3.1 The effect of inclination

We consider the global transport quantities in the case of inclined convection. Inclination of the cell, in the investigated parameter range, usually leads to an increase in the global heat transport until a certain optimal inclination angle $\unicode[STIX]{x1D6FD}^{\ast }$ where the heat transport is maximal. Further inclination leads to a decrease of the global heat transport (see figure 2 a,c,e). For the Reynolds number, $Re$ , based on the kinetic energy (2.5), the behaviour with inclination is similar to that for the heat transport. However, the maximum of the Reynolds number, based on kinetic energy, is usually observed for angles smaller than or equal to $\unicode[STIX]{x1D6FD}^{\ast }$ and the relative decrease of $Re$ is much stronger near the VC case (see figure 2 b,d,f). Also, the relative increase of the Reynolds number is usually less than the relative increase of the Nusselt number (cf. figure 2 c,e with 2 d,f). The only exception to the above observations is the case of $Pr=1$ and $Ra=10^{7}$ , where inclination always leads to a decrease of the heat and momentum transport. The reason for this will be discussed further below.

Figure 2.

The absolute and normalized (with respect to RBC) Nusselt and Reynolds numbers plotted against the inclination angle $\unicode[STIX]{x1D6FD}$ for $Pr=1$ (solid lines) and $Pr=0.1$ (dashed lines). Rayleigh numbers are $10^{6}$ (blue), $10^{7}$ (cyan), $10^{8}$ (magenta) and $10^{9}$ (red). (a) Absolute Nusselt number. (b) Absolute Reynolds number, based on kinetic energy. (c) Normalized Nusselt number for $Pr=1$ . (d) Normalized Reynolds number for $Pr=1$ .(e) Normalized Nusselt number for $Pr=0.1$ . (f) Normalized Reynolds number for $Pr=0.1$ . (g) Normalized large-scale Reynolds number. (h) Absolute small-scale Reynolds number.

Figure 3.

Inclination angle $\unicode[STIX]{x1D6FD}^{\ast }$ for the optimal heat transport, as a function of $Ra$ , for $Pr=1$ (triangles) and $Pr=0.1$ (circles), for the aspect ratios $\unicode[STIX]{x1D6E4}=1/5$ (filled symbols) and $\unicode[STIX]{x1D6E4}=1$ (open symbols). The data for $\unicode[STIX]{x1D6E4}=1$ , $Ra<10^{9}$ are adapted from Shishkina & Horn (Reference Shishkina and Horn2016).

Figure 4.

Vertical cross-sections through the instantaneous flow fields for $Pr=1$ and (a)  $Ra=10^{6}$ , (b) $Ra=10^{7}$ , (c) $Ra=10^{8}$ and (d) $Ra=10^{9}$ . The colour scale is from hot (red) to cold (blue). The numbers are (from top) the Nusselt number (red), Reynolds number (blue) and inclination angle $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}$ (black). The thickness of the black streamlines reflects the normalized velocity magnitude.

Figure 5.

Vertical cross-sections through the instantaneous flow fields for $Pr=0.1$ and (a)  $Ra=10^{6}$ , (b) $Ra=10^{7}$ , (c) $Ra=10^{8}$ and (d) $Ra=10^{9}$ . The numbers are (from top) the Nusselt number (red), Reynolds number (blue) and inclination angle $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}$ (black). Colour scale and streamlines as in figure 4.

Figure 6.

The temporal behaviour of the Nusselt number for $Ra=10^{8}$ , $Pr=1$ , $\unicode[STIX]{x1D6E4}=1/5$ and $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ (green), $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.1$ (orange) and $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.15$ (blue). The time scale is based on $T_{f}=\sqrt{R/(\unicode[STIX]{x1D6FC}g\unicode[STIX]{x1D6E5})}$ . (a) Deviations of the Nusselt number from its mean value (note that the positive and negative scale is logarithmic) versus time. (b) Amplitude of the Fourier spectrum ${\mathcal{F}}$ of the Nusselt number. (c,d) Probability density functions (p.d.f.) of the Nusselt number normalized by the respective maximum. Here the instantaneous Nusselt number $Nu(t)$ is determined by averaging over the entire volume.

