Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-31T01:46:07.989Z Has data issue: false hasContentIssue false
  • English
  • Français

Energy dissipation rate limits for flow through rough channels and tidal flow across topography

Published online by Cambridge University Press:  04 November 2016

R. R. Kerswell
R. R. Kerswell*
Affiliation:
School of Mathematics, Bristol University, Bristol BS8 1TW, UK
*
†Email address for correspondence: R.R.Kerswell@bristol.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

An upper bound on the energy dissipation rate per unit mass, $\unicode[STIX]{x1D700}$, for pressure-driven flow through a channel with rough walls is derived for the first time. For large Reynolds numbers, $Re$, the bound – $\unicode[STIX]{x1D700}\leqslant cU^{3}/h$ where $U$ is the mean flow through the channel, $h$ the channel height and $c$ a numerical prefactor – is independent of $Re$ (i.e. the viscosity) as in the smooth channel case but the numerical prefactor $c$, which is only a function of the surface heights and surface gradients (i.e. not higher derivatives), is increased. Crucially, this new bound captures the correct scaling law of what is observed in rough pipes and demonstrates that while a smooth pipe is a singular limit of the Navier–Stokes equations (data suggest $\unicode[STIX]{x1D700}\sim 1/(\log Re)^{2}U^{3}/h$ as $Re\rightarrow \infty$), it is a regular limit for current bounding techniques. As an application, the bound is extended to oscillatory flow to estimate the energy dissipation rate for tidal flow across bottom topography in the oceans.

JFM classification

Turbulent Flows: Shear layer turbulence Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 808 , 10 December 2016 , pp. 562 - 575
Check for updates
Copyright
© 2016 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.)

References

van den Berg, T. H., Doering, C. R., Lohse, D. & Lathrop, D. P. 2003 Smooth and rough boundaries in turbulent Taylor–Couette flow. Phys. Rev. E 68, 036307.Google Scholar
Busse, F. H. 1969 Bounds on the transport of mass and momentum by turbulent flow between parallel plates. Z. Angew. Math. Phys. 20, 1–14.Google Scholar
Busse, F. H. 1970 Bounds for turbulent shear flows. J. Fluid Mech. 41, 219–240.Google Scholar
Busse, F. H. 1978 The optimal theory of turbulence. Adv. Appl. Mech. 18, 77–121.Google Scholar
Cadot, O., Couder, Y., Daerr, A., Douady, S. & Tsinober, A. 1997 Energy injection in closed turbulent flows: stirring through boundary layers versus inertial stirring. Phys. Rev. E 56, 427–433.Google Scholar
Constantin, P. & Doering, C. R. 1995 Variational bounds on energy dissipation in incompressible flows. II: channel flow. Phys. Rev. E 51, 3192–3198.Google Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69, 1648–1651.Google Scholar
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows. shear flow. Phys. Rev. E 49, 4087–4099.Google Scholar
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III: convection. Phys. Rev. E 53, 5957–5981.Google Scholar
Doering, C. R., Spiegel, E. A. & Worthing, R. A. 2000 Energy dissipation in a shear layer with suction. Phys. Fluids 12, 1955–1968.Google Scholar
Doering, C. R. & Constantin, P. 2001 On upper bounds for infinite Prandtl number convection with and without rotation. J. Math. Phys. 42, 784–795.Google Scholar
Doering, C. R. & Foias, C. 2002 Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289–306.Google Scholar
Frisch, U. 1995 Turbulence. Cambridge University Press.Google Scholar
Goluskin, D. & Doering, C. R. 2016 Bounds for convection between rough boundaries. J. Fluid Mech. 804, 370–386.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.Google Scholar
Hoffmann, N. P. & Vitanov, N. K. 1999 Upper bounds on energy dissipation in Couette–Ekman flow. Phys. Lett. A 255, 277–286.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
Howard, L. N. 1972 Bounds on flow quantities. Annu. Rev. Fluid Mech. 4, 473–494.Google Scholar
Kerswell, R. R. 1996 Upper bounds on the energy dissipation in turbulent precession. J. Fluid Mech. 321, 335–370.Google Scholar
Kerswell, R. R. 1997 Variational bounds on shear-driven turbulence and turbulent Boussinesq convection. Physica D 100, 355–376.Google Scholar
Kerswell, R. R. 1998 Unification of variational principles for turbulent shear flows: the background method of Doering–Constantin and the mean-fluctuation formulation of Howard–Busse. Physica D 101, 178–190.Google Scholar
Kerswell, R. R. 2002 Upper bounds on general dissipation functionals in turbulent shear flows: revisiting the ‘efficiency’ functional. J. Fluid Mech. 461, 239–275.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196–212.Google Scholar
Marchioro, C. 1994 Remark on the energy dissipation in shear driven turbulence. Physica D 74, 395–398.Google Scholar
Melet, A., Legg, S. & Hallberg, R. 2016 Climate impacts of parameterized local and remote tidal mixing. J. Clim. 29, 3473–3500.Google Scholar
Moody, L. F. 1944 Friction factors for pipe flow. Trans. ASME 66, 671–684.Google Scholar
Nicodemus, R., Grossmann, S. & Holthaus, M. 1997 Improved variational principle for bounds on energy dissipation in turbulent shear flow. Physica D 101, 178–190.Google Scholar
Nobili, C. & Otto, F.2016 Limitations of the background field method applied to Rayleigh–Bénard convection. J. Math. Phys. (in press). arXiv:1605:08135.Google Scholar
Otero, J., Wittenberg, R. W., Worthing, R. A. & Doering, C. R. 2002 Bounds on Rayleigh–Bénard convection with an imposed heat flux. J. Fluid Mech. 473, 191–199.Google Scholar
Otto, F. & Seis, C. 2011 Rayleigh–Bénard convection: improved bounds on the Nusselt number. J. Math. Phys. 52, 083702,1–24.Google Scholar
Plasting, S. C. & Kerswell, R. R. 2003 Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse’s problem and the Constantin–Doering–Hopf problem with one-dimensional background field. J. Fluid Mech. 477, 363–379.Google Scholar
Plasting, S. C. & Kerswell, R. R. 2005 A friction factor bound for transitional pipe flow. Phys. Fluids 17, 011706.Google Scholar
Plasting, S. C. & Ierley, G. R. 2005 Infinite-Prandtl number convection. Part 1. Conservative bounds. J. Fluid Mech. 542, 343–363.Google Scholar
Seis, C. 2015 Scaling bounds on dissipation in turbulent flows. J. Fluid Mech. 777, 591–603.CrossRefGoogle Scholar
Waleffe, F., Boonkasame, A. & Smith, L. M. 2015 Heat transport by coherent Rayleigh–Bénard convection. Phys. Fluids 27, 051702.Google Scholar
Wen, B., Chini, G. P., Dianati, N. & Doering, C. R. 2013 Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection. Phys. Lett. A 377, 2931–2938.Google Scholar
Wang, X. 1997 Time averaged energy dissipation rate for shear driven flows in ℝ n . Physica D 99, 555–563.Google Scholar
Whitehead, J. P. & Doering, C. R. 2011 Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed temperature boundaries. Phys. Rev. Lett. 106, 244501.Google Scholar
Whitehead, J. P. & Wittenberg, R. W. 2014 A rigorous bound on the vertical transport of heat in Rayleigh–Bénard convection at infinite Prandtl number with mixed thermal boundary conditions. J. Math. Phys. 55, 093104.Google Scholar
Wittenberg, R. W. 2010 Bounds on Rayleigh–Bénard convection with imperfectly conducting plates. J. Fluid Mech. 665, 158–198.Google Scholar