3.1 Effective scaling of the Nusselt number

Figure 4 shows that the heat transfer increases with increasing $Ra$ and $Re_{w}$ and that for a given $Ra$ number a minimum heat transfer is obtained at some intermediate $Re_{w}$. Scagliarini et al. (Reference Scagliarini, Gylfason and Toschi2014) showed that the minimum is caused by the thermal plumes being swept away by the shear. Figure 5(a) shows cross-sections for constant $Ra$ which reveal that the location of the minimum heat transfer at constant $Ra$ shifts towards higher $Re_{w}$ with increasing $Ra$. For high enough $Re_{w}$, the behaviour of $Nu$ converges towards $Nu\sim 0.0013Re_{w}$. Figure 5(b), where $Nu$ is normalized by the RB value for the respective $Ra$, shows that the drop in $Nu$ becomes less pronounced and is observed at higher $Re_{w}$ when $Ra$ is increased. This is a good indication that the thermal plumes become stronger and therefore harder to disturb by the applied shear. For $Ra=10^{8}$ the decrease in $Nu$ at $Re_{w}=2000$ is only ${\sim}5\,\%$ while the data for other $Ra$ show percentages up to the high twenties. A more detailed analysis would need more data points for low $Re_{w}$, which are difficult to obtain due to the computational time that is required for each simulation.

Figure 3.

Values of (a) $Nu$ and (b) the streamwise velocity fluctuations for simulations at $Ra=10^{8}$ and $Re_{w}=10\,000$ performed using different grid resolutions. The simulations are performed in a box of $2\unicode[STIX]{x03C0}H\times \unicode[STIX]{x03C0}H\times H$ in the streamwise, spanwise and vertical directions, respectively. The displayed resolutions indicate the extrapolated streamwise resolutions that correspond to the full $9\unicode[STIX]{x03C0}H\times 4\unicode[STIX]{x03C0}H\times H$ box, see table 1 for details. Note that the simulation results are converged for the grid resolution used in this study.

Figure 4.

Value of $Nu$ as a function of $Ra$ and $Re_{w}$ in Couette–RB flow.

Figure 5.

Values of (a) $Nu$ and (b) $Nu$ normalized by the RB value $Nu_{Re_{w}=0}$ as a function of $Re_{w}$.

The results indicate that the heat transfer is influenced by the ratio of the buoyancy and shear forces. Therefore, the bulk Richardson number $Ri$ or the above-defined Monin–Obukhov length, which take the ratio of these forces into account, are natural control and response parameters to identify the different flow regimes. Although the Monin–Obukhov theory itself is only valid for shear dominated flow, which does not necessarily exist in all our simulations, we use the Monin–Obukhov length as an objective criterion to distinguish between buoyancy and shear-driven flow. This choice builds on a long and rich tradition of using the Monin–Obukhov length to characterize mixed convective flows, namely in the seminal works by Obukhov (Reference Obukhov1946) and Monin & Obukhov (Reference Monin and Obukhov1954). Its use has significantly advanced the understanding of mixed convective flows. The Monin–Obukhov length is relevant as it characterizes the effects of both friction and buoyancy, the main physical effects in this system, by a single length scale. Also, in this case, we show that the Monin–Obukhov length is a relevant parameter that gives insight into the behaviour of the flow. From the data in table 2, we find $L_{MO}/H\approx 0.16/Ri^{\,0.91}$ (see appendix C). In figure 6(a) the Monin–Obukhov length is compared to the thermal boundary layer thickness $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$, which is determined from $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}=H/(2Nu)$. Since $L_{MO}/H$ is the fraction of the domain in which the shear forcing is dominant $L_{MO}\geqslant 0.5H$ indicates when the flow is completely shear dominated. This allows us to define three different flow regimes, namely a buoyancy dominated regime ($L_{MO}\lesssim \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$), a transitional regime ($0.5H\gtrsim L_{MO}\gtrsim \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$) and a shear dominated regime ($L_{MO}\gtrsim 0.5H$). At the moment we cannot more accurately determine the Monin–Obukhov length at which the transition takes place, but the presented ranges provide a good indication of the required thermal and shear forcing to achieve the different regimes. A similar behaviour has also been observed in convective boundary layers, where Salesky, Chamecki & Bou-Zeid (Reference Salesky, Chamecki and Bou-Zeid2017) find a cell dominated regime for $-z_{i}/L_{MO}\gtrsim 20$, where $z_{i}$ is the convective boundary layer thickness, a cell and roll dominated regime as transitional state, and a roll dominated regime for $-z_{i}/L_{MO}\lesssim 5$.

