Turbulence spectra in smooth- and rough-wall pipe flow at extreme Reynolds numbers

Abstract

Well-resolved streamwise velocity spectra are reported for smooth-and rough-wall turbulent pipe flow over a large range of Reynolds numbers. The turbulence structure far from the wall is seen to be unaffected by the roughness, in accordance with Townsend's Reynolds number similarity hypothesis. Moreover, the energy spectra within the turbulent wall region follow the classical inner and outer scaling behaviour. While an overlap region between the two scalings and the associated k −1 x law are observed near R + ≈ 3000, the k −1x behaviour is obfuscated at higher Reynolds numbers due to the evolving energy content of the large scales (the very-large-scale motions, or VLSMs). We apply a semi-empirical correction (delÁlamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5-26) to the experimental data to estimate how Taylor's frozen field hypothesis distorts the pseudo-spatial spectra inferred from time-resolved measurements. While the correction tends to suppress the long wavelength peak in the logarithmic layer spectrum, the peak nonetheless appears to be a robust feature of pipe flow at high Reynolds number. The inertial subrange develops around R + > 2000 where the characteristic k −5/3x region is evident, which, for high Reynolds numbers, persists in the wake and logarithmic regions. In the logarithmic region, the streamwise wavelength of the VLSM peak scales with distance from the wall, which is in contrast to boundary layers, where the superstructures have been shown to scale with boundary layer thickness throughout the entire shear layer. Moreover, the similarity in the streamwise wavelength scaling of the large-and very-large-scale motions supports the notion that the two are physically interdependent.

The inertial subrange in turbulent pipe flow: centreline

“…The datasets from which the present results derive (250 × 10 3 Re D 6.0 × 10 6 ) are those previously reported by Vallikivi et al (2011);Rosenberg et al (2013); Hultmark et al (2013);Vallikivi (2014). Structure functions (up to third-order moments) and spectra are calculated using time series of either 60 s or 90 s duration: the latter have previously been presented by Rosenberg et al (2013), and their scaling at lower wavenumbers examined by Vallikivi et al (2015). Full details of the experiments appear in Vallikivi (2014).…”

“…Dissipation spectra (see Rosenberg et al 2013, figure 3) show two decades of inertial subrange. All data conform to the criterion for a "first-order" subrange proposed by Bradshaw (1967), Re λ > 100.…”

Spectral analysis of near-wall turbulence in channel flow at Reτ=4200 with emphasis on the attached-eddy hypothesis

“…As a consequence, the energy profile no longer scales with u τ and the logarithmic variation of u ′ u ′ + in the velocity log-law region is lost or becomes highly indistinct, at least at Re τ < 5000. As observed by Vassilicos et al 23 , this behaviour is incompatible with the AEH, and this led them to propose a new spectral range of the form 15 display a tendency for u ′ u ′ + to re-establish a logarithmic-decay variation well beyond the meso-layer of the velocity log-law and also well beyond the position of maximum largescale energy -although with a slope different from that applicable within the velocity log-law layer at much lower Reynolds-number values. This has led to a proposition that the AEH should only apply in the extreme outer layer of the velocity log-law region, on the grounds that "determining the extend of the logarithmic layer from U + alone is difficult because of the slow departure from any log law" (Smits et al 27 ).…”

“…The outer portion of the meso-layer, beyond the region in which the streamwise energy features an inflection region or a (second) maximum , is populated with very large scale motions (VLSMs), and it is this layer that has been the focus of attention in studies of Vassilicos et al 23 , Hutchins at al 13 , Hultmark et al 14 and Rosenberg et al 15 , at Re τ > 7000, who show that the streamwise energy displays a logarithmic-decay variation at the same slope as that in figure 1. The variation of the energy in the outer layer, 500 < y + < 1000, suggests a log-like decay, especially at the high Reynolds value, and the slope of this decay is given in figure 1.…”

Quantifying wall turbulence via a symmetry approach. Part 2. Reynolds stresses

“…where A 11 , A 33 , B 11 , B 12 , B 22 and B 33 are all constants, + denotes values in wall units, and y is the wall distance normalized by half-channel height or pipe radius R. This hypothesis has received acute attention recently (Marusic & Kunkel 2003;Davidson & Krogstad 2009;Meneveau & Marusic 2013;Vassilicos et al 2015;Laval et al 2017), and has been tested and even further developed against more accurate measurements -both DNS and experiment (Morrison, McKeon & Smits 2004;Hultmark et al 2012;Rosenberg et al 2013;Sillero, Jimenez & Moser 2013;Lee & Moser 2015;Willert et al 2017). Although Perry & Chong (1982) and Perry, Henbest & Chong (1986) further developed a derivation of the u u log profile by invoking the k −1 x spectrum, such a spectrum was found only at small Reynolds number (Re τ 3300) in Princeton/ONR Superpipe (Rosenberg et al 2013); in contrast, the log profile of u u is observed only at high Reynolds number (Re τ 20 000) . Also, for channels, Lee & Moser (2015) found the k −1 x spectrum in their simulation (Re τ ≈ 5200), but no log profile of u u .…”