Wavelet analysis of wall turbulence to study large‐scale modulation of small scales

Continuing research on scale interactions in turbulent boundary layers (TBLs) has revealed the modulation of small scales by large-scale motions. These large-scale structures encompass regions of negative and positive fluctuating velocity and are extensively described in the literature (Favre et al. 1967; Blackwelder and Kovasznay 1972; Brown and Thomas 1977; Wark and Nagib 1991; Ganapathisubramani et al. 2003; Tomkins and Adrian 2003; Hutchins and Marusic 2007a, among others). The way by which the amplitude of the large scales influences the small-scale intensity can be considered an amplitude modulation (AM) and was initially identified by Brown and Thomas (1977) and Bandyopadhyay and Hussain (1984). Over the past decade, there has been a considerable refocus on modulation and the underlying scale interactions. It was shown by Hutchins and Marusic (2007a, b) that large-scale modulation is an important feature of high Reynolds number wallbounded flows. Many studies followed in an attempt to fully quantify the modulation (Mathis et al. 2009; Chung and McKeon 2010; Guala et al. 2011; Ganapathisubramani et al. 2012; Duvvuri and McKeon 2015). Generally, in both recent and more dated studies on scale interactions, the turbulent velocity signal has been decomposed into two sets of scales, being the largeand small-scale components; this is conceptually visualized in Fig. 1a for single-point timeresolved data. A single inner-normalized separation scale is typically taken as x = xUτ /ν = 7000 and the local mean velocity, U + = U/Uτ, is used in determining the Abstract Wavelet analysis is employed to examine amplitude and frequency modulations in broadband signals. Of particular interest are the streamwise velocity fluctuations encountered in wall-bounded turbulent flows. Recent studies have shown that an important feature of the nearwall dynamics is the modulation of small scales by largescale motions. Smalland large-scale components of the velocity time series are constructed by employing a spectral separation scale. Wavelet analysis of the small-scale component decomposes the energy in joint time–frequency space. The concept is to construct a low-dimensional representation of the small-scale time-varying spectrum via two new time series: the instantaneous amplitude of the smallscale energy and the instantaneous frequency. Having the latter in a time-continuous representation allows a more thorough analysis of frequency modulation. By correlating the large-scale velocity with the concurrent small-scale amplitude and frequency realizations, both amplitude and frequency modulations are studied. In addition, conditional averages of the small-scale amplitude and frequency realizations depict unique features of the scale interaction. For both modulation phenomena, the much studied time shifts, associated with peak correlations between the large-scale velocity and small-scale amplitude and frequency traces, are addressed. We confirm that the small-scale amplitude signal leads the large-scale fluctuation close to the wall. It is revealed that the time shift in frequency modulation