3.1. Global flow properties

The results of the global flow properties for the different simulations are presented in table 1, where the configurations are arranged according to the parameter triad ($Ra$, $Re_b$, $S/\delta$). The smooth-wall configurations are indicated by $S=\infty$ and the configurations with streamwise-aligned ridges are listed with decreasing $S$. Since the forced and mixed convection cases are run at constant flow rate, the presence of the Gaussian ridges will increase the drag, which translates into an increase of $C_f$ and $Re_\tau$ compared with the respective case with smooth-wall conditions. As can be seen for all considered cases, the steady decrease of $S$ leads to a monotonic increase of $C_f$ and $Re_\tau$ compared with the smooth-wall case. For forced convection the increase of $C_f$ is up to $26\,\%$, while the largest increase is found for $Ri_b = 10$ with $45\,\%$. Due to the changing friction drag, the ridge height in wall units $h_g^{+}$ changes for all cases as well, and is in the range $7 \le h_g^{+} \le 60$ for the current configurations. In the case of pure forced convection, turbulent secondary motions induced by streamwise-aligned ridges are known to increase the global friction as well as the heat transfer of the flow compared with smooth-wall conditions (Stroh et al. Reference Stroh, Schäfer, Forooghi and Frohnapfel2020a). This behaviour is also observed for the present forced convection case with streamwise-aligned Gaussian ridges, where for the densest ridge spacing $S=0.5\delta$ the heat transfer increases $16\,\%$ compared with the smooth-wall case. In a similar range $17\,\%$ is the increase for the natural convection case, and the largest increase is found for $Ri_b=3.2$ with $32\,\%$.

Table 1. List of simulation configurations with flow parameters and resulting global flow properties.

While the skin friction drag increases with decreasing $S$ and increasing wetted surface area, this behaviour is not found for the heat transfer for all present cases. This is illustrated for the two parameter sweeps at constant $Re_b = 2800$ in figure 4(a) and at constant $Ra=10^{7}$ in figure 4(b). The forced convection case $Ra=0$ and weak convective case $Ra=10^{5}$ in figure 4(a) show the successive increase of $Nu$ with decreasing ridge spacing $S$. An increase of $Ra$ or $Ri_b$ introduces an additional buoyant contribution to the vertical mixing, resulting in larger heat transfer and for large Rayleigh numbers ($Ra>10^{6}$), which represent configurations where buoyancy is comparable to shear or even stronger, the monotonic increase of $Nu$ with decreasing $S$ is also found. However, for the particular cases $Ra=7.5\times 10^{5}$ ($Ri_b=0.024$) and $Ra=10^{6}$ ($Ri_b=0.032$) heat transfer does not monotonically increase with decreasing $S$, which is visible in the inset of figure 4(a).

Figure 4. Nusselt number $Nu$ as a function of Rayleigh number $Ra$ in (a) and bulk Reynolds number $Re_b$ in (b) for different ridge spacings $S$. In (a) the bulk Reynolds number $Re_b = 2800$ and in (b) the Rayleigh number $Ra=10^{7}$ is kept constant.

In figure 4(b) the natural convection case is given by $Re_b = 0$ and the influence of buoyancy is successively reduced by increasing $Re_b$. The minimum found for $Nu$ is associated with the break up of the thermal plumes of the Rayleigh–Bénard case, when shear is added, and has been reported for unstable thermal stratification in Poiseuille and Couette flows (Scagliarini, Gylfason & Toschi Reference Scagliarini, Gylfason and Toschi2014; Blass et al. Reference Blass, Zhu, Verzicco, Lohse and Stevens2020). The non-monotonic behaviour of $Nu$ with decreasing $S$ seen in figure 4(a) also occurs for the largest $Re_b = 10\,000$ case ($Ri_b = 0.025$), which is in a similar bulk Richardson number range to the former cases. Thus, all cases which depict a non-monotonic behaviour of $Nu$ with respect to $S$ fall within a range of bulk Richardson values $Ri_b = 0.016\unicode{x2013}0.032$, where shear effects are strong and buoyancy effects are weak. It will be shown that this range of Richardson numbers marks the transition from forced convection structures to streamwise rolls and the ridge spacing $S$ affects and alters this transition.

In the case of mixed convection, turbulence is driven by two generation mechanisms, the production by shear and by buoyancy, and the exact turbulence level cannot be inferred a priori by $Ra$ and $Re_b$. The same holds for the bulk Richardson number $Ri_b$. To rule out effects caused by the different turbulence levels or Reynolds number effects, which will be discussed in § 3.6, we separate those cases with significantly higher turbulence levels from the cases with comparable values. For the subsequent discussion and sections only cases which fall in a range of Reynolds number values $Re_k = 200\unicode{x2013}355$ are considered. This selection includes only those cases of the two parameter sweeps for which $Ra\le 10^{7}$ and $Re_b \le 2800$. Instead of using the pairs of $Ra$ and $Re_b$, the bulk Richardson number $Ri_b$ is used in the following to characterise the relative importance of buoyancy and shear effects. In analogy to $Ri=Ra/(4 Re_b^{2} Pr)$, $Nu$ is replaced by the Stanton number $St=Nu/(Re_b Pr)$. The corresponding results are presented in figure 5(a) and reveal an increase of $St$ with $Ri_b$.

Figure 5. Stanton number $St$ in (a) and ratio of $St$ to $C_f$ in (b) as a function of bulk Richardson number $Ri_b$ for different ridge spacings $S$. The selected cases have values of the Reynolds number $Re_k$ in a similar range.

The relative increase between heat and momentum transfer is characterised by the ratio $St/C_f$, which is shown in figure 5(b). For each $S$ it can be seen that larger values of $Ri_b$ induce larger $St/C_f$ values. This implies that buoyancy effects lead to a larger increase of heat transfer than of momentum transfer. In addition, a consistent influence of the ridges can be observed in this representation. A decrease of $S$ results in lower $St/C_f$ for all $Ri_b$. This is also reflected in the stability parameter $-\delta _{eff}/L$ in table 1, which is another quantity to compare the relative heat and momentum transfer. The most distinct property of figure 5(b) is the strong increase of $St/C_f$ in the range $Ri_b = 0.016\unicode{x2013}0.032$. This increase is delayed to larger $Ri_b$ with decreasing $S$ and can be linked to a reorganisation of the turbulent flow structures, as discussed in the following section.

3.2. Turbulent flow structures

The different flow organisations of mixed convection flows observed over smooth-wall conditions can be visualised by instantaneous velocity or temperature fluctuations in horizontal planes (Salesky et al. Reference Salesky, Chamecki and Bou-Zeid2017). The influence of heterogeneous surfaces on this flow organisation is shown for various pairs of $Ri_b$ and $S$ for the instantaneous temperature fluctuations in wall-parallel planes located at the half-channel height ($y=\delta$) in figure 6 and slightly above the top of the ridges ($y=0.15\delta$) in figure 7. Please note that the discussion for the near-wall region refers to the bottom wall, unless stated otherwise. Both horizontal planes display the same instantaneous realisation of the flow field and comprise cases that fall into a similar range of turbulent Reynolds number $Re_k$. The $Ri_b$ increases from top to bottom, starting with the forced convection case $Ri_b = 0$ ($Re_b=2800$, $Ra=0$) in the top panel and the natural convection case $Ri_b = \infty$ ($Re_b=0$, $Ra=10^{7}$) in the bottom panel. The ridge spacing $S/\delta$ decreases from left to right, with the smooth-wall case displayed in the first column of figures 6 and 7.

Figure 6. Instantaneous temperature fluctuation fields at the half-channel height position $y=\delta$ for varying Richardson number $Ri_b$ and different spanwise spacings $S$ of the Gaussian ridges. The spanwise position of the ridges is indicated by the black lines on the right outer frame of the figures. The horizontal sections show the full simulation domain of size $16\delta \times 8\delta$.

Figure 7. Instantaneous temperature fluctuation fields at $y=0.15\delta$ for varying Richardson number $Ri_b$ and different spanwise spacing $S$ of the Gaussian ridges.

Considering the smooth-wall cases $S=\infty$ first, the flow topology of the forced convection case has a spotty organisation which is also the case for the mild convective case $Ri_b = 0.003$ ($Re_b=2800$, $Ra=10^{5}$) in figure 6. The transition to streamwise rolls takes place for slightly larger buoyant forcing for case $Ri_b = 0.016$ ($Re_b=2800$, $Ra=5\times 10^{5}$), but the rolls still display some patchiness. This transition also results in a change of the near-wall structures as can be seen in figure 7, where strong elongated temperature fluctuations are preferentially located in regions of streamwise roll updrafts, and are less pronounced in the downdraft region. These elongated temperature fluctuations coincide with the near-wall low-speed streaks (not shown here), since for neutral and moderately convective cases the temperature behaves like a passive scalar with strong correlation with the streamwise velocity (Khanna & Brasseur Reference Khanna and Brasseur1998). The formation mechanism of streamwise rolls is strongly linked to localised buoyancy forces, which concentrate in low-speed streaks and thereby create linear updrafts (Khanna & Brasseur Reference Khanna and Brasseur1998). Multiple updrafts can merge to a strong buoyancy-enhanced streak, forming the updraft region of the streamwise roll in figure 6. This updraft reaches the opposing wall and reduces or destroys the coherence of the low-speed streaks there. At the same time, between these impingement regions of the updrafts at the opposite wall, buoyancy-enhanced low-speed streaks can form, which in turn generate a strong localised downdraft and in combination with the updrafts result in a large-scale streamwise roll motion in the cross-section.

In contrast to case $Ri_b = 0.016$, the streamwise roll of case $Ri_b = 0.024$ in figure 6 is more articulated in its structure, which is associated with a sudden increase of $St/C_f$ in figure 5(b). The streamwise rolls persist up to $Ri_b = 1$ in figure 6 with a spanwise wavelength of approximately $8\delta$, such that the chosen domain size is able to accommodate a single pair of counter-rotating rolls as reported by Pirozzoli et al. (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017). For the cases $Ri_b=0.32$ and $Ri_b=1.0$ in figure 6 the rolls show a strong waviness of the thermal up- and downdrafts, which is also seen in the near-wall region where the updraft region encompasses spanwise inclined near-wall streaks in figure 7. When buoyancy forces become more important, the streamwise roll is more disrupted and reduce its streamwise coherence, since thermal plumes become dynamically more important (see case $Ri_b = 10$ in figure 6) (Salesky et al. Reference Salesky, Chamecki and Bou-Zeid2017). The increased influence of buoyancy also modifies the near-wall structures, where cell-like structures appear in the updraft region, which still depict some streamwise coherence, as seen in figure 7. For the Rayleigh–Bénard case $Ri_b=\infty$ ($Re_b=0$, $Ra=10^{7}$) in figure 6, the streamwise roll is not present any more and the flow organises into convective cells. These structures have a preferential roll orientation in the $x$- and $z$-directions, also seen in the near-wall region in figure 7, which is due to the rectangular domain size (Pirozzoli et al. Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017)

The introduction of the ridges displays no significant differences of the flow structures in the channel centre for the forced convection case $Ri_b=0$ and weak buoyancy case $Ri_b=0.003$ compared with the smooth-wall case. However, the elongated high temperature fluctuations in the near-wall region are more coherent in the streamwise direction in the case of ridge spacings $S=\delta, 2\delta, 4\delta$, for which they preferentially occur at the spanwise ridge position. For mild buoyancy effects, these elongated temperature regions collapse with low-speed streaks (not shown here), forming low-momentum pathways directly above the ridges. As such, the preferential position of the low-speed streaks coincides with the mean upward motion of the secondary motions, which will be shown in § 3.3. For the densest ridge spacing $S=0.5\delta$ these elongated structures still occur at the ridges, however, they appear less coherent in the streamwise direction.

