Stability to Finite Disturbances: Energy Method and Landau’s Equation

Abstract

The main part of Chap. 2 and the whole of Chap. 3 were devoted to topics of linear stability theory dealing with the evolution of very small flow disturbances satisfying the linearized fluid dynamics equations. In Chap. 2 it was shown that the classical normal-mode method of the linear theory of hydrodynamic stability often leads to results which strongly disagree with experimental data. It was also indicated there that these disagreements are apparently due to nonlinear effects, which make linearization of the equations of motion physically unjustified. In Chap. 3 it was explained that the necessity for consideration of the full nonlinear dynamic equations often follows from the fact that many solutions of the initial-value problems for linearized fluid dynamics equations grow considerably at small and moderate values of the time t even in the cases when the normal-mode analysis shows that these solutions decay asymptotically (i.e. at t → ∞).The nonlinear theory of hydrodynamic stability has achieved a high level of development. Although the theory is still far from being completed, it has elucidated many formerly mysterious properties of fluid flows which are interesting for physicists and important for engineers. There is now an enormous literature on this subject and only a small part of it, dealing with relatively simple flows of incompressible fluids, will be considered in this book. In the present Chapter two topics from the nonlinear stability theory will be discussed: the energy method of stability analysis (short introductory consideration of this method was included in Sect. 3.4 above) and Landau's approach to the weakly nonlinear stability theory which described the initial period of the nonlinear development of flow disturbances.