Reduced flow models from a stochastic Navier-Stokes representation

Valentin Resseguier 1, 2 Etienne Mémin 1 Bertrand Chapron 2
1 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences
IRSTEA - Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture, Inria Rennes – Bretagne Atlantique
Abstract : In large-scale Fluids Dynamics systems, the velocity lives in a broad range of scales. To be able to simulate its large-scale component, the flow can be decomposed into a finite variation process, which represents a smooth large-scale velocity component, and a martingale part, associated to the highly oscillating small-scale velocities. Within this general framework, a stochastic representation of the Navier-Stokes equations can be derived, based on physical conservation laws. In this equation, a diffusive sub-grid tensor appears naturally and generalizes classical sub-grid tensors. Here, a dimensionally reduced large-scale simulation is performed. A Galerkin projection of our Navier-Stokes equation is done on a Proper Orthogonal Decomposition basis. In our approach of the POD, the resolved temporal modes are differentiable with respect to time, whereas the unresolved temporal modes are assumed to be decorrelated in time. The corresponding reduced stochastic model enables to simulate, at low computational cost, the resolved temporal modes. It allows taking into account the possibly time-dependent, inhomoge-neous and anisotropic covariance of the small scale velocity. We proposed two ways of estimating such contributions in the context of POD-Galerkin. This method has proved successful to reconstruct energetic Chronos for a wake flow at Reynolds 3900, even with a large time step, whereas standard POD-Galerkin diverged systematically. This paper describes the principles of our stochastic Navier-Stokes equation, together with the estimation approaches, elaborated for the model reduction strategy.
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Valentin Resseguier, Etienne Mémin, Bertrand Chapron. Reduced flow models from a stochastic Navier-Stokes representation. Annales de l'ISUP, Publications de l’Institut de Statistique de l’Université de Paris, 2015, 59. ⟨hal-01215301⟩

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