This paper uses a new decomposition of the fluid velocity in terms of a large-scale continuous component with respect to time and a small-scale non continuous random component. Within this general framework, a stochas-tic representation of the Reynolds transport theorem and Navier-Stokes equations can be derived, based on physical conservation laws. This physically relevant stochas-tic model is applied in the context of the POD-Galerkin method. In both the stochastic Navier-Stokes equation and its reduced model, a possibly time-dependent, inhomoge-neous and anisotropic diffusive subgrid tensor appears naturally and generalizes classical subgrid models. We proposed two ways of estimating its parametrization in the context of POD-Galerkin. This method has shown to be able to successfully reconstruct energetic Chronos for a wake flow at Reynolds 3900, whereas standard POD-Galerkin diverged systematically.