What rotation rate maximizes heat transport in rotating Rayleigh-Bénard convection with Prandtl number larger than one?
Yantao Yang, Roberto Verzicco, Detlef Lohse, Richard J.A.M. Stevens
WWhat rotation rate maximizes heat transport in rotating Rayleigh-B´enard convectionwith Prandtl number larger than one?
Yantao Yang
SKLTCS and Department of Mechanics and Engineering Science, BIC-ESAT,College of Engineering, and Institute of Ocean Research, Peking University, Beijing 100871, China
Roberto Verzicco
Physics of Fluids Group, Department of Science and Technology,MESA+ Institute, Max Planck Center Twente for Complex Fluid Dynamics,and J. M. Burgers Center for Fluid Dynamics, University of Twente, 7500 AE Enschede, The NetherlandsDipartimento di Ingegneria Industriale, University of Rome “Tor Vergata”, Via del Politecnico 1, Roma 00133, Italy andGran Sasso Science Institute - Viale F. Crispi 7, 67100 L’Aquila, Italy
Detlef Lohse and Richard J.A.M. Stevens
Physics of Fluids Group, Department of Science and Technology,MESA+ Institute, Max Planck Center Twente for Complex Fluid Dynamics,and J. M. Burgers Center for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands (Dated: April 16, 2020)The heat transfer and flow structure in rotating Rayleigh-B´enard convection are strongly influ-enced by the Rayleigh ( Ra ), Prandtl ( P r ), and Rossby ( Ro ) number. For P r (cid:38) Ra number regime. For Ra (cid:46) × the heat transfer at a given Ra and P r is highest at an optimal rotation rate, atwhich the thickness of the viscous and thermal boundary layer is about equal. From the scalingrelations of the thermal and viscous boundary layer thicknesses, we derive that the optimal rotationrate scales as 1 /Ro opt ≈ . P r / Ra / . In the low Ra regime the heat transfer is similar in aperiodic domain and cylindrical cells with different aspect ratios, i.e. the ratio of diameter to height.This is consistent with the view that the vertically aligned vortices are the dominant flow structure.For Ra (cid:38) × the above scaling for the optimal rotation rate does not hold anymore. It turnsout that in the high Ra regime, the flow structures at the optimal rotation rate are very differentthan for lower Ra . Surprisingly, the heat transfer in the high Ra regime differs significantly for aperiodic domain and cylindrical cells with different aspect ratios, which originates from the sidewallboundary layer dynamics and the corresponding secondary circulation. I. INTRODUCTION
Since the seminal experiments by Rossby [1], rotating Rayleigh-B´enard convection [2, 3], i.e. the buoyancy-drivenflow of a fluid layer heated from below and cooled from above and rotating about the central vertical axis, has beena model system to study the influence of rotation on heat transfer [4]. Improving our understanding of the influenceof rotation on heat transport is crucial from a technological point of view to better understand important industrialprocesses [5]. It is also essential to better understand the effect of rotation on relevant natural processes such as thethermohaline circulation in the oceans [6], atmospheric flows [7], trade winds [8], zonal flows in planets like Jupiter[9], and the effect of rotation on reversals of the Earth’s magnetic field [10].With the development of experimental techniques [11–26] and simulations [15, 20, 24, 27–42], significant progress onour understanding of rotating convection has been realized. Rotating Rayleigh-B´enard convection is characterized byseveral major flow transitions, which strongly influence the heat transport and flow structures [4]. As rotation is knownto have a stabilizing effect on fluid flow a particular intriguing phenomenon is the observation of a substantial heattransport enhancement at moderate rotation rates. The mechanism responsible for this heat transport enhancementis Ekman pumping [1, 14–16, 27, 28, 32, 33, 36, 43], i.e. due to the rotation, rising or falling plumes of hot or coldfluid are stretched into vertically aligned vortices that suck fluid out of the thermal boundary layers adjacent to thebottom and top plates. A better understanding of the transitions between these regimes, and the physics that dictatesthem, is of paramount importance to understand the convection phenomena described above.Zhong et al. [27] and Stevens et al. [28, 32] found in experiments and direct numerical simulations that the Rayleighnumber Ra and the Prandtl number P r , to be defined explicitly below, strongly influence the Ekman pumpingprocess. As a result, the heat transfer enhancement compared to the non-rotating case strongly depends on thesecontrol parameters. They found that at a fixed non-dimensional rotation rate of 1 /Ro , also to be defined below, theheat transport enhancement is highest for intermediate
