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

Within the chaotic flow field of wall-bounded turbulence, there exist structures that are coherent in space and time. Gaining a mechanistic understanding of wall-bounded turbulence necessitates characterizing these coherent structures. According to Townsend’s attached eddy hypothesis (AEH), these coherent structures are self-similar in the log layer. Subsequent models, such as the attached eddy model, propose these structures are also spatially localized.

The most popular method for representing coherent structures in a flow field is proper orthogonal decomposition (POD), which produces a set of energetically ordered basis elements derived from data; however, in statistically homogeneous directions, POD basis elements are Fourier modes, which are undesirable for representing spatially localized structures. On the other hand, a traditional wavelet decomposition (TWD) provides basis elements that are multiscale and spatially localized; however, the basis is not derived from data and has self-similarity built into it.

We combine features of POD and TWD to obtain data-driven wavelet decomposition (DDWD), which extracts energetic and spatially localized structures from data. We apply DDWD to turbulent pipe flow at a friction Reynolds number of 12,400. We find self-similar, spatially localized structures in the streamwise range of 40–450 wall units to 1 pipe radii and the wall-normal range of 350 wall units to 1 pipe radii, which is consistent with other studies.