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We consider the three dimensional incompressible Navier-Stokes equations withnon stationary source terms f chosen in a suitable space. We prove theexistence of Leray-Hopf weak solutions and that it is possible to characterize(up to sub-sequences) their long-time averages, which satisfy the Reynoldsaveraged equations, involving a Reynolds stress. Moreover, we show that theturbulent dissipation is bounded by the sum of the Reynolds stress work and ofthe external turbulent fluxes, without any additional assumption, than that ofdealing with Leray-Hopf weak solutions. Finally, in the last section weconsider ensemble averages of solutions, associated to a set of differentforces and we prove that the fluctuations continue to have a dissipative effecton the mean flow.

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