The inclination angle $\unicode[STIX]{x1D6FD}^{\ast }$ for the optimal heat transport is plotted against the Rayleigh number in figure 3. The figure shows that $\unicode[STIX]{x1D6FD}^{\ast }$ depends non-monotonically on $Ra$ . However, one can see that, at a lower Prandtl number and/or larger aspect ratio, the optimal heat transport is reached at a larger inclination angle.

A closer look at the instantaneous flow fields (see figure 4 for $Pr=1$ and figure 5 for $Pr=0.1$ ) reveals some expected and also some new features, which can be associated with the inclination. Inclining the cylinder forces the direction of the LSC to be fixed, which results in a very structured flow. This is illustrated by those flow fields in figures 4 and 5, where the streamlines align nicely in the case of inclined convection.

At the lowest Rayleigh number, $Ra=10^{6}$ , the spatial temperature distribution is similar to that of the conductive case and there are no single plumes visible for either of the investigated Prandtl numbers. These flow states are steady solutions of the governing equations (2.1) and (2.2) because the Reynolds number based on the velocity fluctuations, $Re_{u^{\prime }}$ , vanishes. Additionally, the streamlines are aligned and show a flow field with a smooth laminar-like LSC (see figures 4 a and 5 a).

While increasing the Rayleigh number and thus the thermal driving ( $Ra=10^{7}$ , $Pr=1$ ), we still observe a laminar-like flow with an LSC but the spatial temperature distribution looks completely different now (figure 4 b). The temperature at the rising/falling side of the LSC is close to the temperature of the respective BL. This picture is reminiscent of hot/cold channels or plume columns which span the whole cylinder and is observed for all inclination angles. Nevertheless, in this case there are no distinguishable plumes apart from the rising/falling ones and the flow is also laminar-like. As we will see, these plume columns appear to play an important role in the increase of the heat transport in inclined convection (cf. § 3.2).

For $Ra=10^{7}$ and $Pr=1$ , the RBC flow provides the optimal heat transport. This can be explained by the fact that the plume columns are already present in the flow configuration without inclination ( $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ ) (see figure 4 b), while the strength of the LSC decreases with increasing inclination (cf. figure 2 c,g).

A further increase of the Rayleigh number to $Ra=10^{8}$ leads to the development of quite a vigorous flow in RBC (see figure 4 c, $\unicode[STIX]{x1D6FD}=0$ ) and in this case the rising and falling plumes appear chaotically, leading to a complicated flow structure (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2018.477). Also, the small-scale Reynolds number ( $Re_{u^{\prime }}=865$ ) is no longer vanishing, which means that there are significant fluctuations in the velocity field. Inclining the cell leads to a decrease in the small-scale Reynolds number (figure 2 h, solid magenta line), and for an inclination angle of $\unicode[STIX]{x1D6FD}\approx 0.2\unicode[STIX]{x03C0}$ , the flow becomes steady. The emergence of the plume columns is observed at $\unicode[STIX]{x1D6FD}=0.1\unicode[STIX]{x03C0}$ , where also the maximal Nusselt number is obtained (cf. figure 4 b,c). Furthermore, the large-scale Reynolds number reaches its maximum there too, while the small-scale Reynolds number decreases by approximately 50 % compared to that in RBC.

Except for the case of highest Rayleigh number, $Ra=10^{9}$ , and lowest Prandtl number, $Pr=0.1$ , we always observe a transition to a steady flow field when the inclination angle is increased. Hence, we will discuss this transition to steady flow using this parameter set of $Ra=10^{8}$ and $Pr=1$ as an example, since it demonstrates all kinds of behaviour, from a turbulent flow in RBC to the formation of plume columns, as well as an oscillatory state and finally, for a sufficiently large inclination angle, a steady flow. Therefore, we pick the five most interesting inclination angles, namely $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ (RBC), $\unicode[STIX]{x1D6FD}^{\ast }/\unicode[STIX]{x03C0}=0.1$ (optimal heat transport), $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.15$ (oscillatory state), $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}\approx 0.2$ (steady flow) and $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/2$ (VC) to discuss the behaviour. The time evolution of the Nusselt number is displayed in figure 6(a), and for a better understanding we also provide the amplitude of the Fourier spectrum (figure 6 b) and the probability density function (p.d.f.) of the Nusselt number (figure 6 c,d).

