Hydrodynamic Stability, Second Edition
Containing a large number of problems that provide exercises as well as solutions, this book offers a clear and unparalleled explanation of the fundamentals of hydrodynamic stability.
TABLE OF CONTENTS Hydrodynamic Stability, Second Edition
Chapter 7: Nonlinear Stability
Nonlinear hydrodynamic stability theory is really concerned, ultimately, with phenomena such as transition to turbulence. In practice, however, that phenomenon is so complex as to defy rational understanding at the present time. A more limited objective is that of gaining some understanding of nonlinear processes in fluid mechanics, perhaps with reference to the early, relatively-simple stages of the evolution of laminar flow to turbulence. Even then, the mathematical problems posed are challenging enough.
J.T.Stuart (1977)
49 Introduction
49.1 Landau s theory
Although Reynolds (1883) appreciated the importance of nonlinear disturbances of Poiseuille flow in a pipe, and Bohr (1909), Noether (1921) and Heisenberg (1924) treated them theoretically for other special problems, it may be said that the foundations of the theory of nonlinear hydrodynamic stability were laid by Landau in 1944 (see Landau & Lifshitz (1959), 27 for a similar account). In a prophetic essay, Landau outlined the development of linear instability towards the onset of turbulence. His ideas have required substantiation or qualification, but his overall vision has been confirmed by subsequent work. His theory is described below from a modern viewpoint, with some of the details filled in.
The linear theory of stability of a steady basic flow quite generally gives a spectrum of independent modes, each with velocity perturbation of the form
for some complex amplitude A ?e st with relative growth rate s= ?+i ?. When R< R c all disturbances are stable with ?<0. (We write...
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