Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-09T08:50:16.120Z Has data issue: false hasContentIssue false
  • English
  • Français

Bound on the drag coefficient for a flat plate in a uniform flow

Published online by Cambridge University Press:  04 August 2020

Anuj Kumar Open the ORCID record for Anuj Kumar [Opens in a new window]  and
Pascale Garaud Open the ORCID record for Pascale Garaud [Opens in a new window]
Anuj Kumar*
Affiliation:
Department of Applied Mathematics, Baskin School of Engineering, University of California, Santa Cruz, CA95064, USA
Pascale Garaud
Affiliation:
Department of Applied Mathematics, Baskin School of Engineering, University of California, Santa Cruz, CA95064, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The background method has been a successful tool in obtaining strict bounds on global quantities such as the rate of energy dissipation and heat transfer in turbulent flows. However, all applications of this method until now have focused on flows confined between solid boundaries. An important class of problems that, by contrast, has received no attention is the class of external flows, i.e. flow past a body. In this context, obtaining the dependence of the drag coefficient on the Reynolds number is of crucial relevance for many engineering applications. In this paper, we consider the classical problem of flow past a flat plate of finite length at zero angle of incidence and use the background method to obtain a bound on the drag coefficient. Assuming a statistically steady state and appropriate far-field decay rates for the flow variables, we show that at large Reynolds numbers, the drag coefficient ($C_D$) is bounded by a constant, a bound that is within a logarithmic factor of experimental data.

JFM classification

Mathematical Foundations: Variational methods Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A6
Copyright
© The Author(s), 2020. Published by 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

