Direct numerical simulations of turbulent channel flow at Reτ = 205 and 932 have been carried out to examine Taylor’s “frozen turbulence” hypothesis. The terms in Taylor’s hypothesis appear in the transport equation for instantaneous momentum (Navier-Stokes) in this flow. The additional terms, i.e., the additional convective acceleration term and the pressure gradient and viscous force terms, act to diminish the validity of Taylor’s hypothesis when they are relatively large compared to the Taylor’s hypothesis terms and are not in balance. A similar analysis has been applied to the transport equation for instantaneous vorticity. The additional terms in this equation, namely, the additional convective rates of change of vorticity terms, the stretching/compression/rotation of vorticity terms, and the viscous diffusion of vorticity terms, similarly act to diminish the validity of Taylor’s hypothesis when they are relatively large compared to the terms in the hypothesis and are not in balance. Where in the channel flow this diminishment occurs, and to what degree, and which of the non-Taylor’s hypothesis terms in the momentum and vorticity equations contribute most to this diminishment are unraveled here.
An energy spectrum is preliminarily characterized by its mean and standard deviation. In this study, we derive exact expressions for the means and bandwidths of space-time energy spectra at fixed frequencies. The mean wave numbers are used to determine the phase velocities that bridge from temporal spectra to space-time spectra. The bandwidths are used to measure the well-known spectral broadening. We show that phase velocities alone are insufficient to determine the bandwidths of energy spectra. As a result, the cross-spectral method predicts narrower bandwidths of energy spectra. Therefore, in addition to phase velocities, amplitudes are used to rescale the space-time energy spectra, leading to the correct bandwidths. Existing data from direct numerical simulations of turbulent channel flows validate the rescaling approach.
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