Roads to chaos

Hydrodynamic systems often show an extremely complicated and apparently erratic flow pattern of the sort shown in figure 1. These turbulent flows are so highly time‐dependent that local measurements of any quantity that describes the flow—one component of the velocity, say—would show a very chaotic behavior. However, there is also an underlying regularity in which the motion can be analyzed (see figure 1 again) as a series of large swirls containing smaller swirls, and so forth. One approach to understanding this turbulence is to ask how it arises. If one puts a body in a stream of a fluid—for example, a piece of a bridge sitting in the stream of a river—then for very low speeds (figure 2a) the fluid flows in a regular and time‐independent fashion, what is called laminar flow. As the speed is increased (figure 2b), the motion gains swirls but remains time‐independent. Then, as the velocity increases still further, the swirls may break away and start moving downstream. This induces a time‐dependent flow pattern—as viewed from the bridge. The velocity measured at a point downstream from the bridge gains a periodic time‐dependence like that shown in figure 2c. The parameter that characterizes these changes in the flow pattern is the dimensionless Reynolds number R which is the product of the velocity and density times a characteristic length (the size of the bridge pier, for example) divided by the viscosity. As R is increased still further, the swirls begin to induce irregular internal swirls as in the flow pattern of figure 2d. In this case, there is a partially periodic and partially irregular velocity history (see the second column of figure 2d). Raise R still further and a very complex velocity field is induced, and the v(t) looks completely chaotic as in figure 2e. The flow shown in figure 1 has this character.