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Stability of plane Couette flow for high Reynolds number

Published online by Cambridge University Press:  17 February 2009

G. B. Davis  and
A. G. Morris
A. G. Morris
Affiliation:
Department of Mathematics, The University of Wollongong, Wollongong, N.S.W.2500.
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Abstract

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Experimental evidence shows that plane Couette flow becomes unstable when the Reynolds number R reaches certain critical values. Linear stability theory does not predict these observations and has been unable to locate these instabilities. A Chebyshev/QR numerical technique is used to investigate much higher values of R than those previously tested. In particular, values of R up to 108 are confidently tested, whereas previously values of R up to only 2 × 104 have been considered.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 3 , January 1983 , pp. 350 - 357
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Davey, A., “On the stability of plane Couette flow to infinitesimal disturbances”, J. Fluid Mech. 57 (1973), 369–380.CrossRefGoogle Scholar
[2]Deardorff, J. W., “On the stability of viscous plane Couette flow”, J. Fluid Mech. 15 (1963), 623–631.CrossRefGoogle Scholar
[3]Ellingsen, T., Gjevik, B. and Palm, E., “On the nonlinear stability of plane Couette flow”, J. Fluid Mech. 40 (1970), 97–112.CrossRefGoogle Scholar
[4]Francis, J. G. F., “The QR transformation, a unitary analogue to the LR transformation”, Parts I and 2, Comput. J. 4 (1961), 265–271, 332–345.CrossRefGoogle Scholar
[5]Gallagher, A. P. and Mercer, A. McD, “On the behaviour of small disturbances in plane Couette flow”, J. Fluid Mech. 13 (1962), 91–100.CrossRefGoogle Scholar
[6]Gallagher, A. P. and Mercer, A. McD, “On the behaviour of small disturbances in plane Couette flow. Part 2. The higher eigenvalues”, J. Fluid Mech. 18 (1964), 350–352.CrossRefGoogle Scholar
[7]Grohne, D., “Über das Spektrum bei Eigenschwingungen ebener Laminarströmungen”, Z. Angew. Math. Mech. 34 (1954), 344–357 (Translated as “On the spectrum of natural oscillations of two-dimensional laminar flows”, Tech Memor. N.A.C.A. Wash. No. 1417).CrossRefGoogle Scholar
[8]Hopf, L., “Decrement of small vibrations in the flow of a viscous fluid”, Ann. Physik 44 (1914), 1–60.CrossRefGoogle Scholar
[9]Jordinson, R., “The flat plate boundary layer, Part I: Numerical integration of the OrrSommerfeld equation”, J. Fluid Mech. 43 (1970), 801–811.CrossRefGoogle Scholar
[10]Lee, L. H. and Reynolds, W. C., “On the approximate and numerical solution of Orr-Sommerfeld problems”, Quart. J. Mech. Appl. Math. 20 (1967), 1–22.CrossRefGoogle Scholar
[11]Morris, A. G., “Eigenvalues by numerical methods”, Ph.D. Thesis, Wollongong University College, University of New South Wales, 1973.Google Scholar
[12]Morris, A. G. and Horner, T. S., “Chebyshev polynomials in the numerical solution of differential equations”, Math. Comp. 31 (1977), 881–891.CrossRefGoogle Scholar
[13]Orszag, S. A., “Accurate solution of the Orr-Sommerfeld stability equation”, J. Fluid Mech. 50 (1971), 689–703.CrossRefGoogle Scholar
[14]Osborne, M. R., “Numerical methods for hydrodynamic stability problems”, SIAM J. Appl. Math. 15 (1967), 539–557.CrossRefGoogle Scholar
[15]Reichardt, H., “Gesetzmässigkeiten der geradlinigen turbulenten Couetteströmung”, Mitteilungen aus dem Max-Planck Institut für Strömungsforschung und der Aerodynamischen Versuchsanstalt 22 (1959).Google Scholar
[16]Robertson, J. M., “On turbulent plane-Couette flow”, Proc. 6th Midwestern Conference on Fluid Mechanics (1959), 169–182.Google Scholar
[17]Squire, H. B., “On the stability of the three-dimensional disturbances of viscous flow between parallel walls”, Proc. Roy. Soc. London Ser. A 142 (1933), 621–628.Google Scholar
[18]Stuart, J. T., “On the nonlinear mechanics of hydrodynamic stability”, J. Fluid Mech. 4 (1958), 1–21.CrossRefGoogle Scholar
[19]Stuart, J. T., “On nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part I. The basic behaviour in plane Poiseuille flow”, J. Fluid Mech. 9 (1960), 353–370.CrossRefGoogle Scholar
[20]Thomas, L. H., “Stability of plane Poiseuille flow”, Phys. Rev. 91 (1953), 780–783.CrossRefGoogle Scholar
[21]van Stijn, Th. L. and van de Vooren, A. I., “An accurate method for solving the Orr-Sommerfeld equation”, J. Engrg. Math. 14 (1980), 17–26.CrossRefGoogle Scholar
[22]Watson, J., “On the non-linear mechanics of wave disturbances in parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow”, J. Fluid Mech. 9 (1960), 371–389.CrossRefGoogle Scholar
[23]Wilkinson, J. H., The algebraic eigenvalue problem (Oxford University Press, London, 1965).Google Scholar