1. Introduction
Buoyancy and shear are crucial processes in fluid dynamics and key for many flow related phenomena in nature and technology. A paradigmatic example of buoyancy driven flow is Rayleigh–Bénard (RB) convection, a system where the fluid is heated from below and cooled from above (Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009; Lohse & Xia Reference Lohse and Xia2010; Chilla & Schumacher Reference Chilla and Schumacher2012; Xia Reference Xia2013). The flow is controlled by the Rayleigh number 
$Ra=\beta gH^{3}\varDelta /(\kappa \nu )$, which quantifies the non-dimensional thermal driving strength between the two horizontal plates. Here, 
$H$ is their distance, 
$\beta$ the thermal expansion coefficient of the fluid, 
$g$ the gravitational acceleration, 
$\varDelta$ the temperature difference across the fluid layer, 
$\kappa$ and 
$\nu$ the thermal diffusivity and kinematic viscosity, respectively. Furthermore, the Prandtl number is defined as 
$Pr=\nu /\kappa$, which is the ratio between momentum and thermal diffusivities. An important output of the flow is the heat transport between the plates, which can be non-dimensionally quantified by the Nusselt number 
$Nu= QH/(\kappa \varDelta )$, with 
$Q = \left \langle w T \right \rangle _{A,t} - \kappa \left \langle \partial _z T \right \rangle _{A,t}$ the mean vertical heat flux, where 
$\left \langle \cdots \right \rangle _{A,t}$ indicates the mean over time and a horizontal plane.
On the other hand, for flows driven by wall shear stress, a commonly used model problem is the Couette flow (Thurlow & Klewicki Reference Thurlow and Klewicki2000; Barkley & Tuckerman Reference Barkley and Tuckerman2005; Tuckerman & Barkley Reference Tuckerman and Barkley2011). We adopt a geometry in which the bottom and top walls slide in opposite directions with a wall-tangential velocity 
$u_w$ and the forcing can be expressed non-dimensionally by the wall Reynolds number 
$Re_w=H u_w/\nu$. The relevant flow output is now the wall friction, quantified by the friction coefficient 
$C_f = 2\tau _w/(\rho u_w^{2})$, with 
$\rho$ the fluid density and 
$\tau _w$ the surface- and time-averaged wall shear stress. Turbulent Couette flow is dominated by large-scale streaks (Lee & Kim Reference Lee and Kim1991; Tsukahara, Kawamura & Shingai Reference Tsukahara, Kawamura and Shingai2006; Kitoh & Umeki Reference Kitoh and Umeki2008; Pirozzoli, Bernardini & Orlandi Reference Pirozzoli, Bernardini and Orlandi2011, Reference Pirozzoli, Bernardini and Orlandi2014; Orlandi, Bernardini & Pirozzoli Reference Orlandi, Bernardini and Pirozzoli2015; Chantry, Tuckerman & Barkley Reference Chantry, Tuckerman and Barkley2017). These remain correlated in the streamwise direction for a length up to approximately 
$160$ times the distance between the plates (Lee & Moser Reference Lee and Moser2018).
Combining the buoyancy and wall shear forcings yields a complex system that is relevant in many applications, especially for atmospheric and oceanic flows (Deardorff Reference Deardorff1972; Moeng Reference Moeng1984; Khanna & Brasseur Reference Khanna and Brasseur1998). Also, in sheared thermal convection, large-scale structures emerge, as experiments have shown (Ingersoll Reference Ingersoll1966; Solomon & Gollub Reference Solomon and Gollub1990). Investigations on channel flows with unstable stratification (Fukui & Nakajima Reference Fukui and Nakajima1985) revealed that temperature fluctuations in the bulk decrease while velocity fluctuations close to the wall increase for stronger unstable stratification.