In RBC we observe strong fluctuations of the Nusselt number, where the standard deviation is approximately 35 % of the mean value and the p.d.f. shows a wide tail at higher Nusselt numbers. In the case of optimal heat transport ( $\unicode[STIX]{x1D6FD}=0.1\unicode[STIX]{x03C0}$ ), the probability density function of the Nusselt number is narrow, almost symmetric, and the standard deviation is only approximately 1.8 % of the mean value. The width of the p.d.f. of the Nusselt number is decreasing with inclination, which is due to the strong stratification of the flow. The third case, as already mentioned, is special, because we observe an oscillatory state that leads to a periodic variation of the Nusselt number. The main frequency is $f_{1}=0.023T_{f}^{-1}$ and the Fourier spectrum (figure 6 b) also shows multiples of this main frequency. Apparently this frequency is also characteristic (strongest) for the less inclined cases, which show a continuous spectrum. Below, the oscillating case will be further discussed. At $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}\approx 0.2$ , the flow becomes steady and the Nusselt number no longer changes in time, so it is of no interest to look at the p.d.f. or spectrum for any larger inclination angles. In VC we observe a temperature distribution which is strongly stratified with respect to the direction of gravity. For large $Ra$ and $Pr=0.1$ , the Nusselt numbers in VC are larger than those in RBC (cf. figure 2 e).

Supplementary movie 2 shows the temperature isosurfaces of the oscillating flow field ( $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.15$ ). There is a torsional oscillation visible in the flow, which is asymmetric in azimuthal direction. Oscillating states were also observed for RBC in an aspect-ratio-one cell between the onset of convection and fully turbulent flow, by Verzicco & Camussi (Reference Verzicco and Camussi1997). This suggests that the route to turbulence in inclined convection is similar to RBC.

The qualitative picture of the flow structure, i.e. the emergence of plume columns, at the optimal inclination angle $\unicode[STIX]{x1D6FD}^{\ast }$ (with respect to the Nusselt number) appears to be similar for both investigated Prandtl numbers and $Ra\geqslant 10^{7}$ . This means that the maximal heat transport is reached as soon as the plume columns are fully developed, i.e. they span the whole cylinder, and the large-scale Reynolds number is also very close to its maximal value. Thus, our simulations evince similar behaviour of the small- and large-scale Reynolds number as observed in the LES by Teimurazov & Frick (Reference Teimurazov and Frick2017). They claimed that these observations are an indicator that the LSC is playing an important role in the heat transport in inclined convection.

As expected, for a lower Prandtl number ( $Pr=0.1$ ) turbulent flows are obtained already at lower Rayleigh numbers and they remain turbulent until larger inclination angles. This is underlined quantitatively by the small-scale Reynolds numbers, which are plotted in figure 2(h). One can see from these data that the transition from a turbulent to a steady flow happens at larger inclination angles for lower Prandtl numbers or higher Rayleigh numbers.

Although the Prandtl number in the experiments by Frick et al. (Reference Frick, Khalilov, Kolesnichenko, Mamykin, Pakholkov, Pavlinov and Rogozhkin2015) and LES by Teimurazov & Frick (Reference Teimurazov and Frick2017) is significantly lower than $Pr=0.1$ , there are certain similarities between our and their results. We obtained a similar relative increase of the heat transport ( $Nu(\unicode[STIX]{x1D6FD}^{\ast })/Nu(0)$ ). The studies showed also that, for otherwise similar set of parameters, a higher Nusselt number is obtained in VC rather than in RBC. In both the LES and our DNS, similar temperature profiles were obtained that demonstrated a non-vanishing gradient of the mean temperature in the bulk in RBC and vanishing gradient in inclined and vertical convection.

The highest relative increase in heat transport is observed for Rayleigh number $Ra=10^{7}$ and Prandtl number $Pr=0.1$ . We see a vigorous flow, $Re_{u^{\prime }}=756$ , and system-sized plume column formation that finally leads to an approximately 2.3 times stronger heat transport in the inclined case ( $\unicode[STIX]{x1D6FD}^{\ast }\approx 0.25\unicode[STIX]{x03C0}$ ) compared to that in RBC.

