5 - Development of Instabilities in Time and Space

Summary

And wisdom and knowledge shall be the stability of thy times….

Isaiah xxxiii 6

More advanced properties of instabilities will be described in this chapter. The development of normal modes in space as well as time, the effect of weak nonlinearity and the energy budget will be explained.

*The Development of Perturbations in Space and Time

For partial differential systems, such as those describing fluid motions, it is valuable to analyse the nature of stability in more detail.

First, note that if a flow is bounded (and, of course, in practice all flows are bounded), then there is in general a countable infinity of normal modes, but that if the flow is unbounded then there is an uncountable infinity of normal modes; for the Poiseuille pipe flow of Example 2.11, which is unbounded in the x-direction, there is a continuum of modes with a continuous wavenumber k as well as discrete wavenumbers for θ- and r-variations, but for flow in a cube there would be three discrete wavenumbers to specify each normal mode. So for an unbounded flow the most unstable mode can be no more than first among equals, but for a bounded flow the growth rate of the most unstable mode will in general be substantially greater than that of the second most unstable mode. For bounded flows of large aspect ratio (or large Reynolds number), the most unstable modes are usually close together and so approximate a continuum.