Coriolis effect on centrifugal buoyancy-driven convection in a thin cylindrical shell

Abstract We study the effect of the Coriolis force on centrifugal buoyancy-driven convection in a rotating cylindrical shell with inner cold wall and outer hot wall. This is done by performing direct numerical simulations for increasing inverse Rossby number $Ro^{-1}$ from zero (no Coriolis force) to $20$ (very large Coriolis force) and for Rayleigh number $Ra$ from $10^{7}$ to $10^{10}$ and Prandtl number $Pr = 0.7$, corresponding to air. We invoke the thin-shell limit, which neglects the curvature and radial variations of the centripetal acceleration. As $Ro^{-1}$ increases from zero, the system forms an azimuthal bidirectional wind that reaches its maximum momentum at an optimal $Ro^{-1}_{opt}$, associated with a maximal skin-friction coefficient $C_f$ and a minimal Nusselt number $Nu$. Just beyond $Ro^{-1}_{opt}$, the wind weakens and an axial, quasi-two-dimensional cyclone, corotating with the system, begins to form. A local ‘turbulence’ inverse Rossby number (non-dimensionalised by the eddy turnover time) determines the onset of cyclone formation for all $Ra$, when its value reaches approximately $4$. At $Ro^{-1} \gg Ro^{-1}_{opt}$, the system falls into the geostrophic regime with a sudden drop in $Nu$. The bidirectional wind for $Ro^{-1} \le Ro^{-1}_{opt}$ is a feature of this system, as it hastens the boundary layer transition from laminar to turbulent, towards the ultimate regime. We see the onset of this transition at $Ra=10^{10}$ and $Ro^{-1}\simeq Ro^{-1}_{opt}$, although the mean flow profile has not yet fully collapsed on the Prandtl–von Kármán (logarithmic) law.


Introduction
Thermal convection is an important transport mechanism in many engineering and geophysical flows. Centrifugal buoyancy-driven convection (figure 1) is a canonical thermal convection system to study some of these flows (table 1). The studies with geophysical interests consider this system as a closed dynamical model for the earth's liquid (outer) core (Busse & Carrigan 1974), or midlatitude atmosphere (Randriamampianina et al. 2006;Read et al. 2008;Von Larcher et al. 2018). The studies with turbomachinery interests consider this system as a model for the compressor cavity (Bohn et al. 1995;King, Wilson & Owen 2007;Owen & Long 2015;Pitz et al. 2017a;Pitz, Marxen & Chew 2017b). The system is a rotating cylindrical shell with inner cold wall and outer hot wall (figure 1). Rotation introduces centrifugal buoyancy (set by centripetal acceleration) and Coriolis forces.
Centrifugal convection differs from rotating Rayleigh-Bénard convection (table 1). In centrifugal convection, the axis of rotation is parallel to the hot and cold walls (normal to the centrifugal buoyancy force), while in rotating Rayleigh-Bénard convection, the axis of rotation is normal to the hot and cold walls (parallel to the gravitational buoyancy force). However, both systems are characterised by the Rayleigh number Ra, inverse Rossby number Ro −1 and Prandtl number Pr (assuming that gravity is neglected in centrifugal convection, figure 1a). In centrifugal convection, these numbers are defined as (1.1a) where U ≡ (Ω 2 RβΔH) 1/2 , the free-fall velocity; Ω is the rotational speed; R is the outer shell radius; H is the shell thickness; Δ ≡ (T H − T L ), the temperature difference; β is the thermal expansion coefficient; κ is the thermal diffusivity; ν is the kinematic viscosity.
For convenience, we only use the inverse Rossby number Ro −1 (rather than Ro), as it is directly linked to the Coriolis force (i.e. higher Ro −1 implies higher Coriolis force).
In figure 2, we compile a (Ra, Ro −1 ) parameter space of the previous studies on centrifugal convection (figure 2a) and rotating Rayleigh-Bénard convection (figure 2b). We list these studies in table 1. We also add two recent sets of data points for centrifugal convection (figure 2a). One is our present data (•, black) and the other one is by our coauthor, Professor C. Sun and his colleagues (♦, dark grey ♦, process blue ♦, dark orange; Jiang et al. (2020)). This figure highlights the importance of studying centrifugal convection, as a system that is explored to a lesser extent than rotating Rayleigh-Bénard convection. Rotating Rayleigh-Bénard convection has been investigated over almost a continuous parameter sweep of 10 4 Ra 10 15 and 0 Ro −1 100. However, until recent studies (•, black; ♦, dark grey; ♦, process blue; ♦, dark orange), centrifugal convection was investigated at limited values of Ra and Ro −1 , and only for Ro −1 2 (large Coriolis force). von Hardenberg et al. (2015) study Ro −1 < 2 ( , red), but they consider Ra = 10 7 . Also, their set-up is free-slip hot and cold plates rotating about a distant axis. This set-up can be perceived as a slice of a thin cylindrical shell with free-slip boundaries (CC_slip in table 1). The previous studies focus on large Ro −1 because experiments are constrained by their working fluid and apparatus, and numerical studies set their parameter space following the experiments. For instance, among the geophysical studies, Read et al. (2008) ( , blue) follow the experiment of Fowlis & Hide (1965), and Von Larcher et al. (2018) ( , black) follow their own experiment. Among the turbomachinery studies, FIGURE 1. Set-up of flow. (a) Centrifugal buoyancy-driven convection in the cylindrical shell with gap H and outer radius R. The shell undergoes solid-body clockwise rotation about its axis ζ , with rotational speed Ω. The outer wall is hotter than the inner core. (b) Our computational domain as a small chunk of centrifugal convection, highlighted with dashed lines in panel (a), with H R, which is rectilinear, and L x and L y are the domain sizes in the streamwise (circumferential) and spanwise (axial) directions. Due to thin-shell approximation, the computational domain is exposed to a uniform centripetal acceleration Ω 2 R g, so that gravity does not matter here. Pitz et al. (2017a,b) ( , red) and King et al. (2007) ( , black) follow the experiment of Bohn et al. (1995) ( , blue).
Here, we perform DNS at Ra = (10 7 , 10 8 , 10 9 , 10 10 ) and Ro −1 = (0, 0.3, 0.5, 0.6, 0.8, 1.0), (•, black in figure 2a). Additionally, we explore (Ra, Ro −1 ) = (10 7 , 2.0) and (Ra, Ro −1 ) = (10 8 , 20.0) that fall into the regimes of interest by the geophysical and turbomachinery studies. Our objectives are to investigate: (i) the flow regimes from no Coriolis force (Ro −1 = 0) to a very large Coriolis force (Ro −1 = 20); (ii) the universality of these regimes from Ra = 10 7 to 10 10 ; and (iii) the analogy between these regimes and those in rotating Rayleigh-Bénard convection. We show how an optimal choice of Ro −1 can exploit the Coriolis force to tune a persistent large-scale wind. On the other hand, we show how large Ro −1 ≥ 1 can cause the Coriolis force to suppress turbulence and laminarise the flow. The organised wind is a feature of this system. It hastens the boundary layer transition from laminar to turbulent, i.e. transition from the classical regime (Grossmann & Lohse 2000) to the ultimate regime (Grossmann & Lohse 2011). In the classical regime, the effective heat-transfer scaling, expressed through an effective power law for the Nusselt number Nu to Ra relationship (Nu ∝ Ra α ), follows an effective power-law exponent α ≤ 1/3 (Grossmann & Lohse 2000). However, in the ultimate regime the heat-transfer scaling follows a steeper gradient with an effective exponent α > 1/3 (Grossmann & Lohse 2011).
