Coherence of temperature and velocity superstructures in turbulent Rayleigh-Bénard flow
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Coherence of temperature and velocitysuperstructuresin turbulent Rayleigh-B´enard flow
Dominik Krug † , Detlef Lohse , and Richard J.A.M. Stevens Physics of Fluids Group and Twente Max Planck Center, Department of Science andTechnology, Mesa+ Institute, and J.M. Burgers Center for Fluid Dynamics, University ofTwente, P.O Box 217, 7500 AE Enschede, The Netherlands Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 G¨ottingen,Germany(Received xx; revised xx; accepted xx)
We investigate the interplay between large-scale patterns, so-called superstructures, in thefluctuation fields of temperature θ and vertical velocity w in turbulent Rayleigh-B´enardconvection at large aspect ratios. Earlier studies suggested that velocity superstructureswere smaller than their thermal counterparts in the center of the domain. However, ascale-by-scale analysis of the correlation between the two fields employing the linearcoherence spectrum reveals that superstructures of the same size exist in both fields,which are almost perfectly correlated. The issue is further clarified by the observationthat in contrast to the temperature, and unlike assumed previously, superstructures inthe vertical velocity field do not result in a peak in the power spectrum of w . The originof this difference is traced back to the production terms of the θ - and w -variance. Theseresults are confirmed for a range of Rayleigh numbers Ra = 10 –10 , the superstructuresize is seen to increase monotonically with Ra . Furthermore, the scale distribution ofparticularly the temperature fluctuations is pronouncedly bimodal. In addition to thelarge-scale peak caused by the superstructures, there exists a strong small-scale peak.This ‘inner peak’ is most intense at a distance of δ θ from the wall and associated withstructures of size ≈ δ θ , where δ θ is the thermal boundary layer thickness. Finally, basedon the vertical coherence relative to a reference height of δ θ , a self-similar structureis identified in the velocity field (vertical and horizontal components) but not in thetemperature. Key words:
1. Introduction
A remarkable feature of turbulent flows is that amid the inherent disorder both in timeand space, they frequently give rise to a surprisingly organized flow motion on very largescales. Such very large-scale structures in the fully turbulent regime have, for example,been reported for turbulent boundary layers (Hutchins & Marusic 2007 a ), plane Couetteflow (Lee & Moser 2018), and Taylor-Couette (Huisman et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] D ec D. Krug et al. (a) (b)
Figure 1.
Snapshots of the temperature (a) and the vertical velocity (b) field at mid-height fora simulation in a Γ = 32 cell with P r = 1 at Ra = 10 . The diameter of the circles in bothpanels indicates the superstructure size ˆ l = 6 . an idealized configuration that is widely used to study thermal convection (Ahlers et al. Ra , while thedimensionless heat transfer is characterized by the Nusselt number N u .Large-scale organization in convective flows is widespread. An astonishing example isthe formation of so-called cloud streets in the atmosphere that can extend for hundreds ofkilometers (e.g. Young et al. Γ . Naturally, this poses a challenge to experiments and simulations.Experimentally (e.g. Fitzjarrald 1976; Sun et al. a , b ; Zhou et al. et al. et al. et al. (2003) and several related studies havesince been presented in the literature (Parodi et al. et al. et al. et al. et al. P r ) byPandey et al. (2018) and in Γ as well as in Ra by Stevens et al. (2018).From these papers, it has become clear that in RBC superstructures, i.e., flowstructures that are significantly larger than the convection rolls at onset (see, e.g. Drazin& Reid 2004) or in the weakly non-linear regime (Morris et al. Ra .It is widely observed that the superstructure size increases with Ra (Fitzjarrald 1976;Hartlep et al. et al. et al. P r -dependenceappears to be more complicated. For the latter, Pandey et al. (2018) report that at Ra = 10 the largest structures are found for P r ≈
7, but
P r -variations over a significantrange at higher Ra have not been reported yet. Stevens et al. (2018) showed thatvery large domain sizes up to Γ = 64 are necessary to fully converge the size of thesuperstructures at Ra = 10 . Finally, Von Hardenberg et al. (2008) and Pandey et al. (2018) demonstrate that superstructures evolve on timescales much longer than the free-fall time scale.There is no consensus yet on how to best extract and quantify the superstructures oherence of superstructures in turbulent Rayleigh-B´enard flow Ra N x × N y × N z Nu Re h Re v Re t ˆ l δ θ = 1 / (2 Nu )1 × × ×
64 4.35 55.7 40.3 68.7 4.4 0.1154 × × ×
64 6.48 111.7 84.3 140.0 4.5 0.0771 × × ×
96 8.34 176.0 131.6 219.8 4.9 0.0604 × × ×
96 12.27 349.7 250.8 430.4 5.4 0.0411 × × ×
128 15.85 547.1 380.1 666.2 5.9 0.0321 × × ×
192 30.94 1660.3 1056.1 1967.8 6.3 0.0161 × × ×
384 61.83 4879.2 2962.3 5708.1 6.6 0.008
Table 1.
