Stochastic Reynolds transport theorem and generalized subgrid tensor

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We propose a representation that allows decomposing the flow velocity in terms of a smooth component and a highly oscillating random component. This decomposion leads through a stochastic representation of the Reynolds transport theorem to a large-scale expression of the Navier-Stokes equations. In this work we show the benefit of such a representation to construct low order dynamical systems that include naturally a dissipative term related to the action of the small-scale random component.