The ultimate regime has been approached or fully reached in several turbulent systems, including Rayleigh-Bénard convection (Roche et al. 2005;Ahlers et al. 2012;He et al. 2012aHe et al. ,b, 2013He, Bodenschatz & Ahlers 2016), the analogue Taylor-Couette flow (see the review by Grossmann, Lohse & Sun (2016)), and vertical natural convection (Ng et al. 2017) or sheared convection (Pirozzoli et al. (2017) and Blass et al. (2020), though by far not reached here). The three latter systems benefit from a persistent wind, similar to centrifugal convection. However, the source of wind formation, i.e. shear, differs among these systems; in Taylor-Couette flow the shear is due to differentially rotating cylinders, in sheared convection the shear is due to differentially moving walls, in vertical convection the shear is due to gravitational buoyancy, and in centrifugal convection the shear is due  rot_RB_wall Julien et al. (1996a,b, 1999  to the Coriolis force. Centrifugal convection is unique among these systems, because the wind forms due to the Coriolis force that does not alter the kinetic energy. Our coauthor, Professor C. Sun, has proposed this system as an ideal opportunity to reach the ultimate turbulent regime, to mitigate possible non-Oberbeck-Boussinesq effects at large Ra in the standard Rayleigh-Bénard geometry, as here the possible ultimate regime is now triggered by centrifugal buoyancy instead of by temperature differences. It was this proposition which triggered the present numerical work. The paper is organised as follows. In § 2 we describe our DNS set-up as well as simulation and grid convergence studies. In the results section ( § 3), first, we identify the overall flow regimes ( § 3.1) through flow visualisation, Nu and skin-friction coefficient C f . Then, we discuss each flow regime from the bidirectional wind formation ( § 3.2) to the flow laminarisation ( § 3.5). In § 3.6 we show the onset of transition to turbulent by the bidirectional wind. The concluding remarks follow in § 4.

Governing equations
The governing equations are derived from the incompressible Navier-Stokes equations governing the flow in a concentric cylindrical annulus with gap H (figure 1a) in the frame rotating in a clockwise direction about its cylindrical axis ζ at constant rotational speed Ω, as described by velocity v = v r e r + v φ e φ + v ζ e ζ and temperature T in cylindrical coordinates (r, φ, ζ ). The boundary conditions in this rotating frame are no-slip and impermeable walls, v(r = R − H) = v(r = R) = 0, corresponding to the inner core and outer wall, respectively, and isothermal walls with the prescribed temperature difference Δ = T H − T L , with T(r = R − H) = T L and T(r = R) = T H , corresponding to an inner colder core and an outer hotter wall. We have invoked the Oberbeck-Boussinesq approximation, which assumes constant fluid properties, ν, κ and β, and that density variations are only dynamically important in the buoyancy term. In the buoyancy term the density variation is (ρ − ρ o ) = −βρ o θ , where ρ o = ρ(T o = (T H + T L )/2), the reference density at temperature T o , and θ = T − T o , the temperature variation relative to T o . For the sake of brevity we refer the reader to Kundu & Cohen (1990), for example, for the equations in the (r, φ, ζ ) coordinate system. Since the equations are presented in a rotating frame, two additional terms appear in the Navier-Stokes equations: the Coriolis force, −2Ωv φ e r + 2Ωv r e φ , and the centripetal acceleration, −βΩ 2 rθ e r .
To further simplify the problem, we consider the thin-shell (unity radius ratio) limit, ≡ H/R 1 (figure 1b). To this end, we transform the equations from (r, φ, ζ ) into curvilinear coordinates (x, y, z) with the origin placed at the outer cylinder. The transformed coordinates will be x = rφ, y = −ζ, z = R − r, and the transformed velocity will be u = v φ , v = −v ζ , w = −v r . Then, we non-dimensionalise the variables using the gap width H, the free-fall velocity U ≡ (Ω 2 RβΔH) 1/2 , and Δ:x = x/H,ỹ = y/H,z = z/H,t = tU/H are the scaled space and time coordinates;ũ = u/U,ṽ = v/U,w = w/U are the scaled velocity components; andp = ( p − ρ o Ω 2 R 2 /2)/(ρ o U 2 ) andθ = θ/Δ are the scaled pressure and temperature variation. Substituting these into the transformed equation, and expanding in small , we obtain, to leading order  figure 1a). Therefore, the transformed (2.1a)-(2.1e) that are identical to the Navier-Stokes equations in the Cartesian coordinate system are solved in a rectilinear box (figure 1b). The transformed boundary conditions in the x-and y-directions are periodic and at the outer and inner walls areũ(z = 0) =ũ(z = 1) = 0,θ(z = 0) = 1/2 andθ(z = 1) = −1/2.
The results are presented in terms of (x, u), (z, w) and ( y, v), the circumferential, (negative) radial and (negative) axial directions of the cylindrical shell, respectively, and are noted as the streamwise, wall-normal and spanwise directions. The inner and outer walls of the cylindrical shell are also noted as the top and bottom walls, respectively. In the entire manuscript, we denote a non-dimensionalised quantity by U and H with tilde (e.g.t = tU/H), an x y-plane and time averaged quantity with overbar (e.g.ū), a volume and time averaged quantity with angle bracket (e.g. u ) and an averaged quantity over a specific direction with superscript (e.g. u y is averaged in the y-direction). Fluctuating quantities are obtained by subtracting x y-plane and time averaged quantities from their instantaneous counterpart (e.g. u = u −ū).

