# Hydrodynamic Instability and Transition to TurbulenceMore About Linear Stability Theory: Studies of The Initial-Value Problem

Hydrodynamic Instability and Transition to Turbulence: More About Linear Stability Theory:... [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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{e}^{-}}^{i\omega t}$$ \end{document} and, in the case of homogeneity in the streamwise direction Ox, also to eikx which grows exponentially as t → ∞ or, in the spatial formulation, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x\to \infty$$ \end{document}), 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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Hydrodynamic Instability and Transition to TurbulenceMore About Linear Stability Theory: Studies of The Initial-Value Problem

Part of the Fluid Mechanics and Its Applications Book Series (volume 100)
Editors: Frisch, Uriel
116 pages      Publisher
Springer Netherlands
ISBN
978-94-007-4236-9
Pages
175 –290
DOI
10.1007/978-94-007-4237-6_3
Publisher site
See Chapter on Publisher Site

### 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{e}^{-}}^{i\omega t}$$ \end{document} and, in the case of homogeneity in the streamwise direction Ox, also to eikx which grows exponentially as t → ∞ or, in the spatial formulation, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x\to \infty$$ \end{document}), 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.]

Published: Dec 17, 2012

Keywords: Plane Parallel Flow; Inviscid Plane Couette Flow; Circular Poiseuille Flow; Vertical Velocity Disturbance; Transient Growth