3.2 Plume columns and the flow structure of enhanced heat transfer

Since the plume columns appear to increase the heat transport as soon as they span the whole cylinder and, therefore, impinge on the opposite boundary layer, one might also compare the situation in inclined convection with that of impinging jets on a surface. When a jet impinges on a hot surface, the heat transport depends on many parameters, such as the Reynolds number, Prandtl number, ratio of plate distance to nozzle diameter, and radial distance from the stagnation point (Jambunathan et al. Reference Jambunathan, Lai, Moss and Button1992). It is known that the heat transport is generally larger in an impinging jet rather than in a parallel flow, but the heat transport decreases with the distance from the stagnation point of the jet. This was shown by various experiments such as those by Huang & El-Genk (Reference Huang and El-Genk1994). Under the assumption that the fully developed plume columns, spanning the whole cylinder, act like a jet impinging on a surface, it is reasonable to assume that this might cause the strongly increased heat transport. Additionally, such impinging zones are clearly seen in figure 7(h,k). From these panels it is apparent that the ratio of the impinging area to the plate area is much smaller in aspect ratio $\unicode[STIX]{x1D6E4}=1$ compared to $\unicode[STIX]{x1D6E4}=1/5$ and the heat transport behaves similarly, i.e. it is higher inside the slender cylinder. For a thorough analysis of the aspect ratios, see § 3.4.

Our slender geometry also shows some similarities of heat transport through an inclined vertical channel. Riedinger et al. (Reference Riedinger, Tisserand, Seychelles, Castaing and Chillà2013) carried out such experiments with a square channel of aspect ratio $1/4$ , where the ends of the channel are connected to a hot/cold reservoir of water. They found similar behaviour in the inclined channel, like stratification due to gravity, and an apparently higher scaling exponent in the transition region to the turbulent regime (as discussed below in § 3.5). Furthermore, they also observed two non-mixing symmetrical jets with inclination angle $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}>0$ .

In general, stratification of the temperature field, which is induced by the cell inclination, supports the formation of the plume columns. On the other hand, high Rayleigh numbers enhance the turbulent mixing and thus break apart the plume columns. All this leads to the conclusion that, for larger $Ra$ , the optimal heat transfer is to be expected at larger values of $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}$ . A careful interpretation of figure 3 supports this trend. Here, one has to keep in mind that the plume columns exist only for $Ra\geqslant 10^{7}$ .

Figure 7.

Cross-sections parallel to the heated plate at the edge of the thermal BL ( $z=\unicode[STIX]{x1D6FF}_{T}=H/(2Nu)$ ) for $Ra=10^{9}$ and $Pr=1$ of (a–f) the instantaneous and (g–l) the time-averaged temperature and velocity fields (colour scale and streamlines as in figure 4). The aspect ratios are (a–c, g–i) $\unicode[STIX]{x1D6E4}=1$ and (d–f, j–l) $\unicode[STIX]{x1D6E4}=1/5$ . All cross-sections are taken for three different inclination angles $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ (first column), $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.25$ (second column) and $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.5$ (third column). The inner circles in (a–c) show the area of the cross-section in the case of $\unicode[STIX]{x1D6E4}=1/5$ for the same height as in (d–f), respectively.

Figure 8.

Vertical cross-sections sketched in (a) through the instantaneous flow fields of (b,c) the single-roll state for $Ra=10^{6}$ and $Pr=0.1$ , (d,e) the double-roll state for $Ra=10^{6}$ and $Pr=0.1$ , and (f,g) the double-roll state for $Ra=10^{7}$ and $Pr=1$ . Vertical cross-sections through (b,d,f) the plane of the rolls and (c,e,g) the plane perpendicular to it. Colour scale and streamlines as in figure 4.

