Three-dimensional shear driven turbulence with noise at the boundary

We consider the incompressible 3D Navier-Stokes equations subject to a shear induced by noisy movement of part of the boundary. The effect of the noise is quantified by upper bounds on the first two moments of the dissipation rate. The expected value estimate is consistent with the Kolmogorov dissipation law, recovering an upper bound as in [15] for the deterministic case. The movement of the boundary is given by an Ornstein-Uhlenbeck process; a potential for over-dissipation is noted if the Ornstein-Uhlenbeck process were replaced by the Wiener process.