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Computers & Fluids 33 (2004) 97–118

Numerical study of Taylor–Couette flow with an axial flow Jong-Yeon Hwang, Kyung-Soo Yang

Department of Mechanical Engineering, Inha University, Incheon 402-020, Republic of Korea Received 20 April 2002; received in revised form 30 October 2002; accepted 23 January 2003

Abstract The flow between two concentric cylinders with the inner one rotating and with an imposed pressuredriven axial flow is studied using numerical simulation. This study considers the identical flow geometry and flow parameters as in the experiments of Wereley and Lueptow [Phys. Fluids 11 (12) (1999) 3637], where particle image velocimetry measurements were carried out to obtain detailed velocity fields in a meridional plane of the annulus. The objectives of this investigation are to numerically verify the experimental results of Wereley and Lueptow and to further study detailed flow fields and bifurcations related to Taylor–Couette flow with an imposed axial flow. The vortices in various flow regimes such as non-wavy laminar vortex, wavy vortex, non-wavy helical vortex, helical wavy vortex and random wavy vortex are all consistently reproduced with their experiments. It is demonstrated that Ôshift-and-reflectÕ symmetry holds in Taylor–Couette flow without an imposed axial flow. In case of Taylor–Couette flow with an imposed axial flow, one can find that the shift-and-reflect symmetry is roughly valid for the remaining velocity field after subtracting the annular Poiseuille flow. The axial flow stabilizes the flow field and decreases the torque required by rotating the inner cylinder at a given speed. Growth rate of the flow instability is defined and used in predicting the type of the vortices. The velocity vector fields obtained also reveal the same vortex characteristics as found in the experiments of Wereley and Lueptow. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Taylor–Couette flow; Axial flow; Instability; Simulation; Vortex; Shift-and-reflect symmetry

1. Introduction The flow between two concentric cylinders with the inner one rotating and the outer one stationary, called Taylor–Couette flow, has been studied by many researchers for decades. With a low rotating speed of the inner cylinder, the exact solution of the laminar velocity field consists of *

Corresponding author. Tel.: +82-32-860-7322; fax: +82-32-863-3997. E-mail address: (K.-S. Yang).

0045-7930/04/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0045-7930(03)00033-1


J.-Y. Hwang, K.-S. Yang / Computers & Fluids 33 (2004) 97–118

vr ¼ vz ¼ 0;

vh ¼ Xi ri

ro =r À r=ro ; ro =ri À ri =ro


where r, h and z represent the radial, azimuthal, and axial directions of the cylindrical coordinate system, respectively. The angular velocity of the inner cylinder is denoted by Xi ; the inner and outer radii are represented by ri and ro , respectively. It should be noted that Eq. (1) is the solution under the idealization of infinitely long cylinders where endwall effects are ignored; this solution is not realized exactly in an physical experiment or simulation that include endwall effects. If the Taylor number (Ta) based on Xi goes over a critical one (Tac1 ), the flow instability caused by the curved streamlines of the main flow produces axisymmetric Taylor vortices. This fact was first noticed by Taylor (1923) in an analytical study of the related flow instability [1]. Since then, many researchers have studied the instability causing Taylor vortices [2–4]. In the early days of studying Taylor vortices, researchersÕ attention was mainly focused on determining Tac1 by experimental or analytical methods. As Ta further increases over a higher threshold value (Tac2 ), the Taylor vortices become unsteady and non-axisymmetric, called wavy vortices [5]. Davey et al. [5] analytically determined the value of Tac2 ; that was subsequently confirmed by EaglesÕ experiment [6]. Many...
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