Significant effects of the ridges can be seen for the transition between forced convection structures and streamwise rolls in figure 6 for cases $Ri_b = 0.016\unicode{x2013}0.032$. As shown before, the transition towards streamwise rolls for smooth-wall conditions takes place at $Ri_b=0.016$ and this transition can be also observed for the coarsest ridge spacing $S=4\delta$, which, however, is more interrupted by individual turbulent spots than the smooth-wall case. This is also reflected in a change of the near-wall structures, where, for case $S=4\delta$, the elongated high temperature fluctuations still favour the updraft region, but in contrast to the smooth-wall case, also occur inside the downdraft region at the ridge position (figure 7). For denser ridge spacing $S<4\delta$ the preferred concentration of low-speed streaks is not observed anymore. Therefore, the streamwise roll is not visible in the channel core and the flow structures resemble those seen for the forced convection and weak buoyancy cases $Ri_b = 0.003$. For slightly larger bulk Richardson number $Ri_b = 0.024$ the streamwise roll is now clearly visible for the two coarsest ridge spacings $S=4\delta$ and $S=2\delta$ in figure 6, and strong enough to reorganise the near-wall structures seen in figure 7, while for the denser ridge spacings the streamwise roll is not present. Eventually, the streamwise roll is observed for all ridge spacings $S$ at $Ri_b = 0.032$, while for the denser ridge spacings still some spot-like structures overlap with the rolls. Considering figure 5(b), the delayed transition between forced convection structures and streamwise rolls with decreasing $S$ can be related to the increased drag introduced by the ridges. Denser ridge spacings introduce more drag and shear in the near-wall region, and in consequence larger buoyancy forces are required to form the streamwise rolls which in turn induce an increase in heat transfer.

The streamwise roll, present for the intermediate Richardson number $Ri_b=0.32$ and $Ri_b=1.0$ cases (figure 6), displays no significant influence of the ridges. This is likewise the case for the near-wall region in figure 7, where it can be seen that the formation of high temperature fluctuations occur inside the updraft region of the roll.

At higher convective conditions for $Ri_b = 10.0$ the streamwise roll is present for the smooth-wall case and the two coarsest ridge spacings $S =2\delta, 4\delta$, while it completely disappears for denser ridge spacings $S \le \delta$. For $S = \delta$ the streamwise roll is replaced by convection cells, resembling that found for the Rayleigh–Bénard case with a spacing of $S=4\delta$. For the densest ridge spacing $S=0.5\delta$, rolls with a preferential orientation in the spanwise direction occur, which have similarities to the densest ridge spacing $S=0.5\delta$ of the Rayleigh–Bénard case. This transition from roll to cell structures is also reflected in a transition of the near-wall structures in figure 7. The roll-to-cell transition is also observed for the lower bulk Richardson case $Ri_b = 3.2$ (higher $Re_k$) for the same ridge spacings $S$ (not shown). This result is remarkable, since the transition from roll to cell structures over homogeneous wall conditions in ABL is expected to begin at larger values of the stability parameter $-z_i/L \approx 26$ (Salesky et al. Reference Salesky, Chamecki and Bou-Zeid2017), while the stability parameter for the cases $Ri_b = 3.2$ and $Ri_b = 10.0$ are in the range $-\delta _{eff}/L = 3.4\unicode{x2013}9.7$. This illustrates that streamwise-aligned ridges reduce significantly the range of $Ri_b$ or $-\delta _{eff}/L$ in which streamwise rolls appear. This suggests that heterogeneous rough surfaces can trigger the roll-to-cell transition with smaller buoyancy forces.

As can be seen for the Rayleigh–Bénard case $Ri_b = \infty$ in figure 6, decreasing the ridge spacings $S$ results in an increasingly preferential orientation of the convective cell towards the spanwise direction $z$. The rolls with orientation in the $x$-direction experience increasing lateral drag as $S$ decreases, and for $S\le 2\delta$ these rolls can eventually no longer emerge and only rolls in the $z$-direction, whose circulation is along the ridge direction, occur. This observation will be discussed further in § 3.4. The increase of $Nu$ for smaller $S$ is also reflected by intensified thermal up- and downdrafts in the channel centre plane. We note that additional simulations for $S=\infty$ and $S=0.5\delta$ in a wider domain ($L_x = L_z = 16\delta$) do not indicate a domain size dependence of the obtained results.

3.3. Mean properties

The effects of ridge spacing $S$ and the relative strength of the shear and buoyancy effects on the time and horizontally averaged mean streamwise velocity and temperature profiles are shown in figure 8. The mean temperature is represented as the difference from the bottom wall temperature $T_b$ and scaled by the friction temperature $T_\tau = Q/\rho c_{p} u_\tau$. Starting from forced convection, the trend of the mean profiles to become flatter with increasing $Ri_b$ is consistent with the results of Pirozzoli et al. (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017). The logarithmic region of $\langle \bar {u} \rangle$ found for a weak convective condition starts to deviate at $Ri_b = 0.032$. The reduction of the spanwise ridge spacing $S$ leads to a decrease of the mean streamwise velocity and temperature profile, which is in agreement with increased surface drag (see table 3 and figure 5b). However, temperature profiles within the transition range from forced convection structures to streamwise rolls show deviations from this behaviour close to the channel core, which is highlighted by the inset of figure 8(b). In this figure, it can be seen for $Ri_b=0.024$ and $Ri_b = 0.032$ that the temperature takes larger values with decreasing $S$ at the channel centre and at the same time the slope of the temperature increases. This indicates that the thermal mixing of the flow is increasingly weakened by the ridges. The transition from streamwise rolls to spot-like structures for case $Ri_b = 0.024$ can be also inferred from the similar slope of $S=\delta$ and $S=0.5\delta$ to that of the forced convection cases. Interestingly, for case $Ri_b =0.032$, where all $S$ feature streamwise rolls, the slope of the temperature profile for $S=0.5\delta$ also resembles that of the forced convection cases, which indicates that the transition point is already close. As can be seen, the influence and effects of the ridge spacing $S$ are of the same order as a change of the bulk Richardson number $Ri_b$.

Figure 8. Effect of Richardson number $Ri_b$ and $S$ on wall-normal profiles of streamwise mean velocity and mean temperature for different ridge spacings $S$ scaled in wall units.

