The logarithmic variance of streamwise velocity and conundrum in wall turbulence

Abstract

The logarithmic dependence of streamwise turbulence intensity has been observed repeatedly in recent experimental and direct numerical simulation data. However, its spectral counterpart, a well-developed $k^{-1}$ spectrum ( $k$ is the spatial wavenumber in a wall-parallel direction), has not been convincingly observed from the same data. In the present study, we revisit the spectrum-based attached eddy model of Perry and co-workers, who proposed the emergence of a $k^{-1}$ spectrum in the inviscid limit, for small but finite $z/\delta$ and for finite Reynolds numbers ( $z$ is the wall-normal coordinate, and $\delta$ is the outer length scale). In the upper logarithmic layer (or inertial sublayer), a reexamination reveals that the intensity of the spectrum must vary with the wall-normal location at order of $z/\delta$ , consistent with the early observation argued with ‘incomplete similarity’. The streamwise turbulence intensity is subsequently calculated, demonstrating that the existence of a well-developed $k^{-1}$ spectrum is not a necessary condition for the approximate logarithmic wall-normal dependence of turbulence intensity – a more general condition is the existence of a premultiplied power-spectral intensity of $O(1)$ for $O(1/\delta ) < k < O(1/z)$ . Furthermore, it is shown that the Townsend–Perry constant must be weakly dependent on the Reynolds number. Finally, the analysis is semi-empirically extended to the lower logarithmic layer (or mesolayer), and a near-wall correction for the turbulence intensity is subsequently proposed. All the predictions of the proposed model and the related analyses/assumptions are validated with high-fidelity experimental data (Samie et al., J. Fluid Mech., vol. 851, 2018, pp. 391–415).

Batchelor Prize Lecture: Measurements in wall-bounded turbulence

“…The first overlap region yields a k −1 variation, and the second a k −5/3 variation, where k is the streamwise wavenumber. These results are in line with 'classical' expectations, but high Reynolds number experiments at Princeton failed to confirm such trends (Morrison et al 2002(Morrison et al , 2004Rosenberg et al 2013;Vallikivi, Ganapathisubramani & Smits 2015a), as did the most recent measurements in the Melbourne tunnel (Hwang, Hutchins & Marusic 2021). It appears that the existence of a well-developed k −1 spectrum is not a necessary condition for the presence of a logarithmic wall-normal dependence of the turbulence intensity.…”

“…2002, 2004; Rosenberg et al. 2013; Vallikivi, Ganapathisubramani & Smits 2015 a ), as did the most recent measurements in the Melbourne tunnel (Hwang, Hutchins & Marusic 2021). It appears that the existence of a well-developed spectrum is not a necessary condition for the presence of a logarithmic wall-normal dependence of the turbulence intensity.…”

“…Hwang et al. (2021) also suggested, based on their analysis of the data, that the Townsend–Perry constant is weakly dependent on Reynolds number.…”

“…It appears that the existence of a well-developed k −1 spectrum is not a necessary condition for the presence of a logarithmic wall-normal dependence of the turbulence intensity. Hwang et al (2021) also suggested, based on their analysis of the data, that the Townsend-Perry constant is weakly dependent on Reynolds number.…”

Self-similar, spatially localized structures in turbulent pipe flow from a data-driven wavelet decomposition

“…Since our Superpipe data has , a power law is not expected to appear clearly. Moreover, Hwang, Hutchins & Marusic (2022) showed that even without the presence of a power law, the framework of the spectral AEM can still successfully yield a log law for , which is a key feature of wall-bounded turbulence. Therefore, within the context of the spectral AEM, the existence of a power law may not be a necessary feature of wall-bounded turbulence.…”