The columns from left to right indicate the Ra number, the numerical resolutionin the horizontal and wall normal directions ( N x × N y × N z ), the Nu number, and thehorizontal ( Re h = (cid:112) (cid:104) v x + v y (cid:105) V (cid:112) P r/Ra ), vertical ( Re w = (cid:112) (cid:104) w (cid:105) V (cid:112) P r/Ra ), and total( Re t = (cid:112) (cid:104) v x + v y + w (cid:105) V (cid:112) P r/Ra ) Reynolds numbers. The length scale ˆ l denotes thesuperstructure scale based on the coherence spectrum γ θw (plotted as triangles in figure 8b)and δ θ is the thermal boundary layer thickness. in RBC. Researchers have relied on peaks in velocity and/or temperature power spec-tra (Von Hardenberg et al. et al. et al. et al. et al. et al. et al. θ ) field are larger than in the vertical velocity w (Pandey et al. et al. w . Theseobservations seem at odds with the notion that superstructures in RBC form large-scaleconvection rolls for which temperature and velocity scales should be of the same size.To address and clarify this issue along with related questions, we use the dataset ofStevens et al. (2018) to assess energy distributions and coherence on a scale-by-scalebasis. Before presenting our results in §
3, we provide the relevant details on the datasetof Stevens et al. (2018), together with the parameters of additional simulations performedfor this study, in §
2. We summarize our findings in §
2. Dataset
We solve the Boussinesq equations with the second-order staggered finite differencecode AFiD. The code has been extensively validated and details of the numerical methodscan be found in Verzicco & Orlandi (1996); Stevens et al. (2010, 2011); van der Poel et al. (2015); Zhu et al. (2018). The governing equations in dimensionless form read: ∂ u ∂t + u · ∇ u = −∇ p + (cid:114) P rRa ∇ u + θ ˆ z, (2.1) ∇ · u = 0 , (2.2) D. Krug et al. ∂θ∂t + u · ∇ θ = 1 √ RaP r ∇ θ, (2.3)where ˆ z is the unit vector pointing in the opposite direction of gravity, u the velocityvector normalized by the free fall velocity √ gα∆H , t the dimensionless time normalizedby (cid:112) H/ ( gα∆ ), θ the temperature normalized by ∆ , and p the pressure normalized by gα∆/H . The control parameters of the system are Ra = αg∆H / ( νκ ) and P r = ν/κ ,where α is the thermal expansion coefficient, g the gravitational acceleration, ∆ thetemperature drop across the container, H the height of the fluid domain, ν the kinematicviscosity, and κ the thermal diffusivity of the fluid. The boundary conditions on the topand bottom plates are no-slip for the velocity and constant for the temperature. Periodicconditions in the horizontal directions are used. In all our simulations, P r is fixed to 1and we analyze data for Γ = L/H = 32, where H is the vertical distance between theplates and L the horizontal extension of the domain. Length scales are normalized by H unless specified otherwise and we set H = 1. Coordinates in the wall-parallel directionare denoted by x and y while the z -axis points along the wall-normal. Horizontal velocitycomponents are denoted v x and v y , respectively. A high spatial resolution in the boundarylayer and bulk has been used to ensure that the resolution criteria set by Stevens et al. (2010) and Shishkina et al. (2010) are fulfilled. Details about the simulations can befound in table 1. The simulations for Ra = 10 and Ra = 10 have been reportedbefore in Stevens et al. (2018), while the simulations for 10 (cid:54) Ra (cid:54) have beenperformed for this study. The horizontal, vertical, and total Reynolds numbers indicatedin table 1 represent the volume and time averages of Re h = (cid:113) (cid:104) v x + v y (cid:105) V (cid:112) P r/Ra , Re w = (cid:112) (cid:104) w (cid:105) V (cid:112) P r/Ra , and Re t = (cid:113) (cid:104) v x + v y + w (cid:105) V (cid:112) P r/Ra , respectively. In thefollowing, we decompose instantaneous quantities ˜ ψ into mean and fluctuating partsaccording to ˜ ψ = Ψ + ψ , where Ψ = (cid:104) ˜ ψ (cid:105) with (cid:104)·(cid:105) denoting an average over a wall-parallelplane and time.
