Heat transport in Rayleigh-Benard convection and angular momentum transport in Taylor-Couette flow: a comparative study
HHeat transport in Rayleigh–B´enard convection and angular momentumtransport in Taylor–Couette flow: a comparative study
Hannes Brauckmann, Bruno Eckhardt,
1, 2 and J¨org Schumacher Fachbereich Physik, Philipps-Universit¨at Marburg, D-35032 Marburg, Germany J. M. Burgerscentrum, Delft University of Technology, 2628 CD Delft, The Netherlands Institut f¨ur Thermo- und Fluiddynamik, Technische Universit¨at Ilmenau,Postfach 100565, D-98684 Ilmenau, Germany (Dated: September 7, 2018)
Abstract
Rayleigh–B´enard convection and Taylor–Couette flow are two canonical flows that have many propertiesin 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 laminarvalues, coincide. We study turbulent Rayleigh–B´enard convection in air at Rayleigh number Ra = 10 andTaylor–Couette flow at shear Reynolds number Re S = 2 × for two different mean rotation rates butthe same Nusselt numbers. For individual pairwise related fields and convective currents, we compare theprobability density functions normalized by the corresponding root mean square values and taken at differentdistances from the wall. We find one rotation number for which there is very good agreement between themean profiles of the two corresponding quantities temperature and angular momentum. Similarly, there isgood agreement between the fluctuations in temperature and velocity components. For the heat and angularmomentum currents, there are differences in the fluctuations outside the boundary layers that increase withoverall 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 revealsthe effects of rotation on the transport. a r X i v : . [ phy s i c s . f l u - dyn ] S e p . INTRODUCTION Convection in layers of fluids heated from below and cooled from above (Rayleigh–B´enard orRB flow) and the flow between two rotating cylinders (Taylor–Couette or TC flow) are among thecanonical flows in fluid mechanics. Studies of their stability properties and the manner in which thelaminar profiles give way to more structured and complicated flows have provided much insight intothe transition to turbulence with linear instabilities [8, 17]. The behavior well above the onset ofturbulence has also been investigated starting with the experiments by Wendt [28]. Many differentflow regimes that are not yet fully explained or explored have been described [16, 23]. It was realizedearly on that despite the differences in the driving forces, there are many similarities, and it is helpfulto draw analogies and to compare the properties of both flows [18]. The intimate relations betweenthe two flows have led Busse [6] to characterize them as the twins of turbulence .A formal analogy between RB and TC flow (and pipe flow as well) was developed and describedin Eckhardt et al. [12, 13] (see also Bradshaw [2] for an earlier approximate relation and Dubrulle &Hersant [11] for a similar analogy). The analogy identifies pairs of equations that describe the totalenergy dissipation and the global transport of heat or angular momentum, respectively, in the twoflows. The equations allow one to relate transport properties, dimensionless parameters and otherquantities, and have been used in particular to study scaling relations in fully developed turbulentflows [16]. The similarity in the equations suggests that a more detailed comparison between the twoflows should be possible.We here explore this option within direct numerical simulations (DNS). We describe the difficultiesone has to overcome in identifying corresponding parameters, and present case studies where detailedcomparisons are possible. In particular, we will compare the turbulent transport currents with respectto their statistical properties. Furthermore, we can relate components of the involved turbulent fieldsto each other and compare their statistical fluctuations at different distances from the walls. The focusof our study is on the general ideas and an illustration for a few examples, but not on a comprehensivestudy for all parameter values. Specifically, we will take one set of data for RB flow and compare it toTC flow cases at two different rotation numbers, which allows us to study the effect of rotation. Thedata are taken from well-resolved DNS of both flows at moderate Rayleigh and Reynolds numbers.The outline of the manuscript is as follows. In section II we present the balance equations, thenumerical methods and discuss the analogy. In section III the choice of corresponding parametersfor the comparison is explained. In section IV the area-averaged mean currents and their probabilitydensity functions (PDFs) as well as other pairwise related properties at different distances from thewall are analyzed. We conclude the work with a short discussion of the particular structures of the2onvective currents and a summary in section V.
