Transition to ultimate RayleighBénard turbulence revealed through extended self similarity scaling analysis of the temperature structure functions
Abstract
In turbulent RayleighBénard (RB) convection, a transition to the socalled ultimate regime, in which the boundary layers (BL) are of turbulent type, has been postulated. Indeed, at very large Rayleigh number a transition in the scaling of the global Nusselt number (the dimensionless heat transfer) and the Reynolds number with has been observed in experiments and very recently in direct numerical simulations (DNS) of twodimensional (2D) RB. In this paper we analyse the local scaling properties of the lateral temperature structure functions in the BLs of this simulation of 2D RB, employing extended selfsimilarity (ESS) (i.e., plotting the structure functions against each other, rather than only against the scale) in the spirit of the attached eddy hypothesis, as we have recently introduced for velocity structure functions in wall turbulence (Krug et al., J. Fluid Mech., vol. 830, 2017, pp. 797819). We find no ESS scaling below the transition and in the near wall region. However, beyond the transition and for large enough wall distance , we find clear ESS behaviour, as expected for a scalar in a turbulent boundary layer. In striking correspondence to the scaling, the ESS scaling region is negligible at and well developed at , thus providing strong evidence that the observed transition in the global Nusselt number at indeed is the transition from a laminar type BL to a turbulent type BL. Our results further show that the relative slopes for scalar structure functions in the ESS scaling regime are the same as for their velocity counterparts extending their previously established universality. The findings are confirmed by comparing to scalar structure functions in 3D turbulent channel flow.
Key words:
1 Introduction
Thermal convection is relevant to a wide range of applications across various fields such as building ventilation (e.g. Linden 1999), atmospheric (e.g. Hartmann et al. 2001) or oceanic (e.g. Rahmstorf 2000) flows. The phenomenon is widely studied in terms of the paradigmatic case of RayleighBénard (RB) convection (see the reviews of Ahlers et al. 2009; Lohse & Xia 2010; Chillà & Schumacher 2012), in which two horizontal plates at a distance are cooled from above and heated from below. In the presence of a gravitational acceleration , a flow is driven with properties (for given Prandtl number and aspect ratio ) that are controlled by the Rayleigh number . Here, and are kinematic viscosity and the thermal expansion coefficient, respectively, is some lateral length scale and denotes the temperature difference between the bottom (held at ) and top () plates. The primary response parameter of the system is the resulting heat flux, which in its nondimensional form is given by the Nusselt number relating the actual heat transfer to the one in the purely conductive case. The behaviour of the system at very high is of interest in many applications, and theoretical work (Kraichnan 1962; Spiegel 1971; Grossmann & Lohse 2000, 2011) predicts the existence of a so called ‘ultimate regime’, in which the scaling switches from the classical (Malkus 1954) to . This transition is related to a change of the boundary layer (BL) structure, of both velocity and temperature, from a laminar to turbulent type. A transition in the scaling —as well as in the Reynolds number scaling— has indeed been observed experimentally, most convincingly by He et al. (2012b, a, 2015) and very recently also in a direct numerical simulation (DNS) of twodimensional (2D) RB by Zhu et al. (2018). Results of the latter are reproduced in figure 1a, where the transition is evident from the change in slope of the (compensated) at . As expected from the theory and indicative of the turbulent nature of the BLs, Zhu et al. (2018) found logarithmic dependencies with respect to the distance from the wall for the mean temperature and velocity BLs. In the following, we will characterize the thermal BLs further by studying their lateral structure functions.
