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Transition in circular Couette flow

Published online by Cambridge University Press:  28 March 2006

Donald Coles
Affiliation:
Harvard University, Cambridge, Massachusetts
On leave of absence from Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California.

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.

Type
Research Article
Copyright
© 1965 Cambridge University Press

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References

Couette, M. 1890 Ann. Chim. Phys. (6), 21, 433510.
Davey, A. 1962 J. Fluid Mech. 14, 33668.
Diprima, R.C. 1961 Phys. Fluids, 4, 7515.
Donnelly, R. J. 1963 Phys. Rev. Letters, 10, 2824.
Fraenkel, L. E. 1956 Proc. Roy. Soc. A, 233, 50626.
Fultz, D. 1961 Adv. Geophys. 7, 1103.
Görtler, H. 1944 Z. angew. Math. Mech. 24, 21014.
Hagerty, W. W. 1946 Ph.D. dissertation, University of Michigan, Ann Arbor.
Kovasznay, L. S. G. 1949 Proc. Heat Transf. Fluid Mech. Inst. 21122.
Lewis, J. W. 1928 Proc. Roy. Soc. A, 117, 388406.
Malkus, W. V. R. & Veronis, G. 1958 J. Fluid Mech. 4, 225.
Mallock, A. 1888 Proc. Roy. Soc. A, 45, 12632.
Mallock, A. 1896 Phil. Trans. A, 187, 4156.
Pai, S.-I. 1943 NACA TN 892.
Rayleigh, Lord 1916 Proc. Roy. Soc. A, 93, 14854.
Roshko, A. 1955 NACA TN 3488.
Schultz-Grunow, F. & Hein, H. 1956 Z. Flugwiss. 4, 2830.
Segel, L. A. 1962 J. Fluid Mech. 14, 97.
Stuart, J. T. 1958 J. Fluid Mech. 4, 121.
Taylor, G. I. 1923 Phil. Trans. 223, 289343.
Taylor, G. I. 1936 Proc. Roy. Soc. A, 157, 54664.
Terada, T. & Hattori, K. 1926 Rep. Tokyo Aeron. Res. Inst. 2 (no. 26), 287326.
Wendt, F. 1933 Ing. Arch. 4, 57795.