On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations
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We consider three-dimensional stochastically forced Navier-Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper and lower bounds of the mean value of the time-averaged energy dissipation rate, E [〈ε〉], are derived directly from the equations. First, we show that for a weak (martingale) solution to the stochastically forced Navier-Stokes equations, E [〈ε〉] ≤ G^2 + (2+ 1/Re)U^3/L, where G^2 is the total energy rate supplied by the random force, U is the root-mean-square velocity, L is the longest length scale in the applied forcing function, and Re is the Reynolds number. Under an additional assumption of energy equality, we also derive a lower bound if the energy rate given by the random force dominates the deterministic behavior of the flow in the sense that G^2 > 2 F U, where F is the amplitude of the deterministic force. We obtain, 1/3 G^2 - 1/3 (2+ 1/Re)U^3/L≤E [〈ε〉] ≤ G^2 + (2+ 1/Re)U^3/L . In particular, under such assumptions, we obtain the zeroth law of turbulence in the absence of the deterministic force as, E [〈ε〉] = 1/2 G^2. Besides, we also obtain variance estimates of the dissipation rate for the model.
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