Reduced flow models from a stochastic Navier-Stokes representation
|Other titles||Modèle réduit d'écoulement à partir d'une représentation stochastique de Navier-Stokes|
|Author(s)||Resseguier Valentin1, 2, Memin Etienne1, Chapron Bertrand2|
|Affiliation(s)||1 : Inria Rennes, Bretagne Atlantique, Fluminance team, 35042 Rennes, France
2 : IFREMER, LOS, 29280 Plouzané, France
|Meeting||Workshop on Stochastic Weather Generators. September 17-19, 2014, Avignon|
|Source||Annales de l'ISUP (1626-1607), 2015 , Vol. 59 , N. 1-2 , P. 57-85|
|Keyword(s)||Stochastic calculus, uid dynamics, large eddy simulation, Proper Orthogonal Decomposition, reduced order model, uncertainty quantification|
|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 de- composed 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 gener- alizes 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 De- composition 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.