The occurrence of secondary motions over streamwise-aligned ridges in forced convection flows is observed in time- and streamwise-averaged velocity fields in the channel cross-section, and is shown for case $Ri_b=0$ in figure 9. The relative strengths of the coherent motion among the different ridge spacings can be directly compared, due to the same scaling of the cross-sectional velocity components in bulk units. The smooth channel flow exhibits no coherent motion in the cross-section (not shown here), while for streamwise-aligned ridges the secondary motions appear in the mean flow field as counter-rotating vortices at each ridge in figure 9. These vortices introduce an upward motion above each ridge and a downward motion is located to the side of each ridge. As can be seen, the secondary motion induces a bulging of the mean streamwise velocity above the ridge, transporting low momentum into the bulk region, and for large spacings $S=4\delta$ and $S=2\delta$ almost reaching the half-channel height. The spacing between the ridges of case $S=4\delta$ in figure 9(a) is large enough, so that a homogeneous region unaffected by the secondary motion can form between the ridges. Decreasing the spanwise spacing $S$, the secondary motions fill almost the entire channel domain for case $S=2\delta$ (figure 9(b) and case $S=\delta$ (figure 9(c). However, for case $S=\delta$ the wall-normal extent of the secondary motions is slightly reduced compare with case $S=2\delta$, which indicates that the secondary motions of adjacent ridges affect each other at a spacing of $S=\delta$. Further decrease of the ridge spacing to $S=0.5\delta$ (figure 9d) shows a significant reduction of the spatial extent of the secondary motion in the wall-normal direction.

Figure 9. Effect of spanwise spacing on mean streamwise velocity for forced convection case $Ri_b = 0$ ($Re_b = 2800$, $Ra=0$). The spanwise spacing of the Gaussian ridges is $S = 4\delta$ (a), $S = 2\delta$ (b), $S = \delta$ (c) and $S = 0.5\delta$ (d). Arrows indicate cross-sectional velocity components and are scaled by bulk velocity.

The investigation of the horizontal fields of the instantaneous temperature have shown that the transition of the flow topology for smooth-wall conditions is affected by the introduction of the streamwise-aligned ridges. This reorganisation is also reflected in the mean streamwise velocity and temperature fields in the cross-section, which are shown in figure 10 for the transition from forced convection structures to streamwise rolls at a ridge spacing of $S=4\delta$. The strengths of the cross-sectional velocity components are represented by the arrows in the mean streamwise velocity field, while their flow topology is visualised by streamlines in the mean temperature field. The weak buoyancy case $Ri_b = 0.003$ is shown in figure 10(a) and, similar to the forced convection case in figure 9, secondary motions are present in the mean velocity field which also leads to a bulging of the mean temperature at the ridges.

Figure 10. Effect of buoyancy on streamwise mean velocity and temperature for constant $Re_b=2800$ and $S=4\delta$ for different Richardson numbers. Arrows indicate cross-sectional velocity components and are scaled by bulk velocity.

The transition towards streamwise rolls has been seen to occur for smooth-wall conditions at $Ri_b=0.016$, while for $S=4\delta$ a slight tendency towards rolls was present. Figure 10(b) shows that secondary motions can still occur at the ridges, although two diminished roll structures emerge that extend to the opposite wall. This in turn replaces the local bulging of the mean temperature at the ridge by a significant wider bulging of the mean temperature. As $Ri_b$ increases to $Ri_b = 0.024$ the secondary motions are replaced by a streamwise roll. The upward and downward motions of the convective rolls are located in the valley between two Gaussian ridges and each roll has a spanwise extent of four half-channel heights. As can be seen in the streamwise mean velocity field, the rolls induce a stronger cross-sectional velocity in the entire channel, which introduces a recirculation zone at the leeward side of the Gaussian ridges. The cross-sectional velocities of the streamwise roll further intensify as $Ri_b$ increases, as can be seen for case $Ri_b = 0.32$ in figure 10(e).

Figure 11 shows the effect of $S$ on the mean temperature and mean streamwise velocity for case $Ri_b =10$, which features the transition between streamwise rolls and convective cells. The bulging of $\bar {T}$ and $\bar {u}$ due to the streamwise roll is found for smooth-wall conditions and $S\ge \delta$, while this is not found for the densest ridge spacing $S=0.5\delta$. For the latter case this reflects the transition from streamwise rolls to convective cells with preferential orientation in the $z$-direction found in the instantaneous temperature fields in figures 6 and 7, which results in the disappearance of the cross-sectional motion in figure 11(e). The up- and downdrafts of the streamwise rolls for $S=4\delta$ and $S=2\delta$ are located in the valleys between adjacent ridges. For the former case the lateral movement of the roll encounters the ridge in the middle between up- and downdrafts, where large spanwise velocities of the roll occur. The ridges close to the up- and downdrafts for case $S=2\delta$ support the wall-normal motion of the roll by the upward deflection at the ridges, which results in stronger bulging of $\bar {u}$ at the up- and downdraft region compared with $S=4\delta$. As can be seen for case of $S=\delta$ in figure 11(d) the strength of the cross-sectional motion is reduced compared with the coarser $S$ cases, since the roll experiences more lateral drag on crossing the ridges due to decreasing $S$.

Figure 11. Effect of ridge spacing $S$ on streamwise mean velocity and temperature for constant $Ri_b = 10$.

3.4. Turbulent properties

The mean velocity and temperature fields presented in the previous section have shown that secondary motions and streamwise rolls manifest as large-scale coherent motion in the cross-sectional plane. The energetics of these structures is further analysed by applying the decomposition procedures of (2.6) and (2.7) to the turbulent kinetic energy $k= 0.5 \overline {u'_i u'_i}$. This separates the turbulent kinetic energy $k$ into its coherent contribution $\tilde {k}$ and random contribution $k''$ given by $k = \tilde {k} + k''$. In order to extract the influence of the cross-sectional motion, the coherent turbulent kinetic energy $\tilde {k}$ is decomposed into its cross-sectional $\tilde {k}_c = 0.5 (\tilde {v}\tilde {v} + \tilde {w}\tilde {w})$ and streamwise $\tilde {k}_s = 0.5 \tilde {u}\tilde {u}$ parts. Since the global mean velocity components $\langle \bar {v} \rangle$ and $\langle \bar {w} \rangle$ are zero, the coherent components $\tilde {v}$ and $\tilde {w}$ represent the mean velocity motion in the cross-sectional plane (seen for instance in figure 10). The coherence of the large-scale motion is quantified by $K_c$, which is the volume average of the coherent turbulent kinetic energy of the cross-sectional components $\tilde {k}_c$ obtained by the integration formula of (2.8). While $K_c$ is a good measure of coherence for the majority of cases considered here, it will be shown later that there are two cases for which $K_c$ is not a useful measure. The first case applies when the coherent motion involves a strong temporal dynamics leading to a reduction of the coherent velocities $\tilde {v}$ and $\tilde {w}$ by long time averages. The second case concerns any coherent motion in the $x$–$y$ plane, so that this coherence is masked in the random velocity variance contribution.