Numerical simulations of wall sheared convection (Hathaway & Somerville Reference Hathaway and Somerville1986; Domaradzki & Metcalfe Reference Domaradzki and Metcalfe1988) have revealed that adding shear to buoyancy increases the heat transport for low 
$Ra$, but also causes the large-scale structures to weaken, thus decreasing the heat transport for 
$Ra \gtrsim 150.000$. Similar phenomena have been observed in Poiseuille–RB, where the wall parallel mean flow is driven by a pressure gradient rather than the wall shear: in this case, the 
$Nu$ decrease was attributed to the disturbance via the longitudinal wind of the thermal plumes (Scagliarini, Gylfason & Toschi Reference Scagliarini, Gylfason and Toschi2014; Scagliarini et al. Reference Scagliarini, Einarsson, Gylfason and Toschi2015; Pirozzoli et al. Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017). This plume-sweeping mechanism, causing a Nusselt number drop, was also observed in Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020), who report very long thin streaks, similar to those of the atmospheric boundary layer where these convection rolls are called cloud streets (Etling & Brown Reference Etling and Brown1993; Kim, Park & Moeng Reference Kim, Park and Moeng2003; Jayaraman & Brasseur Reference Jayaraman and Brasseur2018).
In both flows, Couette–RB and Poiseuille–RB, the ratio between buoyancy and mechanical forcings can be best quantified by the bulk Richardson number
\begin{equation} Ri=\frac{Ra}{Re^{2}_wPr}, \end{equation}
which is a combination of the flow governing parameters 
$Ra$, 
$Re_w$ and 
$Pr$. In the Couette–RB flow of Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020), 
$Ri$ was used to distinguish between three different flow regimes, namely thermal buoyancy dominated, transitional and shear dominated, similarly to the case of stably stratified wall turbulence, where Zonta & Soldati (Reference Zonta and Soldati2018) distinguish between the buoyancy dominated, buoyancy affected and turbulence dominated regimes.
Indeed, sheared stably or unstably stratified flows are present in many different situations involving both liquids and gases. Therefore the fluid properties, as reflected in the Prandtl number, play a major role (Chong et al. Reference Chong, Wagner, Kaczorowski, Shishkina and Xia2018). In the atmosphere it results in 
$Pr={O}(1)$, while in ocean dynamics 
$Pr={O}(10)$. However, a much larger 
$Pr$ variation is found in industrial applications. For example, 
$Pr\approx {O}(10^{-3})$ for liquid metals (Teimurazov & Frick Reference Teimurazov and Frick2017), which are for example in use for cooling applications in nuclear reactors (Usanov et al. Reference Usanov, Pankratov, Popov, Markelov, Ryabaya and Zabrodskaya1999), or 
$Pr\approx {O}(10^{3})$ for molten salts or silicone oils (Vignarooban et al. Reference Vignarooban, Xu, Arvay, Hsu and Kannan2015) for high-performance heat exchangers.
Despite this staggering range of Prandtl numbers encountered in real applications, the vast majority of studies on sheared, thermally stratified flows have been performed only at 
$Pr={O}(1)$. To overcome this limitation, in this paper we extend the work of Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) for 
$Pr=1$ by analysing the parameter space 
$0 \leqslant Re_w \leqslant 4000$ and 
$0.22 \leqslant Pr \leqslant 4.6$ while keeping the Rayleigh number constant at 
$Ra=10^{6}$ (see figure 1 for the complete set of simulations).
Figure 1. Phase diagram of simulation runs. We show two panels to better illustrate our choice of simulation input parameters, which were determined based on 
$Re_w$ (a) and 
$Ri$ (b). Values of 
$Re_w=2000,3000,4000$ were chosen to be consistent with Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) and to cover the shear dominated regime. The squared symbols show the datapoints for 
$Re_w =0$ for completeness and independently of the 
$y$-axis, since they cannot be directly included in the logarithmic scale. To have a sufficient amount of data in the thermal buoyancy dominated regime, we picked 
$Ri=100$ as the most thermal dominated case and then logarithmically spaced three more datapoints.
The present study can be considered similar and complementary to that of Zhou, Taylor & Caulfield (Reference Zhou, Taylor and Caulfield2017) who carried out numerical simulations with a large 
$Pr$ variation for a stably stratified Couette flow.
The manuscript is divided in the following manner. Section 2 briefly reports the numerical method. Section 3 focusses on the global transport properties and § 4 on the boundary layers. The paper ends with conclusions (§ 5).
