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

Published online by Cambridge University Press:  04 November 2016

R. R. Kerswell
R. R. Kerswell*
Affiliation:
School of Mathematics, Bristol University, Bristol BS8 1TW, UK
*
†Email address for correspondence: R.R.Kerswell@bristol.ac.uk
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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.

JFM classification

Turbulent Flows: Shear layer turbulence Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 808 , 10 December 2016 , pp. 562 - 575
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Copyright
© 2016 Cambridge University Press 

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