Author:
Grosch Chester E.,Salwen Harold
Abstract
The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients.For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas.For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance.For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Reference34 articles.
1. Obremski, H. J. & Fejer, A. A. 1967 Transition in oscillating boundary layer flows J. Fluid Mech. 29,93.
2. Stuart, J. T. 1954 On the stability of viscous flow between parallel planes in the presence of a co-planar magnetic field.Proc. Roy. Soc. A221,189.
3. Kelly, R. E. 1965 The stability of an unsteady Kelvin-Helmholtz flow J. Fluid Mech. 22,547.
4. Grohne, D. 1954 Über das Spektrum bei Eigenschwingungen ebener Laminarströmungen Z. angew. Math. Mech. 34,344. (Translated as On the spectrum of natural oscillations of two-dimensional laminar flows. Tech. Memor. Nat. Adv. Comm. Aero. Wash. no. 1417.)
5. Gilbrech, D. A. & Combs, G. D. 1963 Critical Reynolds numbers for incompressible pulsating flow in tubes, in Developments in Theoretical and Applied Mechanics, I .New York:Plenum Press.
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