DNS
The equations are solved using a fully conservative fourth-order finite-difference code, validated in the previous DNS studies of similar flow physics (Ng et al. 2015(Ng et al. , 2017(Ng et al. , 2018. Table 2 lists all the simulation cases and figure 3 shows the visualizations of the production runs. For all cases, Pr = 0.7 and L x /H × L y /H = 1 × 1. We choose a fixed aspect ratio of Γ = 1 to focus on the Coriolis force effect (Ro −1 ) and achieve the highest possible Ra. Nevertheless, we speculate that the essential physics, hence the trend in Nu and skin-friction coefficient C f do not change with Γ . Our conjecture is based on the previous studies on centrifugal convection and similar thermal convection systems. In classical Rayleigh-Bénard convection, the sensitivity of Nu with Γ ≥ 1 is less than 7 % for Ra 2 × 10 7 (figure 4a in Stevens et al. (2018)). The sensitivity of α (Nu ∝ Ra α ) with Γ ≥ 1 is less than 3 % for Ra 10 9 (we analysed figure 3 of Zhou et al. (2012)). In rotating Rayleigh-Bénard convection, Nu is almost insensitive to Γ ≥ 1 for Ro −1 ≥ 0.4 (figure 4 in Stevens et al. (2011)). Even for Ro −1 < 0.4, Nu versus Ro −1 shows the same trend for different Γ . In centrifugal convection, the trend in Nu versus Ro −1 for a full cylindrical shell (figure 2a in Jiang et al. (2020)) is similar to what we report in § 3.1. Also, the flow regimes that we discuss in § 3 are similar to the previous experiments. At low rotational speed, experiments report an axisymmetric flow, circulating parallel to the walls (e.g. figure 2a,b in Fowlis & Hide (1965); figure 13e, f in Hide & Mason (1970)). We observe similar flow regime (bidirectional wind) in § 3.2. At high rotational speed, experiments report geostrophic regime with large-scale quasi-two-dimensional cyclones and anticyclones (e.g. figure 2d-h in Fowlis & Hide (1965); figure 4 in Jiang et al. (2020)). We observe similar flow regime in § 3.5.
We use the same number of grid points N in each direction. The grid points are uniformly distributed in the x-and y-directions, and are stretched in the z-direction. The choice for N and stretching factor are decided a priori based on Shishkina is the maximum local grid size, and η l (z) ≡ (ν 3 /ε u ) 1/4 and η g ≡ (ν 3 / ε u ) 1/4 are the local and global Kolmogorov length scales; ε u nrm = ε u Pr 2 H 4 /[ν 3 (Nu − 1)Ra] and ε θ nrm = ε θ H 2 /(κΔ 2 Nu) measure the global balance between the dissipation rates and Nusselt number, a measure of resolution sufficiency (Calzavarini et al. 2005;Stevens, Verzicco & Lohse 2010). For a perfect resolution, the criterion ε u nrm = ε θ nrm = 1 must be nearly satisfied, regardless of the scheme used to compute ε u and ε θ . See the text for discussion. Ra Ro -1 FIGURE 3. Flow visualisation of the simulation cases. The unframed cases correspond to four Rayleigh numbers Ra = (10 7 , 10 8 , 10 9 , 10 10 ), from the first to the last row, respectively, and six inverse Rossby numbers Ro −1 = (0, 0.3, 0.5, 0.6, 0.8, 1.0), from the first to the last column, respectively. The special framed case is at Ra = 10 8 , Ro −1 = 20.0. Isosurface of θ = −Δ/10 (blue) and θ = +Δ/10 (red).
(36), (42) and (43)) to resolve the Kolmogorov scale both in the bulk and within the boundary layers. In table 2, we call these resolutions standard, as we show here that they are suitable for well-resolved DNS. We also performed calculations at coarser and finer resolutions, and we call them coarse and fine, respectively. In total, four values of Ra are simulated, ranging from 10 7 to 10 10 , and at each Ra, Ro −1 is varied from zero (no Coriolis force) to unity (large Coriolis force). We performed two additional cases: one at Ro −1 = 2.0 for Ra = 10 7 , and the other one at Ro −1 = 20 for Ra = 10 8 , where the Coriolis force is much larger than inertia. We perform an extensive parameter and grid-convergence study (table 2), following the prescriptions by Stevens et al. (2010). For simulation convergence, each case is run for approximately 100 turnover times H/U to discard initial transients, and data is averaged over an additional τ avg 150H/U for statistical convergence. Statistical convergence is evaluated based on the Nusselt number Nu ≡ (H/Δ)|dθ/dz| w , where |dθ/dz| w = (|dθ/dz| z=0 + |dθ/dz| z=H )/2 is the absolute wall temperature gradient, averaged over time, x y-plane and both walls. We call a simulation statistically converged, once the difference between Nu obtained from averaging overτ avg and its counterpart Nu h obtained from averaging over the second half ofτ avg is less than 1 % (see Diff τ avg in table 2). We can also see the statistical convergence in the less than 1 % difference between Nu and its counterpart at the bottom wall Nu bot . Another way for statistical convergence is when the difference between top and bottom wall Nusselt numbers Diff Nu = |Nu top − Nu bot |/Nu is negligibly small (Kunnen et al. 2016). For all cases up to Ra = 10 9 , Diff Nu is less than 2 %.
We perform such an extensive statistical convergence study up to Ra = 10 9 , where the resolution requirement (maximum N = 512) is affordable to run the calculations for at least 250H/U. The cases at Ra = 10 10 require N = 1024 for well-resolved simulation, and are substantially expensive. For example, running each well-resolved case at Ra = 10 9 (N = 512) for 250H/U, demands approximately 0.05 million central processing unit (CPU) core hours, whereas running each well-resolved case at Ra = 10 10 (N = 1024) for 250H/U, demands approximately 2.0 million CPU core hours, 40 times more expensive than Ra = 10 9 . Given the expensive cost at Ra = 10 10 , each case is simulated for at least 60H/U and statistical averaging is performed over τ avg 30H/U. Running these cases to full statistical convergence does not add to the conclusions that we draw by studying the cases up to Ra = 10 9 . Our primary aim by reporting Ra = 10 10 is to demonstrate some evidence of boundary layer transition to turbulent, owing to the favourable role of the Coriolis force.
Grid convergence was evaluated based on three criteria.
Comparison of parameters of interest between sequentially finer grid resolutions.
Criterion (i) was initially prescribed by Grötzbach (1983), suggesting h ≤ πη g , where h = (Δ x Δ y Δ z ) 1/3 is the grid size and η g ≡ (ν 3 / ε u ) 1/4 is the global Kolmogorov scale, based on the volume and time-averaged dissipation rate. Grötzbach (1983) ignored the anisotropy in the grid (by using the geometric mean for h) and heterogeneity in the dissipation rate (by integrating ε u over the domain and time). Stevens et al. (2010) showed that for well-resolved simulations, the anisotropic grid and flow heterogeneity must be taken into account. Therefore, following Stevens et al. (2010) we chose h = max(Δ x , Δ y , Δ z ), and calculated the local Kolmogorov scale η l in each height based on ε u . Also we calculated the global Kolmogorov scale η g , and in table 2 we compare the maximum ratios (h/η g ) max and (h/η l ) max . As observed, resolving η l demands finer resolution than resolving η g . Results of Stevens et al. (2010) and our results in table 2 show that the criterion (h/η g ) max ≤ π is not sufficient for well-resolved simulation. For instance, the simulations at Ra = 10 9 with N 3 = 256 3 , satisfy (h/η g ) max ≤ π but not (h/η l ) max ≤ π, and they poorly satisfy the global exact relations for criterion (ii) ( ε u nrm ε θ nrm 0.90). For a perfect resolution, the criterion ε u nrm = ε θ nrm = 1.0 must be nearly satisfied, even if ε u and ε θ are calculated with a different scheme than the code discretisation scheme. Here, we use a fourth-order kinetic energy conservative and a third-order scalar variance non-conservative code. We compute ε u and ε θ using a second-order central differencing scheme. We obtain ε u nrm ε θ nrm 0.97, for all the standard resolution cases, similar to Stevens et al. (2010). These standard resolution cases also satisfy criterion (i), i.e. (h/η l ) max ≤ π.