2. Numerical method
The three-dimensional incompressible Navier–Stokes equations with the Boussinesq approximation are integrated numerically. Once non-dimensionalised, the equations read
\begin{gather} \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ={-}\boldsymbol{\nabla} P + \left(\frac{Pr}{Ra} \right)^{1/2} \nabla^{2}\boldsymbol{u}+\theta \hat{z}, \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} =0, \end{gather}
\begin{gather} \frac{\partial \theta}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \theta = \frac{1}{(Pr Ra)^{1/2}} \nabla ^{2} \theta, \end{gather}
with 
$\boldsymbol {u}$ the velocity, normalised by 
$\sqrt {g \beta {\rm \Delta} H}$, and 
$\theta$ the temperature, normalised by 
$\varDelta$; 
$t$ is the time normalised by 
$\sqrt {H/(g \beta \varDelta )}$ and 
$P$ the pressure in multiples of 
$g \beta {\rm \Delta} H$.
Equations (2.1a,b) and (2.2) are solved using the AFiD GPU package (Zhu et al. Reference Zhu, Phillips, Arza, Donners, Ruetsch, Romero, Ostilla-Mónico, Yang, Lohse, Verzicco, Fatica and Stevens2018b) which is based on a second-order finite-difference scheme (van der Poel et al. Reference van der Poel, Ostilla-Mónico, Donners and Verzicco2015). The code has been validated and verified several times (Verzicco & Orlandi Reference Verzicco and Orlandi1996; Verzicco & Camussi Reference Verzicco and Camussi1997, Reference Verzicco and Camussi2003; Stevens, Verzicco & Lohse Reference Stevens, Verzicco and Lohse2010; Stevens, Lohse & Verzicco Reference Stevens, Lohse and Verzicco2011; Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014; Kooij et al. Reference Kooij, Botchev, Frederix, Geurts, Horn, Lohse, van der Poel, Shishkina, Stevens and Verzicco2018). We use a uniform discretisation in the horizontal periodic directions and a non-uniform mesh, with an error function-like node distribution in the wall-normal direction. To implement the sheared Couette-type forcing we move the top and bottom walls in opposite directions with velocities 
${\pm } u_w$, i.e. relative velocity 
$2u_w$ between the two plates.
Following Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020), we performed our simulations in a 
$9{\rm \pi} H \times 4{\rm \pi} H \times H$ domain, which are the streamwise, spanwise and wall-normal directions, respectively. The grid resolutions are also based on Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) and then further modified to account for the Prandtl number variation in this study.
3. Flow organisation and global transport properties
3.1. Organisation of turbulent structures
Using as guideline the description of Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) we observe that also in the present case the flow can be classified into buoyancy dominated, transitional and shear dominated regimes (see figure 2 and table 1 for a full overview). As shown in Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020), for 
$Pr=1$ and increasing 
$Re_w$, we observe the thermal buoyancy dominated regime at 
$Re_w =0$ while already at 
$Re_w =1000,2000$ the compact thermal structures elongate into streaks and evidence the transitional regime. Further increasing the wall shear causes the streaks to meander in the spanwise direction, which indicates the shear dominated regime (
$Re_w =3000,4000$).
Figure 2. Snapshots of the temperature field at midheight (
$z/H=0.5$) for a subdomain of the parameter space. The applied wall shear is in the 
$x$-direction, while 
$y$ is the spanwise coordinate.
Table 1. Main simulations considered in this work. The columns from left to right indicate the input and output parameters and the resolution in streamwise, spanwise and wall-normal directions 
$(N_x,N_y,N_z)$. The simulations for 
$0 \leqslant Re_w \leqslant 1000$ were chosen to allow the first non-zero 
$Re_w$ at 
$Ri=100$. The other two 
$Re_w<1000$ simulations for each 
$Pr$ respectively were logarithmically evenly spaced in 
$Re_w$. Data of Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) have been used for 
$Pr=1;Re=0,2000,3000,4000$. The data of the Monin–Obukhov length were added for consistency with Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020), although not specifically discussed in this manuscript.
As 
$Pr=\nu /\kappa$ exceeds unity, kinematic viscosity overtakes thermal diffusivity and the wall shear affects the flow structures in the bulk more easily. In fact, it can be observed that, already for 
$Re_w =1000$, the flow shows meandering behaviour of the shear dominated regime. For 
$Pr =4.6$ and 
$Re_w =4000$ the shear is strong enough to make the effect of the thermal forcing negligible, as confirmed by the flow structures similar to the plane Couette flow.
Conversely, for Prandtl numbers smaller than unity, the shear is less effective for a given 
$Re_w$ and the bulk flow is more dominated by the thermal structures. In the case of 
$Pr =0.22$, a wall shear of 
$Re_w =1000$ is not strong enough to fully disturb the plumes and only the next datapoint at 
$Re_w =2000$ shows signs of elongated streaks.