3. Results
In presenting our results, we initially ( § § Ra = 10 ). A detailed discussion of the Ra -dependence of our findingsis then provided in § Spectral distribution of energy and coherence of temperature and vertical velocity
To evaluate the energy distribution across different scales, we first consider the one-sided power spectra Φ ψψ ( k ), where ψ is a zero-mean quantity (velocity or temperaturehere) and k is the radial wavenumber k = (cid:113) k x + k y . The spectra are computed forhorizontal planes and averaged in time. Results for θ and w at several distances from thewall are presented in figures 2a and 2b, respectively. Data are presented in premultipliedform kΦ ψψ , such that the area under the curve equals the variance when plotted on alogarithmic scale, according to (cid:104) ψ (cid:105) = (cid:90) ∞ Φ ψψ d k = (cid:90) ∞ kΦ ψψ d(log k ) . (3.1)For reference, the wall-normal temperature and vertical velocity variance profiles arepresented in figures 3a and 3b, respectively. The symbols in these figures mark thepositions at which the spectra in figure 2 are computed.First, we focus on the situation at mid-height ( z/H = 0 . oherence of superstructures in turbulent Rayleigh-B´enard flow -3 -3 (a) (b) z z (c) (d) Figure 2.
Premultiplied temperature (a) and vertical velocity (b) power spectra. Thepremultiplied co-spectrum kΦ θw (c) is normalized such that it integrates to the turbulent heatflux. (d) Linear coherence spectrum γ θw , see eq. (3.2). The dashed and dotted vertical linesindicate k = 1 and k = 34, respectively. The grey-shaded area marks the approximate range ofsuperstructure scales k = 1 ± .
4. The results presented here are computed for Ra = 10 . Thecolor of the curves indicates the wall distance according to the legend below the figures. (a) (b) (c) Figure 3.
Wall-normal temperature (a) and vertical velocity (b) variance profiles for Ra = 10 .Panel (c) shows the corresponding normalized turbulent heat flux. Symbols denote the locationof the spectra plotted in figures 2 with corresponding colors. D. Krug et al.the location of the snapshots shown in figure 1. These results are represented by thered lines in figure 2. Figure 2a shows that the temperature spectrum kΦ θθ ( z/H = 0 . k ≈ ± . z/H = 0 .
5, while the remainder of the variance is spread out overa wide range of intermediate and small-scales, which individually carry relatively littleenergy. Figure 2b reveals that the corresponding vertical velocity spectrum kΦ ww ( z/H =0 .
5) spans approximately the same range of scales as its temperature counterpart overall.However, its shape is significantly different as it is much more broadband and has afairly wide peak centered around k ≈ .
5. It is important to note, though, that thereis significant energy in the kΦ ww -spectrum at the scales corresponding to the thermalsuperstructures, which are marked by grey shading in all panels of figure 2. This impliesthat velocity structures of the same size as the temperature superstructures indeed exist.Yet, their contribution is overshadowed by stronger velocity fluctuations at smaller scales.More insight into the correlation between the velocity and temperature structures isobtained by analyzing the one-sided co-spectrum Φ θw = Re( (cid:104)F ( θ ) F ( w ) ∗ (cid:105) ) where F ( · )indicates the Fourier transform in the horizontal plane and ( · ) ∗ the complex conjugate.Figure 2c shows that the temperature-velocity co-spectrum kΦ θw at mid-height featuresa pronounced large-scale peak at k ≈
1. This indicates that a correlation exists betweenthe large-scale structures in θ and w . Further, kΦ θw ( z/H = 0 .
5) decreases with increasing k , but scales smaller than the superstructure size nevertheless contribute significantly tothe turbulent heat transport. Aside from the degree of correlation between θ and w alsotheir magnitudes factor into the co-spectrum at a given scale. In order to focus on thecorrelation aspect only, we analyze the linear coherence spectrum γ θw ( k ) = | Φ θw ( k ) | Φ θθ ( k ) Φ ww ( k ) . (3.2)By definition, 0 (cid:54) γ θw (cid:54) γ θw (cid:62) . γ θw = 0 .
