2.1. 2-D Rayleigh–Bénard

For the 2-D Rayleigh–Bénard system, there are two velocity fields and a temperature field. We take ${x}_1$ as the spanwise (horizontal) direction and impose periodicity in this direction, ${x}_2$ is an additional horizontal direction that is absent in the 2-D case, and ${x}_3$ points in the direction of gravity The velocity field then has components ${\boldsymbol {u}}=({u}({x}_1,{x}_3), {w}({x}_1,{x}_3))$ that are restricted to be incompressible, ${\partial }_1 {u} + {\partial }_3 {w}=0$. The boundary conditions on the velocity field are usually rigid ${u}={w}=0$, or free-slip (${{w}=0}$ and ${\partial }_3 {u} = 0$) at the top and bottom plates. The derivations in this section do not depend on the specific boundary conditions on ${\boldsymbol {u}}$; however, in § 3.3, we will consider free-slip boundary conditions specifically and the implications on maximal heat transport (figure 1).

Figure 1.

$(a)$ The geometry, choice of coordinates, and generic boundary conditions for the Rayleigh–Bénard set-up (particular boundary conditions on $u$ depend on the physical setting, i.e. stress-free versus no-slip). $(b)$ The same for Taylor–Couette.

The dimensional temperature is ${T}={T}({x}_1,{x}_3)$ and it satisfies the boundary conditions ${T}({x}_3=0)=T_0+\delta T$ and ${T}({x}_3=d)= T_0$, i.e. the temperature at the bottom plate is higher by the fixed amount $\delta T$. Using the Boussinesq approximation, the equations of motion are

(2.1a)\begin{gather} {\partial}_t {u} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {u} + 1/\rho {\partial}_1 {p} = \nu {\varDelta} {u}, \end{gather}

(2.1b)\begin{gather} {\partial}_t {w} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {w} + 1/\rho {\partial}_3 {p} = \nu {\varDelta} {w} + g \beta {T}, \end{gather}

(2.1c)\begin{gather} {\partial}_t {T} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {T} = \kappa {\varDelta} {T}, \end{gather}

combined with incompressibility: $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {u} = 0$, where $\boldsymbol {\nabla }=({\partial }_1, {\partial }_3)$, and ${\varDelta } = {\partial }_{11}^2+{\partial }_{33}^2$. The other variables are the kinematic viscosity $\nu$, the thermal diffusivity $\kappa$, the density $\rho$, the expansion coefficient $\beta$ and the gravitational constant $g$. As mentioned above, ${x}_3=0$ is the bottom plate and ${x}_3=d$ the top plate. In the absence of convection, the temperature displays a linear profile, ${T}_0({x}_3)=\delta T\, (1-(x_3/d))$. This non-convective buoyancy is balanced by the pressure field ${p}_0({x}_3) = \beta g \delta T x_3 (1-(x_3/(2d)))$. We decompose the full temperature field according to ${T}({x_1},{x_3},{t})={T}_0({x}_3) + {\theta }({x}_1,{x}_3,{t})$. Decomposing the pressure in a similar manner, but using ${p}$ to now refer to perturbations about the laminar pressure ${p}_0$, we can derive the equations for the deviation ${\theta }$ from the diffusive profile:

(2.2a)\begin{gather} {\partial}_t {u} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {u} + {\partial}_1 {p} = \nu {\varDelta} {u}, \end{gather}

(2.2b)\begin{gather} {\partial}_t {w} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {w} + {\partial}_3 p = \nu {\varDelta} {w} + g \beta {\theta}, \end{gather}

(2.2c)\begin{gather} {\partial}_t {\theta} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {\theta} - \delta T (w/d) = \kappa {\varDelta} {\theta}, \end{gather}

coupled with incompressibility for the velocity field.

In RBC, dimensionless variables for temperature, length and time are based on the temperature difference $\delta T$, the height $d$ and the thermal diffusivity $\kappa$. Then in the non-dimensional setting, we have the system:

(2.3a)\begin{gather} \partial_t u + (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}) u + \partial_1 p= Pr \varDelta u, \end{gather}

(2.3b)\begin{gather} \partial_t w + (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}) w + \partial_3 p= Pr \varDelta w + Pr Ra \theta, \end{gather}

(2.3c)\begin{gather} \partial_t \theta + (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}) \theta = w + \varDelta \theta, \end{gather}

again coupled with incompressibility of $\boldsymbol {u}$, and where the Rayleigh number is given by $Ra=({g \beta \delta T\, d^3})/(\kappa \nu)$, and the Prandtl number $Pr={\nu }/{\kappa }$. We now wish to re-scale the 2-D RPC system so that it has the same functional form as these three equations. As we see, the analogy is exact only when $Pr = 1$, and if we carefully re-scale the azimuthal component of the velocity field so that it appears in the same way as the temperature fluctuations.