From the panels of figure 2 it is evident how 
$Pr$ changes the relative strength of the momentum and thermal diffusivities: a higher Prandtl number, corresponding to a larger kinematic viscosity, increases the momentum transfer from the boundaries to the bulk and the transition to the shear dominated regime occurs at a lower 
$Re_w$ than for a corresponding low 
$Pr$ flow. Vice versa, for small Prandtl numbers, the thermal dominated regime is more persistent and the shear dominated flow features appear only at high 
$Re_w$. These findings are consistent with those of Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) that the Richardson number, 
$Ri$, which is constant for constant 
$Re_w^{2} Pr$ (see (1.1)), determines the flow regime.
3.2. Heat transfer
The Nusselt number 
$Nu$ is plotted in figure 3 as a function of 
$Re_w$, showing a non-monotonic behaviour. The common feature is that, for increasing wall shear, 
$Nu$ first decreases and then increases, as already observed in Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) for 
$Pr=1$. In the present case, however, the specific values are strongly dependent on 
$Pr$, as seen in figure 3(c). The effect of 
$Pr$ is strongly dependent on the amount of shear added to the system. For pure Rayleigh–Bénard convection (
$Re_w=0$), 
$Nu$ increases with 
$Pr$ for 
$Pr<1$ and saturates to a constant value for 
$1<Pr<4.6$, see figure 3(b), in agreement with the findings of van der Poel, Stevens & Lohse (Reference van der Poel, Stevens and Lohse2013) and Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013). For increasing 
$Re_w$, the effect of the wall shear on the heat transfer is more pronounced for increasing 
$Pr$, because of the higher momentum transfer from the boundaries to the bulk. This is confirmed both by the initial 
$Nu$ decrease up to 
$20\,\%$ of the RB value at 
$Pr=4.6$ and the subsequent strong increase by more than 
$50\,\%$ for the highest 
$Re_w$. In both cases the effects of the momentum transfer are enhanced by the high Prandtl number. We mention that the non-monotonic behaviour of the Nusselt number observed here is a frequently occurring feature of flows in which more than one parameter determines the value of the heat transfer; other known cases have been reported by Scagliarini et al. (Reference Scagliarini, Gylfason and Toschi2014) and Pirozzoli et al. (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017) for Poiseuille–RB flow, Yang et al. (Reference Yang, Verzicco, Lohse and Stevens2020) and Wang et al. (Reference Wang, Chong, Stevens, Verzicco and Lohse2020b) for thermal convection with rotation or Chong & Xia (Reference Chong and Xia2016) for severe lateral confinement, although the exact interplays between the forces in these cases are different.
Figure 3. (a) Value of 
$Nu$ versus 
$Re_w$ for varying 
$Pr$. The curves show a more or less pronounced minimum 
$Nu^{min}$ at a certain shear Reynolds number 
$Re_w(Nu^{min})$. (b) Shows 
$Nu(Re_w=0)$ versus 
$Pr$. (c) Shows 
$Re_w(Nu^{min})$ versus 
$Pr$. Note that the error bars for these values are considerable, given our limited resolution in 
$Re_w$. Nonetheless, we include a power-law fit in the figure.
3.3. Flow layering
The initial 
$Nu$ decrease can be understood upon considering that the added wall shear perturbs the thermal RB structures and produces a horizontal flow layering that weakens the vertical heat flux. Once the wall shear is strong enough, however, the flow undergoes a transition to a shear dominated regime and the vertical cross-stream motion generated by the elongated streaks makes up for the suppressed RB structures, thus starting the Nusselt number monotonic increase (Blass et al. Reference Blass, Zhu, Verzicco, Lohse and Stevens2020). To better understand the effect of the horizontal flow layering, we discuss the results of figure 4. In these ‘side views’ (i.e. streamwise cross-sections) of the temperature field snapshots and the corresponding top views of figure 2, we can observe how the flow changes from thermal plumes to straight thin streaks and then to meandering structures. As expected, the increase in wall shear causes the flow to become more turbulent. But the change in the large-scale structures is also very recognisable. Here, the transitional regime displays a more unexpected behaviour. In contrast to what is seen in figure 4(a,c), where the flow structures appear clearly divided into hot and cold columns, in figure 4(b) the structures are more complex. Due to the wall shear and the thereby imposed horizontal flow, the vertical structures are disturbed, the flow is not able to reach the opposite hot/cold wall, but is instead trapped in a warm/cool state in the bulk of the flow. The fluctuations in the flow are not strong enough to mix the bulk and therefore the heat gets insulated in a stably stratified layer in the middle of the flow. This layering causes the total heat transfer to decrease and is the reason for the drop in 
$Nu$ for low 
$Re_w$ in figure 3. Because of the heat entrapment in the bulk layer, relatively cold fluid comes very close to relatively warm fluid and the temperature gradients in the wall-normal direction increase significantly. In the atmosphere, this phenomenon can be observed as cloud streets, which, similar to the high-shear end of the transitional regime observed here, manifests as long streaks of convection rolls (Etling & Brown Reference Etling and Brown1993; Kim et al. Reference Kim, Park and Moeng2003; Jayaraman & Brasseur Reference Jayaraman and Brasseur2018).
