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Non-linear theory of unstable plane Poiseuille flow

Published online by Cambridge University Press:  29 March 2006

E. H. Dowell
E. H. Dowell
Affiliation:
Princeton University
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Abstract

A theoretical study of plane Poiseuille flow is made using the full non-linear Navier-Stokes equations. The mathematical technique employed is to use a Fourier decomposition in the streamwise spatial variable, a Galerkin expansion in the lateral variable and numerical integration with respect to time. By retaining the non-linear terms, the limit cycle oscillations of an unstable (in a linear sense) flow are obtained. A brief investigation of the possibility of instability due to large (non-linear) disturbances is also made. The results are negative for the cases examined. Comparisons with results previously obtained by others from linear theory illustrate the accuracy and efficacy of the method.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 38 , Issue 2 , 3 September 1969 , pp. 401 - 414
Copyright
© 1969 Cambridge University Press

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References

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. 13, 866–872.Google Scholar
Dolph, C. L. & Lewis, D. C. 1958 On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseuille flow Q. Appl. Math. 16, 97–110.Google Scholar
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. New York: Springer Verlag.
Gallagher, A. P. & Mercer, A. MCD. 1962 On the behaviour of small disturbances in plane Couette flow J. Fluid Mech. 13, 91–100.Google Scholar
Grosch, C. E. & Salwan, H. 1968 The stability of steady and time-dependent plane Poiseuille flow J. Fluid Mech. 34, 177–205.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Meksyn, D. & Stuart, J. T. 1951 Stability of viscous motion between parallel flows for finite disturbances. Proc. Roy. Soc A 208, 517–526.Google Scholar
Stuart, J. T. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. 1. The basic behaviour in plane Poiseuille flow J. Fluid Mech. 9, 357–370.Google Scholar
Thomas, L. H. 1953 The stability of plane Poiseuille flow Phys. Rev. 91, 780–783.Google Scholar
Watson, J. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flow. 2. The development of a solution for plane Poiseuille flow and for plane Couette flow J. Fluid Mech. 9, 371–389.Google Scholar