P r . For lower
P r , the efficiency of Ekman pumping is limited a r X i v : . [ phy s i c s . f l u - dyn ] A p r R a Pr FIG. 1: Simulated Ra and P r combinations in a horizontally periodic domain. For each point a series of simulations for various1 /Ro is performed. Further details on the simulations can be found in the appendix. by the heat diffusing out of the vertically aligned vortices due to the high thermal diffusivity. For higher
P r thethermal boundary layer is much thinner than the viscous boundary layer, where the base of the vortices forms, andthis limits the amount of hot fluid that enters the vortices at the base. Furthermore, the effect of Ekman pumpingreduces with increasing Ra . The reason is the increase of the turbulent diffusion, which limits the ability of thevertically aligned vortices to transport heat effectively.In this work, we study the heat transfer and flow structures in rotating Rayleigh-B´enard convection as a functionof the main control parameters of the system. These control parameters are the Rayleigh number Ra = βg ∆ L / ( κν ),where β is the thermal expansion coefficient, g the gravitational acceleration, L and ∆ the distance and temperaturedifference between the bottom and top plates, respectively, and the Prandtl number P r = ν/κ , where ν and κ are thekinematic viscosity and the thermal diffusivity, respectively. The rotation rate Ω is non-dimensionalized in the form ofthe Rossby number Ro = (cid:112) βg ∆ /L/ (2Ω). As Ro varies as an inverse rotation rate, we indicate the non-dimensionalrotation rate as 1 /Ro in this work. Alternatively, the strength of the rotation can be characterized by the Ekmannumber Ek = ν/L Ω or the Taylor number
T a = (2 /Ek ) = Ra/ ( P rRo ) The heat transfer is given by the Nusseltnumber N u , which is the ratio between the convective and conductive heat flux. The Reynolds number Re measuresthe strength of the flow.The effects of the specific geometry of the domain will also be investigated in the current study. We run simulationswith horizontally periodic Cartesian domain and cylindrical cells with two different aspect ratios for a wide rangeof P r and Ra . Experimental data from Refs. [19, 21, 27, 37] are also included for comparison. It has long beenrecognized that domain geometry has strong influences on flow properties for (rotating) Rayleigh-B´enard convection,e.g. see [23, 44, 45]. The aspect ratio of the cylinder can alter the formation of secondary flows in the Ekman andStewartson layers [41, 46]. Recent studies further reveal that the boundary zonal flow close to the sidewall in a slendercylinder exhibits very rich structures and dynamics for both momentum and temperature fields [47, 48]. Here wewill focus on the heat transfer enhancement and its dependence on the domain geometry, i.e., periodic domains andcylinders with different aspect ratios.The remainder of the manuscript is organized as follows. In section II we describe the simulation methods used inthis study. To study the transition from the low to the high Ra regime, we performed simulations in a periodic anda cylindrical domain. In section III we discuss the observation of the low and the high Ra regime based on the heattransfer data obtained from simulations and corresponding experimental data published in literature [27]. In sectionIV we discuss the main flow features in the different regimes. The conclusions and an outlook to future work are givenin section V. II. METHOD
We perform direct numerical simulations by solving the three-dimensional Navier-Stokes equations within theBoussinesq approximation. We consider two different system geometries, namely a cylindrical and a Cartesian hor-izontally periodic domain. We use our in-house code, which has been extensively validated for Rayleigh-B´enardturbulence. The code employs a second-order finite-difference scheme with a fractional-time-step step method. Forsimulations in the cylinder domain, the code is the same as in our previous studies, see e.g. Refs [32, 33]. For sim-ulations in the periodic domain at high Ra and P r , the multiple-resolution method for scalar turbulence is used toimprove the computational efficiency [49]. In this method the momentum equations are solved on a base mesh whilethe scalar field is solved on a refined mesh.Constant temperature and no-slip boundary conditions at the bottom and top plates are employed. For thecylindrical domain, we use an adiabatic sidewall and we consider
P r = 4 .