It is necessary to further discuss the physical processes that support the formation of plume columns, since the reported findings indicate the importance of the emerging plume columns to the increase in heat transport. Therefore, let us consider a hot thermal plume or some small patch of hot fluid that has just detached from the thermal BL and what will happen to it afterwards. There are two relevant time scales: the time that the plume needs to travel to the opposite BL, and the thermal diffusion time, during which the plume loses its thermal energy to the surrounding fluid. First, the thermal plume is collected by the LSC and therefore accelerates to the velocity of the LSC, $U=\sqrt{\langle \langle \boldsymbol{u}\rangle _{t}^{2}\rangle _{V}}$ (as used in (2.9)), which gives the travelling time scale, $\unicode[STIX]{x1D70F}_{u}=H/U$ . The relevant length scale for the travelling time is the height of the cylinder $H$ , whereas we set the radius $R$ as the relevant length scale for the thermal diffusion time, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}=R^{2}/\unicode[STIX]{x1D705}$ , since this is the approximate lateral extent the plume can gain before it will interact with the cold plume channel. The lateral viscous diffusion time, $R^{2}/\unicode[STIX]{x1D708}$ , is longer and hence not important in these considerations for $Pr\leqslant 1$ . Plume columns can be formed if the travelling time is less than or approximately the same as the thermal diffusion time, $H/U\lessapprox R^{2}/\unicode[STIX]{x1D705}$ . From this we can estimate a threshold for our large-scale Reynolds number that is necessary to produce plume columns: $4\unicode[STIX]{x1D6E4}^{2}Re_{U}Pr\gtrapprox 1$ . As already discussed for $Ra=10^{6}$ and both Prandtl numbers, we have a profile that is close to the conductive temperature distribution (no plume columns) and indeed we find $4\unicode[STIX]{x1D6E4}^{2}Re_{U}Pr<1$ . For $Ra=10^{7}$ and $Pr=1$ we observe plume columns and $4\unicode[STIX]{x1D6E4}^{2}Re_{U}Pr\approx 3$ . A further condition for the plume columns to exist is that the velocity fluctuations must not prevail over the mean flow.

3.3 Single-roll and double-roll states in Rayleigh–Bénard convection

For the RBC case of this parameter set, we find two possible states: a single-roll state and a double-roll state (figure 8). The single-roll state is used as the reference case for the relative change of the Nusselt number and Reynolds number, because in inclined convection the single-roll state dominates. As known, for example, from experiments by Xi & Xia (Reference Xi and Xia2008) and Weiss & Ahlers (Reference Weiss and Ahlers2013), the heat flux in the double-roll state is smaller compared to that in a single-roll state. Our simulations confirm this fact, and we find a strong reduction, i.e. the heat transport in the double-roll state is only approximately 80 % of the heat transport in the single-roll state for $Ra=10^{6}$ and $Pr=0.1$ (figure 8 b–e). There is one further case where we observe a single-roll state and a double-roll state, which is $Ra=10^{7}$ and $Pr=1$ (figures 4 b and 8 f,g).

Both states are steady (vanishing velocity fluctuations) but their occurrence depends on the initial flow fields of the simulation. This means that, unlike the experiment by Weiss & Ahlers (Reference Weiss and Ahlers2013), transitions between these states are not to be expected.

For a slightly higher Rayleigh number, $Ra=2\times 10^{7}$ , we tried to find both states as well, but initiating the simulations from the single-roll state leads to the double-roll state. This suggests that the double-roll state might be the preferred state; however, finding the probability of each state is computationally too expensive. At the Rayleigh number $Ra=5\times 10^{7}$ and higher, there is no long-term LSC present any more. This is caused by the small aspect ratio, as we will discuss in § 3.4.

3.4 The effect of confinement

To investigate the influence of the aspect ratio on the transport quantities, as well as on the flow structure, simulation data for $Ra=10^{9}$ , $Pr=1$ and 11 different inclination angles $\unicode[STIX]{x1D6FD}$ are compared for the aspect ratios $\unicode[STIX]{x1D6E4}=1$ and $\unicode[STIX]{x1D6E4}=1/5$ . Thereby, a special focus lies on the cases of RBC ( $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ ), the optimal inclination angle ( $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}^{\ast }$ ) and VC ( $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.5$ ).

In RBC both aspect ratios have similar values of the Nusselt numbers, where $Nu\;(\unicode[STIX]{x1D6E4}=1)$ is less than 2 % larger than $Nu\;(\unicode[STIX]{x1D6E4}=1/5)$ . Although the heat transport shows qualitatively similar behaviour with inclination in both cases, i.e. the heat transport increases with inclination, reaches a maximum at a certain inclination angle (in this case $\unicode[STIX]{x1D6FD}^{\ast }\approx \unicode[STIX]{x03C0}/4$ for both aspect ratios) and then decreases again, there are some important differences. The first and most apparent difference is the higher relative increase in the heat transport for the slender cylinder, which is approximately 40 % compared to only 6 % increase for the cylinder with aspect ratio one (figure 9 a). Furthermore, the small-aspect-ratio cylinder shows a larger heat transport for VC compared to RBC, but for the aspect-ratio-one cylinder VC has a lower heat transport compared to the respective RBC.

