From Time to Space and Back: Convection and Wave Velocities in Turbulent Shear Flows

Abstract

To obtain spatial spectra, experimentalists map temporal spectra using Taylor’s hypothesis [18]. This transfer function relates wavenumber to frequency using a convection velocity, a velocity that proves difficult to define and use. Adding to this difficulty, convection and wave velocities have been treated the same, despite the obvious difference between them. Convection velocities are velocities at which a specific structure moves—group velocity—and wave velocities are velocities at which a single wave moves—phase velocity. The ideal mapping function to go between wavenumber and frequency is the wavenumber-frequency spectrum that shows a range of wave velocities can contribute to a single wavenumber or a single frequency, and this range can differ significantly between them. To try to capture the influence of this range, experimentalists have applied various peak or average wave velocities with varying success. However, none account for the spread in wave velocities directly. In this paper we propose a two-point cross-spectral approach that uses a distribution of wave velocities to reconstruct the wavenumber-frequency plane. This plane can then be integrated to obtain the spatial spectrum. We verify the technique on particle image velocimetry data set of a turbulent boundary layer, and we obtain a transfer function from this data set. The transfer function is applied to hot-wire data at a comparable Reynolds, and comparison of the newly proposed technique with the classic Taylor’s hypothesis approach shows that Taylor’s hypothesis hold for larger frequencies and wavenumbers the smaller frequencies and wavenumbers (long temporal and spatial scales) there are significant differences.