95 at k = 1 (markedby a dashed line in all panels of figure 2). The coherence quickly drops below γ θw = 0 . k . This explains why the overall correlation coefficient between θ and w , whichis essentially an average over the coherence spectrum, is smaller than 0 . et al. (2018).In order to demonstrate also visually how well the large scales of w and θ are correlated,we present the snapshots from figure 1 again in figure 4, but this time with the small-scale contributions removed. More specifically, we obtain the large-scale fields θ L and w L using a spectral low-pass filter where the cut-off wavenumber k cut = 2 . γ θw ( z/H = 0 .
5) drops below 0.5. Figure 4 convincingly shows thatthere is indeed a very good correspondence between patterns at the superstructure scalein temperature and vertical velocity fields, not only in size but also in location.To summarize, we have shown that patterns corresponding to the thermal superstruc-tures also exist in the vertical velocity. For the vertical velocity, though, the contributionof the superstructures in the kΦ ww -spectrum is sub-dominant in the sense that it does notresult in a spectral peak. This has previously led to the notion that the superstructures in the velocity field are smaller than in the temperature field, whereas it is really the sizeof the most energetic structures , as measured by the spectral peak, that is different. Wewill revisit the reasons for the different spectral distributions of θ and w in § oherence of superstructures in turbulent Rayleigh-B´enard flow L w L (a) (b)w L = 0 L = 0.5 Figure 4.
Same snapshots of temperature (a) and vertical velocity (b) at mid-height as presentedin figure 1, but this time filtered with a spectral low-pass filter with cut-off wavenumber k cut = 2 . wall-normal locations that span the full domain down to the thermal boundary layerthickness δ θ = 1 / (2 N u ). Remarkably, curves at all z -positions collapse around the peakat k = 1 for the temperature spectra in figure 2a. This suggests that there is very littleevolution of the large-scale thermal structures along the vertical direction. Similarly, alsothe coherence between θ and w (figure 2d) is almost independent of z at the largestscales. In contrast, there is a pronounced increase in kΦ ww around k ≈ kΦ ww that also drives a growth of the large-scale peakof the co-spectrum as z increases, as shown in figure 2c.What is striking about the kΦ θθ spectra (figure 2a) is that at heights of the order of δ θ there exists a second strong peak in addition to the one caused by the superstructures.This small-scale peak is located at k ≈
34 (indicated by the dotted lines in figure 2), whichcorresponds to a typical small-scale structure size of about 11 δ θ . Upon comparison withfigure 3a, it becomes clear that this peak carries the energy that leads to the maximumof (cid:104) θ (cid:105) at z = δ θ . A similar small-scale peak is also observed for kΦ ww in figure 2b,even though it is located at slightly larger k in this case. For kΦ ww , this peak broadenstowards intermediate scales with increasing z and the increase of (cid:104) w (cid:105) with increasing z (see figure 3b) is mostly associated with increasing energy content at intermediate scales k ≈
10. It is further interesting to note that the spectral decomposition of kΦ θw shiftsfrom small-scale dominated ( z (cid:47) δ θ ) over broadband (0 . H (cid:47) z (cid:47) . H ) to a maximumat large scales for z (cid:39) . H . At the same time, the overall heat transport (cid:104) θw (cid:105) staysapproximately constant beyond z ≈ δ θ (see figure 3c). In connection, these observationsappear consistent with the concept of merging plumes. This was advocated by e.g. Parodi et al. (2004), who found that the structure size increases going away from the wall whilethe flux remains constant.3.2. Production of temperature and vertical-velocity fluctuations
In order to uncover the origin of the different spectral distribution of temperature andvertical velocity that became apparent in figure 2a,b, we now study the variance produc-tion terms of the respective variance budgets. These production terms are (Deardorff &
