Journal of Nonlinear Science

, Volume 24, Issue 1, pp 185–199 | Cite as

Bounds on Surface Stress-Driven Shear Flow

Letter to the Editor

Abstract

The background method is adapted to derive rigorous limits on surface speeds and bulk energy dissipation for shear stress-driven flow in two- and three-dimensional channels. By-products of the analysis are nonlinear energy stability results for plane Couette flow with a shear stress boundary condition: when the applied stress is gauged by a dimensionless Grashoff number \(\operatorname{Gr}\), the critical \(\operatorname{Gr}\) for energy stability is 139.5 in two dimensions, and 51.73 in three dimensions. We derive upper bounds on the friction (a.k.a. dissipation) coefficient \(C_{f} = \tau/\overline{u}^{2}\), where τ is the applied shear stress and \(\overline{u}\) is the mean velocity of the fluid at the surface, for flows at higher \(\operatorname{Gr}\) including developed turbulence: Cf≤1/32 in two dimensions and Cf≤1/8 in three dimensions. This analysis rigorously justifies previously computed numerical estimates.

Keywords

Turbulence Turbulent transport Navier–Stokes equations 

Mathematics Subject Classification

76D05 

Notes

Acknowledgements

The authors gratefully acknowledge the hospitality of the Geophysical Fluid Dynamics Program at Woods Hole Oceanographic Institution, supported by NSF and ONR, where this work was begun. This work was also supported by in part by USDOE Award DE-FG02-ER53223 (GIH) and NSF Awards PHY-0555324, PHY-0855335, and PHY-1205219 (CRD).

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Magneto-Fluids Division, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Departments of Physics and Mathematics, and Center of the Study of Complex SystemsUniversity of MichiganAnn ArborUSA

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