188宝金博页面版

J . Fluid Mrch. (1988). col 191. p p 1-18 Printed in Great Britain 1 Bifurcation phenomena in Taylor-Couette flow in a very short annulus By G. PFISTER,t H. SCHMIDT,? K. A. CLIFFEt AND T. MULLINS Institube of Applied Physics, Univrrsity of Kiel. m'. CkArmany 1 Theoret,ic.al Physics Division. AERE Haruell. Oxford OX 1 ORA. UK !$(:larendon Laboratory, LTniversity of Oxford. Parks Road. Oxford OX 1 BPU. L X (Received 17 November 1986 and in revised form 4 November 1987) We present the results of an experimental and numerical investigation into Taylor-Couettc flow with gap-length to width ratios (r = l/d) ranging from 0.3 to 1.4. Laser-Doppler-velocimetry is used to obtain quantitative information on the bifurcation set experimentally, and novel flow phenomena are uncovered. These results are compared with those obtained using numerical bifurcation techniques applied to a finite-element discretization of the Navier-Stokes equations. In general, the agreement is good and most of the observations are satisfactorily explained. 1. Introduction The importance of end effect's in selecting the steady cellular stat>es of Taylor-Couette flow was recognized by Benjamin (1978a, b). He used Leray- Schauder degree theory to study the general properties of bifurcations among steady solutions of the Navier-Stokes equations. The degree arguments had previously been used by Velte (1964, 1966) to show the exist'ence of solutions for cellular motion in Taylor's (1923) infinite-cylinder model. Benjamin demonstrated the requirement that a minimum of nine solution branches is necessary to explain the exchange of stability between adjacent steady solutions. Thus, in any experiment where the cylinders are moderately long, the number of solutions is very large (Benjamin & Mullin 1982), so that' a rigorous study of the nature of the bifurcations is difficult in practice. Thus the studies of finite-length effects have so far been concentrated on shorter cylinders where well controlled experiments are less difficult to perform because the set of possible solutions is limit'ed. The hysteresis phenomena associated with the exchange of stability between adjacent modes was demonstrated experimentally by Benjamin (1978a, b) and later extended to other modes by Mullin (1982) and Mullin, Pfister & Lorenzen (1983). A model to explain the hysteretic events was proposed by Schaeffer (1980) who used a homotopy parameter, 7, to form a connection between the infinite-cylinder model (7 = 0) and the finite geometry (7 = 1) of the experiment. The model has been worked out in detail for a specific case by Hall (1982). The experimental and theoretical work was later confirmed to a large extent in the numerical work of Cliffe (1984) who also uncovered new aspects of the problem that are related to the reflection symmetry about the midplane. A necessary consequence of both Benjamin's and Schaeffer's work is the exist'ence of other stable, steady solution branches which are generally disconnected from those