## Abstract

Rayleigh–Bénard convection and Taylor–Couette flow are two canonical flows that have many properties in common. We here compare the two flows in detail for parameter values where the Nusselt numbers, i.e. the thermal transport and the angular momentum transport normalized by the corresponding laminar values, coincide. We study turbulent Rayleigh–Bénard convection in air at Rayleigh number *Ra*=10^{7} and Taylor–Couette flow at shear Reynolds number *Re*_{S}=2×10^{4} for two different mean rotation rates but the same Nusselt numbers. For individual pairwise related fields and convective currents, we compare the probability density functions normalized by the corresponding root mean square values and taken at different distances from the wall. We find one rotation number for which there is very good agreement between the mean profiles of the two corresponding quantities temperature and angular momentum. Similarly, there is good agreement between the fluctuations in temperature and velocity components. For the heat and angular momentum currents, there are differences in the fluctuations outside the boundary layers that increase with overall rotation and can be related to differences in the flow structures in the boundary layer and in the bulk. The study extends the similarities between the two flows from global quantities to local quantities and reveals the effects of rotation on the transport.

This article is part of the themed issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

## 1. Introduction

Convection in layers of fluids heated from below and cooled from above (Rayleigh–Bénard, RB flow) and the flow between two rotating cylinders (Taylor–Couette, TC flow) are among the canonical flows in fluid mechanics. Studies of their stability properties and the manner in which the laminar profiles give way to more structured and complicated flows have provided many insights into the transition to turbulence with linear instabilities [1,2]. The behaviour well above the onset of turbulence has also been investigated starting with the experiments by Wendt [3]. Many different flow regimes that are not yet fully explained or explored have been described [4,5]. It was realized early on that despite the differences in the driving forces, there are many similarities, and it is helpful to draw analogies and to compare the properties of both flows [6]. The intimate relations between the two flows have led Busse [7] to characterize them as the *twins of turbulence research*.

A formal analogy between RB and TC flow (and pipe flow as well) was developed and described in Eckhardt *et al.* [8,9] (see also Bradshaw [10] for an earlier approximate relation and Dubrulle & Hersant [11] for a similar analogy). The analogy identifies pairs of equations that describe the total energy dissipation and the global transport of heat or angular momentum in the two flows. The equations allow one to relate transport properties, dimensionless parameters and other quantities, and have been used in particular to study scaling relations in fully developed turbulent flows [5]. The similarity in the equations suggests that a more detailed comparison between the two flows should be possible.

We here explore this option within direct numerical simulations (DNS). We describe the difficulties one has to overcome in identifying corresponding parameters, and present case studies where detailed comparisons are possible. In particular, we compare the turbulent transport currents with respect to their statistical properties. Furthermore, we can relate components of the involved turbulent fields to each other and compare their statistical fluctuations at different distances from the walls. The focus of our study is on the general ideas and an illustration for a few examples, but not on a comprehensive study for all parameter values. Specifically, we take one set of data for RB flow and compare it with TC flow cases at two different rotation numbers, which allows us to study the effect of rotation. The data are taken from well-resolved DNS of both flows at moderate Rayleigh and Reynolds numbers.

The outline of the manuscript is as follows. In §2, we present the balance equations, the numerical methods and discuss the analogy. In §3, the choice of corresponding parameters for the comparison is explained. In §4, the area-averaged mean currents and their probability density functions (PDFs) as well as other pairwise related properties at different distances from the wall are analysed. We conclude the work with a short discussion of the particular structures of the convective currents and a summary in §5.

## 2. Relations for transport currents and dissipation rates

RB flow is modelled by the three-dimensional Boussinesq equations for the velocity field **u** and the temperature field *T* [12,13]. The equations are solved using Nek5000 software [14], a spectral element method [15,16]. The physical system is characterized by the imposed temperature difference between bottom and top plates, Δ, the height *d* of the domain, and the free-fall velocity with the thermal expansion coefficient *α* and the acceleration due to gravity, *g*. The kinematic viscosity *ν* of the fluid and the thermal diffusivity *κ* are combined in the Prandtl number *Pr*=*ν*/*κ*. The flow is confined to a cylinder with insulating sidewalls.

