Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T01:08:47.599Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniformly valid solution of the Orr-Sommerfeld equation by a modified Heisenberg method

Published online by Cambridge University Press:  20 April 2006

Shunichi Tsugé  and
Hiroshi Sakai
Shunichi Tsugé
Affiliation:
Institute of Engineering Mechanics, University of Tsukuba, Sakura. Ibaraki 305. Japan
Hiroshi Sakai
Affiliation:
Institute of Engineering Mechanics, University of Tsukuba, Sakura. Ibaraki 305. Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

The classical Heisenberg method of solving the Orr–Sommerfeld equation is modified in such a way that inner and outer expansions are replaced by a uniformly valid successive approximation in which no data on the second derivative of the parallel shear profile are needed. It is shown that this feature enables us to calculate stability characteristics for wider classes of flows with improved accuracy. As a preliminary check for validity of the method, stability of the Blasius flow is calculated and compared with existing methods. It turns out that the method works for high Reynolds numbers, up to about 105, and that the expressions for the eigenfunctions and the eigenvalue condition are much simpler than those found by existing methods.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 153 , April 1985 , pp. 167 - 183
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

Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Davey, A. 1982 A difficult numerical calculation concerning the stability of the Blasius boundary layer. In Stability in the Mechanics of Continua (ed. F. H. Schroeder), p. 365. Springer.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Heisenberg, W. 1924 Über Stabilität und Turbulenz von Flüssigkeitsströmungen. Ann. d. Phys. 74, 577.Google Scholar
Jordinson, R. 1970 The flat plate boundary layer. Part 1. Numerical investigation of the Orr-Sommerfeld equation. J. Fluid Mech. 43, 801.Google Scholar
Lin, C. C. 1945/46 On the stability of two-dimensional parallel flows. Parts 1–3. Q. Appl. Maths 3, 117, 218, 277.Google Scholar
Radbil, J. R. 1966 A new method for prediction of stability of laminar boundary layers. North American Aviation Rep. C6-1019/020.Google Scholar
Sakai, H. 1983 Program ‘s-T Method For o-s EQ.’ Part of Masters Thesis, Institute of Engineering Mechanics, University of Tsukuba.
Schlichting, H. 1935 Amplitudenverteilung und Energiebilanz der kleinen Störungen bei der Plattengrenzschicht. Nachr. Ges. Wiss. Gött., Math.-Phys. Kl. 1, 47.Google Scholar
Tsugé, S. 1978 Methods of separation of variables in turbulence theory. NASA CR 3054.
Wazzan, A. R., Okamura, T. T. & Smith, A. M. O. 1968 Spatial and temporal stability charts for the Falkner-Skan boundary-layer profiles. McDonnell Douglas Rep. DAC-67086.Google Scholar