The influence of the transition between forced convection structures and streamwise rolls on $K_c$ is illustrated in figure 12(a). As can be seen, the forced convection case $Ri_b = 0$ and weakly convective case $Ri_b = 0.003$ for smooth-wall conditions display no coherent energy in the cross-sectional components due to the missing presence of coherent motion. However, the introduction of the Gaussian ridges and the appearance of secondary motions results in a coherent kinetic energy contribution $K_c$ with the highest value for a ridge spacing of $S=\delta$, consistent with recent studies (Medjnoun et al. Reference Medjnoun, Vanderwel and Ganapathisubramani2020; Wangsawijaya et al. Reference Wangsawijaya, Baidya, Chung, Marusic and Hutchins2020). At $Ri_b = 0.016$ the streamwise rolls emerge for the smooth-wall case, which is reflected by an increase of $K_c$. This increase induced by the streamwise rolls is eventually also present for ridge spacings $S=4\delta$ and $S=2\delta$ for case $Ri_b = 0.024$. As shown in the previous sections, the two densest ridge spacing cases display secondary motions at higher $Ri_b$, which leads to an unchanged and constant value of $K_c$ up to $Ri_b = 0.024$ for $S=\delta$ and $S=0.5\delta$. This behaviour is consistent with the observation of the delayed increase in $Nu$ with increasing $Ra$ for these two cases in figure 4(a), illustrating the importance of the flow structures in the scaling of global quantities. Due to the presence of streamwise rolls for all cases at $Ri_b = 0.032$, $K_c$ also increases for all $S$, whose values almost double compared with $Ri_b = 0.024$.

Figure 12. Volume-averaged coherent turbulent kinetic energy of the cross-sectional components for low $Ri_b$ cases scaled in wall units in (a) and for large $Ri_b$ cases scaled in free-fall units in (b).

The change of coherence due to the transition between streamwise rolls and convective cells with increasing $Ri_b$ is shown in figure 12(b). For this range of values $Ri_b$, $K_c$ is scaled in free-fall units, which eases the comparison with the natural convection case $Ri_b = \infty$. For the smooth-wall condition the coherence increases up to $Ri_b = 1$ and subsequently decreases to the natural convection case. This maximum of $K_c$ occurs for a value of $-\delta _{eff}/L = 1.23$, which is consistent with recent findings in ABL, for which the maximum coherence of streamwise rolls is found at $-z_i/L = 1.08$ (Jayaraman & Brasseur Reference Jayaraman and Brasseur2021). For the rough-wall cases the coherence decreases monotonically with decreasing $S$ only for $Ri_b =0.032$ and $Ri_b=\infty$, while this behaviour is not found for values in between. Comparing with the smooth-wall cases the introduction of coarsely spaced ridges $S=4\delta$ yields a large drop of $K_c$ for $Ri_b = 1$ and $Ri_b =3.2$. The reason for this reduction is the aforementioned temporal variability of the streamwise rolls, which causes the up- and downdrafts to slowly move in the spanwise direction over a long period of time instead of being fixed, thereby reducing averaged values of $\tilde {v}$ and $\tilde {w}$ and thus $K_c$. This will be discussed in more detail in § 3.5.

For $Ri_b = 10$ the value of $K_c$ for $S=4\delta$ is below $S=2\delta$, indicating that the coherence of the streamwise roll is more affected by the coarser ridge spacing. In this case, the reduction of $K_c$ is not related to time variability of the streamwise rolls, but to the relative position of the ridges to the up- and downdrafts. As can be seen in figure 11 the up- and downdrafts for $Ri_b = 10$ occur between the ridges, which for $S=4\delta$ results in the roll encountering a ridge in the middle of its lateral motion, which causes stronger lateral drag and thereby weakens the roll motion. For $S=2\delta$, the ridges do not interfere with the roll motions at their strongest lateral velocity. Instead, the adjacent ridges at the up- and downdrafts support the upward motion of the roll by its wall-normal deflection at the ridges. Even though denser ridge spacings contribute to more drag, the support of the deflections compensates in part for the losses in $K_c$ for $S=2\delta$, while this does not occur for $S=4\delta$.

For $Ri_b = 3.2$ and $Ri_b = 10$ the value of $K_c$ vanishes for the densest ridge spacing $S=0.5\delta$, which reflects that streamwise rolls are not present for these cases, as can be seen in the instantaneous temperature fields in figure 6. Also, for the natural convection case, the value of $K_c$ approaches zero for $S\le 2\delta$, even though the instantaneous temperature fields in figure 6 suggest an increase of the coherence in the $x$-direction due to rolls aligned in the $z$-direction. This reflects the property of $K_c$ that only coherent motion in the $z$-direction can be detected, while any coherence in the $x$-direction is masked. Consequently, for the current ridge cases, a reduction of $S$ leads to a weakening of the coherence in the $z$-direction, which is equivalent to a weakening of rolls with orientation in the $x$-direction. Note that, for a further reduction of $S$ down to the limit of $S\rightarrow 0$, the surface will approach a smooth wall again with a reduced cross-sectional area. Since $Re_b$ and $Ra$ are kept constant while varying $S$ we expect the flow for $S\rightarrow 0$ to be similar to the present smooth-wall case $S=\infty$.

The strength of the different large-scale coherent motions is associated with different wall-normal regions, where coherence is dominant. This is shown for the transition between the forced convection structures and streamwise rolls for the horizontally averaged velocity stresses and temperature variance in figure 13 for three different $Ri_b$ cases. The dashed line represents the Reynolds stresses (e.g. $\langle \overline {u'u'}\rangle$), while the solid line indicates the coherent stress (e.g. $\langle \tilde {u}\tilde {u}\rangle$). The difference of both contributions results in the random stress, e.g. $\langle \overline {u'' u''} \rangle = \langle \overline {u' u'} \rangle - \langle \tilde {u}\tilde {u}\rangle$. For the forced convection case $Ri_b = 0$ and rough-wall condition the coherent streamwise stress $\langle \tilde {u}\tilde {u}\rangle$ is concentrated close to the wall, with increasing peak values with decreasing $S$ down to $S=\delta$. The densest ridge spacing $S=0.5\delta$ has a similar peak value as $S=\delta$, however, extending less into the bulk region, consistent with the reduced spatial extent of the mean secondary motions for this ridge spacing seen in figure 9. This is also reflected in the wall-normal location of the peak values of $\langle \tilde {v}\tilde {v}\rangle$ and $\langle \tilde {w}\tilde {w}\rangle$, which are located closer to the wall for $S=0.5\delta$. The coherent temperature variance displays a similar trend to $\langle \tilde {u}\tilde {u}\rangle$ with respect to $S$, since temperature is a passive scalar for this case resulting in a strong correlation between the streamwise velocity and temperature.

