17th International Couette-Taylor Workshop, Leeds, July 25-27, 2011
Time constant in the onset and decay flow of Taylor-Couette flow around
critical Reynolds number
Takashi Watanabe1
1 Graduate
School of Information Science, Nagoya University, Nagoya 464-01 Japan
Introduction
Benjamin [1, 2] showed that the bifurcation from the Couette flow to Taylor vortex flow is
imperfect in the flow with finite cylinder length. After this study, many flow modes have been
discovered and bifurcation diagram in the space spanned in the Reynolds number and the aspect
ratio has been evaluated [3, 4, 5].
The onset and decay of the axially periodic Taylor-Couette flow was investigate by [6]. They
simplified the growth equation and estimate the onset and the decay of the amplitude of the
radial velocity component from or to the rest of the fluid. Abshagen et al. [7] used the StuartLandau equation and evaluated the time constant at one point of the flow field. They identified
the growing flow from the subcritical Reynolds number of Taylor Couette flow and the decaying
flow the supercritical Reynolds number. The result showed that the decaying time constant is
larger than the onset time constant.
In this paper, we numerically calculate the growth and decay of Taylor Couette flow from the
subcritical state and from the supercritical state, and determine the thime constant in the whole
azimuthal section.
Formulation
In the Taylor Couette system, the radii of the inner and outer cylinders are Ri and Ro , respectively. The radius ratio η = 0.667 and the critical Reynolds number for the onset of the
Taylor Couette flow is 66.5 [8]. The length of the cylinder is L and the aspect ratio Γ is given by
L/(Ro − Ri ). Both end walls of the cylinders are fixed and stationary.
The governing equation is the time-dependent axisymmetric Navier-Stokes equation and the
equation of the continuity represented by the cylindrical coordinate system. The velocity components in the radial, azimuthal and axial directions are u, v and w, respectively. These equations
are solved by the finite difference method. The fractional step method is used to integrate in the
time direction.
Initially, the subcritical flow at Re = 65.5 and the supercritical flow at Re = 67.5 are obtained
as well-developed flows. Then the Reynolds number is increased to 67.5 or it is decreased to
Figure 1: Velocity vector in the azimuthal section at the aspect ratio 4.0, 8.0, 12.0 and the
Reynolds number is 75.5 and 77.5.
65.5, and the onset and the decay of the flow are examined.
The time constant T is evaluate by fitting the calculated time-series data to the equation
A(t) = √
A f et/T
e2t/T + (A f /Ai )2 − 1
where Ai and A f are the initial and final amplitude [7].
Results
The steady state velocity vectors are shown in 1. The color contour shows the value of u2 +w2 .
When the aspect ratio Γ is small, the Ekman vortices induce inner vortex even at the subcritical
Reynolds number. At larger aspect ratio, no vortex is induced at the center in the axial direction.
The time constants of the radial velocity component and the axial velocity component are
shown in Fig.s. 2 and 3. The time begins to be measured when the Reynolds number is changed
in the fully developed flow. In each figure, Tmax shows the maximum of the color map represented by red, and the blue denotes zero time constant.. Please note that the Tmax may be
different in each figure. The time constant of the decay is more than three times larger when
compared the time constant at the onset. The time constant at the inner region is longer than that
of the flow near the end wall of the cylinder where Ekman layer appear (upper and lower ends
Figure 2: Contour of the time constant of the radial velocity component. In the color map, blue
represent the value of 0.0 and the red shows the value of Tmax .
in the figure).. In Fig. 2, the time constant is high near the center of the Taylor vorticies. This
means that, as the increase of the Reynolds number, it takes longer time for the radial velocity
component to reach the plateau near the vortex center where the value of the radial velocity
component is small. In Fig. 3, the time constant is small near the end walls. For the onset of the
flow, the value is also small at the radial position where Taylor vortices appear.
References
[1] T.B. Benjamin, Bifurcation phenomina in steady flow of a viscous fluid I, theory, Proc.
Roy, Soc. London A, 76, 1-26, (1978).
[2] T.B. Benjamin, Bifurcation phenomina in steady flow of a viscous fluid II, experiment,
Proc. Roy, Soc. London A, 76, 27-43, (1978).
Figure 3: Contour of the time constant of the axial velocity component. In the color map, blue
represent the value of 0.0 and the red shows the value of Tmax .
[3] R. Tagg, The Taylor Couette problem, Nonlinear Science Today, 4, 1-25, (1994).
[4] H. Furukawa, T. Watanabe, Y. Toya, I. Nakamura, Flow pattern exchange in the TaylorCouette system with decelerating inner cylinder", Phys. Rev. E, 65, 036306-1-7,(2002).
[5] T. Watanabe, H. Furukawa, I. Nakamura, Nonlinear development of flow patterns in an
annulus with decelerating inner cylinder, Phys.Fluids, 14, 333-341, (2002).
[6] D. Dirshfeld, D.C. Rapaport, Growth of Taylor vortices: a molecular dynamics study, Phys.
Rev. E, 61, R-21-24, (2000).
[7] J. Abshagen, O. Meincke, G. Pfister, K.A.Cliffe, T.Mullin, Transient dynamics at the onset
of Taylor vortices, J. Fluid Mech., 476, 335-343, (2003).
[8] E.L. Koschmieder, Bénard cells and Taylor vorticies, Cambridge Univ. Press. (1993).