Figure 9.

The Nusselt number and the different Reynolds numbers for $Ra=10^{9}$ and the aspect ratio $\unicode[STIX]{x1D6E4}=1$ (open triangles) and $\unicode[STIX]{x1D6E4}=1/5$ (filled triangles) plotted against the inclination angle $\unicode[STIX]{x1D6FD}$ . (a) Absolute Nusselt number. (b) Absolute large-scale Reynolds number $Re_{U}$ .

Figure 10.

The normalized mean temperature profiles, $\unicode[STIX]{x1D6E9}(z)=2(T_{+}-\langle T\rangle _{t,A})/\unicode[STIX]{x1D6E5}$ , for $Ra=10^{9}$ , $Pr=1$ and three different inclination angles $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ (green dash-dotted), $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.25$ (orange solid) and $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0.5$ (blue dashed). The aspect ratios are (a) $\unicode[STIX]{x1D6E4}=1$ and (b) $\unicode[STIX]{x1D6E4}=1/5$ .

In contrast to the heat transport, the Reynolds number, based on the kinetic energy, is higher for all inclination angles in aspect ratio $\unicode[STIX]{x1D6E4}=1$ , whereas the relative increase with inclination is higher for the slender cylinder. This is due to the stabilizing effect of the slender geometry on the flow. A look at the large-scale Reynolds numbers $Re_{U}$ (figure 9 b) reveals that, in aspect ratio $\unicode[STIX]{x1D6E4}=1$ , the maximum is reached already at $\unicode[STIX]{x1D6FD}=0.1\unicode[STIX]{x03C0}$ , which is at a smaller inclination angle than the optimal one ( $\unicode[STIX]{x1D6FD}^{\ast }=\unicode[STIX]{x03C0}/4$ ). Above we discussed that for slender cylinders the points of the maxima of $Re_{U}$ and of the Nusselt number coincide, pointing out that the strength of the LSC is important for the increase in the heat transport (Teimurazov & Frick Reference Teimurazov and Frick2017). Also from the same figure we can see that, in aspect ratio one, there is a dominant LSC in RBC.

In figure 10 we compare the normalized temperature profiles, averaged in time and over cross-sections at distance $z$ from the heated plate, of the two aspect ratios, for the three cases: RBC, the optimal inclination angle and VC. The temperature is normalized in such a way that it is equal to zero at the heated plate and to one in the core part of the domain. The usual expectation of the temperature profile in RBC is that the temperature rises within the thermal BL close to the mean value, then in the bulk region it takes the mean value, hence the temperature gradient is approximately zero in the bulk. In the aspect-ratio-one case, we observe this expected behaviour, but in the slender cylinder, the temperature rises only to approximately $3/4$ of the bulk temperature within the thermal BL. This is accompanied by a non-zero temperature gradient inside the bulk region (see figure 10 b). At an inclination angle of $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/4$ , both aspect ratios show a slight inversion of the bulk temperature gradient, i.e. the sign of the temperature gradient changes compared to the gradient of the conductive profile. This kind of behaviour was previously observed by Scheel & Schumacher (Reference Scheel and Schumacher2016) for a very low Prandtl number, $Pr=0.005$ , in RBC inside an aspect-ratio-one cylinder. For vertical convection there is a slightly colder/warmer region above the hot/cold thermal BL. These regions show up as local extrema in the temperature profile and can also be seen in the vertical slices of the time-averaged fields (figures 11 c,f and 12 e,f). In the aspect ratio $\unicode[STIX]{x1D6E4}=1$ case, the region of the local extremum shows an inversion, which means that the layer of hot fluid above the BL has a temperature larger than the average temperature inside the bulk.

