Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-18T06:12:22.956Z Has data issue: false hasContentIssue false

Buoyancy flux bounds for surface-driven flow

Published online by Cambridge University Press:  26 July 2005

C. P. CAULFIELD
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA

Abstract

I calculate the optimal upper bound, subject to the assumption of streamwise invariance, on the long-time-averaged buoyancy flux ${\cal B}^*$ within the flow of an incompressible stratified viscous fluid of constant kinematic viscosity $\nu$ and depth $h$ driven by a constant surface stress $\tau\,{=}\,\rho u^2_\star$, where $u_\star$ is the friction velocity with a constant statically stable density difference $\Delta \rho$ maintained across the layer. By using the variational ‘background method’ (due to Constantin, Doering and Hopf) and numerical continuation, I generate a rigorous upper bound on the buoyancy flux for arbitrary Grashof numbers $G$, where $G\,{=}\,\tau h^2/(\rho \nu^2)$. As $G \,{\rightarrow}\, \infty$, for flows where horizontal mean momentum balance, horizontally averaged heat balance, total power balance and total entropy flux balance are imposed as constraints, I show numerically that the best possible upper bound for the buoyancy flux is given by ${\cal B}^* \,{\leq}\, {\cal B}^*_{\hbox{\scriptsize max}}\,{=}\,u_{\star}^4/(4\nu)+ O(u_{\star}^3/h)$. This bound is independent of both the overall strength of the stratification and the layer depth to leading order. This bound is associated with a velocity profile that has the scaling characteristics of a somewhat decelerated laminar, linear velocity profile.

Type
Papers
Copyright
© 2005 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)