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Nudging is an important data assimilation technique where partial fieldmeasurements are used to control the evolution of a dynamical system and/or toreconstruct the entire phase-space configuration of the supplied flow. Here, weapply it to the toughest problem in fluid dynamics: three dimensionalhomogeneous and isotropic turbulence. By doing numerical experiments we performa systematic assessment of how well the technique reconstructs large- andsmall-scales features of the flow with respect to the quantity and thequality/type of data supplied to it. The types of data used are: (i) fieldvalues on a fixed number of spatial locations (Eulerian nudging), (ii) Fouriercoefficients of the fields on a fixed range of wavenumbers (Fourier nudging),or (iii) field values along a set of moving probes inside the flow (Lagrangiannudging). We present state-of-the-art quantitative measurements of thescale-by-scale {\it transition to synchronization} and a detailed discussion ofthe probability distribution function of the reconstruction error, by comparingthe nudged field and the {\it truth} point-by-point. Furthermore, we show thatfor more complex flow configurations, like the case of anisotropic rotatingturbulence, the presence of cyclonic and anticyclonic structures leads tounexpectedly better performances of the algorithm. We discuss potential furtherapplications of nudging to a series of applied flow configurations, includingthe problem of field-reconstruction in thermal Rayleigh-B\'enard convection andin magnetohydrodynamics (MHD), and to the determination of optimalparametrisation for small-scale turbulent modeling. Our study fixes thestandard requirements for future applications of nudging to complex turbulentflows.

Synchronization to big-data: nudging the Navier-Stokes equations for data assimilation of turbulent flows