38 in a Γ = 1 cylindrical cell up to Ra = 1 . × . These simulations show excellent agreement with previous measurements performed by Zhong andAhlers [27]. From previous work we have datasets for Γ = 1 / P r = 4 .
38 and for Ra up to Ra = 4 . × )and Γ = 1 (for Ra = 10 and various P r ) available. For the Cartesian periodic domain we performed simulationsfor various Ra and P r combinations as indicated in figure 1. For each pair of Ra and P r we gradually increasedthe rotation rate 1 /Ro from zero to a strong enough rotation to ensure that the heat transfer is lower than for thenon-rotating cases. We used a horizontal domain width that is much larger than the typical size of flow structures inthe bulk to ensure that the periodic boundary condition is appropriate. As the horizontal length scale of the verticallyaligned vortices decreases with increasing Ra and P r , we can reduce the domain width accordingly to save computingresources. Moreover, for most rotating cases the domain size is more than 10 times larger than the most unstablewavelength for convection instability which scales asymptotically as L c = 4 . Ek / [41]. It has been shown that suchdomain size for rotating RB is enough to assure the convergence of the Nusselt number [41]. Further details aboutthe simulations are summarized in the appendices. III. OVERVIEW
Figure 2 shows the heat transfer enhancement with respect to the non-rotating case, i.e.
N u/N u , versus thenon-dimensional rotation rate 1 /Ro for various cases. Figure 2a shows that we obtain excellent agreement betweensimulations and the experiments performed by Zhong and Ahlers [27]. Similarly to previous studies, we find thatthe heat transfer first increases for moderate rotation rates before it quickly decreases for strong rotation rates. Theoptimal rotation rate is defined as the rotation rate 1 /Ro opt for which the heat transfer ( N u ) for that Ra , P r , andaspect ratio Γ is maximal. For Ra (cid:46) × and P r = 4 .
38 the optimal rotation rate increases with increasing Ra before it rapidly decreases with increasing Ra when Ra (cid:38) × . This indicates that the flow dynamics thatdetermine the optimal rotation rate are different in the low and the high Ra regime. Figures 2b and c show resultsfrom the periodic domain simulations for P r = 4 .
38 and
P r = 100 and various Ra . A direct comparison of theheat transfer data for P r = 4 .
38 in figure 3 reveals that for Ra = 10 the heat transfer behaves almost identicallyin a periodic domain and cylinders with different aspect ratios, while there are surprisingly significant differences for Ra = 10 .To reveal the transition between the low and the high Ra regime more clearly, we plot the optimal rotation rate asa function of Ra and P r for all available cases in figure 4. To determine the optimal rotation rate from simulationdata, we perform a second-order polynomial fit around the rotation rate at which the heat transport is highest anddetermine the optimal rotation rate using that polynomial fit. For experimental results, we first determine the largestheat transfer enhancement
N u max − N u , where N u max is the highest heat flux obtained for fixed
P r and Ra anddifferent Ro . Subsequently, we determine the optimal rotation rate from a polynomial fit to all data points for which N u − N u > . N u max − N u ). From figure 4(a) the existence of a low and a high Ra regime and their differentfeatures are immediately apparent. For P r = 4 .
38 and Ra (cid:46) × the optimal rotation rate is well described bythe following scaling (1 /Ro ) opt ≈ . P r / Ra / . (1)The derivation of this scaling law will be given in section IV A. For Ra (cid:46) × the data for the different domaingeometries is well described by the above scaling. However, we find that in contrast to the above prediction theoptimal rotation rate decreases with increasing Ra for Ra (cid:38) × .Although for P r = 4 .
