The spatio-temporal spectrum of turbulent flows.

Identification and extraction of vortical structures and of waves in a disorganised flow is a mayor challenge in the study of turbulence. We present a study of the spatio-temporal behavior of turbulent flows in the presence of different restitutive forces. We show how to compute and analyse the spatio-temporal spectrum from data stemming from numerical simulations and from laboratory experiments. Four cases are considered: homogeneous and isotropic turbulence, rotating turbulence, stratified turbulence, and water wave turbulence. For homogeneous and isotropic turbulence, the spectrum allows identification of sweeping by the large-scale flow. For rotating and for stratified turbulence, the spectrum allows identification of the waves, precise quantification of the energy in the waves and in the turbulent eddies, and identification of physical mechanisms such as Doppler shift and wave absorption in critical layers. Finally, in water wave turbulence the spectrum shows a transition from gravity-capillary waves to bound waves as the amplitude of the forcing is increased.