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Hydrodynamic Instability and Transition to Turbulence pp 175–290Cite as

More About Linear Stability Theory: Studies of The Initial-Value Problem

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Abstract

The normal-mode method of the linear stability theory, which was considered in Chap. 2, deals only with special “wave-like” infinitesimal disturbances of a given laminar flow. This method equates the strict instability of a steady flow to the existence of at least one wave-like disturbance (proportional to \( {{e}^{-}}^{i\omega t} \) and, in the case of homogeneity in the streamwise direction Ox, also to e ikx which grows exponentially as t → ∞ or, in the spatial formulation, as \(x\to \infty \)), and states that ordinary instability means that there exists a wave-like disturbance which is not damped at infinity. (The adjectives “strict” and “ordinary” will be omitted below in all cases where the difference between two types of instability is unimportant or it is clear from context which instability is considered.) However, is this definition of instability always appropriate? Is it not more reasonable to call a flow unstable, if there exists at least one small disturbance of any form which grows without bound after a long-enough time? Moreover, in practice even a bounded but large-enough initial growth of a small disturbance can violate the applicability of the linear stability theory, and make the flow unstable whatever be the asymptotic behavior of this disturbance according to linear theory. In Sect. 2.5 we have already noted in this respect that practical usefulness of the method of normal modes must not be exaggerated. In this chapter this topic will be considered at greater length.

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Notes

  1. 1.

    According to Lyapunov, a trajectory \( {{U}_{0}}(t),0\le t<\infty , \) of a dynamic system in a phase space with a norm \( ||U|| \)is stable, if for any \( \varepsilon>0 \) there exists a number \( \delta (\varepsilon )>0 \) such that for any initial value \( U(0) \) satisfying the inequality \( ||U(0)-{{U}_{0}}(0)||<\delta (\varepsilon ) \) the inequality \( ||U(t)-{{U}_{0}}(t)||<\varepsilon\) is valid for any t. For more details about such stability and discussion of its application to fluid mechanics, see Sect. 4.1 in Chap. 4 of this series. Lyapunov’s stability clearly depends on the selection of the norm \( ||U|| \) which in studies by Dikii included the absolute values of the function and its two derivatives on z.

  2. 2.

    It is true that Eq. (3.13) implies only that if the initial values of \( |w|,\ |w'| \) and \( |w''| \) are small enough, then their root-mean-square values will be bounded by some small constants at any value of t. However, using results of the initial-value-problem investigations, it is possible to prove that in fact the values of these functions of t and z will be uniformly bounded by some small constants for all t > 0 and 0 < z < H; see, e.g., Dikii (1976).

  3. 3.

    They are not identical to normal modes since the functions c(k) do not coincide with the discrete eigenvalues of Rayleigh’s eigenvalue problem, which do not exist in many important cases. In fact, functions c(k) correspond to limits, as \( \operatorname{Re}\to \infty , \) of discrete eigenvalues of the Orr-Sommerfeld eigenvalue problem for given profile U(z); their determination from the Rayleigh equation requires careful examination of the analytic continuation fo this equation into the complex-variable plane.

  4. 4.

    These results were found for the plane-parallel model of the Blasius boundary layer. In reality the thickness of a boundary layer increases with x and this must lead to gradual weakening of the influence of viscosity. This effect was studied by Luchini (1996) who found that in the model of a boundary layer with the thickness depending on x a three-dimensional disturbance can exist which algebraic growth produced by the lift-up effect overcomes the viscous damping. Therefore, within the limits of the linear stability theory and of the model of a boundary layer of infinite streamwise extent, this disturbance is growing at all times.

  5. 5.

    The physical mechanism of the “force effect” is rather simple: If \( U'\ne 0, \) the vertical velocity w leads to vertical displacements of fluid particles transferring their original streamwise velocity to a new height with different mean velocity U, i.e., producing additional disturbances of the streamwise velocity (Landahl’s lift-up effect mentioned in Sect. 3.2). If \( U'\ne 0, \) and \( \partial w/\partial y\ne 0, \) then the lift-up effect will vary with the span wise coordinate y creating regions of non-zero derivative \( \partial u/\partial y \) and hence acting as a source of vertical vorticity. Note also that the first Eq. (3.53) describes forced streamwise velocity oscillations where the force on the right side represents the lift-up effect.

  6. 6.

    Note that the properties of self-adjointness and normality of an operator L depend not only on the operator itself but also on the norm (and scalar product) introduced in the space H. Thus, both these properties can be lost (or gained) when the norm is changed.

  7. 7.

    The situation is different in the case of stability problems four Couette-Taylor flow between two rotating coaxial cylinders and for convection in a layer of fluid heated from below. This fact leads to a fundamental difference between these two problems and stability problems for parallel shear flows, and means that the initial-problem approach is not useful in the case of the former two problems.

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Acknowledgments

This work could not have been accomplished without much help from a number of individuals and institutions. Therefore I am happy to have the possibility to express here my deep gratitude to Massachusetts Institute of Technology (MIT) which gave me an office for everyday work at its Department of Aeronautics and Astronautics (Dept. Head Prof. E.F. Crawley) and to the Center for Turbulence Research (CTR) at Stanford University and NASA Ames Research Center (Director Prof. P. Moin) which provided the main part of the financial support for the work on the series and arranged the publication of its separate chapters as CTR Monographs. Substantial support for my work was also given by the U.S. Army Research Laboratory (Dr. R.E. Meyers); Cambridge Hydrodynamics, Inc.; and by Dr. J.W. Poduska, Sr.

I am particularly indebted to Prof. P. Bradshaw of Stanford University, who read the whole manuscript with great care, corrected many defects of my English and made a number of suggestions leading to the improvement of the text, and to Prof. M.T. Landahl (MIT), who also read the manuscript and made some valuable suggestions. (Of course, for thau remaining deficiencies of language, style and scientific content I take full responsibility.) Many important remarks and useful written material related to the contents of Chap. 3 were received from K.S. Breuer, F.H. Busse, W.O. Criminale, L.A. Dikii, B.F.Farrell, S. Grossmann, F. Hussain, M.V. Morkovin, S.C. Reddy, and L.N. Trefethen. Moreover, K.S. Breuer, B.F. Farrell, and S.C. Reddy also helped me in preparing figures for this chapter. Debra Spinks of CTR regularly helped me with many administrative matters and problems arising in the preparation of the written material for publication, and (Ms.) Ditter Peschcke-Koedt made many improvements to my typing. To all the persons and organizations mentioned I express my sincere thanks.

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    Akiva M. Yaglom

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    Uriel Frisch

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    Uriel Frisch

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Yaglom, A., Frisch, U. (2012). More About Linear Stability Theory: Studies of The Initial-Value Problem. In: Frisch, U. (eds) Hydrodynamic Instability and Transition to Turbulence. Fluid Mechanics and Its Applications, vol 100. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4237-6_3

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