Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-25T06:51:44.539Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear and nonlinear stability of plane stagnation flow

Published online by Cambridge University Press:  21 April 2006

M. J. Lyell  and
P. Huerre
M. J. Lyell
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90089-0192
P. Huerre
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90089-0192
Get access Rights & Permissions [Opens in a new window]

Abstract

Plane stagnation flow is known to be linearly stable to three-dimensional perturbations. The purpose of this theoretical study is to show that the same flow can be destabilized if fluctuation levels are sufficiently high. In the present formulation, finite-amplitude disturbances are expanded in terms of the eigenfunctions pertaining to the linear stability of potential stagnation flow and a Galerkin method is used to derive the nonlinear amplitude equations coupling the different modes. Two- and three-mode interaction models based on the least-damped eigenfunctions of linear theory indicate that three-dimensional fluctuations can be triggered to grow exponentially above a certain critical intensity. The existence of such a threshold is in qualitative agreement with experimental studies of the secondary vortices arising in flows past blunt bodies.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 161 , December 1985 , pp. 295 - 312
Copyright
© 1985 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

Colak-Antic, P. 1971 Visuelle Untersuchungen von Langswirbeln im Staupunktgebiet eines Kreiszylinders bei turbulenter Anströmung. D.L.R. Mitteilung 71–13, Bericht über die DGLR-Fachausschuss-Sitzung, Laminare und Turbulente Grenzschichtung, p. 194. Göttingen.
DiPrima, R. 1967 Vector eigenfunction expansions for the growth of Taylor vortices in the flow between rotating cylinders. In Nonlinear Partial Differential Equations (ed. W. F. Ames), p. 19. Academic.
Finlayson, B. 1972 The Method of Weighted Residuals and Variational Principles. Academic.
Görtler, H. 1955 Dreidimensionale Instabilität der ebenen Staupunkt-Strömung gegenüber wirbelartigen Störungen. In Fifty Years of Boundary Layer Research. (ed. H. Görtler & W. Tollmien), p. 304. Vieweg und Sohn.
Hall, P. 1982 Taylor-Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41.Google Scholar
Hämmerlin, G. 1955 Zur Instabilitätstheorie der ebenen Staupunkströmung. In Fifty Years of Boundary Layer Research (ed. H. Görtler & W. Tollmien), p. 315. Vieweg und Sohn, Braunschweig.
Hassler, H. 1971 Hitzdrahtmessungen von Langswirbelartigen Instabilitätserscheinungen im Staupunktgebiet eines Kreiszylinders in turbulenter Anströmung. D.L.R. Mitteilung 71–13, Bericht über die DGLR-Fachausschuss-Sitzung, Laminare und Turbulente Grenzschichtung, p. 221. Göttingen.
Hodson, P. & Nagib, H. 1975 Longitudinal vortices induced in a stagnation region by wakes - their incipient formation and effects on heat transfer from cylinders. IIT Fluids and Heat Transfer Rep. R75–2.
Iida, S. 1978 Nonlinear stability of a two-dimensional stagnation flow. Bull. Japan Soc. Mech. Engng 21, 992.Google Scholar
Lyell, M. J. 1982 The nonlinear stability of two fluid flows: three-dimensional plane stagnation flow and two-dimensional jets. Ph.D. Thesis, University of Southern California.
Morkovin, M. 1979 On the question of instabilities upstream of cylindrical bodies. IIT Fluids and Heat Transfer Rep. R79–3.
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.
Piercy, N. & Richardson, E. 1928 The variation of velocity with amplitude close to the surface of the cylinder moving through a viscous fluid. Phil. Mag. 6, 970.Google Scholar
Sadeh, W. & Brauer, H. 1980 A visual investigation of turbulence in stagnation flow about a circular cylinder. J. Fluid Mech. 99, 53.Google Scholar
Sadeh, W., Sutera, S. & Maeder, P. 1970 An investigation of vorticity amplification in stagnation flow. Z. angew. Math. Phys. 21, 717.Google Scholar
Schlichting, H. 1979 Boundary Layer Theory. McGraw-Hill.
Stuart, J. T. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. J. Fluid Mech. 9, 353.Google Scholar
Watson, J. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. J. Fluid Mech. 9, 371.Google Scholar
Wilson, S. & Gladwell, I. 1978 The stability of a two-dimensional stagnation flow to three-dimensional disturbances. J. Fluid Mech. 84, 517.Google Scholar