Thermal plumes are an important feature of turbulent thermal convection and, as mentioned above, the formation of plume columns apparently facilitates an increase of the heat transport. Further visualization of the flow field reveals how thermal plumes are influenced by inclination and the aspect ratio. Slices taken at the edge of the thermal BL, $z=\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ , of the instantaneous temperature and velocity fields are shown for both aspect ratios in figure 7(a–f). There is a visual aid (black circle) given to show the diameter of the small-aspect-ratio cylinder scaled by its height. In the case of RBC and aspect ratio one, plenty of thermal plumes are visible, whereas the aspect ratio $\unicode[STIX]{x1D6E4}=1/5$ shows just very few. However, the absolute size of the thermal plumes appears to be similar, since the Rayleigh number and Prandtl number are the same for both aspect ratios. Tilting the aspect-ratio-one cylinder causes the plumes to change their orientation with respect to the mean flow direction: they align perpendicular to the streamlines. On the contrary, the optimally inclined, slender cylinder shows no separate thermal plumes, but only one rising and one falling plume, which span the whole cylinder (cf. figures 7 b,e and 13 a,b).

Figure 11.

Vertical cross-sections of the time-averaged flow fields, for $Ra=10^{9}$ , $Pr=1$ and $\unicode[STIX]{x1D6E4}=1$ , through (a–c) the plane of the LSC and (d–f) the plane perpendicular to the LSC. The inclination angles are (a,d) $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ , (b,e) $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/4$ and (c,f)  $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/2$ . The thick black arrows indicate the strength and direction of gravity. Colour scale and streamlines as in figure 4.

Figure 12.

Vertical cross-sections of the time-averaged flow fields, for $Ra=10^{9}$ , $Pr=1$ and $\unicode[STIX]{x1D6E4}=1/5$ , through two perpendicular planes. The inclination angles are (a,b)  $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x03C0}=0$ , (c,d)  $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/4$ and (e,f)  $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/2$ . Colour scale and streamlines as in figure 4.

Figure 13.

Isosurfaces of the instantaneous temperature field for $Ra=10^{9}$ , $Pr=1$ , $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/4$ . The aspect ratios are (a) $\unicode[STIX]{x1D6E4}=1$ and (b) $\unicode[STIX]{x1D6E4}=1/5$ . Colour scale as in figure 4.

Furthermore, the flow inside the slender cylinder is strongly stratified compared to the aspect-ratio-one case (cf. figure 13). In the case of RBC, buoyancy is the only mechanism to cause convection, but as soon as the cell is tilted, the shear flow becomes important. From inclined layer convection, where the aspect ratio is large or infinite, it is known that these are two different instability mechanisms that govern the fluid motion: one due to buoyancy and the other due to shear. The buoyancy or thermal mechanism leads to the development of longitudinal rolls where the axes are aligned with the shear flow. The onset of the longitudinal rolls is independent of $Pr$ . The other mechanism, i.e. the shear-flow mechanism, determines the development of transverse rolls, the axes of which are transverse to the shear. In the competition between these two mechanisms, the Prandtl number is crucial: for $Pr<0.26$ transverse rolls are always realized, while for larger $Pr$ longitudinal rolls may start first if the inclination angle is not too large. This has been studied in various works – see Chen & Pearlstein (Reference Chen and Pearlstein1989), Fujimura & Kelly (Reference Fujimura and Kelly1993), Bodenschatz et al. (Reference Bodenschatz, Pesch and Ahlers2000), Daniels, Plapp & Bodenschatz (Reference Daniels, Plapp and Bodenschatz2000), Daniels, Wiener & Bodenschatz (Reference Daniels, Wiener and Bodenschatz2003) and references therein. Thus, depending on the aspect ratio, there might be a specific inclination angle at which both mechanisms support the heat transport in such a way that it can explain the strong increase of the heat transport in inclined convection.

Further features of the flow become apparent in the time-averaged fields. These are again taken at the edge of the thermal BL, $z=\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ , as horizontal slices (figure 7 g–l). In RBC inside the aspect-ratio-one cell a long-term LSC is present in the flow, as one can see in figures 7(g) and 11(a). In contrast to this, for $\unicode[STIX]{x1D6E4}=1/5$ , there are no signs of an LSC visible; instead, there are ring vortices close to the plates (figure 12 a,b). The inclined aspect-ratio-one cell shows a symmetric flow field (figure 7 h) with two hot channels at each side and a smaller impinging zone compared to $\unicode[STIX]{x1D6E4}=1/5$ (figure 7 k). For the latter case we observe that the asymmetric flow from the instantaneous field is still present, despite a long averaging time of $t_{avg}=1725T_{f}$ . However, the flow is not steady: the velocity is still fluctuating, which means that the asymmetric flow structure prevails for a long time. This asymmetry can be seen in the vertical slice through the plane that is perpendicular to the LSC as well (figure 12 d). In vertical convection the velocity field in the aspect-ratio-one case is almost parallel over the whole plate, although one can see the influence of the sidewalls (figure 7 c,f). In the averaged fields (figures 11 c,f and 12 e,f) we see a difference in the flow organization in the vertical slice that is perpendicular to the LSC. In the small-aspect-ratio case there is only one saddle point in the velocity vector field, while in the aspect-ratio-one case, there are three saddle points at half height.

