Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-08T17:04:54.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear spatial stability of pipe Poiseuille flow

Published online by Cambridge University Press:  29 March 2006

V. K. Garg  and
W. T. Rouleau
V. K. Garg
Affiliation:
Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh
W. T. Rouleau
Affiliation:
Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical study of the spatial stability of Poiseuille flow in a rigid pipe to infinitesimal disturbances is presented. Both axisymmetric and non-axisymmetric disturbances are considered. The coupled, linear, ordinary differential equations governing the propagation of a disturbance that has a constant frequency and is imposed at a specified location in the fluid are solved numerically for the complex eigenvalues, or wavenumbers, each of which defines a mode of propagation. A series solution for small values of the pipe radius is derived and step-by-step integration to the pipe wall is then performed. In order to ascertain the number of eigenvalues within a closed region, an eigenvalue search technique is used. Results are obtained for Reynolds numbers up to 10000. For these Reynolds numbers it is found that the pipe Poiseuille flow is spatially stable to all infinitesimal disturbances.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 54 , Issue 1 , 11 July 1972 , pp. 113 - 127
Copyright
© 1972 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

Acton, F. S. 1970 Numerical Methods That Work. Harper & Row.
Batchelor, G. K. & Gill, A. E. 1962 J. Fluid Mech. 14, 529.
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Burridge, D. M. 1972 J. Fluid Mech. to be published.
Collatz, L. 1966 The Numerical Treatment of Differential Equations. Springer.
Corcos, G. M. & Sellars, J. R. 1959 J. Fluid Mech. 5, 97.
Delves, L. M. & Lyness, J. N. 1967 Math. Comp. 21, 543.
Fox, J. A., Lessen, M. & Bhat, W. V. 1968 Phys. Fluids, 11, 1.
Garg, V. K. 1971 Ph.D. thesis, Carnegie-Mellon University, Pittsburgh. (See also University Microfilms, Ann Arbor, Michigan, no. 71–24, 852.)
Gill, A. E. 1965 J. Fluid Mech. 21, 145.
Kuethe, A. M. 1956 J. Aero. Sci. 23, 446.
Leite, R. J. 1959 J. Fluid Mech. 4, 81.
Lessen, M., Fox, J. A., Bhat, W. V. & Liu, T. Y. 1964 Phys. Fluids, 7, 1384.
Lessen, M., Sadler, S. G. & Liu, T. Y. 1968 Phys. Fluids, 11, 1404.
Nehari, Z. 1969 Introduction to Complex Analysis. Boston: Allyn & Bacon.
Pekeris, C. L. 1948 Proc. U.S. Natn. Acad. Sci. 34, 285.
Pretsch, J. 1941 Z. angew. Math. Mech. 21, 204.
Ralston, A. 1965 A First Course in Numerical Analysis. McGraw-Hill.
Reynolds, O. 1883 Phil. Trans. 174, 935. (See also Scientific Papers, vol. 2, p. 51.)
Scarton, H. A. & Rouleau, W. T. 1972 J. Fluid Mech. to be published.
Sexl, T. 1927a Ann. Phys. 83, 835.
Sexl, T. 1927b Ann. Phys. 84, 807.