Figure 4. Mean wall-normal temperature profiles (left) and side view snapshots of temperature fields (right), i.e. streamwise cross-sections, for (a) 
$Pr=0.22;Re_w=0$, (b) 
$Pr=0.22; Re_w=4000$ and (c) 
$Pr=1;Re=4000$. For all right panels only 
$x/H=0\text{--}4{\rm \pi}$ is shown for better visibility and 
$y/H=2 {\rm \pi}$ was chosen for the spanwise location at which periodic boundary conditions are employed.
4. Boundary layers
4.1. Boundary layer thicknesses
A complementary way to better understand the 
$Pr$-dependence of the flow dynamics and the transport properties is to study the viscous and thermal boundary layer thicknesses 
$\lambda _u$ and 
$\lambda _\theta$, respectively. Here, we define both 
$\lambda _\theta$ and 
$\lambda _u$ by extrapolating the linear slopes of the mean temperature and mean streamwise velocity close to the walls, similarly to Shishkina et al. (Reference Shishkina, Stevens, Grossmann and Lohse2010). The dependence of 
$\lambda _u$ and 
$\lambda _\theta$ on 
$Ri$ and 
$Pr$ is shown in figure 5. Here, we use as abscissa the Richardson number. Given that 
$Ra=10^{6}$ is constant, we have 
$Ri \propto (PrRe_w^{2})^{-1}$. At every 
$Pr$, for increasing 
$Ri$ – and therefore decreasing shear – 
$\lambda _\theta$ initially grows, then reaches a plateau at 
$Ri \approx 1$ and eventually decreases slowly to converge to the pure RB value (figure 5a). For comparison, we also plot 
$Nu(Ri)$ in figure 5(c). Given that 
$\lambda _\theta \propto (Nu)^{-1}$ to a good approximation, the behaviour of the thermal boundary layer thickness is consistent with the Nusselt number of figures 3 and 5(c). The different flow regimes can be identified either from the different slopes of 
$\lambda _\theta$ versus 
$Ri$ or from those of 
$Nu(Ri)$. The slope is positive in the shear dominated region (small 
$Ri$), approximately zero in the transitional regime and then negative in the thermal buoyancy dominated regime.
Figure 5. (a) Thermal boundary layer thickness 
$\lambda _\theta$ and (b) kinetic boundary layer thickness 
$\lambda _u$ as functions of the 
$Ri$-number for various 
$Pr$-numbers and fixed 
$Ra=10^{6}$. Note that the scale is the same in both (a,b). (c) Value of 
$Nu(Ri)$ compared to 
$H/(2\lambda _\theta (Ri))$. (d) Ratio of thermal and kinetic boundary layer thicknesses vs 
$Ri$.
As the Richardson number indicates the relative strength of buoyancy and shear, the non-monotonic behaviour of the thermal boundary layer can be expected. For 
$Ri\gtrsim 1$, the flow is not dominated by shear, and therefore an increase of 
$Ri$, which is consistent with a decrease of 
$Re_w$ for constant 
$Ra$ and 
$Pr$, strengthens the thermal plumes and therefore the heat transfer, which results in a smaller thermal boundary layer. The reason for the 
$\lambda _\theta$ increase for 
$Ri\lesssim 1$ is that, in this region, the thermal forcing is weak and the flow is mainly driven by the shear. In this case the thermal boundary layer is slaved to the viscous boundary layer which, according to the expectations, monotonically thickens as the wall shear weakens. From figure 5(b) we can see that indeed 
$\lambda _u$ monotonically increases with increasing 
$Ri$.