D. Krug et al. Figure 5.
Cumulative co-spectrum (cid:82) k Φ θw d k (cid:48) for Ra = 10 . The corresponding co-spectra kΦ θw are shown in figure 2c. The normalization is chosen such that the spectra add up to the relativecontribution of turbulent transport to Nu at each wall-height. Willis 1967; Kerr 2001; Togni et al. S θ = − (cid:104) θw (cid:105) d Θ d z (3.3)for (cid:104) θ (cid:105) and S w = (cid:104) θw (cid:105) (3.4)for (cid:104) w (cid:105) . A trivial but nevertheless important implication that arises from comparing(3.3) and (3.4) is that (cid:104) θw (cid:105) generates w -variance directly , while temperature varianceis only produced in the presence of a mean gradient d Θ/ d z . Consequently, S θ > z (cid:47) δ θ ) since a significant mean temperaturegradient exists only there. This close to the wall (cid:104) θw (cid:105) is predominantly a small-scalequantity as evidenced by kΦ θw ( z = δ θ ) in figure 2c, such that S θ is localized not onlyin space but also in scale. On the contrary, (cid:104) θw (cid:105) is almost independent of z outside ofthe thermal boundary layer, see figure 3c. Hence, also S w is widely distributed acrossthe bulk of the flow. To better understand the spectral distribution of S w , we presentthe data from figure 2c in cumulative form in figure 5. This figure reveals that even at z/H = 0 . Φ θw only contributes about 30% of the total flux (cid:104) θw (cid:105) .In the central region of the flow the bulk of the (cid:104) w (cid:105) production occurs at intermediatescales (say 2 (cid:47) k (cid:47) kΦ ww peaks at thesewall distances, see figure 2b. This further explains why the superstructure contributionis not reflected as a spectral peak in the kΦ ww spectrum.While the analysis of the production terms provides essential insight into the reasons forthe different spectral decomposition of (cid:104) θ (cid:105) and (cid:104) w (cid:105) , other aspects cannot be addressedon this basis alone. Specifically, understanding the apparently very efficient organizationof small-scale temperature fluctuations into thermal superstructures requires the analysisof inter-scale energy transfer. Such an undertaking is beyond the scope of the presentwork. We note, however, that an inverse (i.e. from smaller to larger scales) energy transfer,is indeed observed in certain regions of the flow for both velocity and temperature whenhorizontally averaged budgets are considered (see e.g. Togni et al. et al. Wall-normal coherence of superstructures
So far, we have only considered the correlation between vertical velocity and tem-perature at a given wall-normal location. Another important aspect is the wall-normal oherence of superstructures in turbulent Rayleigh-B´enard flow -2 -1 -2 -1 (a) (b)(c) (d) z Figure 6.
Spatial coherence spectra of temperature (a) and vertical velocity (b) with thereference plane at z R = δ θ . The data from (a) and (b) is plotted again in (c) and (d), respectively,as a function of zk instead of k . All results shown are for Ra = 10 . coherence of superstructures. There exists qualitative evidence from comparing snapshotsat different heights (Stevens et al. et al. z and z R according to γ ψψ ( z R ; z, k ) = |(cid:104)F ( ψ ( z R )) F ( ψ ( z )) ∗ (cid:105)| Φ ψψ ( z R , k ) Φ ψψ ( z ; k ) . (3.5)We fix the reference height at z R = δ θ . Consequently, γ ψψ ( z R = δ θ ; z, k ) is a measureof how correlated structures in field ψ at scale k and height z are with fluctuations ofthe same scale at boundary layer height of the same field. Results are presented fortemperature and vertical velocity in figures 6a and 6b, respectively. By definition, theresult at z = δ θ is the correlation with the reference itself and therefore γ ψψ ( z R = δ θ ; δ θ , k ) = 1 trivially. In the superstructure peak ( k = 1 ± .
4, marked by grey shadingin the figure) γ ψψ ( z R = δ θ ; z, k ) is close to one, even at mid-height. This holds for bothtemperature and vertical velocity and implies a very strong degree of spatial coherencefor the largest structures in both fields. Differences between θ and w only occur at smallerscales. Beyond z = 2 δ θ the spatial coherence of θ decreases very quickly as a function of k ,0 D. Krug et al. -2 -1 (a) (b) z Figure 7.
