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

Abstract

An upper bound on the energy dissipation rate per unit mass, $\unicode[STIX]{x1D700}$, for pressure-driven flow through a channel with rough walls is derived for the first time. For large Reynolds numbers, $Re$, the bound – $\unicode[STIX]{x1D700}\leqslant cU^{3}/h$ 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 suggest $\unicode[STIX]{x1D700}\sim 1/(\log Re)^{2}U^{3}/h$ as $Re\rightarrow \infty$), 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.

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

“…This is because the I 4 and I 5 integrals centred at the top boundary are not exactly analogous to their counterparts at the lower boundary, counter to what is written just under equation (2.23) (in Kerswell 2016). Instead, in both cases, due to the roughness, the full volume integrals must be included which scale differently with .…”

“…The upper bound derived in Kerswell (2016) is incorrect. This is because the I 4 and I 5 integrals centred at the top boundary are not exactly analogous to their counterparts at the lower boundary, counter to what is written just under equation (2.23) (in Kerswell 2016).…”

“…It is worthwhile illustrating this connection in the simpler context of the smooth-walled channel flow problem before giving the background velocity field corresponding to the Otto-Seis 'boundary layer' bounding analysis presented in Kerswell (2016). This background field has shears throughout the interior and so cannot ever satisfy the spectral constraint necessary to get a bound in the background approach.…”

“…This background field has shears throughout the interior and so cannot ever satisfy the spectral constraint necessary to get a bound in the background approach. This, unfortunately, makes it clear that there is no simple fix of the flawed bound in Kerswell (2016).…”

“…(see (2.29), Kerswell 2016). Rewriting L in terms of ν and φ, and then integrating by parts, the boundedness of the kinetic energy and the fact that both φ and ν vanish on z = 0 and 1 leads to the simplified expression…”