In figure 4, we show the adequacy of standard resolution based on criterion (iii). We consider Ra = 10 9 and Ro −1 = 0.8, comparing ε u and ε θ between three grid resolutions: N = 256 (coarse), 512 (standard) and 1024 (fine). At the coarse resolution (N = 256), these quantities are slightly lower than the ones at the standard and fine resolutions (consider the insets in figure 4). However, the difference between the standard and fine resolutions is negligible. According to Stevens et al. (2010), in the under-resolved simulations the gradients are smeared out, and ε u and ε θ are underestimated. The results that we present in the rest of this paper (figure 3), correspond to the well-resolved standard resolution cases (table 2).

Effect of the Coriolis force on heat and momentum fluxes
The resulting Nusselt number Nu = (H/Δ)|dθ/dz| w for all cases is compiled in figure 5. When Ro −1 = 0 (×, no Coriolis force), Nu is close to the Grossmann & Lohse (2000) theory (dashed black line), as expected. At each Ra, as Ro −1 increases (i.e. Coriolis force increases), Nu decreases until it reaches a minimum at an optimal Ro −1 opt ; beyond Ro −1 opt , Nu increases. This is better shown in figure 6(a), plotting Nu versus Ro −1 , at each Ra. Additionally, in figure 6(b) we plot the skin-friction coefficient C f = 2ν|dū/dz| w /U 2 versus Ro −1 , at each Ra. Here |dū/dz| w is the modulus of the wall velocity gradient,  To explain the underlying mechanism for the behaviour seen in Nu and C f versus Ro −1 , we study the flow at different values of Ro −1 (figure 7). We focus on Ra = 10 8 , but our conclusions can be generalised to other values of Ra. In figure 7(e-h), we show the instantaneous spanwise averaged temperature field θ y /Δ, overlaid by the instantaneous spanwise averaged velocity vector (u y /U, w y /U). Comparing the flow at an Ro −1 smaller than Ro −1 opt (figure 7a,e), equal to Ro −1 opt (figure 7b, f ), larger than Ro −1 opt (figure 7c,g) and much larger than Ro −1 opt (figure 7d,h), we see that up to Ro −1 opt the hot fluid is mainly driven in the positive x-direction and the cold fluid is mainly driven in the negative x-direction. This is better seen in the mean velocity profiles (solid red line, solid green line) in figure 8(a). In fact, up to Ro −1 opt = 0.8 an antisymmetric bidirectional wind is formed, that drives the flow near the top and bottom walls in the opposite directions. At Ro −1 opt = 0.8 (figure 7b, f ), the wind gains the maximum momentum, hence maximal C f , and the velocity profile in wall units (solid green line in figure 8b) moves closer to the Prandtl-von Kármán (logarithmic) profile. Beyond Ro −1 opt (figure 7c,g), the wind is weakened and becomes asymmetric, hence C f decreases (figure 6b). In appendix A, we conclude that the asymmetric flow is a persistent statistical state. Our conclusion is based on simulating the case in figure 7(c,g) with two different initial conditions and running each calculation for approximately 1200   The bidirectional wind also appears in centrifugal convection with free-slip hot and cold boundaries (von Hardenberg et al. 2015;Novi et al. 2019). In appendix B, we compare this system (CC_slip in table 1) with our system (CC_wall in table 1). We see several differences due to different boundary conditions. In CC_slip, the bidirectional wind never breaks down, but in CC_wall it breaks down. As a result, in CC_slip there is no optimal Ro −1 opt , but in CC_wall there is an Ro −1 opt . Also, in CC_slip the bidirectional wind can have a cyclonic or anticyclonic mean vorticity (von Hardenberg et al. 2015), but in CC_wall the bidirectional wind is always anticyclonic (appendix A).
The different flow regimes caused by changing Ro −1 (Coriolis force) also explains the variations in Nu (figure 6a). The variations in Nu, i.e. heat transfer, is related to the vertical fluid motion between the end walls. The vertical fluid motion is qualitatively observed by the isoline of θ y = 0 in figure 7(e-h). At small Ro −1 , small Coriolis force (figure 7e), the isoline of θ y = 0 (solid red line) highlights the upwelling and downwelling thermal plumes, as seen in Rayleigh-Bénard convection (Ahlers, Grossmann & Lohse 2009) opt , owing to the bidirectional wind, the cold fluid coverage at the edge of the bottom thermal boundary layer reaches minimum (A pl is minimum). Consequently, the heat exchange between the end walls reaches its minimum (Nu is minimal).
Our study shows that the interaction between the Coriolis force and buoyancy force creates different flow regimes. When the buoyancy force is stronger than the Coriolis force (Ro −1 Ro −1 opt ), the flow is similar to Rayleigh-Bénard convection. Vice versa, when the Coriolis force is stronger than buoyancy (Ro −1 Ro −1 opt ), the flow becomes laminar. At an intermediate force balance (Ro −1 opt ), optimal transport occurs with maximal C f and minimal Nu. Similar flow regimes are reported by Jiang et al. (2020) (see their figure 2). They perform DNS of a full cylindrical shell with finite shell thickness (H/R 0.5). Their variation in Nu versus Ro −1 is similar to figure 6(a); a minimal Nu occurs at an optimal Ro −1 .