Spatial coherence spectra of horizontal velocity v as a function of (a) k and (b) zk .The reference height is z R = δ θ and Ra = 10 . but has a limited z -dependence. In contrast, curves for γ ww in figure 6b show significantvariation with z with the decline occurring at progressively smaller k with increasing z .Apart from quantifying the correlation, γ ψψ ( z R = δ θ ) also provides information aboutthe self-similarity of structures that are connected or ‘attached’ to the thermal boundarylayer. This is of interest since previous authors (e.g. He et al. et al. γ ψψ ( z R = δ θ ) should collapse if plotted against zk , that is if the scale isnormalized by the distance from the wall (see Baars et al. et al. w indeed collapse to a reasonable degree for 3 δ θ (cid:47) z (cid:47) . H . To check if self-similarityscaling in this range is a property of the velocity field in generally, we additionally presentresults for γ vv ( z R = δ θ ), where v is the horizontal velocity component, in figure 7. Thevertical coherence of v also exhibits the same superstructure peak as observed for theother quantities, which can be seen in figure 7a. Only its magnitude decreases withincreasing z and is close to zero at mid-height. This is consistent with the roll structuresnot having a horizontal component at z/H ≈ . Φ vv ( k =1 ± .
4) (not shown) is minimal there. As figure 7b shows, γ vv ( z R = δ θ ) displays thesame collapse when plotted versus zk and in the same range of z as previously observedfor w . This means that for the velocity fields in a significant part of the domain (at least3 δ θ (cid:47) z (cid:47) . H ) structures attached to the boundary layer display self-similar behavior.The same trends are observed at different Ra but are not shown here for brevity.3.4. Rayleigh number trends
As a final point, we study the Ra number dependence of the properties discussed in § γ θw evaluated at mid-height for 10 (cid:54) Ra (cid:54) in figure 8a. The magnitude and the shape of the large-scale peak are nearly independentof Ra . However, the peak location shifts towards smaller k with increasing Ra . Thecorresponding increase in the large-scale structure is quantified in figure 8b, wherethe triangles indicate the structure size (ˆ l = 2 π/ ˆ k ) corresponding to the peak in the oherence of superstructures in turbulent Rayleigh-B´enard flow -3 -3 (a) (b) Ra (c) (d) Figure 8. (a) Coherence spectrum at mid-height for 10 (cid:54) Ra (cid:54) ; see panel (b) for thecolor-code. Stars indicate the wavenumber corresponding to 10 η at the respective Ra . (b)Wavelength ˆ l of the spectral peaks of γ θw (triangles), Φ θθ (circles), Φ ww (filled squares), kΦ ww (open squares) and kΦ θw (black dots). The corresponding spectra kΦ θθ and kΦ ww are shown inpanels (c) and (d) with symbols marking the peak locations as described for (b). The inset in(b) additionally shows the aspect ratio dependence of ˆ l based on γ θw at Ra = 10 , see appendixA for details. coherence. Here, the peak location ˆ k is obtained from fitting a parabola to three pointscentered around the peak of γ θw and the results are also listed in table 1. Evidently, ˆ l is significantly larger than the wavelength of the structure at the onset of convection,which is ≈ Φ θθ (circles), Φ ww (filled squares), and kΦ ww (open squares) are includedin figure 8b and the corresponding spectra are shown in panels (c,d) of the same figure.The spectral peak from the temperature spectrum corresponds to slightly larger lengthscales compared to the results based on γ θw , but the differences are quite small. Due toits broadband nature, the spectral peaks for the vertical velocity are found at a differentlocation in the regular and premultiplied spectra. The maximum of Φ ww only agrees withthe results based on coherence and temperature at Ra = 10 . For higher Ra fluctuationsat intermediate length scales dominate the velocity spectrum. Therefore, the use of thevelocity spectra leads to significantly lower estimates for ˆ l than the temperature spectraat higher Ra , as mentioned in § Γ of the periodic domainwas already discussed in Stevens et al. (2018). From their results, it appears that thesuperstructure size based on the peak in Φ θθ increases monotonically (albeit slowly for Γ >
