dashed line shows the minimal channel critical height, $z_{c}=0.4L_{y}$ . The white contour line shows the value $u^{+}=4$ and $\unicode[STIX]{x1D703}^{+}-\unicode[STIX]{x1D703}_{w}^{+}=4$ , to highlight the viscous and thermal diffusive sublayers. The black contour line shows zero velocity (recirculation). Flow is from left to right.

Figure 10 shows instantaneous snapshots of the streamwise velocity (left column) and fluid temperature (right column) in the streamwise–vertical plane for increasing roughness sizes. A white contour line for $u^{+}=4$ and $\unicode[STIX]{x1D703}^{+}-\unicode[STIX]{x1D703}_{w}^{+}=4$ has been selected to provide an indication of the viscous and thermal diffusive sublayers. The smooth-wall flow (figure 10 a) produces thin sublayers close to the wall, for both the velocity and temperature fields. The roughness in the transitionally rough regime (figure 10 b,c) produces much thicker sublayers. Here, the fluid temperature in most of the region below the roughness crests is near its isothermal wall value, suggesting a reduction in the local temperature gradients at the wall and hence a reduced heat-transfer rate in these regions. However, the large temperature gradients localised to the roughness crests (see also figure 9 b) lead to an increase in the overall heat transfer (Stanton number) for these transitional rough-wall cases relative to the smooth wall. The black contour for the streamwise velocity fields corresponds to zero velocity, giving an indication of the region of recirculation residing behind the roughness elements. As the roughness height increases further and becomes nominally fully rough (figure 10 d,e), we see that much of the fluid below the roughness crests remains near zero, with the selected contour line of $u^{+}=4$ residing mostly above the roughness crests, indicating an increasing dominance of pressure drag. In contrast, the thermal diffusive sublayer is seen to be descending into the roughness canopy for these nominally fully rough cases. It appears as a thin sublayer that follows the roughness geometry and resembles that of the smooth wall if the wall was contorted. The fluid temperature below the roughness crests is therefore much larger than what might have been expected from inspection of the velocity fields.

These figures, especially the contrast in viscous and thermal diffusive sublayers in figure 10(d,e), emphasise the dissimilarity of heat and momentum transfer and therefore the breakdown of Reynolds analogy when roughness is present. Moreover, the visual similarity of the fully rough thermal diffusive sublayer to that of a contorted smooth wall offers some explanation for why the rough-wall Stanton number (figure 7 b) has a similar trend to that of the smooth wall. While the present roughness for $\unicode[STIX]{x1D706}/k\approx 7.1$ has a 17.8 % increase in wetted surface area compared to the smooth wall, this is not enough to explain the 230 % increase in Stanton number that the rough wall produced. The substantial changes to the overlying flow dynamics produced by the roughness are therefore likely to account for much of this increase. The present qualitative description related to the viscous and thermal diffusive sublayers is based on the single roughness geometry studied in this work. It will be interesting to see whether this disparity in sublayers is also observed when the roughness is varied, for example with densely packed (short-wavelength) roughness or with sharp-edged cuboid roughness. While not studied here, these geometries can be readily investigated using the minimal-span channel technique.