II. RELATIONS FOR TRANSPORT CURRENTS AND DISSIPATION RATES
RB flow is modeled by the three-dimensional Boussinesq equations for the velocity field u and thetemperature field T [1, 9]. The equations are solved using the Nek5000 software [21], a spectral elementmethod [15, 25]. The physical system is characterized by the imposed temperature difference betweenbottom and top plates, ∆, the height d of the domain, and the free-fall velocity U f = √ gα ∆ d withthe 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 isconfined to a cylinder with insulating sidewalls.The mean heat flux across the layer, i.e., in the z -direction, is given by J T = (cid:104) u z T (cid:105) − κ ∂ (cid:104) T (cid:105) ∂z = Nu T J T,lam , (1)with J T,lam the purely diffusive heat flux below the onset of convective motion, J T,lam = κ ∆ d . (2)Here, (cid:104)·(cid:105) = (cid:104)·(cid:105) A,t denotes an average over horizontal planes at fixed height and over time. Equation(1) already contains the definition of the Nusselt number Nu T which measures the heat transportrelative to the laminar situation. A second relevant equation is that for the mean kinetic energydissipation rate of the velocity field (cid:15) u . It is obtained by multiplying the momentum balance of theNavier–Stokes–Boussinesq equations with the velocity u and integrating over the volume and overtime, (cid:15) u = ν d Pr − Ra ( Nu T − . (3)Here, Ra = αgd ∆ / ( κν ) is the Rayleigh number, the second dimensionless parameter. The data setwhich we use for the comparisons is obtained in a closed cylindrical cell with a unity aspect ratio(diameter=height) at Ra = 10 and Pr = 0 . r and r of the inner and outer cylinder, which rotatewith the angular velocities ω and ω , respectively. The flow between the cylinders is governed bythe 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 aspectral method [20]. In TC flow, the gap width d = r − r and the velocity difference between thecylinders U = 2 r r ( r + r ) − ( ω − ω ) (calculated in a rotating frame of reference [5, 10]) serve ascharacteristic scales for lengths and velocities. We choose a system height of 2 d so that one pair of3aylor vortices fits into the computational domain. Thus, the diameter of a single Taylor vortex issimilar to that of the large-scale circulation in RB flow with aspect ratio one. Further details of thesimulation procedure are discussed in references [3, 5].For the derivation of expressions in TC flow that correspond to (1) and (3) in RB flow, we startwith the azimuthal velocity u ϕ . Averaging the ϕ -component of the Navier–Stokes equation over timeand over cylinders at fixed radii r between the positions of the inner ( r ) and outer ( r ) cylinder, onefinds [12, 13] J ω = r (cid:18) (cid:104) u r ω (cid:105) − ν ∂ (cid:104) ω (cid:105) ∂r (cid:19) = Nu ω J ω,lam , (4)with the angular velocity ω = u ϕ /r and J ω,lam the angular momentum flux in the laminar case, J ω,lam = ν r r r a d ( ω − ω ) . (5)Here, r a = ( r + r ) / J ω is independent of the radius andconserved in time. Physically, it corresponds to the torque needed to keep the cylinders in motion;it corresponds naturally to the heat transport (1) in RB flow, which is why we also introduced aNusselt number Nu ω corresponding to Nu T . Similarly, one can multiply the Navier–Stokes equationwith the velocity u and average over volume and time to obtain the mean kinetic energy dissipationrate, corresponding to (3). However, the dissipation associated with the laminar profile has to betaken out, so that we are led to consider (cid:15) u = (cid:15) − (cid:15) lam = ( ω − ω ) r a d J ω,lam ( Nu ω −
1) = ν d σ − Ta ( Nu ω − . (6)with the geometric parameter σ = r a / ( r r ) denoted as quasi-Prandtl number. The dimensionlessTaylor number is defined as Ta = σd r a ( ω − ω ) /ν . Furthermore, the radius ratio is denoted by η = r /r and the specific angular momentum is defined by L = ru ϕ .To separate the influences of shear and rotation, we adopt the parameters introduced by Dubrulleet al. [10]. With the Reynolds numbers Re = r ω dν and Re = r ω dν . (7)for the inner and outer cylinders, respectively, we form the shear Reynolds number and the rotationnumber Re S = 21 + η ( Re − η Re ) and R Ω = (1 − η ) Re + Re Re − η Re . (8)The relation to the Taylor number is given by Ta = σ Re S .A comparison between (3) and (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 Taylor4 . . . . . . R Ω N u ω (a) . . . . . . R Ω . . . . . E L S C u / E u (b) FIG. 1. (a) Variation of the Nusselt number with the system rotation R Ω for TC flow with η = 0 .
