Amplitude and frequency modulation of the small scales in a jet

Abstract

The present study is an experimental investigation of the relationship between the large- and small-scale motions in the far field of an air jet at high Reynolds number. In the first part of our investigation, the analysis is based on time series of hot-wire anemometry (HWA), which are converted into space series after applying the Taylor hypothesis. By using a spectral filter, two signals are constructed, one representative of the large-scale motions ($2{\it\lambda}_{T}-L$, where ${\it\lambda}_{T}$ is the Taylor length scale, and $L$ is the integral length scale) and the other representative of the small-scale motions ($1.5{-}5{\it\eta}$, where ${\it\eta}$ is the Kolmogorov length scale). The small-scale signal is found to be modulated both in amplitude and in frequency by the energy-containing scales in an analogous way, both at the centreline and around the centreline. In particular, for positive fluctuations of the large-scale signal, the small-scale signal is locally stronger in amplitude (amplitude modulation), and it locally exhibits a higher number of local maxima and minima (frequency modulation). The extent of this modulation is quantified based on the strength of the large-scale fluctuations. The response of the small-scale motions to amplitude modulation can be considered instantaneous, being on the order of one Kolmogorov time scale. In the second part of our investigation we use long-range ${\it\mu}$PIV to study the behaviour of the small-scale motions in relation to their position in either high-speed or low-speed regions of the flow. The spatially resolved velocity vector fields allow us to quantify amplitude modulation directly in physical space. From this direct estimation in physical space, amplitude modulation is only 25 % of the value measured from HWA. The remaining 75 % comes from the fixed spectral band filter used to obtain the large- and small-scale signals, which does not consider the local convection velocity. A very similar overestimation of amplitude modulation when quantified in the time-frame is also obtained analytically. Furthermore, the size of the structures of intense vorticity does not change significantly in relation to the large-scale velocity fluctuation, meaning that there is no significant spatial frequency modulation. The remaining amplitude modulation in space can be explained as a statistical coupling between the strength of the structures of vorticity and their preferential location inside large-scale high-velocity regions. Finally, the implications that the present findings have on amplitude and frequency modulation in turbulent boundary layers (TBLs) are discussed.