The mean heat flux across the layer, i.e. in the *z*-direction, is given by
2.1
with *J*_{T,lam} the purely diffusive heat flux below the onset of convective motion,
2.2
Here, 〈⋅〉=〈⋅〉_{A,t} denotes an average over horizontal planes at fixed height and over time. Equation (2.1) already contains the definition of the Nusselt number *Nu*_{T}, which measures the heat transport relative to the laminar situation. A second relevant equation is that for the mean kinetic energy dissipation rate of the velocity field *ϵ*_{u}. It is obtained by multiplying the momentum balance of the Navier–Stokes–Boussinesq equations with the velocity *u* and integrating over the volume and over time,
2.3
Here, *Ra*=*αgd*^{3}Δ/(*κν*) is the Rayleigh number, the second dimensionless parameter. The dataset which we use for the comparisons is obtained in a closed cylindrical cell with a unity aspect ratio (diameter=height) at *Ra*=10^{7} and *Pr*=0.7.

The TC system is characterized by the radii *r*_{1} and *r*_{2} of the inner and outer cylinder, which rotate with the angular velocities *ω*_{1} and *ω*_{2}, respectively. The flow between the cylinders is governed by the incompressible Navier–Stokes equations for the velocity **u**=(*u*_{r},*u*_{φ},*u*_{z}) in cylindrical coordinates (*r*,*φ*,*z*). We solve the equations with periodic boundary conditions in the axial direction using a spectral method [17]. In TC flow, the gap width *d*=*r*_{2}−*r*_{1} and the velocity difference between the cylinders *U*_{0}=2*r*_{1}*r*_{2}(*r*_{1}+*r*_{2})^{−1}(*ω*_{1}−*ω*_{2}) (calculated in a rotating frame of reference [18,19]) serve as characteristic scales for lengths and velocities. We choose a system height of 2*d*, so that one pair of Taylor vortices fits into the computational domain. Thus, the diameter of a single Taylor vortex is similar to that of the large-scale circulation in RB flow with aspect ratio one. Further details of the simulation procedure are discussed in references [20,19].

For the derivation of expressions in TC flow that correspond to (2.1) and (2.3) in RB flow, we start with the azimuthal velocity *u*_{φ}. Averaging the *φ*-component of the Navier–Stokes equation over time and over cylinders at fixed radii *r* between the positions of the inner (*r*_{1}) and outer (*r*_{2}) cylinder, one finds [8,9]
2.4
with the angular velocity *ω*=*u*_{φ}/*r* and *J*_{ω,lam} the angular momentum flux in the laminar case
2.5
Here, *r*_{a}=(*r*_{1}+*r*_{2})/2 is the mean radius. The averaged current *J*_{ω} is independent of the radius and conserved in time. Physically, it corresponds to the torque needed to keep the cylinders in motion; it corresponds naturally to the heat transport (2.1) in RB flow, which is why we also introduced a Nusselt number *Nu*_{ω} corresponding to *Nu*_{T}. Similarly, one can multiply the Navier–Stokes equation with the velocity **u** and average over volume and time to obtain the mean kinetic energy dissipation rate, corresponding to (2.3). However, the dissipation associated with the laminar profile has to be taken out, so that we are led to consider
2.6
with the geometric parameter denoted as quasi-Prandtl number. The dimensionless Taylor number is defined as . Furthermore, the radius ratio is denoted by *η*=*r*_{1}/*r*_{2}, and the specific angular momentum is defined by .

To separate the influences of shear and rotation, we adopt the parameters introduced by Dubrulle *et al.* [18]. With the Reynolds numbers
2.7
for the inner and outer cylinders, respectively, we form the shear Reynolds number and the rotation number
2.8
The relation to the Taylor number is given by .