16) with increasing Γ . The inset of figure 8b shows that ˆ l based on γ θw decreasesslightly if Γ is increased from 16 to 64. This difference is not rooted in the fact that adifferent metric is employed here, but is caused by an error in the computation of the2 D. Krug et al.spectra presented in Stevens et al. (2018). We plot the recomputed spectra in appendixA and these show that the temperature spectral peak indeed exhibits the same trend.The concept of a convection roll, i.e. a thermally driven velocity structure, suggeststo define the superstructure size in RBC as the scale where the correlation betweentemperature and velocity fields is maximum. We therefore argue that conceptually themost straightforward way to define the superstructure size is via the coherence spectrum.It should be noted that the coherence peak is not necessarily coincident with the peak ofthe co-spectrum due to the different distributions of Φ θθ and Φ ww . In practice, however,the peaks of γ θw and kΦ θw coincide within measurement accuracy for the cases presentedhere (see figure 8b). This seems to be a consequence of the sharp drop-off of γ θw and Φ θθ with increasing k that outweighs the increase in Φ ww . The situation may changehowever, e.g. for different P r numbers. Some caution is therefore advised in this matter.For a case in point, we note that the small-scale peak in kΦ θw at z = δ θ (figure 2c) iswithout counterpart in γ θw (figure 2c). This indicates that the peak in turbulent transportis predominantly driven by magnitude, not coherence. The peaks of the Φ ww and the kΦ ww spectra may be misleading as indicators for superstructure size since the velocityspectra are dominated by motions at intermediate length scales.As an aside, we discuss the increase of γ θw that is seen to occur at high k in figure 8a.This increase at small-scales occurs for lower values of k and is stronger for the lower Ra .A comparison with figure 8c,d reveals that there is only minimal energy at these smallscales. These observations are consistent with the notion that the higher values of γ θw mark the transition to a viscous dominated regime. In the viscous regime, the correlationbetween θ and w is high because the balance is predominantly between buoyancy andviscous forces. This is similar to the situation at the onset of convection, where thecorrelation between velocity and temperature fluctuations is very high (Bodenschatz et al. η asa reference scale for the viscous regime in the figure. Here, η = ( ν / (cid:104) ε (cid:105) V ) / is theKolmogorov length-scale and (cid:104) ε (cid:105) V is the volume-averaged dissipation rate obtained fromthe identity (cid:104) ε (cid:105) V = ( N u − / √ RaP r . It is seen in figure 8a that the scale at which thehigh-wavenumber increase of γ θw occurs roughly coincides with 10 η for Ra > , justas expected from the above. The agreement is less good for the (marginally turbulent)cases at even lower Ra , where the increase in γ θw starts at scales significantly larger than10 η .The Ra dependence of the near-wall characteristics of the temperature field aredisplayed in figure 9 in which kΦ θθ is plotted at z = δ θ for each Ra . This figure showsthat the small-scale peak contributes an increasingly larger part of the total energywith increasing Ra . At the same time, the scale separation between the small-scales andthe large-scale superstructures increases with increasing thermal driving. We define thelength scale of the small-scale structures as ˆ l δ = 2 π/ ˆ k δ , where ˆ k δ is the location of thehigh- k peak. Figure 9b shows that ˆ l δ is approximately constant for 10 (cid:54) Ra (cid:54) , whennormalized with the boundary layer thickness δ θ . The magnitude of the ratio ˆ l δ /δ θ differsslightly depending on the quantity considered. The most energetic small-scale structuresfor the temperature are about 11 . δ θ , for w it is about 8 . δ θ , and kΦ θw peaks at about10 δ θ .
4. Conclusion
Contrary to what prior analysis (Stevens et al. et al. the same size exist in thetemperature and vertical velocity fields in large-aspect ratio Rayleigh-B´enard flow. These oherence of superstructures in turbulent Rayleigh-B´enard flow -3 (a) (b) Ra Figure 9. (a) Premultiplied temperature power spectra kΦ θθ at z = δ θ for 10 (cid:54) Ra (cid:54) .Symbols mark the location of the small-scale peak determined as the maximum of kΦ θθ ( k ) for k >
2. Note that at the lower Ra , this peak does not correspond to a global maximum of kΦ θθ .(b) Length scale ˆ l δ associated with the small-scale peak of kΦ θθ (circles), kΦ ww (squares) and kΦ θw (triangles) normalized with the thermal boundary layer thickness δ θ . The dashed line isat 11.5 for reference. result in a very significant large-scale peak in the linear coherence spectrum of θ and w that signifies almost perfect correlation at the large length scales. Unlike it is the casefor θ , we find that the superstructures in w do not correspond to a spectral peak in thepower spectrum of w . This difference has previously led to the above-mentioned confusionregarding potentially different sizes of the largest structures in θ and w . The fact that themost energetic motions, as measured by the peak in the spectra, occur at intermediatescales for w , but at the superstructure scale for θ , can be explained by differences inthe production terms of the respective variance budgets. In particular, temperatureproduction is confined to the boundary layer and small-scales, while buoyancy forcingacts at intermediate scales and throughout the entire bulk of the flow. Furthermore,we find that the superstructure scale increases with Ra for 10 (cid:54) Ra (cid:54) , i.e.the full range