3.5 Limiting cases: Rayleigh–Bénard and vertical convection

Figure 14.

(a,c) The Nusselt number and (b,d) the Reynolds number, compensated with respect to the Rayleigh and Prandtl number scaling relations in the regime $I_{l}$ (a,b) in RBC (Grossmann & Lohse Reference Grossmann and Lohse2000) and (c,d) in VC (Shishkina Reference Shishkina2016) for Prandtl numbers $Pr=1$ (triangles) and $Pr=0.1$ (circles). Flows in a single-roll state are marked with filled symbols and flows in double-roll state are marked with open symbols.

Global heat and momentum transport scaling relations $Nu\sim Ra^{a}Pr^{b}$ and $Re\sim Ra^{c}Pr^{d}$ , respectively, are of common interest in all configurations of convective flows and here we will briefly discuss the two limiting cases of inclined convection, which are RBC ( $\unicode[STIX]{x1D6FD}=0$ ) and VC ( $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/2$ ). The scaling relations for these cases are presented in figure 14 in compensated plots with the scaling exponents according to the boundary-layer-determined low-Prandtl-number regime $I_{l}$ (Grossmann & Lohse Reference Grossmann and Lohse2000; Shishkina Reference Shishkina2016). Here the wind-based Reynolds number (2.7) is used, since the GLT assumes a large-scale wind above the BL, which is best represented by the velocity along the plate.

In the regime $I_{l}$ , the kinetic and thermal energy dissipation rates are both dominated by their BL contributions. In the vicinity of the plates, the kinetic BL is embedded in the thermal one, while the kinetic BL, which is attached to the adiabatic sidewall of the cylinder, is located outside the thermal BL. In a slender container, the kinetic BL at the adiabatic sidewall occupies a larger fraction of the total volume and this strongly influences the scaling relations.

In all considered cases we observe the exponents ( $a$ and $c$ ) in the $Nu$ versus $Ra$ and in the $Re$ versus $Ra$ scaling relations, which are larger than those predicted for fully developed, unconfined convective flows. The reason for this discrepancy is the strong lateral geometric confinement (small $\unicode[STIX]{x1D6E4}$ ) that shifts the onset of convection to higher critical Rayleigh numbers. Since for fully developed convective flows the scaling exponents $a$ and $c$ should be the same for all $\unicode[STIX]{x1D6E4}$ , in the case $\unicode[STIX]{x1D6E4}<1$ there exists a transitional $Ra$ region with higher scaling exponents than those predicted by the theory for infinite plates. In other words, for the considered Rayleigh numbers, the size of the heated/cooled plates is not sufficiently large to observe the scaling relations proposed for infinite plates. This kind of transition was observed previously, for example in DNS of RBC inside an aspect-ratio-one cylinder, where the onset of convection and the transition to turbulence in mercury were investigated (cf. figure 11 in Verzicco & Camussi (Reference Verzicco and Camussi1997)). The above discussed influence of the geometrical confinement on the scaling relations is stronger for smaller Rayleigh numbers.

Finally we note that, although the obtained exponent $a$ in the $Nu$ versus $Ra$ scaling relation in VC is slightly larger than that proposed in Shishkina (Reference Shishkina2016) ( $a=1/4$ ), it is significantly smaller than that proposed in Castaing et al. (Reference Castaing, Rusaouën, Salort and Chillà2017) for small-Prandtl-number soft-turbulence regime ( $a=2$ ) and significantly larger than that proposed in Castaing et al. (Reference Castaing, Rusaouën, Salort and Chillà2017) for large-Prandtl-number soft-turbulence regime ( $a=1/5$ ).