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Accurate solution of the Orr–Sommerfeld stability equation

Published online by Cambridge University Press:  29 March 2006

Steven A. Orszag
Steven A. Orszag
Affiliation:
Department of Mathematics Massachusetts Institute of Technology
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Abstract

The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 50 , Issue 4 , 29 December 1971 , pp. 689 - 703
Copyright
© 1971 Cambridge University Press

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References

Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Cooley, J. W., Lewis, P. A. W. & Welch, P. D. 1970 The fast Fourier transform algorithm: programming considerations in the calculation of sine, cosine, and Laplace transforms. J. Sound Vib. 12, 315–337.Google Scholar
Davey, A. & Drazin, P. O. 1969 The stability of Poiseuille flow in a pipe. J. Fluid Mech. 36, 209–218.Google Scholar
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701–720.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. Quart. Appl. Math. 16, 97–110.Google Scholar
Fox, L. 1962 Chebyshev methods for ordinary differential equations. Computer J. 4, 318–331.Google Scholar
Fox, L. & Parker, I. B. 1968 Chebyshev Polynomials in Numerical Analysis. Oxford University Press.
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
Gary, J. & Helgason, R. 1970 A matrix method for ordinary differential eigenvalue problems. J. Comp. Phys. 5, 169–187.Google Scholar
Grosch, C. E. & Salwen, H. 1968 The stability of steady and time-dependent plane Poiseuille flow. J. Fluid Mech. 34, 177–205.Google Scholar
Hamming, R. W. 1962 Numerical Methods for Scientists and Engineers. McGraw-Hill.
Lanczos, C. 1956 Applied Analysis. Prentice-Hall.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Nachtsheim, P. R. 1964 An initial value method for the numerical treatment of the Orr-Sommerfeld equation for the case of plane Poiseuille flow. N.A.S.A. Tech. Note, D-2414.
Orszag, S. A. 1971a Galerkin approximations to flows within slabs, spheres, and cylinders. Phys. Rev. Lett. 26, 1100-1103.
Orszag, S. A. 1971b Numerical simulation of incompressible flows within simple boundaries I. Galerkin (spectral) representations. Studies in Appl. Math. 50, 293-327.
Shen, S. F. 1954 Calculated amplified oscillations in plane Poiseuille and Blasius flows. J. Aero. sci. 21, 62–64.Google Scholar
Thomas, L. H. 1953 The stability of plane Poiseuille flow. Phys. Rev. 91, 780–783.Google Scholar
Wilkinson, J. H. 1965 The Algebraic Eigenvalue Problem. Oxford University Press.
Wright, K. 1964 Chebyshev collocation methods for ordinary differential equations. Computer J. 6, 358–363.Google Scholar