A comparison between (2.3) and (2.6) suggests an association *Pr*≡*σ* between the Prandtl number *Pr* and the quasi-Prandtl number *σ*, and *Ra*≡*Ta* between the Rayleigh number *Ra* and the Taylor number *Ta*. However, there are various reasons why this is not sufficient. For example, a direct comparison between (2.3) and (2.6) suggests equality of the combinations *Pr*^{−2}*Ra* and *σ*^{−2}*Ta* only and does not relate *Ra* and *Ta* directly. Moreover, TC flow has two Reynolds numbers, and the Taylor number captures only their difference. The overall rotation, as measured by the rotation number *R*_{Ω}, does not enter, but it is known that the torque varies non-monotonically with *R*_{Ω} [21,22,20,23,24]. Similarly, critical values for the onset of instability are given by *Ra*_{c}=1708 [1] for RB flow and by *Ta*_{c}=1708/[*R*_{Ω}(1−*R*_{Ω})] [18] for TC flow (in the limit where *Ta*=*Re*^{2}_{S}), again highlighting the significance of the rotation number. We therefore have to look for alternatives on how to relate the two flows.

## 3. Choice of reference point for comparison

In the following, we discuss how the reference point for the one-to-one comparison is chosen. As described above, equating *Ra* and *Ta* directly is not possible because of ambiguities in their definitions. A meaningful comparison can be based on the Nusselt number, because it defines the boundary layer thickness and hence the mean profiles. Similarly, the Reynolds stresses, when normalized by *NuJ*_{lam}, should fluctuate with mean value 1 in regions where the viscous contributions to the transport are small. This allows for an absolute comparison of PDFs because the dimensionless version with *Nu* scaled out has the same mean and, as we will show for one of the cases here, also the same variance.

We also have to select the curvature parameter *η* in TC flow, which has no counterpart in RB flow because the heated and cooled plates are planar. We therefore take *η*=0.99, because the curvature effects disappear for *η*-values close to 1. Finally, the mean system rotation in TC flow which is defined by *R*_{Ω} has to be selected for the direct comparison. Again, an analogous parameter is missing in RB convection. In figure 1*a*, a curve *Nu*_{ω}(*R*_{Ω}) at *η*=0.99 is shown for *Re*_{S}=2×10^{4}. A Nusselt number *Nu*_{ω} that is comparable to the RB flow value of *Nu*_{T}=16.7 at *Ra*=10^{7} (solid horizontal line) was obtained for *R*_{Ω}=0.023 and *R*_{Ω}=0.241 (open circles). These two runs are denoted as case 1 and case 2, respectively, and are used to study the effects of the rotation number. In both cases, the cylinders are co-rotating with angular velocity ratios *μ*=*ω*_{2}/*ω*_{1}=0.40 and *μ*=0.92 for *R*_{Ω}=0.023 and *R*_{Ω}=0.241, respectively. Decreasing the Rayleigh number gives other crossing points, as indicated by the dashed line and the open squares in figure 1*a* for *Ra*=5×10^{6}. For this lower *Ra*, the RB and TC flows differ notably, so that we subsequently focus on the cases 1 and 2 only. We furthermore note that in the first case the relative distance to the linear instability *Ta*/*Ta*_{c}≈5.35×10^{3} is close to the corresponding RB value *Ra*/*Ra*_{c}≈5.86×10^{3} for *Ra*=10^{7}, whereas in the second case the ratio *Ta*/*Ta*_{c}≈4.29×10^{4} is much higher.

Figure 1*b* shows the ratio
3.1
of the energy contained in the mean vortical motion to the energy of the total cross-flow. It measures the relative strengths of temporally and streamwise-averaged Taylor vortices, which are analogous to the large-scale circulation in RB flow. The vortex strength varies with rotation, and the curve shows that case 2 is more strongly dominated by the large-scale vortices than case 1. In RB flow with *Ra*=10^{7}, the corresponding energy ratio of approximately 0.4 is of similar magnitude and lies between the two TC cases.