Note that the viscous boundary layer thickness has a stronger dependence on 
$Pr$ than the thermal boundary layer thickness. Qualitatively, larger 
$Pr$ reflects stronger momentum diffusivity and therefore a thicker viscous boundary layer. Note that part of this strong increase of 
$\lambda _u$ with 
$Pr$ simply reflects that 
$Ri$ is kept constant, because, to achieve this, 
$Re_w$ has to decrease as 
$\propto Pr^{-1/2}$ to keep 
$Ri$ fixed, see (1.1). However, in the shear dominated regime (high 
$Pr$ or low 
$Ri$), 
$\lambda _u$ grows faster than in the other regimes and this is especially true for the flows with higher 
$Pr$. In fact, in these cases the thermal boundary layer is nested within the viscous one and the dynamics of the latter is not sensitive to the former. This is not the case for small 
$Pr < 1$ because then 
$\lambda _u$ evolves inside 
$\lambda _\theta$ whose thinning with increasing 
$Ri$ counteracts the thickening of the viscous boundary layer.
To further stress the importance of the relative thicknesses of the thermal and the viscous boundary layer, we show their ratio versus 
$Ri$ in figure 5(d). We can see that 
$\lambda _\theta /\lambda _u$ increases for decreasing 
$Pr$ at fixed 
$Ri$ since the kinetic boundary layer thickness is driven by the momentum diffusivity. At fixed 
$Pr$ the behaviour of the boundary layer ratio is more complex: it always shows a decreasing trend in the high end of 
$Ri$ which is due to the thinning of the thermal boundary layer. On the other hand, at the low end of 
$Ri$ one can observe an increase only for 
$Pr > 1$, which is due to the steep growth of 
$\lambda _\theta$ with 
$Ri$ observed in figure 5(a).
Due to the limited number of datapoints, we cannot show a more detailed behaviour in the extreme case of pure shear forcing. In contrast, in the limit of pure Rayleigh–Bénard convection we do observe the asymptotic trend for 
$\lambda _\theta /\lambda _u$; there, the effect of the shear becomes very small (no imposed shear, all shear due to natural convection roll) and the ratio depends on 
$Pr$ only. This saturation occurs earlier for smaller 
$Pr$, because the thermal forcing dominates over the shear forcing at smaller 
$Ra$.
4.2. Velocity and temperature wall profiles
For strong enough shear the boundary layers, which are first of laminar type, will eventually become turbulent, considerably enhancing the heat transport. However, for most of the values of the control parameters (
$Re_w$ and 
$Pr$) of this paper this is hardly the case. This can best be judged from the velocity profiles, which we show in figure 6(a–c) for three different values of 
$Pr$ and various 
$Re_w$. Only in the high-
$Pr$ range, towards the limit of plane Couette flow, can we see that 
$u^{+}$ evolves towards the well-known Prandtl–von Kármán logarithmic behaviour 
$u^{+}(z^{+})=\kappa ^{-1} \log z^{+} +B$ for high 
$Re_w$. Since the shear strongly affects the flow, the boundary layers can undergo the transition to turbulence earlier than without shear. Note that the large 
$Pr$ number enhance the shear in the boundary layer. In fact, at 
$Pr=4.6$ already the flow at 
$Re_w=3000$ shows the onset of a log-law behaviour, in spite of the quite low 
$Ra = 10^{6}$. This occurrence of the log layer for large 
$Pr=4.6$ goes hand in hand with an increase in the Nusselt number as a function of 
$Re_w$, see figure 3(a). It resembles the onset of a log-law behaviour for the velocity boundary layer profile in two-dimensional RB simulations at very large Rayleigh numbers 
$Ra \geqslant 10^{13}$ (Zhu et al. Reference Zhu, Mathai, Stevens, Verzicco and Lohse2018a), which also coincides with an enhanced Nusselt number and which has been associated with the onset of the ultimate regime. The same coincidence of the development of a log layer and an enhanced heat transfer had also been found by Wang, Zhou & Sun (Reference Wang, Zhou and Sun2020a) for high frequency horizontal vibration of the RB cell. Here, in our present simulations, the more 
$Pr$ is decreased, the harder it becomes for the wall shear to disturb the thermal plumes and, as a result, at 
$Re_w \leqslant 4000$ and 
$Pr \leqslant 2.2$, the log scaling cannot be attained in our simulations.