38 the transition to the high Ra number regime is very pronounced, such a transition cannot be seen clearly in the high P r number data. Studying the transition at higher
P r would require more high Ra number simulations. However, unfortunately, such simulations are too time-consuming to be performed. In thefollowing discussion, for the low Ra regime, we include all cases with Ra ≤ × and all P r ’s. While for the high Ra regime, we will limit ourselves to P r = 4 .
38. We explain the different behaviors of the optimal rotation rate inthe low and high Ra regime, and the physical mechanism behind it. -1 (a) N u / N u × × -1 (b) N u / N u × × -2 -1 (c) N u / N u FIG. 2:
Nu/Nu versus 1 /Ro for different Ra as indicated in the legend. (a) Experimental (open symbols; [27]) and simulation(solid symbols) results for P r = 4 .
38 and in a Γ = 1 cylinder. Panel (b) and (c) show simulation results for
P r = 4 .
38 and
P r = 100, respectively, obtained in a periodic domain. -1 (a) N u Periodic Γ =1 Γ =0.5 6070 10 -1 (b) N u Periodic Γ =1 Γ =0.5 FIG. 3: Comparison of Nu obtained in a periodic and in cylindrical domains with Γ = 1 / P r = 4 .
38 at (a) Ra = 10 and (b) Ra = 10 . The Γ = 1 / IV. OPTIMAL HEAT TRANSFER IN THE LOW AND HIGH Ra REGIMEA. The low Ra regime ( Ra ≤ × ) It has been conjectured that the heat transfer reaches a maximum when the viscous and thermal boundary layershave a similar thickness [12, 16, 17, 38, 50–52]. The importance of this boundary layer transition has been recognizedin several previous works. For example, King et al. [53] used it to derive a scaling law to describe the transition tothe geostrophic convection regime [39, 50, 53–55]. In rotating Rayleigh-B´enard convection the Ekman boundary layerthickness scales as [16, 24, 38, 53, 56] λ u /L ∼ /T a / ∼ (1 /Ro ) − / P r / Ra − / . (2) (a) ( / R o ) op t Ra Nums PeriodicNums Γ =0.5 Nums Γ =1.0Exps Γ =1.0 Exps Γ =0.510 (a) ( / R o ) op t Ra Nums PeriodicNums Γ =0.5 Nums Γ =1.0Exps Γ =1.0 Exps Γ =0.5 10 (b) ( / R o ) op t Ra Pr=4.38Pr=6.4 Pr=25Pr=10010 (b) ( / R o ) op t Ra Pr=4.38Pr=6.4 Pr=25Pr=10010 (b) ( / R o ) op t Ra Pr=4.38Pr=6.4 Pr=25Pr=10010 (b) ( / R o ) op t Ra Pr=4.38Pr=6.4 Pr=25Pr=10010 (b) ( / R o ) op t Ra Pr=4.38Pr=6.4 Pr=25Pr=100
FIG. 4: The optimal rotation rate for different Ra for (a) P r = 4 .
38 and different geometries, i.e. a periodic domain andcylindrical domains with Γ = 0 . P r in a periodic domain. The vertical line marks thetransition from the low to high Ra regime. The dashed lines indicate the boundary layer scaling law 1 /Ro ≈ . P r / Ra / .The experimental results are from Refs. [19, 21, 27] and the Γ = 1 / The thickness of the thermal boundary layer is related to the scaling of the
N u number. For non-rotating convectionthe
N u number as function of Ra and P r is well described by the unifying theory for thermal convection [57–59]. For intermediate
P r and before the onset of the ultimate regime [2] one obtains that the thermal boundary layer thickness λ θ can be approximated as λ θ /L ∼ (1 /Ro ) P r Ra − / . (3)Obviously, this relation is a simplification that is used in the analysis [16, 24, 38, 53], which does not do full justiceto the Rayleigh-B´enard dynamics as for non-rotating convection and P r ∼ γ in N u ∼ Ra γ depends on Ra as described by the unifying theory [57–59]. For intermediate P r and before the onset of the ultimateregime the effective scaling exponent typically is in the range 0 . − .