As additional measures for the comparison of both flows, we analyse the boundary layer thicknesses. In analogy to the thermal boundary layer thickness *δ*_{T}=−Δ/(2∂_{z}〈*T*〉|_{z=0,d})=*d*/(2*Nu*_{T}) in RB flow, we define the boundary layer thicknesses at the inner cylinder (*r*=*r*_{1})
3.2
for the angular velocity and angular momentum profiles in TC flow with the total differences Δ_{ω}=*ω*_{1}−*ω*_{2} and . In the low-curvature case *η*=0.99 analysed here, we also have *δ*_{ω}≈*d*/(2*Nu*_{ω}), whereas such a relation does not exist for . However, for strongly co-rotating cylinders *δ*_{ω} overestimates the width of the boundary layer region since then the angular velocity profiles have a significant slope in the bulk [20,23]. As the angular momentum profile generally becomes almost flat in the bulk [19], the thickness provides a better approximation to the size of the boundary layer region and will therefore be used here. In figure 2, the boundary layer thickness is plotted for the same data as in figure 1*a*. It is observed that the boundary layer thickness of case 1 matches almost perfectly with the thermal boundary layer thickness *δ*_{T} of the RB flow at *Ra*=10^{7}. In case 2, the differences in the thickness scales are larger; here, the thickness is smaller than *δ*_{ω}≈*δ*_{T}=*d*/(2*Nu*_{T}), because the angular velocity profile is not flat in the central region. As we see in the following, these differences affect the statistical properties of the TC flows and thus the agreement with RB flow.

## 4. Statistical properties

### (a) Mean vertical profiles

In figure 3, we compare the mean profiles of temperature to the mean profiles of the angular velocity *ω*=*u*_{φ}/*r* and the angular momentum . The top row displays the comparison with the TC flow at the first local maximum at *R*_{Ω}=0.023 (case 1) (figure 1*a*). The bottom row repeats this comparison for the TC flow at the second local maximum in the *Nu*_{ω}−*R*_{Ω} relation at *R*_{Ω}=0.241 (case 2). While the agreement with case 1 is very good, there are differences for case 2. Here, the angular velocity profile has a notable gradient in the central region, whereas the angular momentum is well mixed and lies closer to the temperature profile. From this comparison, one can conclude that *T* is more closely associated with than with *ω*. Therefore, we compare the root mean square profiles of *T* and fluctuations in figure 3*c* and *f*. The fluctuating fields are obtained by
4.1
and
4.2
For case 1, the peaks in are broader than in case 2, which is consistent with the observation that the boundary layers are turbulent for case 1, but not for case 2 [25]. In the RB flow case, the boundary layer dynamics is close to laminar, and the peaks in *θ*_{rms} are narrower. Furthermore, we note that the shape of the root mean square angular velocity profile (not shown here) hardly differs from because the radius only varies by 1% for *η*=0.99.

The heat flux in RB flow is decomposed into a convective and diffusive contribution which results in
4.3
A similar decomposition into a convective and viscous contribution in the TC flow case leads to
4.4
Figure 4 displays the vertical (radial) profiles. Figure 4*a* compares with case 1, whereas figure 4*b* compares with case 2. As expected the dissipative contributions are significant in the boundary layers and become small in the bulk. The convective parts dominate the bulk and drop to zero at the walls owing to the no-slip boundary conditions. Furthermore, the sum of both transport contributions remains constant across the whole layer in both systems. It can be seen again that profiles of case 1 show better agreement with RB flow than the profiles of case 2. Because, in the latter case, the angular velocity profile is not flat in the central region (cf. figure 3*d*), the viscous contribution is larger than the corresponding diffusive part , which additionally results in a smaller convective contribution .

### (b) Probability density functions

We now refine the analysis and report the statistics of the convective currents in averaging surfaces at different distance from the inner (bottom) wall. First, it is important to note that only the temperature and angular momentum fluctuations *θ* and that deviate from the corresponding mean profile contribute to the net transport through the averaging surfaces, because 〈*u*_{z}〉=〈*u*_{r}〉=0 by incompressibility, and therefore 〈*u*_{z}*T*〉=〈*u*_{z}*θ*〉 and . Therefore, we study the local convective currents based on the fluctuations *θ* and instead of the total fields. Figure 5 compares the PDFs of the local convective angular momentum current for cases 1 (top row) and 2 (bottom row) with the local convective heat current for planes at different distances from the wall. All quantities are normalized by the corresponding mean currents *J*_{ω} and *J*_{T}. In all cases, it is observed that the skewness of the distributions increases with the distance of the analysis surface from the inner (bottom) wall. The net convective transport has to be positive, and its share of the total transport increases towards the bulk. It is also observed that the tails of the PDFs of TC flow for case 2 deviate strongly from the ones for RB flow away from the boundary layer.