Figure 13. Velocity and temperature variances scaled in inner units for low bulk Richardson number $Ri_b$ cases at transition from forced convection structures to streamwise rolls. Black vertical dotted line indicates the height of the Gaussian ridges.

As discussed before, streamwise rolls are present for case $Ri_b = 0.024$ and $S=\infty, 4\delta, 2\delta$, which can be seen most clearly by increased values of $\langle \tilde {T}\tilde {T}\rangle$ and $\langle \overline {T' T'} \rangle$ within the entire bulk region in figure 13(k). For these cases the coherent temperature variance $\langle \tilde {T}\tilde {T}\rangle$ contributes a large fraction of the temperature variance $\langle \overline {T' T'} \rangle$, which reflects the strong bulging of the mean temperature seen in figure 10(c). The induced coherence by streamwise rolls is also seen in the coherent velocity stresses but is less pronounced. Among them, this is most noticeable for the spanwise coherent velocity stress $\langle \tilde {w}\tilde {w}\rangle$ in figure 13(h), where stronger spanwise coherent stresses are observed with respect to the forced convection cases (figure 13g). A slight increase of $\langle \tilde {u}\tilde {u}\rangle$ and $\langle \tilde {v}\tilde {v}\rangle$ can be also found for these cases in the bulk region, which is illustrated by the insets in figure 13(b,e). Although the coherent velocity stresses of the streamwise rolls are for this case rather weak, this motion is sufficient to cause a strong imprint in the coherent temperature variance. For the two densest ridge spacings $S=\delta$ and $S=0.5\delta$ where secondary motions occur, the velocity stresses and temperature variances remain similar to the forced convection cases at $Ri_b = 0$. This further supports the fact that the increased drag and vertical mixing due to the ridges for these two cases are strong enough to inhibit the formation of streamwise rolls.

For case $Ri_b = 0.032$, when all ridge configurations exhibit streamwise rolls, the two densest ridge spacings $S$ now also show a significant increase in the coherent temperature variance $\langle \tilde {T}\tilde {T}\rangle$, although their values are lower than for the streamwise roll cases of $Ri_b = 0.024$. Likewise, the increase of coherent velocity stresses, which was initiated at $Ri_b=0.024$, continues, which is clearly seen by $\langle \tilde {w}\tilde {w}\rangle$ in figure 13(i). As can be seen, the successive reduction of the ridge spacing $S$ results in a decrease of the coherent velocity stresses, indicating that the streamwise rolls are damped by the presence of the ridges. For $S=0.5\delta$ only a mild increase of $\langle \tilde {u}\tilde {u}\rangle$ and $\langle \tilde {v}\tilde {v}\rangle$ is found in the bulk region (figure 13c,f), and the near-wall peak of $\langle \tilde {v}\tilde {v}\rangle$, introduced by the secondary motions, is still visible. The persistence of stronger wall-normal coherent motions near the wall for $S=0.5\delta$, similar to the forced convection cases, is consistent with the streamwise rolls to appear more spot like, as seen in figure 6.

Figure 14 shows the velocity stresses and temperature variance for the transition from streamwise rolls to natural convection. For case $Ri_b = 1$ the streamwise roll displays the strongest coherent cross-sectional motion, which is illustrated by larger values of $\langle \tilde {w}\tilde {w}\rangle$ compared with the $\langle \tilde {u}\tilde {u}\rangle$. At the same time the wall-normal Reynolds stresses have comparable magnitude to the streamwise Reynolds stress in the channel centre region. The decrease of coherent velocity stresses with decreasing $S$ is found for the three densest ridge spacings, while case $S=4\delta$ displays significantly lower values due to the time variability of the streamwise rolls. At the same time, the coherent temperature variance in the bulk region increases with decreasing $S$ (inset figure 14j) and $\langle \overline {T' T'} \rangle$ exhibits larger values in the near-wall region compared with the smooth-wall case.

Figure 14. Velocity and temperature variances scaled in free-fall units for high bulk Richardson number $Ri_b$ cases at transition from streamwise rolls to natural convection.

As discussed in relation to figure 12, the coherence of streamwise rolls is reduced when increasing $Ri_b$ beyond values of $Ri_b \approx 1$, which is reflected in a reduction of the coherent velocity stresses for $Ri_b = 10$. For $Ri_b = 10$ and dense ridge spacings $S=\delta$ and $S=0.5\delta$ the instantaneous temperature fields in figure 6 show a transition from the streamwise rolls to convection cells, which is reflected here in significant lower coherent stresses $\langle \tilde {v}\tilde {v}\rangle$ and $\langle \tilde {w}\tilde {w}\rangle$ compared with the smooth-wall case. The increased peak value of $\langle \overline {T'T'} \rangle$ with decreasing $S$ indicates that ridges are more efficient in mixing temperature close to the wall. For the densest ridge spacing $S=0.5\delta$ the convection cells are oriented along the spanwise direction, which is in agreement with the observation of zero coherent spanwise stresses $\langle \tilde {w}\tilde {w}\rangle$ in figure 14(h).