32 [2], which is close to the approximation usedabove, and sufficiently accurate to not affect the discussion presented below.Combining equations (2) and (3) gives that the ratio between the boundary layer thicknesses scales as λ θ λ u ∼ (1 /Ro ) / P r − / Ra − / . (4)This is equivalent to λ θ /λ u ∼ RaEk / [53].Figure 4 shows that this scaling indeed captures the position of the optimal rotation rate in the low Ra regime. Tofurther verify the assumptions in the above analysis, we show the viscous and thermal boundary layer thickness as afunction of 1 /Ro for Ra = 10 and P r = 4 .
38 in figure 5a. The viscous boundary layer thickness λ u is determinedby the peak location of the root-mean-square (rms) value of the horizontal velocities, and the thermal boundarylayer thickness λ θ by the height of the first peak location of the temperature rms profile. The figure shows that theviscous boundary layer thickness decreases monotonically with increasing rotation rate. The thermal boundary layerthickness first decreases slightly with increasing rotation before it increases rapidly with increasing rotation. Theviscous and thermal boundary layer thicknesses are about equal at the optimal rotation rate of 1 /Ro ≈
10. Figure5b shows that in the low Ra regime the highest heat transfer is obtained for λ θ /λ u ≈ . λ θ /λ u ≈ .
0. For the cylindricalcase, the viscous boundary layer thickness was determined as twice the height where the horizontally averaged valueof (cid:15) (cid:48)(cid:48) u := (cid:104) u · ∇ u (cid:105) h is highest, as in Ref. [33]. Thus in the low Ra regime the optimal rotation rate is determinedby the ratio of the thermal and viscous boundary layer thickness. That the scaling relation (1 /Ro ) ∝ P r / Ra / is appropriate to capture the peak is further illustrated in figure 6. Figure 6a shows the dependence of N u/N u on1 /Ro for different combinations of Ra and P r obtained in a periodic domain. The onset of heat transfer enhancementis independent of Ra and P r for
P r > /Ro , P r , and Ra . However, figure 6b shows that the peak for N u/N u nicely collapses for allcombinations of Ra and P r when the above scaling is applied. Figure 6c shows that this scaling argument also worksfor data obtained in a cylindrical cell. The data in this last figure are from Ref. [60] and have been supplemented herewith data from additional simulations to ensure that smooth curves can be plotted for all
P r . -2 -1 -1 (a) B L t h i c kn e ss λ θ λ u -1 (b) N u / N u λ θ / λ u (4.38,10 )(4.38,10 )(6.4,10 )(6.4,10 ) (25,10 )(25,10 )(100,10 )(100,10 ) λ θ / λ u (0.7,10 )(3.05,10 )(6.4,10 )(20,10 )(55,10 ) FIG. 5: (a) The thermal λ θ and viscous λ u boundary layer thickness versus 1 /Ro for P r = 4 .
38 and Ra = 10 obtained in aperiodic domain. (b) Nu/Nu as function of λ θ /λ u for eight different combinations of Ra and P r in the low Ra regime; notethat the optimal rotation rate is found at λ θ /λ u ≈ .
8, see the dashed vertical line. (c) Same as panel (b), but now for dataobtained in a cylindrical domain. -2 -1 (a) N u / N u (4.38,10 )(4.38,10 )(6.4,10 )(6.4,10 ) (25,10 )(25,10 )(100,10 )(100,10 ) -2 -1 (b) Ro -1/2 Pr -1/4 Ra -1/12 (4.38,10 )(4.38,10 )(6.4,10 )(6.4,10 ) (25,10 )(25,10 )(100,10 )(100,10 ) -2 -1 (c) Ro -1/2 Pr -1/4 Ra -1/12 (0.7,10 )(3.05,10 )(6.4,10 )(20,10 )(55,10 ) FIG. 6:
Nu/Nu as function of (a) 1 /Ro and (b) Ro − / P r − / Ra − / , which determines the locations of the optimal rotationrate in the low Ra regime, for eight combinations of Ra and P r . The vertical line in (b) and (c) marks the location of themaximum heat transfer at (1 /Ro ) / P r − / Ra − / ≈ .
