Lowering dissipation bounds for turbulent shear flows using a smoothness constraint

Abstract

The effect of a physically plausible `smoothness' condition is examined on existing upper bounds ϵ bound for the viscous dissipation rate ϵ in turbulent Plane Couette flow. We find that if the minimum lengthscale allowed parallel to the plates (separation distance h) is ℓ= λRe 2 α−1 h, then (0.308/ λ ) Re −α<ϵ bound <(0.562/ λ ) Re −α as Re→∞ (measured in units of V 0 3/ h where Re= V 0 h/ ν, V 0 is the velocity differential and ν is the kinematic viscosity). This implies that transverse lengthscales down to O( Re −1 h) must be included in numerical simulations to avoid prejudicing dissipation scalings. In particular, if ℓ is taken as the order of the Kolmogorov scale ( ν 3/ ϵ) 1/4, then the energy dissipation cannot be more than O( Re −1/7) in wall-bounded shear flow.