Statistical behaviour of self-similar structures in canonical wall turbulence

Abstract

Townsend's attached-eddy hypothesis (AEH) provides a theoretical description of turbulence statistics in the logarithmic region in terms of coherent motions that are self-similar with the wall-normal distance (y). Here, we show the self-similar behaviour of turbulence motions contained within wall-attached structures of streamwise velocity fluctuations using the direct numerical simulation dataset of turbulent boundary layer, channel, and pipe flows ($R{e}_{\tau }\approx 1000$) The physical sizes of the identified structures are geometrically self-similar in terms of height, and the associated turbulence intensity follows the logarithmic variation in all three flows. Moreover, the corresponding two-dimensional energy spectra are aligned along a linear relationship between the streamwise and spanwise wavelengths (${\lambda }_x$ and ${\lambda }_z$, respectively) in the large-scale range ($12y<{\lambda }_x<$ 3--4$\delta$), which is reminiscent of self-similarity. Consequently, one-dimensional spectra obtained by integrating the two-dimensional spectra over the self-similar range show some evidence for self-similar scaling ${\lambda }_x\sim {\lambda }_z$ and the possible existence of $k_x^{-1}$ and $k_z^{-1}$ scaling regions in a similar subrange. The present results reveal that the asymptotic behaviours can be obtained by identifying the self-similar coherent structures in canonical wall turbulence, albeit in low Reynolds number flows.

arXiv comment24 pages, Journal of Fluid Mechanics (in press)

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• Wall-attached structures of velocity fluctuations in a turbulent boundary layer
• Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 1. Energy spectra
• Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 2. Integrated energy and $A_1$
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Cited by in corpus (5)

• Active and inactive components of the streamwise velocity in wall-bounded turbulence
• Self-similar mechanisms in wall turbulence studied using of resolvent analysis
• Data-driven enhancement of coherent structure-based models for predicting instantaneous wall turbulence
• Model-based spectral coherence analysis
• Constructing wall turbulence using hierarchical hairpin vortices