The trend is different for the comparison of case 1 with RB flow. While the largest differences arise for the data at , the agreement is very good for the data taken at and *r*_{a}. The region just above the boundary layer thickness is dominated by rising plumes and recirculations next to the plumes. It is sometimes also denoted as the plume mixing layer [26]. The reason for the differences in the width of the tails in figure 5*b* and *e* could therefore be related to the shape of the plumes and the frequency of their detachment, which differ between RB and TC flow as will be shown in the next section.

The observation that the local fluctuations in case 2 are enhanced compared to case 1 can be understood by analysing the components that form . Because the fluctuation amplitude of varies little between both cases (cf. figure 3*c*,*f*), and the radius *r* remains unchanged, the difference must occur in the radial velocity *u*_{r}. In [19], it was shown that the fluctuation amplitude (*u*_{r})_{rms} varies with mean rotation (*R*_{Ω}) and in case 2, it is twice as large as in case 1. This increase is partly caused by a strengthening of the mean Taylor vortices, cf. figure 1*b*. The stronger *u*_{r} fluctuations result in wider tails in figure 5*e*,*f* but do not significantly affect the distribution at where the radial velocity is restricted owing to the proximity of the cylinder wall.

In figure 6, we compare individual components of the transport currents of angular momentum and heat. They are *u*_{z} and *θ* in the RB case, and the radial velocity *u*_{r} and angular momentum in TC flow. It can be observed that the agreement between case 1 and the RB flow is good. For case 2 (not shown here), the deviations of the individual PDFs of *θ* and were larger. In a turbulent flow, one expects Gaussian statistics for the individual components of the velocity field. In figure 6*a*, exponential tails are observed for the PDFs of *u*_{z} and *u*_{r} at the height of (thermal) boundary layer thickness. This is a clear statistical fingerprint for an enhanced intermittency in the near-wall region which is connected with the plume formation. In addition, in figure 6*b*, a fatter tail for the radial component is detected which confirms our observation in figure 5*b*.

The distributions of the temperature and angular momentum fluctuations (see bottom row of figure 6) are skewed and take a symmetric shape in the midplane only as shown in figure 6*f*. For all three distances, the tails of both PDFs are in very good agreement. The distribution in the midplane is again not Gaussian which has been reported already in [27]. The specific cusp-like form around the origin and the pronounced exponential tails in the PDF of temperature fluctuations have been discussed, for example by Yakhot, as an effect of the bursting plumes and the large-scale circulation [28]. Interestingly, even such specific details of the small-scale statistics prevail in our comparison between RB and TC flow.

### (c) Relation between transport and flow structures

In the last sections, we identified small differences in the statistical properties of the RB flow and case 1 of TC flow and attributed them to differences in the flow structures. The spatial organization of currents is shown in figure 7, where isosurfaces of the convective current (the top row of figure 5) for the levels *j*^{(c)}/*J*=±3 are plotted. Figure 7*a* and *c* shows the full simulation domains in both cases. The magnification in figure 7*b* displays a section of the TC flow with the same aspect ratio as the RB domain in figure 7*a*. Because the net transport current is positive, on average, there is a larger volume fraction of red isosurfaces than blue ones. The large coherent regions of high convective current, which occur near the sidewall in figure 7*a* and near the left and right surfaces in figure 7*b*, coincide with the upward and downward motion of the large-scale circulation in RB and a Taylor vortex in TC (see white arrows in figure 7*a*,*b*). Consequently, the positive tails in figure 5*c* are related to this large-scale motion. The similarity in the large-scale organization of the currents explains why the differences in geometry (cylindrical versus rectangular domain) have a minor influence on the statistical properties in the middle, cf. figure 5*c*. The isosurfaces of the TC flow are more fragmented and less smooth than for the RB flow, which indicates a higher level of fluid turbulence in the system. Quantitatively, we find in RB flow for the large-scale Reynolds number *Re*_{rms}=*u*_{rms}*d*/*ν*=675, with the root mean square velocity *u*_{rms} calculated from all three velocity components in the entire cell [29]. This is significantly smaller than the corresponding Reynolds numbers in TC flow (with the mean rotation subtracted), which are *Re*_{rms}=2251 and *Re*_{rms}=2712 for the cases 1 and 2, respectively. It can thus be expected that the boundary layers in the RB case are still close to laminar, whereas the ones in case 1 of TC flow are already turbulent [25]. Specifically, we find for the boundary-layer Reynolds numbers, defined based on the boundary layer thickness and shear across the boundary layer, values of approximately 30 for RB flow [29] and of approximately 300 and 200 for the TC cases 1 and 2, respectively [25]. The turbulent fluctuations in the TC boundary layer account for the deviations in the tails of the PDFs and the slight deviations in the area-averaged profiles, in particular at the heights of *z*=4*δ*_{T} and *y*=4*δ*_{ℒ}, respectively, in figure 5*b*.

