Although major advances have been achieved over the past decades for the reduction and identification of linear systems, deriving nonlinear low-order models still is a challenging task. In this work, we develop a new data-driven framework to identify nonlinear reduced-order models of a fluid by combining dimensionality reductions techniques (e.g. proper orthogonal decomposition) and sparse regression techniques from machine learning. In particular, we extend the sparse identification of nonlinear dynamics (SINDy) algorithm to enforce physical constraints in the regression, namely energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a full-order or high-fidelity solver to project the Navier-Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed measurement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of Galerkin regression is demonstrated on two different flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favorably against reduced-order models obtained from a standard Galerkin projection procedure. Present results highlight the importance of cubic nonlinearities in the construction of accurate nonlinear low-dimensional approximations of the flow systems, something which cannot be readily obtained using a standard Galerkin projection of the Navier-Stokes equations. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online. arXiv:1611.03271v2 [physics.flu-dyn]

. Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2014, 760, pp.175-211. 10.1017/jfm.2014.589. hal-01086744 Science Arts & Métiers (SAM)is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. The linear global instability and resulting transition to turbulence induced by an isolated cylindrical roughness element of height h and diameter d immersed within an incompressible boundary layer flow along a flat plate is investigated using the joint application of direct numerical simulations and fully three-dimensional global stability analyses. For the range of parameters investigated, base flow computations show that the roughness element induces a wake composed of a central low-speed region surrounded by a threedimensional shear layer and a pair of low-and high-speed streaks on each of its sides. Results from the global stability analyses highlight the unstable nature of the central low-speed region and its crucial importance in the laminar-turbulent transition process. It is able to sustain two different global instabilities: a sinuous and a varicose one. Each of these globally unstable modes are related to a different physical mechanism. While the varicose mode finds its root in the instability of the whole three-dimensional shear layer surrounding the central low-speed region, the sinuous instability turns out to be similar to the von Kármán instability in the two-dimensional cylinder wake and finds its root in the lateral shear layers of the separated zone. The aspect ratio of the roughness element plays a key role on the selection of the dominant instability. Indeed, whereas the flow over thin cylindrical roughness elements transitions due to a sinuous instability of the near-wake, for larger roughness element the varicose instability of the central low-speed region turns out to be the dominant one. Direct numerical simulations of the flow past an aspect ratio η = 1 (with η = d/h) roughness element sustaining only the sinuous instability have revealed that the bifurcation occurring in this particular case is supercritical. Finally, comparison of the transition thresholds predicted by global linear stability analyses with the von Doenhoff-Braslow transition diagram provides qualitatively good agreements.