8 - Transient dynamics

Summary

The initial-value problem

The fundamental needs for specifying an initial-value problem for stability investigations are not in any way different from those that have long since been established in the theory of partial differential equations. This is especially true in view of the fact that the governing equations are linear. Thus, by knowing the boundary conditions as well as the particular initial specification, the problem is, in principle, complete. Unfortunately, in this respect, classical theory deals almost exclusively with second order systems and, as such, few problems in this area can be cast in terms of well known orthogonal functions. For the equations that are the bases of shear flow instability, however, it is only the inviscid problem that is second order (Rayleigh equation) and even this limiting equation does not have a detailed set of known functional solutions. The more serious case where viscous effects are retained, then the minimum requirement is an equation that is fourth order (Orr-Sommerfeld equation) and even this, as previously noted, is fortuitous. An a priori inspection would have led one to believe that the full three-dimensional system should be sixth order, such as that discussed for the case of the Ekman boundary layer, for example. The net result is one where there are neither known closed solutions nor mutual orthogonality. It is only the accompanying Squire equation, where the solutions are coupled to those of the Orr-Sommerfeld equation, that eventually makes for sixth order.