The observation that the coherence of convective cells oriented along the $x$-direction is significantly reduced for $S\le 2\delta$ is also reflected in the coherent velocity stresses and coherent temperature variance. The coherent contribution of $\langle \tilde {w}\tilde {w}\rangle$ for $S=4\delta$ is significantly reduced compared with smooth-wall conditions, as can be seen in figure 14(i). The only increase of coherence can be found for $\langle \tilde {T}\tilde {T}\rangle$ in the near-wall region with decreasing $S$ in figure 14(l), while it vanishes in the bulk region for dense spacings. The preferred orientation of the convective cells in the spanwise direction, as seen in figure 6, needs to result in larger streamwise motion, which is reflected in the steady increase of $\langle \overline {u'u'}\rangle$ with $S$ (figure 14c). This is also accompanied by a steady increase of $\langle \overline {T'T'}\rangle$ with decreasing $S$, which suggests that the aligned ridges induce stronger thermal plumes. Note that, in a square domain with smooth-wall conditions, the streamwise and spanwise stresses have the same distributions due to the directional invariance of the cells (Pandey, Scheel & Schumacher Reference Pandey, Scheel and Schumacher2018). However, the smooth-wall case already displays slightly larger values for $\langle \overline {u' u'} \rangle$ than $\langle \overline {w' w'} \rangle$ and this indicates that the convection cells are slightly more oriented in the spanwise direction before introducing the aligned ridges.

3.5. Variability of streamwise rolls

In the previous section it was found that the coherence of the streamwise rolls for coarse ridge spacings $S=4\delta$ drops significantly for case $Ri_b = 1$ and $Ri_b = 3.2$ compared with the denser values of $S$, indicating weaker cross-sectional motion of the roll. However, the instantaneous temperature visualisations in figures 6 and 7 for this specific case do not indicate weaker streamwise rolls, suggesting that a time-varying behaviour of the streamwise rolls might be present. For this purpose, the volume-averaged coherent turbulent kinetic energy of the cross-sectional components $K_c$, which is based on the average of the entire time series, is now averaged for shorter time windows. The short-time-averaged coherent turbulent kinetic energy $K_c^{s}$ (superscript $s$ indicates the short-time average) is computed over a time range of ${\rm \Delta} t_s \approx 3.4t_f$. Note that the value of the short-time average is the shortest available data for the present simulations. The time evolution of $K_c^{s}$ for consecutive short-time intervals is shown for case $Ri_b = 1$ in figure 15. As can be seen, all cases feature a relatively slow dynamics and for $S=\infty$ and $S\le 2\delta$ the time variations vary mildly around their full time-averaged values $K_c$ in figure 12(b). For $S=4\delta$ the time variation is more pronounced and the dynamics shows a clearly visible periodic reduction of $K_c^{s}$ with a period of $O(t) \approx 100t_f$, which corresponds to $O(t) \approx 200 t_b$. For this analysis an additional simulation with $S=8\delta$ is performed and it shows similar time variation compared with $S=4\delta$, however, with smaller amplitude. This observed dynamics is significantly slower than observations in ABL where the dynamics of the flow reaches a statistically quasi-steady state in roughly $6t_f$ (Moeng & Sullivan Reference Moeng and Sullivan1994). The time mean value of $K_c^{s}$ is significantly larger than the value $K_c$, which suggests that the streamwise rolls of $S=4\delta$ and $S=8\delta$ feature some time variability, which is masked by considering quantities based on the average of the entire time series. This time-varying behaviour is also found for $Ri_b =3.2$ and $S=4\delta$, which also displayed a significant reduction of $K_c$ in figure 12.

Figure 15. Short-time-averaged coherent kinetic energy $K^{s}_c$ of case $Ri_b = 1$ and different ridge spacings over time. The values of $K^{s}_c$ are averaged for time intervals ${\rm \Delta} t_s \approx 3.4t_f$.

The variability of the streamwise rolls is illustrated by the time evolution of the short-time- and streamwise-averaged temperature $\overline {T}^{s}$ at the wall-normal channel centre $y=\delta$ along the spanwise direction $z$ in figure 16. The spanwise position of the thermal up- and downdrafts of the streamwise roll is represented by the higher and lower temperature values, respectively. While the spanwise location of the up- and downdrafts remain at the same position for $S=\infty$ and $S\le 2\delta$, the spanwise location of the up- and downdrafts of case $S=4\delta$ and $S=8\delta$ is strongly varying in time. The up- and downdrafts for these two cases exhibit strong lateral movement. While this movement is almost periodic for $S=8\delta$ and remains between the ridges, the up- and downdrafts of $S=4\delta$ are able to cross the ridges at some time instances, e.g. $t/t_f \approx 100$, while they are not able to cross them at other time instances, e.g. $t/t_f \approx 350$. The large values of $K_c^{s}$ of $S=4\delta$ and $S=8\delta$ in figure 15 correspond to occasions when the up- and downdrafts are located close to or directly at the ridges, e.g. $t = 700\unicode{x2013}800t_f$, while small values of $K_c^{s}$ correspond to locations of the up- and downdrafts in between the ridges. This increase of $K_c^{s}$ can be interpreted by the formation mechanism of streamwise rolls proposed by Khanna & Brasseur (Reference Khanna and Brasseur1998), which relates them to the organisation of localised buoyancy forces within near-wall streaks. When the up- and downdrafts are located at ridges, the ridges support the formation of strong localised buoyancy forces, leading to strong local up- and downdrafts. Due to the symmetric arrangement of the ridges at both walls, the up- and downdrafts impinge at another ridge on the opposite wall, which is supposed to counteract the impinging roll by the formation of localised buoyancy forces with opposing direction. Thus, for $S=4\delta$ the ridges support the formation of strong up- and downdrafts, however, they cannot remain at the spanwise location due to the counteraction of the opposing ridge, leading to the lateral evasion. For case $S=8\delta$ this enhancement at the ridges occurs only for the up- or the downdraft and not simultaneously as for $S=4\delta$, which might be an explanation for the observation that up- and downdrafts are not able to cross the ridges.

Figure 16. Streamwise- and short-time-averaged temperature $\overline {T}^{s}$ over time and spanwise position at the wall-normal channel centre location $y=\delta$ for cases $Ri_b = 1$. The spanwise position of the ridges is indicated by the black lines on the top panels.