Such an analysis originated from Davidson et al. (2006), who pointed out that the spectral scaling ( being the streamwise wavenumber) of the streamwise velocity power spectrum predicted by the attachededdy hypothesis (Townsend 1976; Perry & Chong 1982), is equivalent to —and more readily observed as— a scaling of the second order longitudinal structure function. Here and in the following, denotes the streamwise separation distance and is the distance off the wall. de Silva et al. (2015) found that the scaling also applies to velocity structure functions of arbitrary even order such that for the socalled energycontaining range ( being the boundary layer thickness)
(1.0) 
Here is the velocity increment between two points at a distance off the wall separated by along the streamwise direction, is the mean friction velocity and , are constants. We use superscript to denote quantities relating to the velocity and when referring to the scalar later on. While the direct scaling according to (1) is only observed at relatively large Reynolds numbers , de Silva et al. (2017) and Krug et al. (2017) have shown that a relative scaling is evident at much smaller if the so called extended selfsimilarity (ESS) framework is employed. In this case, the scaling is not analysed as a function of but — in the spirit of the original ESS hypothesis by Benzi et al. (1993, 1995) — relative to a structure function of different order. For an arbitrary reference order this results in the ’ESS form’ of (1), namely
(1.0) 
de Silva et al. (2017) and Krug et al. (2017) found that the linear scaling of (1) could not only be observed at relatively low well within the capabilities of current DNS, but also that the relative slopes exhibit nontrivial, i.e. nonGaussian, universal behaviour across various flow geometries such as flat plate boundary layers, pipe and channel flow and even TaylorCouette flow. From our experience in TaylorCouette flow (Krug et al. 2017) we learned that ESSscaling according to (1) is not observed if large scale structures in the bulk (such as the Taylor rolls) contribute to the velocity component under investigation. This is also the case for the wall parallel velocity component in RB, as can be seen from the snapshots in figures 1(a,b). Since no equivalent scaling exists for the wallnormal component (Perry & Chong 1982), the 2D velocity field in RB is ruled out as a suitable candidate for this scaling analysis. Therefore the current study instead focuses on structure functions of the temperature field
(1.0) 
Here, and are analogous to and , respectively. ESSscaling of scalar structure functions in the energy containing range has not been considered yet. It is however conceivable that the situation will be similar to velocity structure functions: The theory underlying (1) and (1) is based on an inertial assumption, which implies that momentum transport is dominated by turbulent eddies that are larger than and therefore constrained by (where is the Kolmogorov length scale) but smaller than . In the spirit of the Reynolds analogy and for these eddies can be assumed to affect momentum and scalar similarly. Consequently, the relevant arguments based on the attached eddy hypothesis (see de Silva et al. 2015; Krug et al. 2017) can be expected to transfer to the scalar field as well. It is therefore our objective here to investigate whether and at what the thermal boundary layers in RB convection exhibit an ESS scaling according to
(1.0) 
The expectation is that such a scaling regime should originate coinciding with the scaling transition, which has been link to the emergence of logarithmic boundary layers (Grossmann & Lohse 2011). Before moving ahead, we would like to mention that the attached eddy framework, from which (1) and (1) can be derived, is not new in the RB context. It has already been referred to by Ahlers et al. (2014), who investigate logarithmic dependencies of temperature profiles, and by He et al. (2014) in a study of temperature power spectra scalings the spatial equivalent to the scaling) in order to interpret the observations made.
Since the data set of Zhu et al. (2018) that will be employed for this endeavour is 2D, we will also check our results by using a three dimensional channel simulation as a reference and for comparison. A brief overview over both datasets will given in §2, before presenting our results in §3 and conclusions in §4.
2 Datasets
The simulations of Zhu et al. (2018) where performed using the secondorder finitedifference code AFiD (van der Poel et al. 2015) on a twodimensional domain with periodic boundary conditions in the lateral direction with and . While the original dataset spans six decades , only four data points for are used here (see figure 1a). We refer to the original publication for further details on the numerical setup and validation of the results. One of the particularities of the 2D setup is that the large scale structures, which can be observed in the vector maps of figures 1(a,b), remain almost fixed in place allowing for simple temporal averaging. Since the temporal mean velocity gradient changes sign along the plate, we take the spatial mean of its absolute value when computing in RB convection. For the present simulations we obtain at , respectively.