35. (a) and (b) show the cases in periodic domain, while (c) in cylinderdomain, respectively.
B. The high Ra regime ( Ra (cid:38) × ) According to the boundary layer scaling arguments discussed above the optimal rotation rate should increase withincreasing Ra . However, figure 4 shows that for P r = 4 .
38 the high Ra regime sets in around Ra = 5 × . In thishigh Ra regime the optimal rotation rate decreases with increasing Ra , which shows that the boundary layer scalingargument cannot hold anymore, and the flow dynamics at the optimal rotation rate must be different. To investigatethis transition, we show in figure 7 the volume renderings of the temperature for Ra = 10 and Ra = 2 . × , which,according to figure 4, are respectively in the low and the high Ra regime. The difference between the two cases isvery distinct. For Ra = 10 vertically aligned vortices, which extend over almost the full domain height, are visible.For Ra = 2 . × these vortices are much less pronounced and much shorter. Clearly, the flow structures at theoptimal rotation rate are much more coherent in the low Ra regime than in the high Ra regime.In an attempt to quantify the above observation we characterize the coherence of the flow structures by calculatingthe following cross-correlation function: C ( δz ) = (cid:104) w ( x, y, λ u ) w ( x, y, λ u + δz ) (cid:105) x,y (cid:104) ( w ( x, y, λ u )) (cid:105) x,y , (5)where (cid:104)·(cid:105) x,y indicates the average in horizontal direction. As we are interested in the vertically aligned vortices, wecalculate the correlation using the horizontal plane at the viscous boundary layer height ( z = λ u ), where the base (a) (b) FIG. 7: Volume renderings of the temperature field at the optimal rotation rate for simulations with
P r = 4 .
38 at (a) Ra = 10 and (b) Ra = 2 . × . The colormap in both panels is identical. The figure shows that the flow structure at the optimalrotation rate is very different in the low and high Ra regime. z C z C (a) (b) FIG. 8: Correlation C as function of δz for different rotation rates in periodic domains in the (a) low ( Ra = 10 and P r = 4 . Ra regime ( Ra = 2 . × and P r = 4 . /Ro . The lines for eachsuccessive 1 /Ro are shifted upward by 1 for visibility. The black line indicates the optimal inverse Rossby number. of the vortices forms, as reference. Figure 8 shows the correlation C as a function of the distance from the viscousboundary layer δz for different rotation rates in the low and high Ra regime. For all cases the correlation C firstincreases from 1 for δz = 0 to some maximum value before it decreases below 1 further away from the boundary layer.The reason for the maximum is that the vertical velocities are higher at some distance above the viscous boundarylayer height than at z = λ u .We take the δz value at which the maximum occurs as a measure for the height of the vertically aligned vorticesand the magnitude C m of the peak of C ( δz ) as a measure of the coherence of the vortices. Figure 9 shows that thevariation of C m with 1 /Ro is similar for both Ra , which suggests that the flow coherence mainly depends on therotation rate. However, the height of the plumes or vertically aligned vortices, which is indicated by δz m , is verydifferent in the low and the high Ra regime. In the low Ra regime the height of the vertically aligned vortices increasesfor smaller 1 /Ro than in the high Ra regime. For Ra = 2 . × the height at which the vertical coherence is highestis similar for the non-rotating case and the optimal rotation rate. In contrast, the vertical coherence at the optimalrotation rate is significantly higher than for the non-rotating case at Ra = 10 .It is well known that strong rotation not only induces vertically aligned vortices but also suppresses the verticalfluid motion. This is illustrated in figure 10a, which shows that the vertical Reynolds number Re z , defined by therms of vertical velocity, decreases with increasing rotation rate. The reduction of the vertical velocity between thenon-rotating case and the optimal rotation rate is stronger for Ra = 10 than for Ra = 2 . × . Figure 10b shows -2 -1 -1 (a) δ z m Ra=10 Ra=2.3 × -1 (b) C m Ra=10 Ra=2.3 × FIG. 9: The (a) location δz m and (b) magnitude C m for the peak of the correlation function C ( δz ) versus 1 /Ro for P r = 4 . Ra in periodic domains. The optimal rotation rate for Ra = 10 and Ra = 2 . × is indicated by the solidand dashed line, respectively. -1 (a) R e z Ra=10 Ra=2.3 × -1 (b) R e z / R e h Ra=10 Ra=2.3 × FIG. 10: (a) The Reynolds number defined by the rms of the vertical velocity, and (b) the ratio of the vertical and horizontalrms velocities as function of 1 /Ro for
P r = 4 .