Optimal transport also occurs in rotating Rayleigh-Bénard convection (see the review by Stevens, Clercx & Lohse (2013a)), but only in rot_RB_wall and rot_RB_cyl with hot and cold walls (table 1). For rot_RB_wall see Julien et al. (1996b), King et al. (2009King et al. ( , 2012, Pieri et al. (2016) and Chong et al. (2017); and for rot_RB_cyl see Kunnen et al. (2008) Zhang et al. (2020). In these systems, optimal transport roughly coincides with the transition between vertically coherent columns (Taylor columns) and vertically spinning plumes (Julien et al. 2012;Stellmach et al. 2014;Cheng et al. 2015;Kunnen et al. 2016). These structures are emanated from the Ekman layers at the walls. As a result, optimal vertical transport (maximal Nu) occurs, as opposed to optimal horizontal transport (minimal Nu) in centrifugal convection. This difference is due to different axis of rotation between rotating Rayleigh-Bénard convection and centrifugal convection. Novi et al. (2019) changed the angle between the buoyancy force and axis of rotation from zero (rotating Rayleigh-Bénard convection) to 90 • (centrifugal convection). They observed that the flow structures evolve from columnar vortices to bidirectional wind. In rot_RB_slip with free-slip hot and cold boundaries (table 1), no Ekman layer forms. Therefore, no optimal transport occurs (Julien et al. 1996b;Schmitz & Tilgner 2009. Optimal transport occurs in other systems where buoyancy (as a destabilising mechanism) interacts with a stabilising mechanism (Chong et al. 2017). Stabilising mechanisms are confinement in confined Rayleigh-Bénard convection (Chong et al. 2015), salinity in double diffusive convection (Yang, Verzicco & Lohse 2016) or Lorentz force in magnetoconvection (Lim et al. 2019). In these flows, the interplay between the stabilising and destabilising mechanisms forms coherent structures (e.g. Taylor columns). These structures manipulate the flow towards the optimal transport (Chong et al. 2017).

Formation of bidirectional wind
The bidirectional wind is formed once we rotate the system, hence a combination of the Coriolis force and buoyancy force generates the wind. To investigate the underlying mechanism, we study the momentum equations ((2.1b)-(2.1d)) averaged over time and the x y-plane, where (3.1a,b) are the averaged u-and w-momentum equations, respectively. If we integrate (3.1a) fromz = 0 to an arbitraryz, we obtainũ w = (Ra/Pr) −1/2 dũ/dz − (Ra/Pr) −1/2 dũ/dz|z =0 . We can draw two conclusions from this equation. First, considering this equation atz = 1 (top wall), we obtain dũ/dz|z =0 = dũ/dz|z =1 , hence the wall shear stresses at the top and bottom walls have equal magnitudes but opposite signs (τ w top = −ρ o ν dū/dz| z=H ,τ w bot = ρ o ν dū/dz| z=0 ). Second, if we further integrate this equation from z = 0 to 1, we obtain C f top = C f bot = 2| 1 0ũ w dz|, hence the wind momentum is adjusted by the integral of Reynolds shear stress. These two conclusions must be valid for all values of Ro −1 (3.1a,b). The first conclusion can be confirmed from figure 8(b), comparing the mean velocity profiles near the bottom wall (solid line) and top wall (dot-circled line). The profiles within a distance of 20 wall units from the end walls are identical to each other, even at Ro −1 = 1 (solid black line), where the antisymmetric wind is broken. From the second conclusion, the Coriolis and buoyancy forces must generate the wind through u w . Inspecting (3.1b) shows that the Coriolis force (Ro −1ū ) and buoyancy force (θ ) can modify the wall-normal Reynolds stress w w , which in turn modifies u w . At Ro −1 opt = 0.8 (solid green line), where the wind momentum reaches maximum, u w is maximum at all heights (figure 8c). At Ro −1 = 20 (solid grey line), the flow is laminar and u w is due to the flow unsteadiness (u = (u −ū) is the unsteady component). Nevertheless, | H 0 u w dz| at Ro −1 = 20 is still smaller than its counterpart at Ro −1 opt = 0.8.

3.3.
Flow behaviour beyond the optimal Coriolis force As discussed in § 3.1, beyond Ro −1 opt (figure 7c,g), the strong bidirectional wind is weakened and loses its antisymmetric nature. We visualise this case (Ra = 10 8 and Ro −1 = 1.0 > Ro −1 opt = 0.8) in figure 9 for a period of approximately 4H/U (indicated in the x y-plane averaged wall shear-stress τ xy w history). We see that wind breaking is coincident with  the formation of coherent large-scale circulations. These circulations are identified by the y-averaged spanwise vorticity ω y y = (∂u y /∂z − ∂w y /∂ x). Regions of high vorticity, hence strong circulation coincide with the regions of high Coriolis force. We can show this by taking the divergence of momentum equation (2.1b)-(2.1d), Equation (3.2) shows that the Coriolis term Ro −1ω y is strong anywhere that ω y is large. In total, two circulations are formed: a strong cyclone near the top wall (marked with ×, green, corotating with the system) and a weak anticyclone in the bulk (marked with +, red, counter-rotating to the system). In figure 9(h-m), we observe that a cyclone is a quasi-two-dimensional roller that elongates in the spanwise direction. The z-location of the cyclone does not change with time, but it travels in the x-direction. The terms cyclone and anticyclone are widely used in rotating flows to identify coherent corotating or counter-rotating structures. However, they do not represent the shape of structures, e.g. columnar, plume-like or roll-like. The flow structures that we observe here as a large-scale concentrated cyclone accompanied by a field of weak anticyclones are also seen in rot_RB_slip with free-slip boundaries (Favier et al. 2014;Guervilly et al. 2014). Also, recent study of rot_RB_wall with no-slip boundaries (Aguirre Guzmán et al. 2020)  Bartello et al. (1994), Morize et al. (2005) and Favier et al. (2014) studied the probability density function of vorticity. They concluded that the prevalence of cyclones is due to the dominance of cyclonic small scales.

Local 'turbulence' Rossby number
In our system, cyclones and anticyclones are formed at different transitional Ro −1 , depending on Ra. Also, at a fixed Ra, the transitional Ro −1 depends on the rotating system. For example, in Favier et al. (2014) at Ra = 10 7 the transitional Ro −1 is 3.1 (while it is 1.2 in our system), and at Ra = 10 8 it is 1.8 (and 1.0 in our system). In both systems, the transitional Ro −1 decreases as Ra increases, but the transitional values are different. Favier et al. (2014) consider rotating Rayleigh-Bénard convection with free-slip hot and cold boundaries (rot_RB_slip in table 1), while we consider centrifugal convection (figure 1). The Ra and system dependency of transitional Ro −1 is because Ro −1 ≡ 2ΩH/U characterises the Coriolis force over the entire system. However, the Coriolis force is locally distributed differently, depending on Ra and the rotating system. Therefore, the formation of a cyclone as a local phenomenon, must be characterised based on a local measure of inverse Rossby number (Ro −1 L ).