A DNS of channel flow was performed using the fourthorder code described in Chung et al. (2014) at , where is half the channel height. The periodic (in streamwise and spanwise directions) box of size was discretised by grid points in the streamwise, spanwise, and wallnormal directions, respectively. A passive scalar with was added with values fixed to at the bottom and at the top wall as boundary conditions.
Convergence of the statistics computed from both datasets was checked by plotting the premultiplied probability density function of at various and found acceptable up to tenth order.
3 Results
3.1 Direct analysis of the scaling in the energycontaining range in RB convection
We begin by plotting structure functions of second, fourth and tenth order from 2D RB convection as a function of in figure 2. For all cases, we present results at three distances from the wall, namely (figure 2ac), (figure 2df) and (figure 2gi), where as usual the superscript indicates normalization by . Clearly, there is no scaling according to a scalar equivalent of (1) for either at the position closest to the wall (figure 2ac). However, in the other cases an approximately linear region appears for making it tempting to fit the slopes directly. A fitting range (indicated by crosses in the figure) captures this region quite well for all and the resulting fits are shown as red dashed lines. No fits are computed at the lowest where the fitting range becomes prohibitively small.
A detailed comparison of the values of obtained in this way (figure 3a) reveals that the results depend on both and in the investigated range. Generally, values of are higher at and this difference becomes larger with increasing . In most cases, is very low and it is only at and (and small ) that the values of are of comparable magnitude to results for in highRe turbulent boundary layers (TBLs). We point out that the decrease of with increasing observed in some cases at higher is unphysical and likely related to the insufficient fitting range at higher orders. We further emphasize that even at high a direct match between and is not necessarily expected. It is well established (see e.g. Warhaft 2000, for passive scalars) that intermittency in the inertial range is higher for scalars as compared to the velocity itself. This translates to lower scaling exponents in the corresponding scaling relation as compared to the velocity counterpart . From matching the inertial scaling with (1) at de Silva et al. (2015) semiempirically derived a linear relationship between and , suggesting that (disregarding other dependencies) these differences may persist in the energycontaining range investigated here. However, a definitive answer to this question will have to be based on high data of the scalar field in wallbounded turbulent flows.
While the direct analysis of the slopes remains inconclusive, additional insight can be gained from studying the second order structure function . In this case, the difference between the structure function at two different separation distances and can be interpreted as the contribution of eddies with sizes in the range between and to the overall (scalar) energy. Consequently, when using the bounds of the loglinear scaling regime for , as indicated in figure 2d, this increment , characterizes the contribution of the energycontaining range to the overall energy. Note that here we adopted the bounds of the fitting region used above for for simplicity, but other reasonable choices give qualitatively similar results. Figure 3(b), where is normalized by the result obtained at , shows that this quantity only increases mildly at low . However, it rises steeply between and consistent with transitional behaviour in this range. No significant differences arise between and in this case.
3.2 Scalar ESS scaling in RB convection
We now focus on the relative scalings according to (1). The ESS framework has been demonstrated to extend the scaling regime not only to low but also to a wider range of wallnormal distances. In particular, Krug et al. (2017) found convincing scaling for as low as . From figure 4ac, it is clear that the same does not hold for the scalar structure functions in RB convection. Even at the highest , there is no linear relationship between and for all orders considered at this location.



The situation improves at (figure 4df), where particularly at the highest and low orders the curve begin to exhibit an approximately linear range. However, deviations become more apparent with increasing and results at the higher orders in figures 4(e,f) demonstrate that the ESS scaling is not yet fully attained at this position. It is only at (figures 4gi) that a convincing ESS scaling according to (1) is obtained up to tenth order. The scaling range is well established at , already significantly decreases in size at and is basically nonexistent at . This behaviour is in very good correspondence to the changes in the scaling in figure 1a, corroborating that the change in scaling observed there is indeed due to a transition in the BL structure from laminar (no scaling) to turbulent (with ESS scaling). It should be noted that, just as for velocity structure functions, ESS scaling at is observed for for all , i.e. the spatial scaling range is the same. However, consistent with the results in figures 2 and 3(b), there is hardly any energy in this range at below transition to the ultimate regime. Remarkably, the relative slopes attained for the temperature structure functions in 2D RB at appear to be the same as those measured for their velocity counterparts and reported in de Silva et al. (2017). This is remarkable since as pointed out in §3.1, the directly measured slopes and need not be the same. So this feature seems to be one of scalar fields.
