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Hydrodynamic stability in plane Poiseuille flow with finite amplitude disturbances

Published online by Cambridge University Press:  29 March 2006

W. D. George  and
J. D. Hellums
W. D. George
Affiliation:
Rice University, Houston, Texas Present address: Getty Oil Company, Houston, Texas.
J. D. Hellums
Affiliation:
Rice University, Houston, Texas
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Abstract

A general method for studying two-dimensional problems in hydrodynamic stability is presented and applied to the classical problem of predicting instability in plane Poiseuille flow. The disturbance stream function is expanded in a Fourier series in the axial space dimension which, on substitution into the Navier-Stokes equation, leads to a system of parabolic partial differential equations in the coefficient functions. An efficient, stable and accurate numerical method is presented for solving these equations. It is demonstrated that the numerical process is capable of accurate reproduction of known results from the linear theory of hydrodynamic stability.

Disturbances that are stable according to linear theory are shown to become unstable with the addition of finite amplitude effects. This seems to be the first work of quantitative value for disturbances of moderate and larger amplitudes. A relationship between critical amplitude and Reynolds number is reported, the form of which indicates the existence of an absolute critical Reynolds number below which an arbitrary disturbance cannot be made unstable, no matter how large its initial amplitude. The critical curve shows significantly less effect of amplitude than do those obtained by earlier workers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 51 , Issue 4 , 22 February 1972 , pp. 687 - 704
Copyright
© 1972 Cambridge University Press

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