38 and two different Ra in periodic domains. The optimal rotation rate for Ra = 10 and Ra = 2 . × is indicated by the solid and dashed line, respectively. -2 -1 -1 (a) δ z m periodic cylinder Γ =1 1.02.03.0 10 -1 (b) C m periodic cylinder Γ =1 FIG. 11: A comparison of (a) the location δz m and (b) magnitude C m for the peak of correlation function C ( δz ) versus 1 /Ro obtained from simulations performed in a periodic and cylindrical domain (Γ = 1) for P r = 4 .
38 and Ra = 2 . × . For thecylinder domain the flow data from the region close to the sidewall is excluded from the analysis. that, as a result, the ratio of the vertical to horizontal Reynolds number at the optimal rotation rate is larger in thehigh Ra regime than in the low Ra regime.The above analysis reveals that in the low Ra regime the maximum heat transfer is observed when there is a strongcoherence in the vertically aligned vortices, and in this regime the boundary layer structure controls the optimalrotation rate. In contrast, in the high Ra regime, the maximum heat transfer occurs when the vertical motion isstronger than the horizontal motion, and the ratio between the viscous and thermal boundary layer thickness doesnot determine the optimal rotation rate anymore.Figure 4 shows that periodic domain and cylindrical cells with different aspect ratios generate similar heat transportin the low Ra regime. In contrast, it turns out that in the high Ra regime the transport depends very strongly onthe specific geometry of the domain. Previous studies demonstrated the importance of finite size effects in rotatingRayleigh-B´enard by showing that the rotation rate at which heat transport sets in, and the formation of secondaryflows in the Ekman and Stewartson layers [41, 46], depends on the aspect ratio of the domain. However, the extremedependence of the heat transport on the domain geometry is surprising considering the small horizontal length scaleof the vertically aligned vortices that are a dominant feature of the flow. In figure 11 we compare the results for thecorrelation function (equation 5) for the periodic and the cylindrical domain in an attempt to explain the origin ofthis difference. To eliminate the effects of the sidewall and the secondary circulation, we excluded the sidewall region0 . < r/L < . δz m and C m are verysimilar for the cylindrical and the periodic domain over the whole range of rotation rates. Thus in the bulk region,the characteristic height and the coherence of flow structures are hardly affected by the sidewall. Hence, we concludethat the difference in the heat transfer is strongly related to the sidewall boundary layer or an effect of the secondaryflow circulation. V. CONCLUSIONS
To summarize, we systematically studied the heat flux enhancement in rotating Rayleigh-B´enard convection for awide range of control parameters. Based on the available data, it becomes clear that there is a low and a high Ra regime in rotating Rayleigh-B´enard. In the low Ra regime, the bulk is dominated by long vertically aligned vortices,and due to the strong vertical coherence a pronounced heat transport enhancement compared to the non-rotatingcase is observed. The optimal rotation rate occurs when the viscous and thermal boundary layer thickness are aboutequal. According to this argument the optimal rotation rate scale as 1 /Ro opt ≈ . P r / Ra / , which is equivalentto Ek opt ≈ . Ra / , where the numerical values have been determined by fitting experimental and simulation data.In the high Ra regime, the optimal rotation rate decreases with increasing Ra . This means that the trend isopposite to the one observed in the low Ra regime, which implies that the optimal rotation rate is not obtained whenthe viscous and thermal boundary layer thickness are similar. In the low Ra regime the flow structure at the optimalrotation rate is characterized by pronounced vertically aligned vortices. However, this is not the case in the high Ra regime. Instead, we find that in the high Ra regime the ratio of the vertical to horizontal velocity is much larger atthe optimal rotation rate than in the low Ra regime. Furthermore, we