Hostname: page-component-5c6d5d7d68-pkt8n Total loading time: 0 Render date: 2024-08-08T05:04:23.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional transverse instabilities in detached boundary layers

Published online by Cambridge University Press:  04 January 2007

FRANÇOIS GALLAIRE ,
MATTHIEU MARQUILLIE  and
UWE EHRENSTEIN
FRANÇOIS GALLAIRE
Affiliation:
Laboratoire J. A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 02, France
MATTHIEU MARQUILLIE
Affiliation:
Laboratoire de Mécanique de Lille, Université des Sciences et Technologies de Lille, Boulevard Paul Langevin, F-59655 Villeneuve d'Ascq Cédex, France
UWE EHRENSTEIN
Affiliation:
Laboratoire J. A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 02, France
Get access Rights & Permissions [Opens in a new window]

Abstract

A direct numerical simulation of the incompressible Navier–Stokes equations of the flow over a bump shows a stationary longitudinal instability at a Reynolds number of Re = 400. A three-dimensional global mode linear analysis is used to interpret these results and shows that the most unstable eigenmode is steady and localized in the recirculation bubble, with spanwise wavelength of approximately ten bump heights. An inviscid geometrical optics analysis along closed streamlines is then proposed and modified to account for viscous effects. This motivates a final discussion regarding the physical origin of the observed instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 571 , 25 January 2007 , pp. 221 - 233
Copyright
Copyright © Cambridge University Press 2007

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

Armaly, B., Durst, F., Pereira, J. & Schonung, B. 1983 Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech. 127, 473–496.CrossRefGoogle Scholar
Barkley, D., Gomes, M. & Henderson, D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–189.Google Scholar
Bayly, B. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.CrossRefGoogle ScholarPubMed
Bayly, B. 1988 Three-dimensional centrifugal type instability in an inviscid two-dimensional flow. Phys. Fluids 31, 56–64.CrossRefGoogle Scholar
Beaudoin, J.-F., Cadot, O., Aider, J.-L. & Wesfreid, J. 2004 Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech. B/Fluids 23, 147–155.CrossRefGoogle Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218.Google Scholar
Farrell, B. & Ioannou, P. J. 1996 Generalized stability theory part ii: non-autonomous operators. J. Atmos. Sci. 53, 2041–2053.Google Scholar
Godeferd, F., Cambon, C. & Leblanc, S. 2001 Zonal approach to centrifugal, elliptic and hyperbolic instabilities in Stuart vortices with system rotation. J. Fluid Mech. 449, 1–37.CrossRefGoogle Scholar
Hammond, D. & Redekopp, L. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. B/Fluids 17, 145–164.CrossRefGoogle Scholar
Kaiktsis, L., Karniadakis, G. & Orszag, S. 1991 Onset of three-dimensionality equilibria and early transition in flow over a backward-facing step. J. Fluid Mech. 231, 501–528.CrossRefGoogle Scholar
Kaiktsis, L., Karniadakis, G. & Orszag, S. 1996 Unsteadiness and convective instabilities in two-dimensional flow over a backward facing step. Eur. J. Mech. B/Fluids 321, 157–187.CrossRefGoogle Scholar
Landman, M. J. & SaS0022112006002898_inlinean, P. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 2339–2342.CrossRefGoogle Scholar
Leblanc, S. & Godeferd, F. S. 1999 An illustration of the link between ribs and hyperbolic instability. Phys. Fluids 11, 497–499.CrossRefGoogle Scholar
Lifshitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3, 2644–2651.CrossRefGoogle Scholar
Marquillie, M. & Ehrenstein, U. 2002 Numerical simulation of separating boundary-layer flow. Computers Fluids 31, 683–693.CrossRefGoogle Scholar
Marquillie, M. & Ehrenstein, U. 2003 On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech. 490, 169–188.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Sipp, D. & Jacquin, L. 1998 Elliptic instability in two-dimensional flattened Taylor-Green vortices. Phys. Fluids 10, 839–849.CrossRefGoogle Scholar
Sipp, D. & Jacquin, L. 2000 Three-dimensional centrifugal-type instabilities of two-dimensional flows in rotating systems. Phys. Fluids 12, 1740–1748.CrossRefGoogle Scholar
Sipp, D., Lauga, E. & Jacquin, L. 1999 Vortices in rotating systems: centrifugal, elliptic and hyperbolic type instabilities. Phys. Fluids 11, 3716–3728.Google Scholar
Synge, J. L. 1933 The stability of heterogeneous liquids. Trans. R. Soc. Canada 27, 1–18.Google Scholar
Theofilis, V., Hein, S. & Dallmann, U. 2000 On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Soc. Lond. A 358, 3229–3246.Google Scholar
Williams, P. & Baker, A. 1997 Numerical simulations of laminar flow over a 3D backward facing step. Intl J. Numer. Meth. Fluids 24, 1159–1183.3.0.CO;2-R>CrossRefGoogle Scholar