Spatial and temporal characteristics of modulated waves in the circular Couette system

We have used flow-visualization and spectral techniques to study the spatial and temporal properties of the flow that precedes the onset of weak turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating (the circular Couette system). The first three flow regimes encountered as the Reynolds number is increased from zero are well-known – Couette flow, Taylor-vortex flow, and wavyvortex flow. The present study concerns the doubly periodic regime that follows the (singly periodic) wavy-vortex-flow regime. Wavy-vortex flow is characterized by a single frequency f 1 , which is the frequency of travelling azimuthal waves passing a point of observation in the laboratory. The doubly periodic regime was discovered in studies of power spectra several years ago, but the fluid motion corresponding to the second frequency f 2 was not identified. We have found that f 2 corresponds to a modulation of the azimuthal waves; the modulation can be observed visually as a periodic flattening of the wavy-vortex outflow boundaries. Moreover, in addition to the previously observed doubly periodic flow state, we have discovered 11 more doubly periodic flow states. Each state can be labelled with two integers m and k , which are simply related to physical characteristics of the flow: m is the number of azimuthal waves, and k is related to the phase angle between the modulation of successive azimuthal waves by Δϕ = 2π k / m . This expression for the phase angle was first conjectured from the flow-visualization measurements and then tested to an accuracy of 0·01π in spectral measurements. Recently Rand (1981) has used dynamical-systems concepts and symmetry considerations to derive predictions about the space–time symmetry of doubly periodic flows in circularly symmetric systems. He predicted that only flows with certain space–time symmetries should be allowed. The observed flow states are in agreement with this theory.