Fluid Flow as a Function of Reynolds Number

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

Before we discuss some conjectures as to the onset of turbulence, let us first of all get a qualitative feel for the different stages of behavior as the Reynolds number is increased. Consider, for example, the viscous flow around a circular cylinder ([feyn64], [bat67]). 

For very low Reynolds numbers, 7 . 1, the flow lines wrap smoothly around the cylinder, as shown in figure 9.1. 

Increasing TZ to values 1 ( 10 - 20) induces a distinct change in behavior a circulation appears, in the form of two symmetrically positioned vortices behind the cylinder (see figure 9.2). It is believed that the circulation pattern appears suddenly at some threshold value of TZ,. Also notice that the fluid velocity at any given point in space is still constant. 

An altogether different behavior emerges for slightly larger values of 7 ( 40 -tSee also our discus.sion of spatiotemporal chaos in coupled-map lattices in section 8.2. 

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