Synchronization to Big Data: Nudging the Navier-Stokes Equations for Data Assimilation of Turbulent Flows

Abstract

Nudging is an important data assimilation technique where partial field measurements are used to control the evolution of a dynamical system and/or to reconstruct the entire phase-space configuration of the supplied flow. Here, we apply it to the canonical problem of fluid dynamics: three-dimensional homogeneous and isotropic turbulence. By doing numerical experiments we perform a systematic assessment of how well the technique reconstructs large-and small-scale features of the flow with respect to the quantity and the quality or type of data supplied to it. The types of data used are (i) field values on a fixed number of spatial locations (Eulerian nudging), (ii) Fourier coefficients of the fields on a fixed range of wave numbers (Fourier nudging), or (iii) field values along a set of moving probes inside the flow (Lagrangian nudging). We present state-of-the-art quantitative measurements of the scale-by-scale transition to synchronization and a detailed discussion of the probability distribution function of the reconstruction error, by comparing the nudged field and the truth point by point. Furthermore, we show that for more complex flow configurations, like the case of anisotropic rotating turbulence, the presence of cyclonic and anticyclonic structures leads to unexpectedly better performances of the algorithm. We discuss potential further applications of nudging to a series of applied flow configurations, including the problem of field reconstruction in thermal Rayleigh-Bénard convection and in magnetohydrodynamics, and to the determination of optimal parametrization for small-scale turbulent modeling. Our study fixes the standard requirements for future applications of nudging to complex turbulent flows.

Synchronization of low Reynolds number plane Couette turbulence

“…dissipation lengths can be slaved by the larger-scale motions. Similar estimates were obtained by Yoshida, Yamaguchi & Kaneda (2005) and Clark Di Leoni, Mazzino & Biferale (2020) in their investigations of data assimilation of small scales from knowledge of the large scales in IHT. Henshaw et al (2003) also conjectured that the disparity found in the size of the synchronizable subspaces should be attributed to the enhanced dissipation that occurs in the strong deformation regions forming in the flow, which is not captured in the analysis of Foias & Prodi (1967) and Ladyzhenskaya (1975).…”

The synchronisation of intense vorticity in isotropic turbulence

“…The synchronisation of chaotic systems is a well-known and widely studied phenomenon with important consequences and applications in the natural sciences and engineering (Pecora & Carroll 1990). See Di Leoni, Mazzino & Biferale (2020) for a recent application to turbulence. In general, two or more chaotic systems are said to be synchronised when all, or some, of their state variables evolve towards a common spatio-temporal pattern under the action of a driving force or coupling.…”

State estimation in turbulent channel flow from limited observations

“…if more information about the flow state is available from PIV data. Recent state estimation tests in homogeneous isotropic turbulence (Yoshida, Yamaguchi & Kaneda 2005;Lalescu, Meneveau & Eyink 2013;Di Leoni, Mazzino & Biferale 2019;Li et al 2020) demonstrated that the reconstruction of turbulence is successful only when the highest wavenumber k m of velocity data satisfies k m η > 0.2, where η is the Kolmogorov scale of the flow. In wall turbulence, however, flow inhomogeneity in the wall-normal direction, the wall-normal dependence of mean advection and the turbulence production all preclude adopting the same criterion from the homogeneous case.…”