**Abstract** : Classical approaches of turbulence often use the Boussinesq assumption or deal with a Navier-Stokes equation forced by a random perturbation. In both case, a force is added in a rather empirical way. Here, we follow another way, developed by Mémin (2014), where the flow is decomposed into a differentiable smooth component, which represents a large-scale flow component, and a random, time uncorrelated part, associated to the highly oscillating small-scale velocities. Within this general framework, stochastic versions of the Reynolds transport theorem, mass conservation and Navier-Stokes equations can be respectively derived. In this new version of the Navier-Stokes equation, a sub-grid tensor, appearing naturally without using any additional hypothesis, generalizes classical sub-grid tensors in exhibiting general time-dependant inhomogeneous and anisotropic diffusions. This model can be used for a large-eddy-simulation purpose but also for physically based uncertainty quantification, filtering, mixing diagnostic, downscaling approaches, and other stochastic inverse problems. If the Eulerian smallest-scale velocity is Gaussian and its covariance is known, the sub-grid tensor is specified and the stochastic Navier-Stokes model is closed. The estimation of this, eventually time-dependant, non-stationary spatial covariance is a main issue in our framework.
Here, a dimensionally reduced large-scale simulation is performed. A Galerkin projection of our Navier-Stokes theorem is done on a Proper Orthogonal Decomposition basis. This POD basis, containing spatial modes so-called Topos, is obtained from a PCA on a set of fluid velocity snapshots. The coefficients of the velocity, in this reduced basis, are time-dependant and called Chronos. The Galerkin projection leads to a finite set of coupled ODEs, which describe the evolution of Chronos.
In our approach of the POD, the large-scale velocity, w, is assumed to be deterministic and represented by the resolved modes, meaning the ones kept in the reduced basis. Therefore, the residual velocity, orthogonal to the reduced basis, represents the random and time-uncorrelated smallest-scale velocity. Thus, Stochastic Calculus methods (Rao 1999) can be used to estimate its variance-covariance matrix. This matrix field, depending of space and time, gives informative description of the turbulence, especially of its energy and anisotropy. Knowing this matrix field and the resolved Topos, the coefficients of the resolved-Chronos evolution equations can be computed. This method successfully reconstructs energetic Chronos for a wake flow at Reynolds 3900, whereas the deterministic POD method diverges quickly.
The talk will describe the principles of our stochastic Navier-Stokes equation and the considered model reduction and estimation approaches.