### 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*^{2}î, where u* is the friction velocity. It is well known that ε ≤ ε_{Stokes} = 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 = τh^{2}/(ρν^{2}) is the Grashof number, and E = ν/fh^{2} 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 ≤ ε_{max} = u*^{4}/ν - 2.93u*^{2}f, an improved upper bound from the Stokes dissipation, and ε ≥ ε_{min} = 2.795u*^{3}/h, a lower bound which is independent of the kinematic viscosity ν.

Original language | English (US) |
---|---|

Pages (from-to) | 373-391 |

Number of pages | 19 |

Journal | Journal of Fluid Mechanics |

Volume | 540 |

DOIs | |

State | Published - Oct 10 2005 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Journal of Fluid Mechanics*,

*540*, 373-391. https://doi.org/10.1017/S0022112005005926

**Bounds on dissipation in stress-driven flow in a rotating frame.** / Tang, Wenbo; Caulfield, C. P.; Young, W. R.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 540, pp. 373-391. https://doi.org/10.1017/S0022112005005926

}

TY - JOUR

T1 - Bounds on dissipation in stress-driven flow in a rotating frame

AU - Tang, Wenbo

AU - Caulfield, C. P.

AU - Young, W. R.

PY - 2005/10/10

Y1 - 2005/10/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=26944454746&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26944454746&partnerID=8YFLogxK

U2 - 10.1017/S0022112005005926

DO - 10.1017/S0022112005005926

M3 - Article

VL - 540

SP - 373

EP - 391

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -