On the Reynolds time-averaged equations and the long-time behavior of Leray-Hopf weak solutions, with applications to ensemble averages

We consider the three dimensional incompressible Navier-Stokes equations with a non stationary source term f, chosen in a suitable space. We prove the existence of global Leray-Hopf weak solutions and also that it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations, involving the additional Reynolds stress. Moreover, we show that the turbulent dissipation is bounded by the sum of the Reynolds stress work and the external turbulent fluxes, without any additional assumption, other than that of using Leray-Hopf weak solutions. Finally, in the last section we consider ensemble averages of solutions, associated to a set of different forces and we prove that the fluctuations continue to have a dissipative effect on the mean flow.