## 5. Conclusion

In this work, we discussed a direct comparison of the statistical properties of RB convection and TC flow. The comparison is motivated by analogies of dimensionless system parameters (such as Rayleigh and Taylor numbers), the same form of the energy balances, (2.3) and (2.6), and the similarities in the currents of heat and angular momentum (see also references [8,9,10,11]).

Our study shows that the operating point for a specific comparison between TC and RB flows can be determined by choosing corresponding values of Nusselt numbers because the Nusselt number defines the boundary layer thickness and hence the transport properties. We also find that a better characterization of TC flow can be based on the pair of shear Reynolds and rotation numbers, (*Re*_{S},*R*_{Ω}), than on Taylor and quasi-Prandtl numbers, (*Ta*,*σ*), because the latter do not reflect the mean rotation of the cylinders. We demonstrated that for sufficiently large shear Reynolds number *Re*_{S}, multiple TC flow cases at different rotation numbers can have the same Nusselt number as RB convection, i.e. the same amount of angular momentum is transported between the cylinders in TC flow as heat from the bottom to the top in the RB case. The comparison also shows that the case with the smaller rotation number *R*_{Ω} (case 1) provides a better agreement with RB flow than the case of larger rotation number. For this pair of flows, a remarkable agreement between mean profiles as well as PDFs of fluctuating quantities is found.

Studies of the mean profiles and the PDFs of the convective currents show that the differences between RB flow and TC flow case 1 are most pronounced in the mixing layer above the (thermal) boundary layer. They can be attributed to the strong fluctuations in this region which are connected with the detachment of plumes and other differences in the dynamics: the boundary layers in the convection case are still very close to being laminar, but in the TC system, they are already turbulent. The differences should, therefore, become smaller when the boundary layers in RB also become turbulent.

The TC flow case 2, which is characterized by a larger mean rotation (*R*_{Ω}), shows greater differences to the RB case. As a consequence of rotation, the angular velocity profile has a significant gradient in the central region, which results in a higher (lower) dissipative (convective) transport current than in the RB case. Furthermore, enhanced radial velocity fluctuations and stronger mean Taylor vortices occur for case 2 and lead to broader PDFs of the convective current away from the boundary layer, which differ from the heat flux distributions in RB flow. This demonstrates that the mean rotation determines how well the transport characteristics of TC and RB flow are comparable.

The comparison presented here shows that for judiciously chosen pairs of parameters in RB and TC flow one can actually relate their transport properties in detail, both in the mean and in the fluctuations, thereby confirming the analogies between the twins of turbulence research [7] for a larger set of properties.

## Authors' contributions

H.J.B. carried out the simulations of Taylor–Couette flow, J.S. carried out the simulations of Rayleigh–Bénard flow, H.J.B. and J.S. performed the data analysis, H.J.B., B.E. and J.S. designed the study and drafted the manuscript. All authors read the manuscript and gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This work is supported by the Deutsche Forschungsgemeinschaft within the Research Unit FOR 1182.

## Disclaimer

Not applicable.

## Acknowledgements

We thank M. S. Emran and R. du Puits for scientific discussions at the beginning of this work. J.S. acknowledges computational resources provided by the John von Neumann Institute for Computing within Supercomputing Grant HIL09. H.J.B. and B.E. thank M. Avila for providing the code used for the TC simulations and acknowledge computational resources at the LOEWE-CSC in Frankfurt. The paper was written during a workshop at the Lake Arrowhead Conference Center, and B.E. and J.S. thank the Institute of Pure and Applied Mathematics (IPAM) of the University of California Los Angeles for financial support.

## Footnotes

One contribution of 14 to a theme issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

- Accepted September 8, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.