Bounds on dissipation in stress-driven flow in a rotating frame

W. Tang, C. P. Caulfield, W. R. Young

Research output: Contribution to journalArticle

Abstract

We calculate a rigorous dual bound on the long-time-averaged mechanical energy dissipation rate ε within a channel of an incompressible viscous fluid of constant kinematic viscosity v, depth h and rotation rate f, driven by a constant surface stress τ = ρu*2î, where u* is the friction velocity. It is well known that ε ≤ εStokes = u*4/ν, i.e. the dissipation is bounded above by the dissipation associated with the Stokes flow. Using an approach similar to the variational 'background method' (due to Constantin, Doering & Hopf), we generate a rigorous dual bound, subject to the constraints of total power balance and mean horizontal momentum balance, in the inviscid limit ν → 0 for fixed values of the friction Rossby number Ro* = u*/(fh) = √GE, where G = τh2/(ρν2) is the Grashof number, and E = ν/fh2 is the Ekman number. By assuming that the horizontal dimensions are much larger than the vertical dimension of the channel, and restricting our attention to particular, analytically tractable, classes of Lagrange multipliers imposing mean horizontal momentum balance analogous to the ones used in Tang, Caulfield & Young (2004), we show that ε ≤ εmax = u*4/ν - 2.93u*2f, an improved upper bound from the Stokes dissipation, and ε ≥ εmin = 2.795u*3/h, a lower bound which is independent of the kinematic viscosity ν.

Original languageEnglish (US)
Pages (from-to)373-391
Number of pages19
Journaljournal of fluid mechanics
Volume540
DOIs
StatePublished - Oct 10 2005
Externally publishedYes

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

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