Transition in circular Couette flow

Abstract

Two distinct kinds of transition have been identified in Couette flow between concentric rotating cylinders. The first, which will be called transition by spectral evolution, is characteristic of the motion when the inner cylinder has a larger angular velocity than the outer one. As the speed increases, a succession of secondary modes is excited; the first is the Taylor motion (periodic in the axial direction), and the second is a pattern of travelling waves (periodic in the circumferential direction). Higher modes correspond to harmonics of the two fundamental frequencies of the doubly-periodic flow. This kind of transition may be viewed as a cascade process in which energy is transferred by non-linear interactions through a discrete spectrum to progressively higher frequencies in a two-dimensional wave-number space. ,At sufficiently large Reynolds numbers the discrete spectrum changes gradually and reversibly to a continuous one by broadening of the initially sharp spectral lines.These periodic flows are not uniquely determined by the Reynolds number. For the case of the inner cylinder rotating and the outer cylinder at rest, as many as 20 or 25 different states (each state being defined by the number of Taylor cells and the number of tangential waves) have been observed at a given speed. As the speed changes, theso states replace each other in a repeatable but irreversible pattern of transitions; vortices appear or disappear in pairs, and waves are added or subtracted. More than 70 such transitions have been found in the speed range up to about 10 times the first critical speed. Regardless of the state, however, the angular velocity of the tangential waves is nearly constant at 0.34 times the angular velocity of the inner cylinder.The second kind of transition, which will be called catastrophic transition, is characteristic of the motion when the outer cylinder has a larger angular velocity than the inner one. At a fixed Reynolds number, the fluid is divided into distinct regions of laminar and turbulent flow, and these regions are separated by interfacial surfaces which may be propagating in either direction. Under some conditions the turbulent regions may appear and disappear in a random way; under other conditions they may form quite regular patterns. One common pattern of particular interest is a spiral band of turbulence which rotates at very nearly the mean angular velocity of the two walls without any change in shape except possibly an occasional shift from a right-hand to a left-hand pattern. One example of this spiral turbulence is being studied in some detail in an attempt to clarify the role played in transition by interfaces and intermittency.~~~~

The rise of fully turbulent flow

“…8,9 The sudden appearance of localised turbulent patches in an otherwise quiescent flow was first observed by Osborne Reynolds for pipe flow 1 and has since been found to be the starting point of turbulence in most shear flows. 2,4,[10][11][12][13][14][15] Curiously, in this regime it is impossible to maintain turbulence over extended regions as it automatically 16,17 reduces to patches of characteristic size, called puffs in pipe flow (see Fig. 1a).…”

Shear instabilities in Taylor-Couette flow

“…The H 1 and H 2 bulleted curves shown in Fig. 2c correspond to experimental hysteretic SPT boundaries when decreasing R i from above in [2] and [1], respectively. In particular, we found …”

“…This has been done for values of R i and R o for which experiments reported laminar-turbulent spiral coherence, also termed as spiral turbulence [2,5,4]. Numerical simulation with moderate resolution have managed to reproduce such a flow (see Fig.…”

Experiments in a rotating source-sink annulus

“…Similarly, Coles (1965) in a Couette flow study found that the transition to turbulence occurred at a higher Rossby number than that at whi ch it woul d di sappear when condi tions were reversed. Since a finite amplitude disturbance can exist and be unstable in a flow Reynolds number less than the initial value for production of infinitesimal disturbances, the present hysteresis results are explainable.…”