FIGURE 11. Profiles of the plane and time averaged streamwise velocityū (a-d) and local inverse Rossby number Ro −1 L ≡ 2ΩK/ε u (e-h), at Ra = 10 7 (a,e), 10 8 (b, f ), 10 9 (c,g) and 10 10 (d,h). Here Ro −1 = 0.3 (solid red line), 0.5 (solid blue line), 0.6 (solid magenta line), 0.8 (solid green line), 1.0 (solid black line) and 2.0 (dashed blue line). The vertical dashed-dotted grey line in the bottom row locates the transitional Ro −1 L = 4.0, beyond which the cyclone is formed. All theū profiles that are deformed from their antisymmetric shape are marked with (•, blue; , hot magenta), as well as their Ro −1 L profiles. Those marked with a blue • are partially deformed (still preserve their S shape). Those marked with a hot magenta are completely deformed. The markers are placed at the maximum Ro −1 L . Hopfinger et al. (1982) defined Ro −1 L based on local turbulent velocity scale and integral length scale. They observed the formation of cyclones when locally Ro −1 L increases to approximately 5. Following Hopfinger et al. (1982), other studies (table 3) made the same observation at nearly the same transitional Ro −1 L . The studies in table 3 are both experimental and numerical, and consider different rotating systems. However, they all report a transitional Ro −1 L close to 5. In the rest of this subsection, first, we examine whether there is a unified transitional Ro −1 L for our rotating system, independent of Ra. Then, by taking the advantage of this unified value, we attempt to arrive at a relation between Ro −1 opt (before the cyclone formation) and Ra. We define local 'turbulence' inverse Rossby number Ro −1 L ≡ (2ΩK)/ε u based on eddy turnover timeK/ε u , whereK is the turbulent kinetic energy.
In figure 11 we plot the profiles of Ro −1 L at Ra = 10 7 (figure 11e) to 10 10 (figure 11h), and Ro −1 = 0.3 (solid red line) to 2.0 (dashed blue line). We also add the plane and time averaged streamwise velocity profilesū ( figure 11a-d). We aim to see if there is any relation between Ro −1 L and wind breaking, i.e. when the antisymmetricū profile is deformed. At each Ra, we mark the deformedū profiles and their corresponding Ro −1 L profiles. The profiles marked with a blue • indicate the stage where theū profile still preserves its (S) shape, while the profiles marked with a hot magenta indicate the stage where theū profile is completely deformed. We compare these two stages at Ra = 10 10 in figure 12. These stages correspond to the solid green line and solid black line in figure 11(d,h). The stage marked with a blue • (figure 12a) is slightly beyond Ro −1 opt and cyclone is near the wall. The stage marked with a hot magenta (figure 12b) is further beyond Ro −1 opt and the cyclone has migrated to the bulk. As a result, theū profile direction is reversed (solid black line in figure 11d). We see similar trend in theū profile at Ra = 10 8 (figure 8a). The reversal inū from Ro −1 = 1 (solid black line) to 20 (solid grey line) is due to the migration of cyclone to the bulk.
Considering all the marked Ro −1 L profiles in figure 11(e-h), wind breaking starts (cyclone appears) once Ro −1 L near the core of the cyclone. Slightly beyond Ro −1 opt (marked with •, blue in figure 11), the maximum Ro −1 L also coincides with the peak ofū. Because the cyclone is small (figure 12a), hence its core (the maximum Ro −1 L ) and its edge (the peak of u) are close to each other. Further beyond Ro −1 opt (marked with , hot magenta in figure 11), the maximum Ro −1 L does not coincide with the peak ofū. Because the cyclone is large (figure 12b), hence its core and its edge are distant from each other.    figure 13a,b). We can support this power-law behaviour through scaling arguments for Kū max andεū max (Ro −1 Lū max ≡ 2ΩKū max /εū max ). Deardorff (1970) found w * ≡ (gβκΔNu) 1/3 = κ/H(Pr Nu Ra) 1/3 as a suitable velocity scale for the r.m.s. of wall-normal velocity fluctuations w rms in Rayleigh-Bénard convection. Xie, Hu & Xia (2019) observed that w rms /w * weakly depends on Ra (∼ Ra 0.07±0.02 ). Here, our data ofKū max scale well with w 2 * , without Ra correction (figure 13c). One possibility could be due to our unbounded (periodic) domain compared with the bounded (cylindrical or cubic) domains in Xie et al. (2019). Another possibility could be due to the Coriolis force that is absent in Rayleigh-Bénard convection of Xie et al. (2019). Our data ofεū max scale well with w 3 * /H = ν 3 H −4 NuRaPr −2 (figure 13d). Because the volume and time averaged dissipation rate is ε u w 3 * /H in the inertia-dominated regime (Nu 1), see § 2.2. With these scales forKū max andεū max , we obtain Ro −1 Lū max ∝ Pr 1/6 Ra 1/6 Ro −1 Nu −1/3 . Considering Nu versus Ra (figure 5) for Ro −1 < 1, it nearly falls into the classical regime scaling, i.e. Nu ∝ Ra 0.29 (Scheel & Schumacher 2016). Substituting for Nu and Pr = 0.7, we obtain Ro −1 Lū max ∝ 0.94Ra 0.07 Ro −1 . Figures 13(a), dashed lines, and 13(b), solid lines, show the good agreement between this power-law relation and the simulation data up to Ro −1 L ≤ 4. We can predict Ro −1 opt at each Ra using this power-law. The data points at Ro −1 opt (filled with black dots in figure 13a), fall at Ro −1 Lū max 3.0. After substituting for Ro −1 Lū max = 3.0 and Ro −1 = Ro −1 opt in the power-law relation and some recasting, we arrive at Ro −1 opt 3.19Ra −0.07 . This relation confirms the slight variation of Ro −1 opt with Ra (figure 6a). Our approach in this subsection is similar to the one by King et al. (2009); to identify the transition from the buoyancy (inertia) dominated to the Coriolis dominated regime. King et al. (2009) focus on rot_RB_wall (table 1), while we consider centrifugal convection. Both in King et al. (2009) and here, we relate the transition to a local phenomenon. In King et al. (2009) the transition occurs in the Ekman layer. They explain the transition through the competition between the Ekman layer and thermal boundary layer. Here, the transition occurs in the bulk, i.e. the locale of cyclone. Therefore, we explain the transition through Ro −1 L ≡ 2ΩK/ε u , i.e. the competition between the system rotation time scale and the turbulence time scale.

Flow behaviour at very large Coriolis force
At the very large Ro −1 = 20 (figure 7d,h), turbulence is suppressed and the flow becomes two-dimensional and laminar-like. In figure 14, we visualise this case over a period of 150H/U. In the top row, we visualiseθ overlaid by the (ũ,w) streamlines. In the bottom row, we visualise the vorticity fieldω y = (∂zũ − ∂xw). We locate the core of cyclones ( , green) and anticyclones (•, red) based on the core of streamlines, i.e. (ũ,w) (0, 0). Figure 14 shows that at the very large Ro −1 , the cyclone in the bulk and anticyclones near the walls become equally strong. The stabilisation of both cyclones and anticyclones at very large Ro −1 is observed in rotating Rayleigh-Bénard convection with free-slip boundaries (rot_RB_slip). For instance, refer to figure 3 in Stellmach et al. (2014) (Ra 2 × 10 11 , Ro −1 23) or figure 2 in Guervilly et al. (2014) (Ra 8 × 10 8 , Ro −1 7). However, in those studies turbulence is not completely suppressed. This difference might be due to their different flow set-up (rot_RB_slip in table 1) compared with our set-up (centrifugal convection).