2.2. 2D3C plane Couette flow

The full Taylor–Couette system consists of three velocity components that are linked by incompressibility. In order to split off a temperature-like component, we consider a restricted geometry where the fields do not depend on the azimuthal spatial coordinate, and we investigate the limit of large radius relative to the gap size, i.e. the limit of zero curvature. In this set-up, the azimuthal velocity component decouples and can be re-scaled to resemble the temperature fluctuations from RBC.

The radial direction in the TCF system is analogous to the gravitational direction in the RBC system, so that the ${x}_3$ direction becomes the radial one. The neutral direction for the flow is the axial one, which, therefore, is referred to as the ${x}_1$ direction, and in which we assume the flow to be periodic. Finally, the azimuthal component becomes the ${x}_2$ direction. Invariance of the azimuthal position then implies that the velocity field has dependencies $({u}({x}_1,{x}_3), {v} ({x}_1,{x}_3), {w}({x}_1,{x}_3))$. In a frame of reference rotating with frequency $\varOmega$ around the $x_1$-axis, the system then becomes

(2.4a)\begin{gather} {\partial}_t {u} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {u} + {\partial}_1 {p} = \nu {\varDelta} {u}, \end{gather}

(2.4b)\begin{gather} {\partial}_t {w} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {w} + {\partial}_3 {p} = \nu {\varDelta} {w} + 2\varOmega {v}, \end{gather}

(2.4c)\begin{gather} {\partial}_t {v} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {v} = -2\varOmega {w} + \nu {\varDelta} {v}, \end{gather}

together with incompressibility, ${\partial }_1 {u} + {\partial }_3 {w} = 0$.

The domain is bounded by two walls, i.e. the inner and outer cylinders, which we denote as ${x}_3=0$ and ${x}_3=d$, that are parallel to the ${x}_1-{x}_2$ plane. Between the plates, there is a mean shear, maintained by moving the walls at constant speed $U$ in the $x_2$-direction. This yields the boundary conditions ${v}({x}_3=0)=U$ and ${v}({x}_3=d)=0$, which gives rise to a linear laminar velocity profile, ${v}({x}_3)=U(1-(x_3/d))$. The typical physically motivated boundary condition on the other components of velocity is the no-slip condition, meaning that the other components of the velocity field vanish identically at these walls. In § 3.3, we discuss the effect of considering a slippery boundary for ${v}$ and ${w}$, a condition that is not as physically relevant but is more conducive to analysis.

As in the case of RBC, we are primarily concerned with deviations $v'$ from the linear profile, that is

(2.5)\begin{equation} {v}({x}_1,{x}_3,{t})=U(1-(x_3/d)) + v'({x}_1,{x}_3,{t}), \end{equation}

where $v'$ is the dimensional form of the deviations. The linear part in ${v}$ is absorbed in the pressure (compensating the centrifugal forces) as was done for RBC, so that only the fluctuations remain. The equation for the azimuthal component then becomes

(2.6)\begin{equation} {\partial}_t v' + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) v' - (U/d) {w} = -2\varOmega {w} + \nu {\varDelta} v'. \end{equation}

The contribution from the normal velocity has a prefactor proportional to $(U-2\varOmega d)$ that can be absorbed in the azimuthal component $v'$ with the rescaling

(2.7)\begin{equation} v'=(U-2\varOmega d) \theta. \end{equation}

Note that this scaling introduces a partial asymmetry between the velocity components since it affects only one and not all three components. It is, therefore, not reasonable for the full 3-D system.