Figure 6.

(a) The Monin–Obukhov length as a function of $Ra$ for different $Re_{w}$. The Monin–Obukhov length (solid lines) is compared to the thermal boundary layer thickness (dashed lines) and to $0.5$ to define the flow regime (buoyancy dominated, transitional, shear dominated) of each simulation, see details in the text. Open symbols indicate $L_{MO}<0.5H$. (b) $Nu$ as a function of $Ra$. The numbers indicate the scaling exponent $\unicode[STIX]{x1D6FC}$ in $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$. The $\unicode[STIX]{x1D6FC}=0.37$ effective scaling line is plotted for visual reference only.

Figure 6(b) shows that the heat transfer in the buoyancy dominated regime scales as $Nu\simeq Ra^{0.30}$, which is in agreement with results for classical RB convection ($Re_{w}=0$, Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009)). For the shear dominated regime we find that the effective scaling exponent $\unicode[STIX]{x1D6FC}$ in $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ is $\unicode[STIX]{x1D6FC}\ll 1/3$ and in the transitional regime we find $\unicode[STIX]{x1D6FC}>1/3$. An effective scaling exponent larger than $1/3$ is one of the characteristics of the ultimate regime. It should occur when the boundary layers have transitioned to the turbulent state, which is indicated by their logarithmic profiles. Our analysis in § 4 shows that this is not yet the case in this transitional regime. Instead, for intermediate shear, the heat transfer is decreased with respect to the RB case. The locally larger effective scaling exponent simply is a consequence of the fact that with increasing $Ra$ the heat transfer, which was decreased at intermediate shear, must again converge to the RB case.

3.2 Skin friction

In figure 7 we compare the measured skin friction coefficient for different $Re_{w}$ and $Ra$ with Prandtl’s turbulent friction law (Schlichting & Gersten Reference Schlichting and Gersten2000)

(3.1)$$\begin{eqnarray}\sqrt{\frac{2}{C_{f}}}=\frac{1}{{\mathcal{K}}}\log \left(Re_{w}\sqrt{\frac{C_{f}}{2}}\right)+C.\end{eqnarray}$$

Figure 7.

(a) Skin friction coefficient $C_{f}$ as a function of $Re_{w}$. (b) Zoom in of the grey area shown in panel (a), now on a logarithmic scale, showing the data for pure Couette flow ($Ra=0$, stars) and $Ra=10^{6}$. Note that $C_{f}(Ra=0)$ follows the expected laminar result (- - -) until $Re_{w}=650{-}700$ and then jumps to the turbulent curve ($\cdot \cdot \cdot$). For Couette–RB, i.e. the up-pointing triangle, no jump is observed.

Figure 8.

Instantaneous snapshots of temperature fields at mid-height for a subdomain of the parameter space, see figure 2(a) and table 2, focusing on $Ra=1.0\times 10^{6}{-}2.2\times 10^{7}$ and $Re_{w}=0{-}4000$. The panels have coloured borders depending on the flow regime they display: buoyancy dominated (white), transitional (blue) and shear dominated (orange) regime. For a more detailed quantification of the different flow fields in the presented snapshots, we refer to the values for the Monin–Obukhov length in table 2. An overview of temperature fields over the whole domain can be found in appendix A. The colour range for the snapshots in this figure and in figures 9–11 and 14 is adjusted such that the most important thermal structures are clearly visible.

Figure 9.

(a) $Ri$ versus $Re_{w}$ for different $Ra$. Open symbols indicate the presence of thin straight elongated streaks (see third snapshot from top). The dashed line indicates $L_{MO}=0.5H$. In (b) instantaneous snapshots of temperature fields at mid-height. (c) $Nu$ as a function of $Ri$ for different $Re_{w}$. An indication of the effective scaling exponent $\unicode[STIX]{x1D6FE}$ in $Nu\sim Ri^{\unicode[STIX]{x1D6FE}}$ in the different regimes is also given. For a more detailed quantification of the different flow regimes in the presented snapshots, we refer to the Monin–Obukhov lengths documented in table 2.