In figure 14(b-g), the cyclone has almost a uniform zero temperature. Also, the anticyclones near the bottom and top walls have a uniform hot and cold temperature, respectively. To explain this phenomenon, in figure 15 we study different terms of u-and w-momentum equations. The diffusion terms are negligible and the momentum balance is primarily between the pressure gradient (figure 15c,g) and Coriolis force ( figure 15d,h). This balance is known as the geostrophic balance (Greenspan 1968), leading to the geostrophic regime. In this regime, there is a drop in Nu (King et al. 2009;Schmitz & Tilgner 2009;Ecke & Niemela 2014), also seen in our case at Ra = 10 8 from Ro −1 = 1 to 20 (•, black to , olive green, in figure 5). This regime also appears in rapidly rotating Rayleigh-Bénard convection, in all the common set-ups of this flow (table 1)   for rot_RB_cyl see Vorobieff & Ecke (2002), Ecke & Niemela (2014), Ecke (2015) and Rajaei, Kunnen & Clercx (2017). However, the flow phenomenology evolves differently among these three systems (Kunnen et al. 2016;Zhang et al. 2020). In planetary flows, such as atmospheric or ocean circulations, Ro −1 can go beyond 10 (Kundu & Cohen 1990) and geostrophic balance occurs. Although the geostrophic balance is between the Coriolis force and pressure gradient, in our system we cannot ignore the transient (figure 15a,e) and advection terms (figure 15b, f ), as figure 14 shows that this is an unsteady problem. Also we cannot ignore the buoyancy term (figure 15i), as there would be no flow without this term. We also assessed the terms of the temperature transport equation (2.1e). Similar to momentum, the diffusion terms are orders of magnitude smaller than the transient and advection terms. Therefore, the governing equations ((2.1a)-(2.1e)) are simplified to (3.3a-d) where Dt · = ∂t · +ũ∂x · +w∂z · is the material derivative. The material derivative of a property implies the rate of change of that property as we track a cyclone or anticyclone. We recast (3.3a-d) in terms of stream functionψ (ũ = ∂zψ,w = −∂xψ) and ω y as follows: A similar set of equations is solved for quasi-geostrophic flows to study the interaction of cyclones and anticyclones with themselves (Reinaud, Dritschel & Koudella 2003;Dritschel, Reinaud & McKiver 2004;Reinaud & Carton 2016)  3.6. Towards logarithmic boundary layers In § 3.1 we showed the remarkable feature of this centrifugal system in forming a bidirectional wind with the maximum momentum at Ro −1 opt , presumably helping the boundary layer transition to the turbulent Prandtl-von Kármán type, as hypothesised by Kraichnan (1962). In this subsection, we demonstrate this feature by studying the structure of the wind as Ra increases. The value of Ro −1 opt changes with Ra, from 1.0 at Ra = 10 7 , to 0.6 at Ra = 10 10 (figure 6b). Here, we fix Ro −1 at 0.8, a value slightly smaller or larger than Ro −1 opt , so that the system still generates a high momentum wind. In figure 16, we study the mean velocity and temperature profiles and r.m.s. of their fluctuations from the top wall to the domain midheight, as Ra increases. To study the distance to a turbulent boundary layer, in figure 16 in addition to the law of the wall for Prandtl-von Kármán type boundary layer (dashed-dotted blue line), we overlay the mean and r.m.s. velocity profiles by their counterparts from the DNS of turbulent channel flow (solid red line) at Re τ 180 (Lee & Moser 2015), based on channel half-height, and we overlay the temperature profiles by their counterparts from the DNS of turbulent channel flow with passive scalar (dotted red line) at Re τ = 180 and Pr = 0.71 (Kim & Moin 1989). We choose this value of Re τ for comparison, because in our system at the highest Ra = 10 10 (solid black line in figure 16a), Re τ reaches approximately 155 based on the velocity boundary-layer thickness. We define the boundary-layer thickness where |ū| + is maximum. Theū + profiles (figure 16a) progress towards Prandtl-von Kármán (logarithmic) behaviour as Ra increases. Nevertheless, full collapse on the logarithmic law, corresponding to a fully turbulent wall-bounded flow, is not reached yet at Ra = 10 10 (solid black line in figure 16a). A narrow logarithmic region (dashed magenta line) with a slope of 1/0.41 starts to appear from Ra = 10 9 (dashed-dotted black line in figure 16a). This is consistent with the two-dimensional Rayleigh-Bénard simulation of Zhu et al. (2018) who observed the emergence of the logarithmic slope 1/0.41 in the mean velocity profiles before the ultimate regime (Ra < 10 13 ). They also observed a much shallower logarithmic slope (approximately 1/4) in their temperature profiles compared with what is expected in a fully turbulent boundary layer (0.84/0.41 1/0.48). We observe similar behaviour in theθ + profiles in figure 16(b), considering the fitting lines (dashed magenta line) with a slope of 1/4.0. From figure 16(a,b), it appears that the mean velocity profiles approach the fully turbulent counterpart faster than the mean temperature profiles. Similar behaviour is seen in the r.m.s. profiles (figure 16c,d). Both u rms and θ rms profiles yield an inner peak near the wall. The inner peak in the u + rms profiles (figure 16c) is approximately 3.0 at all values of Ra, which is close to the DNS of channel flow (solid red line). As Ra increases, the location of the u + rms inner peak approaches the one corresponding to the fully turbulent channel flow (solid red line). On the other hand, the inner peak of the θ + rms profiles (figure 16d) is smaller than the fully turbulent counterpart (dotted red line), even at the highest value of Ra = 10 10 (solid black line in figure 16d).
Our observations from figure 16 give indications for the onset of the transition to turbulence in the velocity boundary layer at Ra = 10 10 . We visualise this transition in figure 17, showing the instantaneous field of u at 15 wall units away from the top wall. We observe the streak-like structures at Ra = 10 10 (figure 17d), similar to a fully turbulent boundary layer. The green square with an area of 500 × 500 wall units highlights the approximately 100 wall units spanwise spacing between the near-wall streaks, as in a turbulent wall-bounded flow (Kline et al. 1967). We also see the transition to turbulence in the spectral distribution of Reynolds shear-stress near the top wall (figure 18). We plot the spectrograms of u w at Ro −1 = 0.8 and Ra from 10 7 (figure 18a) to 10 10 (figure 18d). In figure 18(d) we overlay the spectrogram by the one from the DNS of turbulent channel flow (Lee & Moser 2015) at Re τ 180 (solid red line). As we observe, the spectral distribution of the near-wall region at Ra = 10 10 is similar to that of a turbulent channel flow. The spacing of 100 wall units between the energetic near-wall streaks (figure 17d) can also be interpreted from the near-wall energetic mode (marked by '+' in figure 18d), at (H − z) + 20 and λ + y 100. This subsection supports our earlier conjecture that the bidirectional wind helps the boundary layer transition to turbulent, hence the transition to the ultimate regime.