The identification of this renormalization of the azimuthal velocity fluctuations to the variable $\theta$ is intentional, as this normalized dependent variable is identified with the temperature fluctuations for the 2-D RBC system described above. With this definition of $\theta$, the dimensional equation for the azimuthal velocity becomes

(2.8)\begin{equation} {\partial}_t \theta + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) \theta = {d}/{w} + \nu {\varDelta} \theta \end{equation}

and the evolution of ${u}$ and ${w}$ in RPC become

(2.9a)\begin{gather} {\partial}_t {u} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {u} + {\partial}_1 {p} = \nu {\varDelta} {u}, \end{gather}

(2.9b)\begin{gather} {\partial}_t {w} + ({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}) {w} + {\partial}_3 {p} = \nu {\varDelta} {w} + 2\Omega ((U/d)-2\Omega)\theta \end{gather}

Now we introduce dimensionless variables for the rest of the system using the gap $d$ and the viscosity $\nu$ to generate spatial and temporal scales (and, hence, velocity as well), which give the shear Reynolds number $Re_S=Ud/\nu$, and the rotation number, $R_{\varOmega }=2\varOmega d/U$. Then the full non-dimensional equations for RPC become

(2.10a)\begin{gather} \partial_t u + (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}) u + \partial_1 p= \varDelta u, \end{gather}

(2.10b)\begin{gather} \partial_t w + (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}) w + \partial_3 p= \varDelta w + Re_S^2 R_{\varOmega} (1-R_{\varOmega}) \theta, \end{gather}

(2.10c)\begin{gather} \partial_t \theta + (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}) \theta = w + \varDelta \theta, \end{gather}

which is formally identical to the RBC case identifying $Pr=1$ and the Taylor number $Ta$ with the Rayleigh number, $Ra=Re_S^2 R_{\varOmega } (1-R_{\varOmega }).$ This derivation did not use the boundary conditions on the velocity field so that it is valid for both rigid and free-slip boundary conditions or any mixture thereof.

The exact relationships between the RPC and RBC systems considered here are summarized in table 1.

Table 1.

Exact relations between the azimuthally independent RPC system and 2-D RBC.

3.1. Heat and momentum transport

The first set of consequences we discuss here stem from the heat and angular momentum transport, and will be valid for all boundary conditions on the velocity fields $u$ and $w$. In RBC, the temperature difference drives convection, which enhances the heat transport between the plates. The Nusselt number measures the heat transport in units of the diffusive heat transport, and is computed as $Nu_T = 1 + \overline {w \theta }$, where $\bar {{\cdot }}$ refers to an average in time, and across the $x_1$ direction.

For RPC, the azimuthal momentum transport is derived by averaging the dimensional equation for the azimuthal velocity in the ${x}_1$ direction and in time: ${\partial }_3 \overline {{w}{v}} = 2\varOmega \bar {{w}} + \nu {\partial }_{33}^2 \bar {{v}}$. Integrating this once in ${x}_3$ then gives the momentum current

(3.1)\begin{equation} J = \overline{{w}{v}} - \nu {\partial}_3\bar{{v}} = -\nu {\partial}_3\left.\bar{{v}}\right|_{{x}_3=0}, \end{equation}

which is independent of the position ${x}_3$ between the plates. Using the laminar profile and (2.7) for the azimuthal velocity and dividing $J$ by the linear viscous drag $\nu U/d$, one arrives at the equivalent of the Nusselt number in RPC for the non-dimensional variables

(3.2)\begin{align} Nu_S &= 1 + \left(1-\frac{2\varOmega d}{U}\right) \overline{w\theta} = 1+\left(1-R_{\varOmega}\right) \overline{w\theta}, \end{align}

(3.3)\begin{align} &\Rightarrow Nu_S(Re_S, R_{\varOmega}) -1 = (1-R_{\varOmega}) (Nu_T (Ra)-1). \end{align}

Thus, while the relation between the parameters $Ra$ in RBC and $Ta$ in RPC is symmetric under the exchange $R_{\Omega }$ to $1-R_{\Omega }$, the relation between heat and momentum transport is not. Remarkably, this connection predicts that $Nu_S$ approaches the laminar value $Nu_S=1$ for $R_{\Omega }$ approaching $1$, for any value of $Re_S$.

The transport of momentum as a function of rotation number has been studied for different parameter values in both TCF and RPC. Assuming that the Nusselt number in RBC follows a scaling law $Nu_T \sim c Ra^{\alpha }$ with an as yet undetermined exponent $\alpha$ and some constant $c$, we can obtain a scaling for $Nu_S$ in terms of both $Re_S$ and $R_{\varOmega }$ and determine the maximal $R_{\Omega }$. First, we observe that

(3.4)\begin{align} Nu_S -1 &\sim c Re_S^{2\alpha}R_{\varOmega}^{\alpha}(1-R_{\varOmega})^{1+\alpha} - 1 + R_{\varOmega}\nonumber\\ \Rightarrow\quad Nu_S & \sim R_{\varOmega} + c Re_S^{2\alpha} R_{\varOmega}^{\alpha}(1-R_{\varOmega})^{1+\alpha}. \end{align}

