Publication Date:
2005-09-27
Description:
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. © 2005 Cambridge University Press.
Print ISSN:
0022-1120
Electronic ISSN:
1469-7645
Topics:
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
,
Physics
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