In this chapter, we shall draw together some general features of the onset of chaos and turbulence. The theory of dynamical systems, and in particular the theories of bifurcation and chaos, provide a mathematical framework with which we may interpret qualitatively the transition to turbulence without having to clutter our minds with a lot of detail. This framework can be used together with physical arguments of the mechanics of transition to understand the essence of instability of flows which may be so complicated geometrically as to defy solution except in numerical terms. However, the dynamics of fluids is very diverse, and the details of transition to turbulence depend on the details of the flow undergoing transition, and therefore can only be found by careful experiments and computational fluid dynamics of each case.

Evolution of Flows as the Reynolds Number Increases

The details of transition to turbulence not only are complicated but also vary greatly from flow to flow, so there is no possibility of a short summary of all transition. However, there are some unifying themes in the theory, and a few routes to turbulence essentially shared by many flows, even though the physical mechanisms of the same route may differ from one flow to another sharing the same route.