For large $Re_S$, the first part can be neglected and the relation reduces to

(3.5)\begin{equation} Nu_S \sim c' R_{\varOmega}^{\alpha} (1-R_{\varOmega})^{1+\alpha}, \end{equation}

away from $R_{\varOmega } = 0$ or $R_{\varOmega } = 1$. Maximizing this transport over the rotation rate $R_{\varOmega }$ (for $Re_S \gg 1$) leads to

(3.6)\begin{equation} R_{\varOmega,m} \sim \frac{\alpha}{1+2\alpha}. \end{equation}

In one conjectured asymptotic regime of thermal convection (Spiegel Reference Spiegel1963), it is expected that $\alpha ={1}/{2}$ so that the maximal momentum transport would occur for $R_{\varOmega ,m} \sim {1}/{4}$. The Reynolds numbers in numerical simulations and even in experiments are not high enough to reach this regime, and one resorts to Reynolds number dependent local scaling exponents. For RPC and $Re_S = 2\times 10^4$, Salewski & Eckhardt (Reference Salewski and Eckhardt2015) find a maximum near $R_{\varOmega ,m}\approx 0.2$. The transition from TCF to RPC is discussed in Brauckmann et al. (Reference Brauckmann, Salewski and Eckhardt2016), where it is shown that the maximum again appears near $R_{\varOmega ,m}\approx 0.2$ when the ratio of the radius of the inner cylinder to that of the outer cylinder is $\eta = 0.99$. This amounts to a ratio of angular velocities of $\mu = 0.9043$, which is remarkably close to $\mu =0.9801$, which is the Rayleigh criterion for $R_{\varOmega } = 1$ for this value of $\eta$. This observation is further cemented by the recent numerical and experimental results reported in Ezeta et al. (Reference Ezeta, Sacco, Bakhuis, Huisman, Ostilla-Mónico, Verzicco, Sun and Lohse2020). Both sets of data (for TCF and RPC) are shown in figure 2 (blue symbols). This agreement with observations is remarkable because the derivation here is valid only for the 2-D setting with a rescaling of $v$ that violates the natural scaling of the velocity fields, whereas the reported maxima of $R_{\Omega } \sim 0.2$ comes from numerical studies of the full 3-D flow. However, since rotation reduces the transverse components and enhances the azimuthally invariant parts, Brauckmann et al. (Reference Brauckmann, Salewski and Eckhardt2016) also isolated the contribution of the azimuthally invariant 2-D component of the flow, shown in figure 2 as the green data points. They show a maximum near $R_{\varOmega } \approx 0.18$. Moreover, a fit of the 2-D contributions to the functional form (3.5) with $\alpha =0.26$ approximates these data very well over essentially the entire range $0 < R_{\varOmega } < 0.6$ for which data are available. This confirms that the 2-D analysis presented here adequately explains features of the 3-D flow, in this case, the momentum transport of the azimuthally invariant part of the flow. We note that $\alpha =0.26$ is significantly less than either of the prevailing theoretical predictions for turbulent convection ($\alpha =1/2$ and $\alpha =1/3$). This may be due to the restriction to azimuthally invariant parts of the flow (2-D flow, see Whitehead & Doering (Reference Whitehead and Doering2011), for example), or it may be because $Re_S = 2\times 10^4$ is not in the asymptotic regime where one of these expected laws takes hold. In either case, the general form of the fit from (3.5) is remarkably accurate.

Figure 2.

Nusselt number $Nu_S$ for TCF and RPC versus rotation number $R_{\varOmega }$ for $Re_S = 2 \times 10^4$, redrawn from figure 8(f) of Brauckmann, Salewski & Eckhardt (Reference Brauckmann, Salewski and Eckhardt2016). The full symbols are for TCF at radius ratio $\eta =0.99$, the open symbols are for RPC. The blue data are for the full 3-D flow; they have a broad maximum at $R_{\varOmega } \approx 0.22$, and a narrow one for smaller $R_{\varOmega }$ of a different origin (see Salewski & Eckhardt Reference Salewski and Eckhardt2015; Brauckmann et al. Reference Brauckmann, Salewski and Eckhardt2016). The green data are obtained from the azimuthally invariant 2-D part of the flow. The continuous red line is a fit of the 2-D part to expression (3.5), with optimal parameters $\alpha =0.26$ and $c=37$ and a maximum near $R_{\varOmega } \approx 0.18$, very close to the maximum obtained from (3.6).