Figure 6. Velocity and temperature wall profiles for 
$Pr=0.22$ (left), 
$Pr=1$ (middle) and 
$Pr=4.6$ (right) for various 
$Re_w$. (a–c) Mean streamwise velocity and (d–f) mean temperature profiles. Here, 
$u^{+} =u/u_\tau$ and 
$T^{+}=T/T_\tau$, with the friction temperature 
$T_\tau =Q/u_\tau$. The dashed lines in (a–c) show the linear profile for 
$z^{+}\ll 10$ and the Prandtl–von Kármán log law of the wall 
$u^{+}(z^{+})=\kappa ^{-1} \log z^{+} +B$, with 
$\kappa = 0.41$ and 
$B=5$.
Figure 6(d–f) shows a similar behaviour for the mean temperature profiles as for the velocity profiles. One can observe that the temperature profiles converge earlier towards some type of logarithmic behaviour. For 
$Pr=1$, we can see such behaviour for 
$Re_w=4000$, whereas at larger 
$Pr=4.6$, it already shows up even at 
$Re_w=2000$. From the shown temperature profiles, we can also identify the flow layering that was previously discussed in § 3.3. When the flow layering occurs, heat gets entrapped in the bulk flow. Since now an additional layer of warm and cool fluid exists in between of the cold and hot regions, 
$T^{+}$ shows a non-monotonic behaviour with a drop after the initial peak. This can most prominently be seen in figure 6(d) (
$Pr=0.22$) for the strongest shear 
$Re_w=4000$.
5. Conclusion
In this manuscript we performed direct numerical simulations of wall sheared thermal convection with 
$0 \leqslant Re_w \leqslant 4000$ and 
$0.22 \leqslant Pr \leqslant 4.6$ at constant Rayleigh number 
$Ra=10^{6}$. Similarly to Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020), who analysed the 
$Ra$-dependence of wall sheared thermal convection, we found three flow regimes and quantified them by using the bulk Richardson number and a visual analysis of two-dimensional cross-sectional snapshots. The flow undergoes a transition from the thermal buoyancy dominated to the transitional state when 
$Ri \lessapprox 10$. We found that the meandering streaks of the shear dominated regime start to emerge at 
$Ri \lessapprox 0.1$. Also the behaviour of the Nusselt number strongly depends on 
$Pr$. For high Prandtl number, the momentum transfer from the walls to the flow is increased and therefore the flow can more easily reach the shear dominated regime where the heat transfer is again increased. We analysed both the thermal and the kinetic boundary layer thicknesses to better understand the transitions of the flow between its different regimes. We found that the thermal boundary layer thickness 
$\lambda _\theta$ shows a peak in the transitional regime and decreases for both lower and higher 
$Ri$. The kinetic boundary layer thickness 
$\lambda _u$ increases with increasing 
$Ri$ and increasing 
$Pr$. For very strong 
$Re_w$ and in particular large 
$Pr$ we notice the appearance of logarithmic boundary layer profiles, signalling the onset of turbulent boundary layer dynamics, leading to an enhanced heat transport.
Together with the results of Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020), we now have analysed two orthogonal cross-sections of the three-dimensional parameter space 
$(Ra,Pr,Re_w)$. More specifically, we have determined 
$Nu(Ra,Pr,Re_w)$ for the two cross-sections 
$Nu(Ra,Pr=1,Re_w)$ in Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) and 
$Nu(Ra=10^{6},Pr,Re_w)$ here. From standard RB without shear we of course know 
$Nu(Ra,Pr,Re_w=0)$, which is perfectly described by the unifying theory of thermal convection by Grossmann & Lohse (Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001) and Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013). The knowledge of the two new cross-sections in parameter space may enable us to extend this unifying theory to sheared convection.
Acknowledgements
We thank P. Berghout, K.L. Chong and O. Shishkina for fruitful discussions. The simulations were supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s713, s802, and s874. This work was financially supported by NWO and the Priority Programme SPP 1881 ‘Turbulent Superstructures’ of the Deutsche Forschungsgemeinschaft. We also acknowledge the Dutch national e-infrastructure SURFsara with the support of SURF cooperative.
Declaration of interests
The authors report no conflict of interest.
Open access