99 and Re S = 2 × (circles). (b) Ratio of the energy contained in the large-scale circulation E LSCu to the energyof the total cross-flow E u as given in eq. (9). The horizontal lines in (a,b) indicate the corresponding valuesfrom RB convection with Pr = 0 . Ra = 10 (solid) and Ra = 5 × (dashed). The four possible TCdata points for the comparison (with Nu ω close to a RB Nusselt number) are marked as open symbols. number Ta . However, there are various reasons why this is not sufficient. For example, a directcomparison between (3) and (6) suggests equality of the combinations Pr − Ra and σ − Ta , only anddoes not relate Ra and Ta directly. Moreover, TC flow has two Reynolds numbers, and the Taylornumber 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 [3, 19, 22, 24, 27].Similarly, critical values for the onset of instability are given by Ra c = 1708 [8] for RB flow and by Ta c = 1708 / [ R Ω (1 − R Ω )] [10] for TC flow (in the limit η → Ta = Re S ), again highlightingthe significance of the rotation number. We therefore have to look for alternatives on how to relatethe two flows. III. CHOICE OF REFERENCE POINT FOR COMPARISON
In the following, we discuss how the reference point for the one-to-one comparison is chosen. Asdescribed 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 layerthickness and hence the mean profiles. Similarly, the Reynolds stresses, when normalized by Nu J lam ,5hould fluctuate with mean value 1 in regions where the viscous contributions to the transport aresmall. This allows for an absolute comparison of probability density functions since the dimensionlessversion with Nu scaled out has the same mean and, as we will show for one of the cases here, also thesame variance.We also have to select the curvature parameter η in TC flow, which has no counterpart in RBflow because the heated and cooled plates are planar. We therefore take η = 0 .
99 since the curvatureeffects disappear for η -values close to 1. Finally, the mean system rotation in TC flow which is definedby R Ω has to be selected for the direct comparison. Again, an analogous parameter is missing inRB convection. In figure 1(a) a curve Nu ω ( R Ω ) at η = 0 .
99 is shown for Re S = 2 × . A Nusseltnumber Nu ω that is comparable to the RB flow value of Nu T = 16 . Ra = 10 (solid horizontalline) was obtained for R Ω = 0 .
023 and R Ω = 0 .
241 (open circles). These two runs will be denotedas case 1 and case 2, respectively, and will be used to study the effects of the rotation number. Inboth cases the cylinders are co-rotating with angular velocity ratios µ = ω /ω = 0 .
40 and µ = 0 . R Ω = 0 .