Following Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014) we use a von Kármán constant ${\mathcal{K}}=0.41$ and $C=5$. The figure shows that the skin friction increases with $Ra$ and decreases with $Re_{w}$. At fixed $Ra$ the relative strength of the thermal forcing decreases for high $Re_{w}$, and therefore the obtained friction coefficient converges to the Prandtl law. This agrees very well with the findings of Scagliarini et al. (Reference Scagliarini, Einarsson, Gylfason and Toschi2015) and Pirozzoli et al. (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017) for Poiseuille–RB flow. In figure 7(b) we focus on the data for small $Re_{w}$. The skin friction in pure Couette flow follows the expected laminar result $C_{f}=4/Re_{w}$ (Pope Reference Pope2000) until a transition to the turbulent state occurs around $Re_{w}=650{-}700$. Cerbus et al. (Reference Cerbus, Liu, Gioia and Chakraborty2018) discuss that in pipe flow this jump is caused by the formation of puffs and slugs. Brethouwer, Duguet & Schlatter (Reference Brethouwer, Duguet and Schlatter2012) attribute this discontinuous jump in $C_{f}$ to the lack of restoring forces in plane Couette flow (similar to pipe, channel, and boundary layer flows). For the Couette–RB case we do not observe such a discontinuous jump. Instead, this sheared RB case is another example, next to the application of Coriolis, buoyancy and Lorentz forces discussed by Brethouwer et al. (Reference Brethouwer, Duguet and Schlatter2012), which shows that restoring forces can prevent a discontinuous jump in $C_{f}(Re_{w})$. Chantry et al. (Reference Chantry, Tuckerman and Barkley2017), on the other hand, claim that all transitions to turbulence should be continuous if the used box size is large enough. From this figure we can also judge whether the boundary layer is turbulent or not. When the slope of $C_{f}$ approaches the one of pure Couette flow the boundary layers are turbulent. We consider the boundary layer as non-turbulent when this slope starts to strongly deviate from the Prandtl law.

Figure 10.

Instantaneous near-wall snapshots at $z^{+}\approx 0.5$ of the temperature (a) and streamwise velocity (b) for $Ra=4.6\times 10^{6}$. $Re_{w}$ increases from top to bottom. For a more detailed quantification of the different flow fields in the presented snapshots, we refer to the values for the Monin–Obukhov length in table 2. The colour range for the snapshots in this figure is adjusted such that the most important thermal structures are clearly visible.

4.1 Organization of turbulent structures

To further investigate the dynamics of the different regimes we show visualizations of the temperature field for all simulations in figure 8 and appendix A. We decided to show the flow at mid-height because there the flow is least affected by the walls. In the buoyancy dominated regime the primary flow structure resembles the large-scale flow found in RB convection (Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018). In the transitional regime ($L_{MO}\lesssim 0.5H$), the thermal forcing dominates part of the bulk where large elongated thermal plumes transform into thin straight elongated streaks when $L_{MO}$ approaches $0.5H$. In figure 8 and in appendix A this manifests itself as a very visual line diagonally through the diagram, splitting the more thermal and the more shear dominated cases. In the shear dominated regime ($L_{MO}\gtrsim 0.5H$) we find large-scale meandering structures, similar to the ones found in pressure-driven channel flow with unstable stratification (Pirozzoli et al. Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017). This significant change in flow structure can be linked to the minimum in $Nu$ in figure 5. The reason for the minimum is that at intermediate shear the thermal convection rolls are broken up, while the shear is not yet strong enough to increase the heat transfer directly. This observation is in agreement with earlier works described above (Domaradzki & Metcalfe Reference Domaradzki and Metcalfe1988; Scagliarini et al. Reference Scagliarini, Gylfason and Toschi2014, Reference Scagliarini, Einarsson, Gylfason and Toschi2015; Pirozzoli et al. Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017).

In figure 9 we present a clear overview of the behaviour of the flow structures versus the flow control parameters combined in the bulk Richardson number. In panel (a), we compare the different values of $Ri$ with the visually observed flow structures. We find a range of $Ri$ in which the flow undergoes a change from the transitional to the shear dominated regime. This happens in a range of $0.2\lesssim Ri\lesssim 0.7$. In panel (c), we can also detect this trend, where the effective scaling exponent $\unicode[STIX]{x1D6FE}$ in $Nu\sim Ri^{\unicode[STIX]{x1D6FE}}$ changes from $\unicode[STIX]{x1D6FE}=0.05\pm 0.01$ to $\unicode[STIX]{x1D6FE}=0.37\pm 0.02$. We note that more data points would be necessary to define the transition point more accurately.

