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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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References

Betchov, R. & Criminalf, W. O. 1967 Stability of Parallel Flows. Academic.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, p. 634. Oxford University Press.
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701–720.Google Scholar
Dixon, T. N. & Hellums, J. D. 1967 A study on stability and incipient turbulence in Poiseuille and plane Poiseuille flow by numerical finite-difference simulation. A.I.Ch.E. J. 13, 866–872.Google Scholar
Dowell, E. H. 1969 Non-linear theory of unstable plane Poiseuille flow. J. Fluid Mech. 38, 401–414.Google Scholar
Eckhaus, W. 1965 Studies in Non-linear Stability Theory. Springer.
George, W. D. 1970 Application of kite-difference method to the study of non-linear problems in hydrodynamic stability. Ph.D. Thesis in Chemical Engineering, Rice University, Houston, Texas.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Meksyn, D. & Stuart, J. T. 1951 Stability of viscous motion between parallel plates for finite disturbances. Proc. Roy. Soc. A 208, 517–526.Google Scholar
Pekeris, C. L. & Shkoller, B. 1969a Stability of plane Poiseuille flow to periodic disturbances of finite amplitude. J. Fluid Mech., 39, 611–627.Google Scholar
Pekeris, C. L. & Shkoller, B. 1969b The neutral curves for periodic perturbations of finite amplitude of plane Poiseuille flow. J. Fluid Mech. 39, 629–639.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear waves. J. Fluid Mech. 27, 465–492.Google Scholar
Thomas, L. H. 1953 The stability of plane Poiseuille flow. Phys. Rev. 91, 780–783.Google Scholar