Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-01T18:50:56.241Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability of steady and time-dependent plane Poiseuille flow

Published online by Cambridge University Press:  28 March 2006

Chester E. Grosch  and
Harold Salwen
Chester E. Grosch
Affiliation:
Hudson Laboratories of Columbia University, Dobbs Ferry, New York 10522
Harold Salwen
Affiliation:
Stevens Institute of Technology, Hoboken, New Jersey, 07030 and Hudson Laboratories of Columbia University, Dobbs Ferry, New York 10522
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 34 , Issue 1 , 16 October 1968 , pp. 177 - 205
Copyright
© 1968 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

Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Oxford University Press.
Coddington, E. A. & Levenson, N. 1955 Theory of Ordinary Differential Equations. New York: McGraw-Hill.
Conrad, P. W. & Criminale, W. O. 1965a The stability of time-dependent laminar flow: parallel flows. Z. angew. Math. Phys. 16, 233.Google Scholar
Conrad, P. W. & Criminale, W. O. 1965b The stability of time-dependent laminar flow: flow with curved streamlines. Z. angew. Math. Phys. 16, 569.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.Google Scholar
Donnelly, R. J. 1964 Experiments on the stability of viscous flow between rotating cylinders, III. Enhancement of stability by modulation. Proc. Roy. Soc. A 281, 130.Google Scholar
Donnelly, R. J., Reif, F. & Suhl, H. 1962 Enhancement of hydrodynamic stability by modulation Phys. Rev. Lett. 9, 363.Google Scholar
Gallagher, A. P. & Mercer, A. MCD. 1962 On the behaviour of small disturbances in plane Couette flow J. Fluid Mech. 13, 91.Google Scholar
Gallagher, A. P. & Mercer, A. MCD. 1964 On the behaviour of small disturbances in plane Couette flow; Part 2, The higher eigenvalues J. Fluid Mech. 18, 350.Google Scholar
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.
Greenspan, H. P. & Benney, D. J. 1963 On shear-layer instability, breakdown and transition J. Fluid Mech. 15, 133.Google Scholar
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.)Google Scholar
Grosch, C. E. & Salwen, H. 1967 Hydrodynamic stability of a modulated shear flow Phys. Rev. Lett. 18, 946.Google Scholar
Haupt, O. 1912 Über die Entwicklung einer willkürlichen Funktion nach den Eigenfunktionen des Turbulenzproblems Sber. bayer. Akad. Wiss. 2, 289.Google Scholar
Kelly, R. E. 1965 The stability of an unsteady Kelvin-Helmholtz flow J. Fluid Mech. 22, 547.Google Scholar
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time J. Fluid Mech. 27, 657.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lock, R. C. 1956 The stability of an electrically conducting fluid between parallel planes under a transverse magnetic field. Proc. Roy. Soc. A 233, 105.Google Scholar
Miller, J. A. & Fejer, A. A. 1964 Transition phenomena in oscillating boundary-layer flows J. Fluid Mech. 18, 438.Google Scholar
Obremski, H. J. & Fejer, A. A. 1967 Transition in oscillating boundary layer flows J. Fluid Mech. 29, 93.Google Scholar
Parlett, B. N. 1967 The LU and QR algorithms, in Mathematical Methods for Digital Computers, II, Ed. A. Ralston and H. S. Wilf. New York: John Wiley.
Pekeris, C. L. 1948 Stability of the laminar flow through a straight pipe of circular cross-section to infinitesimal disturbances which are symmetrical about the axis of the pipe Proc. Natl. Acad. Sci. U.S. 34, 285.Google Scholar
Ralston, A. 1962 Runge-Kutta methods with minimum error bounds Math. Comp. 16, 431.Google Scholar
Reid, W. H. 1965 The stability of parallel flows, in Basic Developments in Fluid Dynamics, Ed. M. Holt. New York: Academic Press.
Schensted, I. V. 1960 Contributions to the theory of hydrodynamic stability. Ph.D. dissertation, University of Michigan.
Shen, S. F. 1954 Calculated amplified oscillations in the plane Poiseuille and Blasius flows J. Aero. Sci. 21, 62.Google Scholar
Shen, S. F. 1961 Some considerations on the laminar stability of time-dependent basic flows J. Aerospace Sci. 28, 397.Google Scholar
Shen, S. F. 1964 Stability of laminar flows. In Theory of Laminar Flows, Ed. F. K. Morse. Princeton: Princeton University Press.
Southwell, R. V. & Chitty, L. 1930 On the problem of hydrodynamic stability. I. Uniform shearing motion in a viscous fluid. Phil. Trans. A 229, 205.Google Scholar
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. A 221, 189.Google Scholar
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layers, Ed. L. Rosenhead. Oxford University Press.
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. A 223, 289.Google Scholar
Thomas, L. H. 1953 The stability of plane Poiseuille flow Phys. Rev. 91, 780.Google Scholar
Wilkinson, J. H. 1965 The Algebraic Eigenvalue Problem. Oxford University Press.
Supplementary material: PDF

Grosch and Salwen supplementary material

Supplementary Tables A-B

Download Grosch and Salwen supplementary material(PDF)
PDF 936.9 KB