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: email@example.com (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 . 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 . Davey et al.  analytically determined the value of Tac2 ; that was subsequently conﬁrmed by EaglesÕ experiment . Many...