investigated here, when the structure size is based on the coherencespectrum as suggested. It should be noted that integral length scales of temperatureand turbulent kinetic energy as used in Stevens et al. (2018) do not accurately capturethis growth, which shows the importance of selecting the appropriate metric to quantifysuperstructures in RBC.In agreement with previous observations of superstructure footprints in the boundarylayer region, we find an almost perfect spatial correlation of the superstructure scales fromthe boundary-layer height δ θ up to mid-height, for both θ and w . Also, the temperaturespectra, as well as γ θw , are seen to collapse at different heights. Hence, there is nonoticeable dependence of the superstructure scale on z , effectively ruling out a significantgrowth of the thermal structures due to horizontal transport while they are travellingupward as was suggested by Pandey et al. (2018). The decrease of spatial correlation(quantified by the linear coherence spectrum) at intermediate scales when increasing thedistance to the reference height δ θ is observed to follow a self-similar trend for w and thehorizontal component v , but not for θ . The reason for this difference remains unclear butwarrants further investigation.Moreover, we find that the energy distribution of the temperature field is bimodal.Besides the z -independent large-scale contribution of the superstructures, premultipliedspectra reveal the existence of a pronounced small-scale peak at boundary-layer height.The two peaks are separated by a spectral gap that increases with Ra , which is also4 D. Krug et al.visible in the co-spectra of θ and w . However, kΦ ww displays a small-scale peak only nearthe wall and is broadband otherwise. For the temperature fluctuations, the small-scalepeak carries the energy that leads to the maximum of (cid:104) θ (cid:105) at z = δ θ (see e.g. Wang et al. l δ ≈ δ θ .It is interesting to note that the situation described here has a close resemblance tofindings in turbulent boundary layers. There an ‘inner peak’ is observed that is fixed at15 viscous units ( l visc ) away from the wall and with typical streamwise length scales ofabout 1000 l visc (Hutchins & Marusic 2007 b ). The scale separation between the ‘innerpeak’ and the large-scale structures is, however, significantly stronger in RBC. Thisappears to suggest distinctly different processes, as was already pointed out in Pandey et al. (2018) in a different context, and raises questions about their interaction. A betterunderstanding of these aspects will be very insightful for modelling approaches.The authors acknowledge stimulating discussions with Woutijn Baars and thankAlexander Blass for help with the data. This work is supported by the Twente Max-Planck Center, the German Science Foundation (DFG) via program SSP 1881, and theERC (the European Research Council) Starting Grant No. 804283 UltimateRB.The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.( ) for funding this project by providing computing time on theGCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre ( ).Part of the work was carried out on the national e-infrastructure of SURFsara, asubsidiary of SURF cooperation, the collaborative ICT organization for Dutch educationand research.Declaration of Interests. The authors report no conflict of interest. Appendix A. Aspect ratio dependence of the superstructure size
The aspect ratio dependence of the superstructures has been studied in Stevens et al. (2018) before. However, there was a bug in the radial averaging of the spectra, whichled to a somewhat altered spectral distribution, especially at the largest scales. In figure10, we present results at Ra = 10 for aspects ratios varying between 3 and 64. Up to Γ = 8 there is no clear peak as the highest values are attained for the smallest k for allquantities displayed. Distinct large-scale peaks emerge for Γ (cid:62)
16 for kΦ θθ (figure 10a), kΦ θw (figure 10c), and γ θw (figure 10d). The locations of these peaks (again obtainedby fitting a parabola over three points) are shown in figure 10. Results based on kΦ θθ and γ θw decrease monotonically with increasing Γ , while ˆ l based on the co-spectrumincreases between Γ = 16 and Γ = 32. The peaks in kΦ ww , and remarkably these spectrain general, exhibit only a minor sensitivity to Γ . The peaks of the velocity spectra remain,however, at intermediate scales significantly smaller than the superstructure size.The conclusion of Stevens et al. (2018) that very large domains are needed to fullyconverge the superstructure size remains valid, even though, at least for the valueschecked here, the trend is opposite (decreasing size) to what was previously believed.The scale ˆ l still varies by about 10% between the cases with Γ = 32 and Γ = 64. Themore basic requirement is however that Γ (cid:39)
16 because only then the large-scale peakis actually resolved.
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