Figure 9 combines these findings with the above observation that in the shear dominated regime the effective scaling exponent $\unicode[STIX]{x1D6FC}$ in $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ is much smaller than $1/3$, in the transitional regime $\unicode[STIX]{x1D6FC}>1/3$, and in the buoyancy dominated regime $\unicode[STIX]{x1D6FC}\simeq 0.30$. When we compare the regime transitions with the results in figure 5 it becomes clear that the lowest heat transfer for a given $Ra$ occurs at the end of the transitional regime before the emergence of the thin straight elongated streaks. Due to the large computational time that is required for each simulation the number of considered cases is limited, which makes it difficult to pinpoint exactly when the heat transfer is minimal and what the flow structure looks like in that case. However, we note that the onset of the shear dominant regime corresponds to the point where the heat transfer starts to increase as the additional shear can then more effectively enhance the overall heat transport.

To get more insight into the boundary layer dynamics in the different regimes, we show the temperature and streamwise velocity at the boundary layer height for$Ra=4.6\times 10^{6}$ in figure 10. At this Rayleigh number the flow is in the transitional regime for $Re_{w}=2000$ and $Re_{w}=3000$, and in the shear dominant regime for $Re_{w}\geqslant 4000$. For all cases we observe a clear imprint of the large-scale structures observed at mid-height, see figure 8 and appendix A. This indicates that the large-scale dynamics has a pronounced influence on the flow structures in the boundary layers (Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018). The figure also reveals that in the transitional and shear dominated regimes the lowest temperatures at boundary layer height are observed in the high-speed streak regions, which indicates that the regions with the highest shear contribute most to the overall heat flux.

4.2 Flow statistics

We now present the streamwise temperature variance spectra $E_{\unicode[STIX]{x1D703}}(k)$ in figure 11 to analyse the size of the large-scale structures as function of the Monin–Obukhov length. The position of the peak in the temperature spectrum indicates the wavelength of the most prominent thermal structure (Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018). In panels (b,c) we plot the evolution of these wavelengths versus the absolute size of the flow domain. Therefore we define $k_{x_{peak}}$ and $k_{y_{peak}}$ as the wavenumbers of the peak in the respective energy spectrum and $\ell _{x}=2\unicode[STIX]{x03C0}/k_{x_{peak}}$ and $\ell _{y}=2\unicode[STIX]{x03C0}/k_{y_{peak}}$ as the respective wavelengths. If the spectrum does not show a clear peak but keeps growing for small $k$, the structure size is limited by the box size, which is $\unicode[STIX]{x1D6E4}_{x}=9\unicode[STIX]{x03C0}$ in streamwise (figure 11b) and $\unicode[STIX]{x1D6E4}_{y}=4\unicode[STIX]{x03C0}$ in spanwise direction (figure 11c).

Figure 11.

(a) Overview of streamwise temperature variance spectra at mid-height at $Ra=4.6\times 10^{6}$ for the different flow regimes. Panels (b,c) show the evolution of peaks in streamwise and spanwise temperature variance spectra, respectively, versus the Monin–Obukhov length. The spectra were evaluated on three-dimensional snapshots and are subsequently time averaged.

For $L_{MO}\rightarrow \infty$, $\ell _{x}\approx \unicode[STIX]{x1D6E4}_{x}H$, which is expected since for pure Couette flow, structures much larger than $9\unicode[STIX]{x03C0}$ are expected (Lee & Moser Reference Lee and Moser2018). $\ell _{y}\approx 0.8\unicode[STIX]{x03C0}H=0.2\unicode[STIX]{x1D6E4}_{y}H$ for the highest shear case, but here more data points are needed for a clearer determination of its behaviour. In the other limit of $L_{MO}\rightarrow 0$, i.e. in the transitional regime as the RB case (buoyancy dominated regime) is not shown due to the logarithmic axis, the large-scale structures are elongated over the whole streamwise length, which is consistent with figures 8 and 14. For pure RB convection, where $L_{MO}=0$, $\ell _{x}$ decreases to $\ell _{x}\approx 2\unicode[STIX]{x03C0}H$, which is in agreement with Stevens et al. (Reference Stevens, Blass, Zhu, Verzicco and Lohse2018). In the spanwise direction, the flow converges already much earlier to the RB case where $\ell _{y}\approx 2\unicode[STIX]{x03C0}H=0.5\unicode[STIX]{x1D6E4}_{y}H$. In the shear dominated regime, where the flow meanders, the structure size in streamwise direction drops to about half the box length. In the spanwise direction, this flow regime is present as a local peak in panel (c). Due to the minimal number of data points, it is not possible to fully assess the behaviour of $\ell _{x}$ and $\ell _{y}$ versus $L_{MO}$ for all $Ra$ and $Re_{w}$. Nevertheless, the minimum in $\ell _{x}$ and peak in $\ell _{y}$ in the shear dominated regime are very distinct.

