Spatio-temporal spectra in the logarithmic layer of wall turbulence:   large-eddy simulations and simple models
  Motivated by the need to characterize the spatio-temporal structure of turbulence in wall-bounded flows, we study wavenumber-frequency spectra of the streamwise velocity component based on large-eddy simulation (LES) data. The LES data are used to measure spectra as a function of the two wall-parallel wavenumbers and the frequency in the equilibrium (logarithmic) layer. We then reformulate one of the simplest models that is able to reproduce the observations: the random sweeping model with a Gaussian large-scale fluctuating velocity and with additional mean flow. Comparison with LES data shows that the model captures the observed temporal decorrelation, which is related to the Doppler broadening of frequencies. We furthermore introduce a parameterization for the entire wavenumber-frequency spectrum $E_{11}(k_1,k_2,\omega;z)$, where $k_1$, $k_2$ are the streamwise and spanwise wavenumbers, $\omega$ is the frequency and $z$ is the distance to the wall. The results are found to be in good agreement with LES data. 
Michael Wilczek, Richard J.A.M. Stevens and Charles Meneveau
physics.flu-dyn