023 and R Ω = 0 . Ra = 5 × . For thislower Ra the RB and TC flows differ noticeably, so that we will subsequently focus on the cases 1and 2 only. We furthermore note that in the first case the relative distance to the linear instability Ta / Ta c ≈ . × is close to the corresponding RB value Ra / Ra c ≈ . × for Ra = 10 ,whereas in the second case the ratio Ta / Ta c ≈ . × is much higher.Figure 1(b) shows the ratio E LSCu E u = (cid:104)(cid:104) u r (cid:105) ϕ,t + (cid:104) u z (cid:105) ϕ,t (cid:105) r,z (cid:104) u r + u z (cid:105) V,t , (9)of the energy contained in the mean vortical motion to the energy of the total cross-flow. It measuresthe relative strengths of temporally and streamwise-averaged Taylor vortices, which are analogousto the large-scale circulation in RB flow. The vortex strength varies with rotation, and the curveshows that case 2 is more strongly dominated by the large-scale vortices than case 1. In RB flowwith Ra = 10 , the corresponding energy ratio of approximately 0 . δ T = − ∆ / (2 ∂ z (cid:104) T (cid:105)| z =0 ,d ) = d/ (2 Nu T ) in RB flow,we define the boundary layer thicknesses at the inner cylinder ( r = r ) δ ω = ∆ ω − ∂ r (cid:104) ω (cid:105)| and δ L = ∆ L − ∂ r (cid:104)L(cid:105)| (10)for the angular velocity and angular momentum profiles in TC flow with the total differences ∆ ω = ω − ω and ∆ L = L − L . In the low-curvature case η = 0 .
99 analysed here, we also have δ ω ≈ . . . . . . R Ω . . . . δ / d FIG. 2. The boundary layer thickness δ L which is based on the slope of the angular momentum profile at thewall is displayed versus R Ω . The corresponding thermal boundary layer thickness δ T = d/ (2 Nu T ) of the RBflow is indicated by the horizontal lines. Parameter values are η = 0 .
99 with Re S = 2 × for TC flow and Pr = 0 . Ra = 5 × (dashed) and Ra = 10 (solid) for RB convection. The four possible TC datapoints for the comparison are again marked as open symbols. d/ (2 Nu ω ), whereas such a relation does not exist for δ L . However, for strongly co-rotating cylinders δ ω overestimates the width of the boundary layer region since then the angular velocity profiles havea significant slope in the bulk [3, 22]. As the angular momentum profile generally becomes almostflat in the bulk [5], the thickness δ L provides a better approximation to the size of the boundary layerregion and will therefore be used here. In figure 2 the boundary layer thickness δ L is plotted for thesame data as in figure 1(a). It is observed that the boundary layer thickness of case 1 matches almostperfectly with the thermal boundary layer thickness δ T of the RB flow at Ra = 10 . In case 2, thedifferences in the thickness scales are larger; here, the thickness δ L is smaller than δ ω ≈ δ T = d/ (2 Nu T )since the angular velocity profile is not flat in the central region. As we will see in the following, thesedifferences will affect the statistical properties of the TC flows and thus the agreement with RB flow. IV. STATISTICAL PROPERTIESA. 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 L = ru ϕ . The upper row displays the comparison with the TCflow at the first local maximum at R Ω = 0 .
023 (case 1) (see figure 1a). The lower row repeats this7 . . . . . . (a) h T i / ∆( h ω i − ω ) / ∆ ω ( hLi − L ) / ∆ L . . . . (b) . . . . . . . (c) θ rms / ∆1 . L ′ rms / ∆ L . . . z/d , y/d . . . (d) h T i / ∆( h ω i − ω ) / ∆ ω ( hLi − L ) / ∆ L z/δ T , y/δ ω , y/δ L . . . . (e) . . . z/d , y/d . . . . (f) θ rms / ∆1 . L ′ rms / ∆ L FIG. 3. (a) Profiles of the mean temperature (cid:104) T (cid:105) for Pr = 0 . Ra = 10 and of the mean angular velocity (cid:104) ω (cid:105) and angular momentum (cid:104)L(cid:105) for η = 0 . Re S = 2 × and R Ω = 0 .
023 (case 1). All profiles are rescaledto the interval [0 ,
1] using ∆ ω = ω − ω and ∆ L = L − L . The coordinate y = r − r is used for the TC flow.(b) Magnification of the near-wall region. All coordinates are rescaled by the corresponding boundary layerthicknesses. (c) Profiles of the root mean square temperature and the root mean square angular momentum.The latter is rescaled by a factor of 1 . R Ω = 0 .
241 (case 2). comparison for the TC flow at the second local maximum in the Nu ω − R Ω relation at R Ω = 0 .
