Energy dissipation rate limits for flow through rough channels and tidal flow across topography

An upper bound on the energy dissipation rate per unit mass, ε, for pressure-driven flow through a channel with rough walls is derived for the first time. For large Reynolds numbers, Re, the bound - ε ≤ cU³/h where U is the mean flow through the channel, h the channel height and c a numerical prefactor - is independent of Re (i.e. the viscosity) as in the smooth channel case but the numerical prefactor c, which is only a function of the surface heights and surface gradients (i.e. not higher derivatives), is increased. Crucially, this new bound captures the correct scaling law of what is observed in rough pipes and demonstrates that while a smooth pipe is a singular limit of the Navier-Stokes
equations (data suggests ε ∼ 1/(log Re)²U³/h as Re → ∞), it is a regular limit for
current bounding techniques. As an application, the bound is extended to oscillatory flow to estimate the energy dissipation rate for tidal flow across bottom topography in the oceans.