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When studying the asymptotic stability of the basic flow, it is possible to simplify the problem essentially by reducing the consideration of nonlinear equations of motion to the analysis of linearized equations for disturbances. In this chapter, various aspects of the corresponding theory for the so-called parallel shear flows are expounded beginning from formulation of linear hydrodynamic stability problems in time and space for wavy (modal) disturbances. The classical Gaster’s transformation between these two approaches is explained. The dual role of viscosity for flow instability is outlined. Then, relevance of oblique waves in instability is discussed. Finally, such special issues, important in the following chapters, as bi-orthogonality of modes and completeness of their set are introduced.
KeywordsShear Layer Linear Stability Critical Reynolds Number Homogeneous Boundary Condition Neutral Stability
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- Fjørtoft R (1950) Amplification of integral theorems in deriving criteria for instability for laminar flows and the baroclinic circular vortex. Geofus Publ 17(6):1–52 Google Scholar
- Orr WM (1907a) The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part 1: A perfect liquid. Proc R Irish Acad A 27:9–68 Google Scholar
- Orr WM (1907b) The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part 2: A viscous liquid. Proc R Irish Acad A 27:69–138 Google Scholar
- Prandtl L (1922) Bemerkungen über die Entstehung der Turbulenz. Phys Z 23:19–25 Google Scholar
- Lord Rayleigh (1880) On the stability, or instability, of certain fluid motions. Proc Lond Math Soc 9:57–70 Google Scholar
- Schensted IV (1960) Contributions to the theory of hydrodynamic stability. PhD thesis, University of Michigan Google Scholar
- Sommerfeld A (1908) Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegungen. In: Atti IV Congr. Internaz. Mat., Acad. dei Lincei, Rome, vol 3, pp 116–124 Google Scholar
- Tollmien W (1935) Ein algemeines kriterium der instabilität laminarer geschwindigkeitsverteilungen. Nachr Ges Wiss Göttingen (Matematik) 1(5):79–114, translated as NACA TM 792, 1936 Google Scholar
- Yudovich VV (1965) On stability of stationary flows of viscous incompressible fluid. Dokl Akad Nauk 161(5):1037–1040 Google Scholar
- Zhigulev VN, Tumin AM (1987) Onset of turbulence. Dynamical theory of excitation and development of instabilities in boundary layers. Nauka. Sib. Otd., Novosibirsk, In Russian. Google Scholar
- Prandtl L (1935) The mechanics of viscous fluids. In: Durand WF (ed) Aerodynamic Theory, vol 3, Springer–Verlag, Berlin Google Scholar