demonstrate that the domain geometry hasa surprisingly pronounced influence on heat transport for higher Ra . Our current analyses suggest that the sidewallboundary layer in the cylindrical domain is responsible for the difference, since the flow structure in the bulk showalmost the same behavior for the periodic domain and the aspect ratio 1 cylinder. Our analysis shows that the flowstructure in the bulk is almost the same in a periodic domain and the central region of an aspect ratio 1 cylinder. Thissuggests that the sidewall Stewartson boundary layers cause the difference between the simulations in the periodicand cylindrical domain.Finally, although we discuss various aspects for the transition from the low Ra to the high Ra regime in rotatingRayleigh-B´enard at P r = 4 .
38, many questions on this new transition remain. For example, the differences betweenthe low and high Ra regimes are not fully explored, especially for high P r . While the data and a simplified analysissuggest that the maximum heat transfer in the low Ra number regime is obtained when the thermal and viscousboundary layer thickness is equal, it is not clear what physical mechanism determines the optimal rotation rate inthe high Ra number regime. Another aspect that needs further clarification is the role of the sidewall Stewartsonboundary layers. It is namely unclear why the heat transport does not depend on the system geometry in the low Ra number regime, while there is a strong geometry dependence in the high Ra number regime. These and other aspectsneed further investigation to understand better the flow dynamics in the newly discovered high Ra number rotatingRayleigh-B´enard regime. Acknowledgements:
Appendix A: Details simulations performed in cylindrical domain
TABLE I: Direct numerical simulations in cylindrical domain for
P r = 4 .
38 and Γ = 1, matching the experiments by Zhongand Ahlers [19, 27]. The columns from left to right indicate the Ra number, the used numerical resolution in azimuthal, radial,and axial direction, the number of points in the thermal boundary layer (for the non-rotating case), the 1 /Ro range consideredin the simulations, and the number of considered cases. Ra N θ × N r × N z N θ BL /Ro [min − max] Cases1 . × × ×
192 14 0 −
10 141 . × × ×
256 15 0 −
10 141 . × × ×
512 22 0 −
10 142 . × × ×
512 19 0 −
10 131 . × × × − . Appendix B: Details simulations performed in periodic domain
In this section, we give the numerical details of our simulations of the horizontally periodic domain. In each followingtable the columns from left to right show the rotation rate 1 /Ro , the aspect ratio Γ (width/height) of the domain,the number of grid points (and the refinement factor) for the horizontal N x ( m x ) and vertical direction N z ( m z ), theNusselt number N u , the Reynolds number Re defined by the rms value of velocity magnitude, and the viscous λ u and thermal λ θ boundary layer thickness, respectively. The same width and discretization is used in both horizontaldirections. Each table shows the simulation data for one combination of Ra and P r . TABLE II:
P r = 4 .
38 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE III:
P r = 4 .
38 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE IV:
P r = 4 .
38 and Ra = 5 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE V:
P r = 4 .
38 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE VI:
P r = 4 .
38 and Ra = 2 . × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE VII:
P r = 6 . Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE VIII:
P r = 6 . Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE IX:
P r = 6 . Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE X:
P r = 25 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE XI:
P r = 25 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE XII:
P r = 25 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE XIII:
P r = 100 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE XIV:
P r = 100 and Ra = 1 × /Ro Γ N x ( m x ) N z ( m z ) Nu Re λ u /L λ θ /L . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE XV:
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