For case $S=2\delta$ a short-time interval $t \approx 120t_f$ with large values of $K_c^{s}$ is present, which corresponds to a time interval in which the up- and downdrafts are located above ridges. However, most of the time the up- and downdrafts remain in between adjacent ridges, as has been shown in figure 11, and only a slight meandering within this range is observed. The examination of time series of cases $Ri_b = 0.32$ and $Ri_b = 10$ does not reveal this strong lateral movement of the up- and downdraft locations, while for $Ri_b = 3.2$ and $S=4\delta$ a similar lateral movement is found. The results suggest that the dynamics of the streamwise rolls is very sensitive to ridge spacings of the order of the spanwise rolls’ width, as seen for $S=4\delta$ and $S=8\delta$. For denser ridge spacings $S\le 2\delta$ several adjacent ridges contribute by localised buoyancy forces to the formation of the up- and downdrafts which might be strong enough to inhibit disturbances by the opposing ridges and thereby prevent lateral movement of the streamwise rolls. Future investigations with a staggered ridge arrangement or ridges placed only at the bottom or top wall might further shed light on the influence of ridges on the roll formation. For instance, the comparison of a symmetric and staggered arrangement of streamwise-aligned ridges in forced convection flows (Stroh et al. Reference Stroh, Schäfer, Forooghi and Frohnapfel2020a) has shown that a staggered arrangement promotes the coherence of the large-scale secondary motion, which might also be valid for rolls and may lead to a fixation of the rolls for $S=4\delta$.

3.6. Reynolds number effects

In the previous section the flow organisation of mixed convection flows was considered in terms of varying bulk Richardson number $Ri_b$ and ridge spacing $S$, while the Reynolds number $Re_k$ was approximately constant. The effect of $Re_k$ on the transition between forced convection structures and streamwise rolls is shown in figure 17, which presents the cross-sectional mean temperature and flow topology for case $Ri_b=0.024$ with values of $Re_k = 252\unicode{x2013}263$ and case $Ri_b=0.025$ with threefold larger values of $Re_k = 826\unicode{x2013}841$. As can be seen figure 17(a) both cases feature a streamwise roll down to a ridge spacing of $S=2\delta$ (not shown for $S=\infty$ and $S=4\delta$). The comparison of the temperature fields between the low and high $Re_k$ cases depicts that the thermal boundary layer is reduced for higher $Re_k$ due to the more efficient mixing of the flow in the near-wall region. As shown in the previous sections, the streamwise rolls are replaced by secondary motions for the lower $Re_k$ cases with $Ri_b = 0.024$ and $S\le \delta$, while for the larger $Re_k$ cases the streamwise roll remains for these ridge spacings. However, the streamwise roll appears more distorted and affected by the ridges, as can be seen for $S=\delta$ and $S=0.5\delta$. In addition, secondary motion in the form of one pair of counter-rotating vortices emerges at one ridge at the bottom wall lying in the downdraft region of the roll. This illustrates that the ridges on the opposing wall of the up- and downdraft regions are able to form coherent structures that counteract the large-scale roll formation. For the densest ridge spacing $S=0.5\delta$ in figure 17(c) the streamwise roll for the large $Re_k$ case is now confined to a smaller wall-normal region in the bulk flow. This is associated with the recirculation zones at the leeward side of the ridges, which are connected between the up- and downdraft regions, thereby forming a roughness sublayer which inhibits the attachment of the lateral movement of the streamwise roll at the wall. These results suggest that the transition range between forced convection structures and streamwise rolls with heterogeneous rough surfaces is not solely determined by the pair of $Ri_b$ and $S$, but also by the value of the Reynolds number $Re_k$. Due to the increased turbulent mixing for larger values of $Re_k$ the streamwise rolls can counteract the additional shear by the ridges, such that the transition between forced convection structures and streamwise rolls is shifted towards smaller values of $Ri_b$.

Figure 17. Effect of turbulent Reynolds number $Re_k$ and spanwise ridge spacing $S$ on mean temperature for case $Ri_b=0.024,\ Re_k = 252\unicode{x2013}263$ ($Ra=7.5\times 10^{5}$, $Re_b=2800$) on the left side and case $Ri_b=0.025,\ Re_k = 826\unicode{x2013}841$ ($Ra=10^{7}$, $Re_b=10\,000$) on the right side. The spanwise spacing of the Gaussian ridges is $S = 2\delta$ (a), $S = \delta$ (b) and $S=0.5\delta$ (c).

The influence of $Re_k$ on the roll-to-cell transition is illustrated for two cases with $Ri_b = 3.2$ and different $Re_k$ by the temperature fluctuation in the horizontal mid-plane in figure 18. As can be seen, both cases exhibit streamwise rolls for smooth-wall conditions and $S\ge 2\delta$, while differences in the flow structures start to appear at $S=\delta$. At this ridge spacing the lower $Re_k$ case still shows streamwise rolls (figure 18a), while for the larger $Re_k$ case the streamwise rolls are disturbed by strong thermal plumes spanning almost the entire spanwise domain. For the lower $Re_k$ case with $S=0.5\delta$ no coherent streamwise rolls can be observed and, similar to $S=\delta$, for the larger $Re_k$ case thermal plumes emerge, which indicates the beginning of the transition to convective cells. This shows that increasing $Re_k$ has a comparable effect on the flow organisation to the reduction of the ridge spacing $S$. While smaller values of $S$ increase the friction of the flow and weaken the lateral motion of the streamwise rolls, the higher $Re_k$ increases the thermal mixing near the wall, and both effects promote the formation of thermal plumes. The loss of coherence of the streamwise rolls for case $S=0.5\delta$ in figure 18(a) is also supported by a vanishing value of $K_c$ in figure 12(b). Also the inspection of the time series of $K_c^{s}$ reveals only a very weak contribution, which is an order magnitude lower than for case $S=\delta$. The observation that only convective cells with preferential orientation in the $z$-direction occur for the natural convection cases $S\le 2\delta$ and for $Ri_b=10$ with $S=0.5\delta$ is also seen for the large $Re_k$ case with $S=0.5\delta$. The influence of $Re_k$ on the transition between streamwise rolls and convective cells is such that higher values of $Re_k$ initiate this transition to convective cells at smaller $Ri_b$, thus reducing the range of streamwise rolls.

Figure 18. Instantaneous temperature fluctuation fields at the half-channel height position $y=\delta$ for cases $Ri_b = 3.2$ with $Re_k = 236\unicode{x2013}263$ ($Ra=10^{7}$, $Re_b=885$) in (a) and $Re_k = 695\unicode{x2013}792$ ($Ra=10^{8}$, $Re_b=2800$) in (b) for different spanwise ridge spacings $S$.