Statistics of turbulence in the energy-containing range of Taylor–Couette compared to canonical wall-bounded flows

Abstract

Considering structure functions of the streamwise velocity component in a framework akin to the extended self-similarity hypothesis (ESS), de Silva et al. (J. Fluid Mech., vol. 823,2017, pp. 498-510) observed that remarkably the large-scale (energy-containing range) statistics in canonical wall bounded flows exhibit universal behaviour. In the present study, we extend this universality, which was seen to encompass also flows at moderate Reynolds number, to Taylor-Couette flow. In doing so, we find that also the transversal structure function of the spanwise velocity component exhibits the same universal behaviour across all flow types considered. We further demonstrate that these observations are consistent with predictions developed based on an attached-eddy hypothesis. These considerations also yield a possible explanation for the efficacy of the ESS framework by showing that it relaxes the self-similarity assumption for the attached eddy contributions. By taking the effect of streamwise alignment into account, the attached eddy model predicts different behaviour for structure functions in the streamwise and in the spanwise directions and that this effect cancels in the ESS-framework -both consistent with the data. Moreover, it is demonstrated here that also the additive constants, which were previously believed to be flow dependent, are indeed universal at least in turbulent boundary layers and pipe flow where high-Reynolds number data are currently available.

Transition to ultimate Rayleigh–Bénard turbulence revealed through extended self-similarity scaling analysis of the temperature structure functions

“…Even though the RB structure functions exhibit log-linear scaling for r/z 1 when plotted against separation distance, the slopes remain small at low Ra and they were found to vary significantly with both Ra and z + , rendering the analysis inconclusive in this point. While a dependence of the slopes on Re τ and wall-normal position is also observed for velocity structure functions (see Krug et al 2017), typically the log-linear scaling is much less evident in these cases compared to figure 2 at z + 100. 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 log-linear regimes in figure 2 are a consequence of the 2D setup.…”

“…The ESS framework has been demonstrated to extend the scaling regime not only to low Re but also to a wider range of wall-normal distances. In particular, Krug et al (2017) found convincing scaling for D u p /D u 1 as low as z + = 30. From figure 4a-c, it is clear that the same does not hold for the scalar structure functions in RB convection.…”

“…We use superscript u to denote quantities relating to the velocity and θ when referring to the scalar later on. While the direct scaling according to (1.1) is only observed at relatively large Reynolds numbers Re τ = δU τ /ν ∼ O(10 4 ), de Silva et al (2017) and Krug et al (2017) have shown that a relative scaling is evident at much smaller Re τ if the so called extended self-similarity (ESS) framework is employed. In this case, the scaling is not analysed as a function of r but -in the spirit of the original ESS hypothesis by Benzi et al (1993Benzi et al ( , 1995 -relative to a structure function of different order.…”

“…In the spirit of the Reynolds analogy and for P r ≈ 1 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 Ra the thermal boundary layers in RB convection exhibit an ESS scaling according to…”

Hierarchical random additive model for the spanwise and wall-normal velocities in wall-bounded flows at high Reynolds numbers

“…For example, del Alamo et al [20] found evidence of wall-scaling in two-dimensional spectra for all three velocity components. Furthermore, Krug et al [21] showed universality across a wide range of flows for the streamwise and spanwise velocities using an extended form (i.e. the ratio between two structure functions of different orders).…”