241 (case2). While the agreement with case 1 is very good, there are differences for case 2. Here, the angularvelocity profile has a noticeable gradient in the central region, whereas the angular momentum is wellmixed and lies closer to the temperature profile. From this comparison one can conclude that T ismore closely associated with L than with ω . Therefore, we compare the root mean square profiles of8 . . . . . . z/d , y/d J / J l a m (a) J ( c ) T J ( d ) T J ( c ) ω J ( v ) ω . . . . . . z/d , y/d (b) J ( c ) T J ( d ) T J ( c ) ω J ( v ) ω FIG. 4. Comparison between the contributions to the currents for (a) case 1 and (b) case 2 with theparameters described in figure 3. The convective and diffusive currents for the heat transport in RB areshown as continuous lines, the corresponding convective and viscous currents for the angular momentumtransport in TC as dashed lines. T and L fluctuations in panels (c) and (f) of figure 3. The fluctuating fields are obtained by θ ( x , t ) = T ( x , t ) − (cid:104) T (cid:105) ( z ) , (11) L (cid:48) ( x , t ) = L ( x , t ) − (cid:104)L(cid:105) ( r ) . (12)For case 1, the peaks in L (cid:48) rms are broader than in case 2, which is consistent with the observation thatthe boundary layers are turbulent for case 1, but not for case 2 [4]. In the RB flow case, the boundarylayer dynamics is close to laminar, and the peaks in θ rms are narrower. Furthermore, we note that theshape of the root mean square angular velocity profile ω (cid:48) rms = L (cid:48) rms /r (not shown here) hardly differsfrom L (cid:48) rms since the radius only varies by 1% for η = 0 . J T = J ( c ) T ( z ) + J ( d ) T ( z ) = (cid:104) u z T (cid:105) − κ ∂ (cid:104) T (cid:105) ∂z = Nu T J T,lam . (13)A similar decomposition into a convective and viscous contribution in the TC flow case leads to J ω = J ( c ) ω ( r ) + J ( v ) ω ( r ) = r (cid:104) u r L(cid:105) − νr ∂ (cid:104) ω (cid:105) ∂r = Nu ω J ω,lam . (14)Figure 4 displays the vertical (radial) profiles. Panel (a) compares with case 1 while panel (b) compareswith case 2. As expected the dissipative contributions are significant in the boundary layers and9ecome small in the bulk. The convective parts dominate the bulk and drop to zero at the walls dueto the no-slip boundary conditions. Furthermore, the sum of both transport contributions remainsconstant across the whole layer in both systems. It can be seen again that profiles of case 1 showbetter agreement with RB flow than the profiles of case 2. Since in the latter case the angular velocityprofile is not flat in the central region (cf. figure 3d), the viscous contribution J ( v ) ω is larger than thecorresponding diffusive part J ( d ) T , which additionally results in a smaller convective contribution J ( c ) ω . B. Probability density functions
We now refine the analysis and report the statistics of the convective currents in averaging surfacesat different distance from the inner (bottom) wall. First, it is important to note that only thetemperature and angular momentum fluctuations θ and L (cid:48) that deviate from the corresponding meanprofile contribute to the net transport through the averaging surfaces, since (cid:104) u z (cid:105) = (cid:104) u r (cid:105) = 0 byincompressibility, and therefore (cid:104) u z T (cid:105) = (cid:104) u z θ (cid:105) and r (cid:104) u r L(cid:105) = r (cid:104) u r L (cid:48) (cid:105) . Therefore, we study the localconvective currents based on the fluctuations θ and L (cid:48) instead of the total fields. Figure 5 compares theprobability density functions (PDFs) of the local convective angular momentum current j ( c ) ω = ru r L (cid:48) for cases 1 (top row) and 2 (bottom row) with the local convective heat current j ( c ) T = u z θ for planesat 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 distanceof 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 thePDFs 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 arisefor the data at 4 δ L , the agreement is very good for the data taken at δ L and r a . The region just abovethe boundary layer thickness is dominated by rising plumes and recirculations next to the plumes. Itis sometimes also denoted as the plume mixing layer [7]. The reason for the differences in the widthof the tails in panels (b) and (e) of figure 5 could therefore be related to the shape of the plumes andthe frequency of their detachment, which differ between RB and TC flow as will be shown in the nextsubsection.The observation that the local fluctuations in case 2 are enhanced compared to case 1 can beunderstood by analysing the components that form j ( c ) ω = ru r L (cid:48) . Since the fluctuation amplitude of L (cid:48) varies little between both cases (cf. figure 3(c,f)), and the radius r remains unchanged, the differencemust occur in the radial velocity u r . In [5] it was shown that the fluctuation amplitude ( u r ) rms varieswith mean rotation ( R Ω ) and in case 2 it is twice as large as in case 1. This increase is partly caused10 IG. 5. Probability density functions (PDFs) of the local convective currents j ( c ) T = u z θ (filled circles) and j ( c ) ω = ru r L (cid:48) (solid lines) computed at the boundary layer thickness δ T ( δ L ) in panels (a, d), at 4 δ T (4 δ L ) inpanels (b, e) and at the centre d/ r a ) in panels (c, f). The local currents are normalized by the correspondingtotal mean currents J T and J ω . Panels (a, b, c) are for case 1 and panels (d, e, f) are for case 2 for thesame parameters as in figure 3. Here θ = T − (cid:104) T (cid:105) and L (cid:48) = L − (cid:104)L(cid:105) denote the fluctuations around thecorresponding time- and area-averaged profiles. by a strengthening of the mean Taylor vortices, cf. figure 1(b). The stronger u r fluctuations resultin wider tails in figure 5(e,f) but do not significantly affect the distribution at δ L where the radialvelocity is restricted due to the proximity of the cylinder wall.In figure 6 we compare individual components of the transport currents of angular momentum andheat. They are u z and θ in the RB case, and the radial velocity u r and angular momentum L (cid:48) inTC flow. It can be observed that the agreement between case 1 and the RB flow is good. For case11 IG. 6. Comparison of the PDFs of individual components of the turbulent fields. All data are for case 1and the RB flow. In the upper row, the PDFs of the axial velocity u z (RB, filled circles) and radial velocity u r (TC, solid lines) are compared, in the lower row the PDFs of the temperature θ (RB, filled circles) andangular momentum L (cid:48) (TC, solid lines). Data are obtained for δ T ( δ L ) in panels (a, d), at 4 δ T (4 δ L ) in panels(b, e) and at the centre d/ r a ) in panels (c, f). Each PDF is normalized by its corresponding root meansquare value. θ and L (cid:48) were larger. In a turbulentflow one expects Gaussian statistics for the individual components of the velocity field. In figure6(a) exponential tails are observed for the PDFs of u z and u r at the height of (thermal) boundarylayer thickness. This is a clear statistical fingerprint for an enhanced intermittency in the near-wallregion which is connected with the plume formation. Also, in figure 6(b) a fatter tail for the radialcomponent is detected which confirms our observation in figure 5(b).12 a) (b) (c) FIG. 7. Isosurface snapshots of the convective transport currents j ( c ) T for RB flow and j ( c ) ω for TC flow (case1). Isolevels j ( c ) /J = − d in each direction, to match the aspect ratio of the RB flow domain. (c) Simulated fraction of the annulardomain. The arrows indicate the large-scale circulation in (a), the Taylor vortex in (b) and the velocity profilein (c). The distributions of the temperature and angular momentum fluctuations (see lower row of figure6) are skewed and take a symmetric shape in the midplane only as shown in figure 6(f). For allthree distances the tails of both PDFs are in very good agreement. The distribution in the midplaneis again not Gaussian which has been reported already in [14]. The specific cusp-like form aroundthe origin and the increasingly pronounced exponential tails in the PDF of temperature fluctuationshave been discussed, for example, by Yakhot on the basis of a hierarchy of momentum equations forthe temperature fluctuations [29]. Interestingly, even such specific details of the small-scale statisticsprevail in our comparison between RB and TC flow.
