Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-09T06:12:44.197Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of the laminar flow in a rectangular duct

Published online by Cambridge University Press:  26 April 2006

Tomomasa Tatsumi  and
Takahiro Yoshimura
Tomomasa Tatsumi
Affiliation:
Kyoto Institute of Technology, Kyoto 606, Japan
Takahiro Yoshimura
Affiliation:
Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of the laminar flow in a rectangular duct of an arbitrary aspect ratio is investigated numerically by expanding the flow fields of both the main flow and the disturbance into series of Legendre polynomials and solving the eigenvalue problem of the resulting matrix equation. The stability of the flow is found to depend upon the aspect ratio of the duct and the mode of the disturbance. The flow is unstable to two of the four possible modes of different parity and stable to the other two. With respect to the most unstable mode, the flow is stable or unstable according as the aspect ratio is below or above a critical value of 3.2 respectively, and the critical Reynolds number decreases monotonically with increasing aspect ratio towards the known value of 5772 for plane Poiseuille flow. The flow field of the disturbance shows the existence of strong vortex layers along the critical layer at which the velocity equals the phase velocity of the disturbance.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 212 , March 1990 , pp. 437 - 449
Copyright
© 1990 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

Cornish, R. J.: 1928 Flow in a pipe of rectangular cross-section. Proc. R. Soc. Lond. A 120, 691–700.Google Scholar
Davies, S. J. & White, C. M., 1928 An experimental study of the flow of water in pipes of rectangular section. Proc. R. Soc. Lond. A 119, 92–107.Google Scholar
Drazin, P. G. & Reid, W. H., 1981 Hydrodynamic Stability, pp. 216–221. Cambridge University Press.
Kao, T. W. & Park, C., 1970 Experimental investigations of the stability of channel flows. Part 1. Flow of a single liquid in a rectangular channel. J. Fluid Mech. 43, 145–164.Google Scholar
Orszag, S.: 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Rosenhead, L. (ed.) 1963 Laminar Boundary Laters, pp. 492–579. Clarendon.
De Saint-Venant, B. 1855 Mémoire sur la torsion des prismes. Mémorires de l'Academie des Sciences des Savants Etrangers. 14, 233–560.Google Scholar
Schiller, L.: 1923 Uber den Strömungswiderstand von Rohren verschiedenen Querschnitts und Rauhigkeitsgrades. Z. Angew. Math. Mech. 3, 2–13.Google Scholar
Schlichting, H.: 1933 Zur Entstehung der Turbulenz bei der Plattenströmung. Nachr. Ges. Wiss. Göttingen, Math. – phys. Kl. 181–208; Z. Angew. Math. Mech. 13, 171–174.
Schubauer, G. B. & Skramstad, H. K., 1947 Laminar boundary layer oscillations and transition on a flat plate. J. Res. Natl Bur. Stand. 38, 251–292; J. Aero. Sci. 14, 69–78.Google Scholar
Tollmien, W.: 1929 Uber die entstehung der Turbulenz. Nachr. Ges. Wiss. Göttingen, 21–44.Google Scholar