3.2. Mean profiles and the maximum principle

In turbulent RBC flows, one expects the temperature to be well mixed in the interior, so that the profile consists, to a good approximation, of steep boundary layers near the top and bottom plates, and a region of constant temperature in the middle. We anticipate the same behaviour will hold for the azimuthal velocity $v$ relative to the ${x}_3$ direction. Moreover, since (2.1c) is an explicit advection–diffusion equation, the temperature satisfies a maximum principle, i.e. ${T}$ is bounded by its values at the walls,

(3.7)\begin{equation} T_0\le {T}\le T_0+\delta T, \end{equation}

at any point in the volume. This arises because near a maximum the first derivative vanishes and the second is negative, so diffusion will act to reduce it. For RBC, this indicates that heat cannot pile up at the walls beyond that supplied by the boundary condition. Translated to RPC, the corresponding statement will address the relation between the downstream velocity and its value at the walls. A velocity field that lies outside the values provided by the boundary conditions is referred to in the literature as a backflow event. Existence of such backflow events has been a matter of controversy which was settled with the explicit demonstration of such events by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012).

To see how the maximum principle applies to RPC, we let ${v}=-2 \varOmega {x}_3 + \tilde {v}$, so that $\tilde {v}$ satisfies the advection–diffusion equation ${\partial }_t \tilde {v} + ({\boldsymbol {u}}\boldsymbol {\cdot }\boldsymbol {\nabla }) \tilde {v} = \nu {\varDelta } \tilde {v}$. This perturbative velocity field $\tilde {v}$ satisfies a maximum principle, i.e. its extreme values occur only at ${x}_3=0$ or ${x}_3 = d$. This can be stated succinctly as $2\varOmega d \leq \tilde {v} \leq U$, or $0\leq {v} \leq U$ for the original azimuthal velocity component (assuming a positive $U$). This implies that in this setting there is no backflow, i.e. the azimuthal velocity has a constant sign, and events like the ones observed by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012) cannot occur in 2D3C RPC. It would be of significant interest to determine if such events can occur for 3-D RPC or even for the full TCF system. If so, one must question whether these events are present only for full 3-D flows or if the current restriction in RPC is unique.

3.3. Free-slip boundary conditions

Although the no-slip boundary condition is clearly the physically motivated choice for TCF, and, hence, for RPC as well, there is still some interest in considering the free-slip condition, although there is less immediate physical motivation for this. Indeed, Rayleigh (Reference Rayleigh1916) invoked free-slip conditions for mathematical convenience:

Thus, in the interest of reducing such ‘complication’ due to the no-slip condition, we will consider stress-free (free-slip) boundary conditions on the plates for $u$ and $w$. As mentioned previously, this is presented in the form $w=0$ and $\partial _3 u = 0$ at $x_3=0$ and $1$. Incompressibility then implies that the vorticity in the $x_2$ direction defined by $\omega _2 = \partial _1w-\partial _3u$ vanishes at these boundaries as well. This leads to an enstrophy balance, where the enstrophy is defined as the square of the L$^2$ norm of $\omega _2$, i.e. $\|\omega _2\|_2^2$. This restriction which we consider in this section only, does not reflect on any other aspect of the system as described above.

As shown by Whitehead & Doering (Reference Whitehead and Doering2011), this enstrophy balance, coupled with a uniform bound on $w(x_3)$ near the boundary and a piece-wise linear, monotonic temperature profile, will yield a bound on the Nusselt number in RBC of the form $Nu_T \leq c Ra^{5/12}$. Translated into the RPC system, this becomes

(3.8)\begin{equation} Nu_S \lesssim R_{\varOmega} + Re_S^{5/6}(1-R_{\varOmega})^{17/12}R_{\varOmega}^{5/12}, \end{equation}

which will have a maximum as $Re_S \rightarrow \infty$ for $R_{\varOmega ,m} = {5}/{22} \sim 0.227$ by (3.6). This is the only setting to date for which the scaling of $Nu_S$ is sublinear with respect to $Re_S$, deviating from the anticipated $Re_S^{1}$ scaling, and, hence, there is no anomalous dissipation in this setting. Remarkably this scaling affects the $Re_S$-dependence, but it has little influence on the location of the maxima in $R_{\Omega }$.