3.3 Scalar ESS scaling in channel flow
The question remains why ESS scaling is only observed at a larger distance from the wall in the present case as compared to previous findings for velocity structure functions. At this point, possible explanations to be considered are that this might be either a consequence of the 2D setup, a property of RB convection or a feature of the scalar field. To address this, we present ESS results of scalar structure functions in turbulent channel flow in figure 5(ac) at different orders. Indeed, the results are very similar to what was observed for 2D RB before in that there is no ESS scaling for and . And again ESS scaling is recovered at with relative slopes matching those measured for velocity structure functions mirroring the observations made for RB convection.
4 Discussion and Conclusions
We have analysed the temperature boundary layers in the energycontaining range of 2D RB convection by means of temperature structure functions. Even though the RB structure functions exhibit loglinear scaling for when plotted against separation distance, the slopes remain small at low and they were found to vary significantly with both and , rendering the analysis inconclusive in this point. While a dependence of the slopes on and wallnormal position is also observed for velocity structure functions (see Krug et al. 2017), typically the loglinear scaling is much less evident in these cases compared to figure 2 at . Also for the scalar in channel flow investigated here (plots not shown) a direct scaling regime is not discernible such that it appears likely that the more prominent loglinear regimes in figure 2 are a consequence of the 2D setup. An important point remains however, that the contribution of the energycontaining range to the total energy increases significantly beyond , which coincides with the transition in vs. scaling.
The main finding of the paper is the clear evidence that the temperature structure function in the BLs of turbulent RB flows exhibit ESS scaling in the energycontaining range for large enough wall distances . The extent of the scaling range thereby reflects the behaviour of very well in increasing from nonexistent at (well in the classical regime) to considerable beyond the scaling transition at . This provides further evidence that said transition is indeed related to a switch from laminar type to turbulent type thermal BLs.
Moreover, we establish that the relative slopes for scalar structure functions in the ESSform are the same as those previously obtained when analysing velocity BLs. This is confirmed by comparing the RB results to those obtained in a planar channel geometry. In both cases, ESS scaling is only established at which is different from velocity structure functions and appears to be a feature of the scalar field. Interestingly enough, already Perry & Chong (1982) point out differences between the scalar and the velocity field. They argue that at the end of the ‘lifetime’ of an eddy, the vorticity contributions in the two rods of the assumed hairpin cancel such that no induced velocity field remains. However, such a cancellation does not apply for the scalar transported by the eddy, such that the ‘debris’, as they call it, of past eddies sets up a scalar background profile. If and how exactly this is related to the observations made here remains however unclear.
At least for the present cases, there also appears to be no difference between active (in RB flow) and passive (in the channel flow) scalars. In other regards, the deviations close to the wall underline in addition to the fact the observed values of are subGaussian (Krug et al. 2017) that the ESS scaling and the values of the relative slopes are indeed nontrivial.
The work was financially supported by NWOI and the Netherlands Center for Multiscale Catalytic Energy Conversion (MCEC), both sponsored by the Netherlands Organization for Scientific Research (NWO). Part of the simulations were carried out on the Dutch national einfrastructure with the support of SURF Cooperative. We also acknowledge PRACE for awarding us access to Marconi based in Italy at CINECA under PRACE Project No. 2016143351 and the DECI resource Archer based in the United Kingdom at Edinburgh with support from the PRACE aisbl under Project No. 13DECI0246. We also acknowledge the support of the Australian Research Council and the McKenzie Fellowship program at the University of Melbourne.
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