REFERENCES

Blasius, H. 1908 Grenzschichten in Flüssigkeiten mit kleiner Reibung. B.G. Teubner.Google Scholar
Busse, F. H. 1969 On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37 (3), 457–477.CrossRefGoogle Scholar
Busse, F. H. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41 (1), 219–240.CrossRefGoogle Scholar
Caulfield, C. P. 2005 Buoyancy flux bounds for surface-driven flow. J. Fluid Mech. 536, 367–376.CrossRefGoogle Scholar
Caulfield, C. P. & Kerswell, R. R. 2001 Maximal mixing rate in turbulent stably stratified couette flow. Phys. Fluids 13 (4), 894–900.CrossRefGoogle Scholar
Constantin, P. & Doering, C. R. 1995 Variational bounds on energy dissipation in incompressible flows. Part 2. Channel flow. Phys. Rev. E 51 (4), 3192–3198.CrossRefGoogle Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69 (11), 1648–1651.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49 (5), 4087–4099.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. Part 3. Convection. Phys. Rev. E 53 (6), 5957–5981.CrossRefGoogle Scholar
Doering, C. R. & Constantin, P. 2001 On upper bounds for infinite Prandtl number convection with or without rotation. J. Math. Phys. 42 (2), 784–795.CrossRefGoogle Scholar
Doering, C. R., Otto, F. & Reznikoff, M. G. 2006 Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Bénard convection. J. Fluid Mech. 560, 229–241.CrossRefGoogle Scholar
Fantuzzi, G. 2018 Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux. J. Fluid Mech. 837, R5.CrossRefGoogle Scholar
Fantuzzi, G., Nobili, C. & Wynn, A. 2020 New bounds on the vertical heat transport for Bénard–Marangoni convection at infinite Prandtl number. J. Fluid Mech. 885, R4.CrossRefGoogle Scholar
Fantuzzi, G., Pershin, A. & Wynn, A. 2018 Bounds on heat transfer for Bénard–Marangoni convection at infinite Prandtl number. J. Fluid Mech. 837, 562–596.CrossRefGoogle Scholar
Fantuzzi, G. & Wynn, A. 2015 Construction of an optimal background profile for the Kuramoto–Sivashinsky equation using semidefinite programming. Phys. Lett. A 379 (1–2), 23–32.CrossRefGoogle Scholar
Fantuzzi, G. & Wynn, A. 2016 Optimal bounds with semidefinite programming: an application to stress-driven shear flows. Phys. Rev. E 93 (4), 043308.CrossRefGoogle ScholarPubMed
Folland, G. B. 2002 Advanced Calculus. Recording for the Blind & Dyslexic. Pearson.Google Scholar
Goluskin, D. 2015 Internally heated convection beneath a poor conductor. J. Fluid Mech. 771, 36–56.CrossRefGoogle Scholar
Goluskin, D. & Doering, C. R. 2016 Bounds for convection between rough boundaries. J. Fluid Mech. 804, 370–386.CrossRefGoogle Scholar
Hagstrom, G. & Doering, C. R. 2010 Bounds on heat transport in Bénard–Marangoni convection. Phys. Rev. E 81 (4), 047301.CrossRefGoogle ScholarPubMed
Hagstrom, G. I. & Doering, C. R. 2014 Bounds on surface stress-driven shear flow. J. Nonlinear Sci. 24 (1), 185–199.CrossRefGoogle Scholar
Harper, J. F. & Moore, D. W. 1968 The motion of a spherical liquid drop at high Reynolds number. J.Fluid Mech. 32 (2), 367–391.CrossRefGoogle Scholar
Hoffmann, N. P. & Vitanov, N. K. 1999 Upper bounds on energy dissipation in Couette–Ekman flow. Phys. Lett. A 255 (4–6), 277–286.CrossRefGoogle Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17 (3), 405–432.CrossRefGoogle Scholar
Howard, L. N. 1972 Bounds on flow quantities. Annu. Rev. Fluid Mech. 4, 473–494.CrossRefGoogle Scholar
Jobe, C. E. & Burggraf, O. R. 1974 The numerical solution of the asymptotic equations of trailing edge flow. Proc. R. Soc. Lond. A 340 (1620), 91–111.Google Scholar
Kerswell, R. R. 1997 Variational bounds on shear-driven turbulence and turbulent Boussinesq convection. Physica D 100 (3–4), 355–376.CrossRefGoogle 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 121 (1–2), 175–192.CrossRefGoogle Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Malkus, M. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225 (1161), 196–212.Google Scholar
Marchioro, C. 1994 Remark on the energy dissipation in shear driven turbulence. Physica D 74 (3–4), 395–398.CrossRefGoogle Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18 (1), 241–257.CrossRefGoogle Scholar
Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16 (2), 161–176.CrossRefGoogle Scholar
Nicodemus, R., Grossmann, S. & Holthaus, M. 1997 Improved variational principle for bounds on energy dissipation in turbulent shear flow. Physica D 101 (1–2), 178–190.CrossRefGoogle 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.CrossRefGoogle Scholar
Otto, F. & Seis, C. 2011 Rayleigh–Bénard convection: improved bounds on the Nusselt number. J. Math. Phys. 52 (8), 083702.CrossRefGoogle Scholar
Plasting, S. C. & Ierley, G. R. 2005 Infinite-Prandtl-number convection. Part 1. Conservative bounds. J. Fluid Mech. 542, 343–363.CrossRefGoogle 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.CrossRefGoogle Scholar
Prandtl, L. 1904 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des dritten internationalen Mathematiker-Kongresses. Springer.Google Scholar
Roshko, A. 1993 Perspectives on bluff body aerodynamics. J. Wind Engng Ind. Aerodyn. 49 (1–3), 79–100.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2016 Boundary-Layer Theory, 9th edn. Springer.Google Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate II. Mathematika 16 (1), 106–121.CrossRefGoogle Scholar
Tang, W., Caulfield, C. P. & Young, W. R. 2004 Bounds on dissipation in stress-driven flow. J. Fluid Mech. 510, 333–352.CrossRefGoogle Scholar
Tilgner, A. 2017 Bounds on poloidal kinetic energy in plane layer convection. Phys. Rev. Fluids 2 (12), 123502.CrossRefGoogle Scholar
Tilgner, A. 2019 Time evolution equation for advective heat transport as a constraint for optimal bounds in Rayleigh–Bénard convection. Phys. Rev. Fluids 4 (1), 014601.CrossRefGoogle Scholar
Wang, X. 1997 Time averaged energy dissipation rate for shear driven flows in $\mathbb {R}^{n}$. Physica D 99 (4), 555–563.CrossRefGoogle Scholar
Wen, B., Chini, G., 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 (41), 2931–2938.CrossRefGoogle Scholar
Wen, B., Chini, G. P., Kerswell, R. R. & Doering, C. R. 2015 Time-stepping approach for solving upper-bound problems: application to two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 92 (4), 043012.CrossRefGoogle ScholarPubMed
Whitehead, J. P. & Doering, C. R. 2011 a Internal heating driven convection at infinite Prandtl number. J.Math. Phys. 52 (9), 093101.CrossRefGoogle Scholar
Whitehead, J. P. & Doering, C. R. 2011 b Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106 (24), 244501.CrossRefGoogle ScholarPubMed
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 (9), 093104.CrossRefGoogle Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28 (1), 477–539.CrossRefGoogle Scholar
Wittenberg, R. W. 2010 Bounds on Rayleigh–Bénard convection with imperfectly conducting plates. J.Fluid Mech. 665, 158–198.CrossRefGoogle Scholar