To further quantify the cases shown in figure 10, in figure 12 we show the streamwise velocity and temperature profiles for fixed $Ra=4.6\times 10^{6}$ and increasing $Re_{w}$ from $2000$ to $10\,000$. As can be seen, for the wall Reynolds number in the transitional range up to $Re_{w}\approx 4000$ (see corresponding curve in figure 9a) neither the temperature nor the streamwise velocity profiles are logarithmic. This indicates that the boundary layers are not yet turbulent in this state. Hence, the higher $Nu$ scaling in this transitional regime is not caused by a triggered transition to the ultimate regime. Note that spatio-temporally chaotic flow with thermal plume detachment from the walls must not be confused with turbulent flow. The temporal fluctuations in this regime can in fact be considerable, but given the small shear Reynolds numbers, the flow is still not turbulent. Here we use the definition of turbulence as absence of large-scale coherence in space and time.

Figure 12.

(a) Mean streamwise velocity and (b) temperature profiles, where $u^{+}=u/u_{\unicode[STIX]{x1D70F}}$ and, following Pirozzoli et al. (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), $T^{+}=T/T_{\unicode[STIX]{x1D70F}}$ with the friction temperature $T_{\unicode[STIX]{x1D70F}}=Q/u_{\unicode[STIX]{x1D70F}}$ for $Ra=4.6\times 10^{6}$.

In contrast, in the shear dominated regime (beyond $Re_{w}\approx 4000$ for this $Ra=4.6\times 10^{6}$) the streamwise velocity and temperature profiles do start to converge to a logarithmic profile with further increasing $Re_{w}$, reflecting the onset of the ultimate regime. This observation is in agreement with previous findings for Couette–RB (Liu Reference Liu2003; Choi, Chung & Kim Reference Choi, Chung and Kim2004; Debusschere & Rutland Reference Debusschere and Rutland2004; Le & Papavassiliou Reference Le and Papavassiliou2006) and Poiseuille–RB (Scagliarini et al. Reference Scagliarini, Einarsson, Gylfason and Toschi2015; Pirozzoli et al. Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017).

In figure 13 we show the same statistical quantities as in figure 12, but now for fixed $Re_{w}=6000$. For $Ra\gtrsim 10^{7}$ the flow is in the transitional regime and for $Ra\lesssim 10^{7}$ the flow undergoes a transition into the shear dominated regime. Just as in figure 12 we observe that the temperature and streamwise velocity profiles are not logarithmic in the transitional regime. As the Richardson number decreases with decreasing $Ra$, we see that the profiles converge towards a logarithmic behaviour. From a comparison with table 2 we find that $Ri\lesssim 0.2$ seems to be required to achieve logarithmic temperature and velocity profiles. A comparison with the results shown in figure 9 confirms that $Ri\approx 0.2$ is indeed the threshold where the flow undergoes its transition to the shear dominated regime. This is also consistent with the work of Pirozzoli et al. (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), who report a regime with the increased importance of friction at $Ri\approx 0.1$. For the parameter regime under investigation, the effective scaling exponent $\unicode[STIX]{x1D6FC}$ in the shear dominated regime is well below $1/3$. In both figures we can detect a non-monotonic behaviour of both $u^{+}$ and $T^{+}$ for low $Re_{w}$ and high $Ra$. The non-monotonic temperature profile indicates the formation of different flow layers, i.e. heat that is carried by hot plumes originating from the bottom plate gets entrapped somewhere in the middle of the domain. Similarly, some of the cold plumes originating from the top plate also get trapped. Several further statistical quantities are presented in appendix B.

Figure 13.

(a) Mean streamwise velocity and (b) temperature profiles, where $u^{+}=u/u_{\unicode[STIX]{x1D70F}}$ and $T^{+}=T/T_{\unicode[STIX]{x1D70F}}$ with $T_{\unicode[STIX]{x1D70F}}=Q/u_{\unicode[STIX]{x1D70F}}$ for $Re_{w}=6000$. $T_{Ra=0}^{+}$ was determined through a passive-scalar temperature field.