C. Relation between transport and flow structures
In the last subsections, we identified small differences in the statistical properties of the RB flow andcase 1 of TC flow and attributed them to differences in the flow structures. The spatial organizationof currents is shown in figure 7, where isosurfaces of the convective current (the top row of figure 5) forthe levels j ( c ) /J = ± Re rms = u rms d/ν = 675, withthe root mean square velocity u rms calculated from all three velocity components in the entire cell[26]. This is significantly smaller than the corresponding Reynolds numbers in TC flow (with the meanrotation subtracted), which are Re rms = 2251 and Re rms = 2712 for the cases 1 and 2, respectively. Itcan thus be expected that the boundary layers in the RB case are still close to laminar, while the onesin case 1 of TC flow are already turbulent [4]. Specifically, we find for the boundary-layer Reynoldsnumbers, defined based on the boundary layer thickness and shear across the boundary layer, valuesof ∼
30 for RB flow [26] and of ∼
300 and ∼
200 for the TC cases 1 and 2, respectively [4]. Theturbulent fluctuations in the TC boundary layer account for the deviations in the tails of the PDFsand 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). V. CONCLUSIONS
In the present work we discussed a direct comparison of the statistical properties of Rayleigh–B´enard (RB) convection and Taylor–Couette (TC) flow. The comparison is motivated by analogies ofdimensionless system parameters (such as Rayleigh and Taylor numbers), the same form of the energybalances, (3) and (6), and the similarities in the currents of heat and angular momentum (see alsoreferences [2, 11–13]).Our study shows that the operating point for a specific comparison between TC and RB flows canbe determined by choosing corresponding values of Nusselt numbers since the Nusselt number definesthe boundary layer thickness and hence the transport properties. We also find that a better charac-terization 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 ), since the latter do not reflect the mean rotationof the cylinders. We demonstrated that for sufficiently large shear Reynolds number Re S , multipleTC flow cases at different rotation numbers can have the same Nusselt number as RB convection, i.e.14he same amount of angular momentum is transported between the cylinders in TC flow as heat fromthe bottom to the top in the RB case. The comparison also shows that the case with the smallerrotation number R Ω (case 1) provides a better agreement with RB flow than the case of larger rotationnumber. For this pair of flows, a remarkable agreement between mean profiles as well as probabilitydensity functions of fluctuating quantities is found.Studies of the mean profiles and the PDFs of the convective currents show that the differencesbetween 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 connectedwith the detachment of plumes and other differences in the dynamics: the boundary layers in theconvection 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 become turbulentas well.The TC flow case 2, which is characterized by a larger mean rotation ( R Ω ), shows greater differencesto the RB case. As a consequence of rotation, the angular velocity profile has a significant gradient inthe central region, which results in a higher (lower) dissipative (convective) transport current than inthe RB case. Furthermore, enhanced radial velocity fluctuations and stronger mean Taylor vorticesoccur 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 rotationdetermines 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 andTC flow one can actually relate their transport properties in detail, both in the mean and in thefluctuations, thereby confirming the analogies between the twins of turbulence [6] for a larger set ofproperties. ACKNOWLEDGMENTS
We would like to thank M. S. Emran and R. du Puits for scientific discussions at the beginningof this work. JS acknowledges computational resources provided by the John von Neumann Institutefor Computing within Supercomputing Grant HIL09. HB and BE thank M. Avila for providing thecode used for the TC simulations and acknowledge computational resources at the LOEWE-CSC inFrankfurt. The paper was written during a workshop at the Lake Arrowhead Conference Center,and BE and JS would like to thank the Institute of Pure and Applied Mathematics (IPAM) of the15niversity of California Los Angeles for financial support. [1] G. Ahlers, S. Grossmann, and D. Lohse. Heat transfer and large scale dynamics in turbulent Rayleigh–B´enard convection.
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