Conclusions
Detailed DNS and analysis of centrifugal buoyancy-driven convection has been carried out with a focus on the effect of the Coriolis force. The inverse Rossby number Ro −1 was varied from zero to 20, the Rayleigh number Ra from 10 7 to 10 10 and the Prandtl number was fixed at Pr = 0.7, corresponding to air. Our range of Ro −1 covered the regimes related to the turbomachinery, Ro −1 ∼ O(1), and geophysical flows, Ro −1 1. The results show that below an optimal Ro −1 opt , the Coriolis and buoyancy force interaction form a bidirectional wind. The value of Ro −1 opt decreases from approximately 1.0 to 0.6, as Ra increases from 10 7 to 10 10 . At Ro −1 opt , the wind momentum reaches its maximum, leading to a maximal C f and a minimal Nu. Just beyond Ro −1 opt , the Coriolis force locally dominates. It weakens the wind and forms a large-scale cyclone. At Ro −1 Ro −1 opt , the Coriolis force fully dominates. It balances the pressure gradient (geostrophic balance), laminarises the flow and stabilises both cyclones and anticyclones.
The flow regimes have both similarities and differences to those in rotating Rayleigh-Bénard convection. The differences are due to the different axis of rotation and boundary conditions. In rotating Rayleigh-Bénard convection, optimal transport (Ro −1 opt ) occurs only if the hot and cold boundaries are no-slip (rot_RB_wall and rot_RB_cyl in table 1). The presence of the wall, hence the Ekman layer, forms coherent columnar or plume-like structures. These structures enhance vertical transport, leading to a maximal Nu. On the other hand, large-scale quasi-two-dimensional cyclone has only been observed in rot_RB_slip and rot_RB_wall so far. At Ro −1 1, the geostrophic regime occurs for all the rotating Rayleigh-Bénard systems, but the flow evolution depends on the boundary conditions.
Our study highlights that with centrifugal convection we can control the wind at Ro −1 opt to generate a shear boundary layer, providing the opportunity to hasten transition to turbulent. Turbulent shear boundary layer (Marusic et al. 2010) is the main assumption for the ultimate regime (Kraichnan 1962;Grossmann & Lohse 2011). By our highest Ra = 10 10 , we see transitional behaviour in the boundary layer. In particular, the boundary layer yields streak-like structures with a spectral distribution similar to a canonical turbulent boundary layer. Yet, the mean flow does not reach the Prandtl-von Kármán (logarithmic) behaviour. Recently, Iyer et al. (2020) performed Rayleigh-Bénard simulation up to Ra = 10 15 in a slender cylindrical cell of aspect ratio 1/10. They did not see a departure from the classical regime, because the boundary layer structure differed from a unidirectional canonical turbulent boundary layer. As they discuss, thermal plumes block the boundary layer from development. We also conjecture that the slender cylinder does not allow the boundary layer to develop.
From an experimental perspective, the bidirectional wind is a bonus to reach the ultimate regime, in addition to mitigating non-Oberbeck-Boussinesq effects. Experiments are conducted in a vertical cylindrical annulus with closed end walls. As a result, some additional phenomena may affect the flow, including gravitational buoyancy, Ekman and Stewartson layers (Jacoby et al. 2011;Pitz et al. 2017b;Von Larcher et al. 2018). However, these effects can be minimised by adjusting the working fluid, operating conditions and geometry. The gravitational buoyancy effect can be minimised by making RΩ 2 g. For instance, Jiang et al. (2020) could experimentally achieve RΩ 2 60g. The Ekman layer effect can be minimised by making L/H Ek −1/2 (Busse 1970), where L is the annulus height and Ek ≡ ν/(ΩL 2 ) is the Ekman number. Busse & Carrigan (1974) could experimentally mitigate the Ekman layer effect by making L/H 50. Alonso et al. (1999) show numerically that Ekman and Stewartson layer effects can be minimised by increasing Pr, R/H, L/R or Taylor number Ta ≡ (2ΩH 2 /ν) 2 ≡ RaRo −2 /Pr. To our knowledge, most of the recent experiments consider R, H and L in the same order (Von Larcher & Egbers 2005;Von Larcher et al. 2018;Jiang et al. 2020). Nevertheless, Jiang et al. (2020) show small differences in Nu between their experimental rig (closed cavity) and DNS (open ended shell). However, all the experiments so far operate at Ro −1 > 2 (figure 2a). Therefore, the effects of Ekman and Stewartson layers on the bidirectional wind (at Ro −1 opt 1) and how effective the mitigating prescriptions are remain to be investigated. Answering the above questions lead us to several future directions. These directions also address some important aspects of turbomachinery studies. One direction is to enclose the boundaries normal to the rotation axis (closed cavity). Therefore, we can study the effects of Ekman and Stewartson layers on the bidirectional wind. Additionally, our system will be more similar to a compressor cavity (figure 1 in Owen & Long (2015)). Another direction is to systematically change the ratios R/H and L/R. Therefore, we can study how these geometrical parameters alter the interaction between the bidirectional wind and viscous layers. Also, we can evaluate the prescriptions (Busse 1970;Alonso et al. 1999) for mitigating the effects of viscous layers. This direction is also valuable from a turbomachinery perspective. In compressor cavities (Atkins & Kanjirakkad 2014), H/R varies from approximately 0.9 (Farthing et al. 1992) to 0.5 (Bohn et al. 1995). history. The fourth row shows the flow fields at the same times. They show the spanwise averagedθ y ≡ θ y /Δ overlaid by the streamlines of (ũ y ,w y ) ≡ (u y /U, w y /U). The circled red arrow indicates the system rotation direction (clockwise). The Nu for CC_slip (dashed black line) at Ro −1 0.38 (filled red square), 1.13 (filled black square), 1.88 (filled blue square), 3.00 (filled cyan square) and 3.75 (filled forest green square), and CC_wall (solid black line) at Ro −1 = 0.3 ( , red), 0.5 ( , blue), 0.6 (+, magenta), 0.8 (*, green), 1.0 (•, black) and 2.0 ( , blue). convection with free-slip hot and cold boundaries (CC_slip in table 1). We compare these two systems in terms ofū (figure 20a) and Nu (figure 20b). We consider Pr 0.7, Ra = 10 7 and 0.3 Ro −1 4.0, where there is an overlap between our parameter space and the one by von Hardenberg et al. (2015) (•, black; , red in figure 2a). In CC_slip, the bidirectional wind can have a cyclonic or anticyclonic mean vorticity (von Hardenberg et al. 2015). Here, we only consider the anticyclonic wind, because in CC_wall the bidirectional wind is always anticyclonic. In CC_slip (dashed-dotted lines in figure 20a), the bidirectional wind never breaks down. Its strength increases up to