We present numerical simulations to demonstrate the existence of laminar spiral flows between both finite and infinite length concentric cylinders and finite truncated cones where only the inner wall rotates. The velocities and pressure are calculated by a spectral element/Fourier method. Different gap ratios are investigated. Convergence of the numerical results is shown with reference to flows between infinite cylinders. The presence of top and bottom endplates results in vortex dislocations that are observed at the frontiers between the Ekman vortices present at each end and the spiral vortices

Blackburn, H. M.

Noui-Mehidi, M. N.

Ohmura, N.

University of Queensland eSpace

16th Australasian Fluid Mechanics Conference 
Crown Plaza, Gold Coast, Australia 
2-7 December 2007 
 
Laminar Spirals in the Outer Stationary Cylinder  
Couette-Taylor System 
 
M.N. Noui-Mehidi1, H.M. Blackburn2 and N. Ohmura3 
1CSIRO Manufacturing & Materials Technology 
PO Box 56, Highett, Victoria, 3190 AUSTRALIA 
    2Department of Mechanical Engineering 
Monash University, Victoria, 3800 AUSTRALIA 
3Department of Chemical Engineering 
Kobe University, Kobe, 657-8501 JAPAN 
 
 
Abstract 
We present numerical simulations to demonstrate the existence 
of laminar spiral flows between both finite and infinite length 
concentric cylinders and finite truncated cones where only the 
inner wall rotates.  The velocities and pressure are calculated by 
a spectral element/Fourier method.  Different gap ratios are 
investigated.  Convergence of the numerical results is shown 
with reference to flows between infinite cylinders.  The presence 
of top and bottom endplates results in vortex dislocations that 
are observed at the frontiers between the Ekman vortices present 
at each end and the spiral vortices. 
 
Introduction  
The last century has seen hundreds of works devoted to the flow 
confined between circular cylinders since the pioneering 
investigation of Taylor [1], but even now, new flow behaviours 
are reported both experimentally and theoretically [2-4].  
Laminar spirals have been studied extensively and were reported 
when both cylinders were oppositely rotated [5-10].  With such 
an experimental system, Andereck et al. [5] obtained a stability 
diagram where laminar spirals occupied a very narrow region.  In 
the Couette system the basic flow is naturally unstable to non-
axisymmetric spiral vortices near the inner cylinder when the 
outer cylinder rotates in the opposite direction as reported by 
Coles [11], Krueger et al. [12] and Snyder [13-14].  Spiral 
vortices are strongly dependent on initial conditions, as they exist 
in different flow modes, and may even co-exist with steady 
Taylor vortices [5].   Laminar spirals have been also observed in 
the system of conical cylinders.  Wimmer [15] found that helical 
flow structures could be generated between the conical cylinders 
when only the inner body was rotated.  These structures were 
very sensitive to initial conditions.  Noui-Mehidi et al. [16] have 
reported that the helical structure between conical cylinders could 
be generated only for very slow acceleration rates of the inner 
body.  
 
Andereck et al. [5], working in a system with counter-rotating 
finite cylinders, related the presence of laminar spirals to the 
presence of top and bottom boundaries.  However, Antonijoan et 
al. [4] predicted via numerical simulation that spiral regimes may 
exist in infinite Taylor-Couette flow.  They reported that spiral 
regimes (which in general are temporally periodic in a fixed 
frame of reference) arose through a Hopf bifurcation from the 
circular Couette flow in a system with counter-rotating cylinders.  
Their results also included solutions with laminar spirals in a 
system where only the inner cylinder could rotate, but this 
possibility was not dealt with in detail. 
 
The present paper reports on numerical simulations of laminar 
spirals in circular Couette flow. Motivation for this work initially 
emanated from an investigation of the helical flow between finite 
conical cylinders of same apex angle, and where only the inner 
cylinder rotates. Study of the effect of the conical apex angle on 
the stability of the helical structure showed that spiral vortices 
were maintained even at very small apex angle.  Then, the 
numerical projection of the solution obtained between conical 
cylinders into a cylindrical annular gap showed that spiral 
vortices were also maintained in a configuration of circular 
Couette flow where the outer cylinder is stationary, and where 
endwalls are present.  In agreement with the results of Antonijoan 
et al. [4], spiral vortices are also found to be stable when the 
cylinders are infinite in length for both counter-rotating cylinders, 
and when the outer cylinder is fixed.  Thus the results show that 
laminar spirals can exist between both infinite and finite circular 
cylinders when the outer cylinder is stationary and that these 
spirals are stable over a range of Reynolds numbers. 
 
 
Numerical method 
The direct numerical simulation formulation used here has been 
described in detail by Blackburn & Sherwin [17] and is reported 
briefly in the following.  In the cylindrical geometry the velocity 
u (z, r, θ) is projected by Fourier transformation in the azimuthal 
direction onto a set of two-dimensional complex modes ŭk (z, r).  
This Fourier basis is coupled with a spectral element 
discretization in the meridional plane.  Time integration is carried 
out with a second-order semi-implicit scheme.  The meridional 
computational domain was discretized into 231 spectral elements 
with Gauss-Lobatto-Legendre (GLL) nodal basis functions. 
Through numerical experiments, it was found that 16 planes of 
data in the azimuthal direction were sufficient to generate a stable 
and smooth solution. The accuracy of the results is also checked 
with regards of the number of element np as shown in Table 1.  A 
tensor-product polynomial order of 7 within each spectral 
element was found sufficient to achieve a good accuracy.  
 
 
np 3 5 7 9 
Umax 0.56513 0.56574 0.56566 0.56563 
 
Table 1. Comparative analysis of the accuracy of the numerical results: np 
is the GLL polynomial order in each direction within each spectral 
element, Umax is the averaged value of the maximum axial velocity 
component, normalized by Ωd.  
411
Results 
In our investigation, in order to validate the numerical results 
against those of Antonijoan et al. [4], calculations with a gap 
ratio of η=0.8 were conducted in the case of infinite counter-
rotating cylinders. The outer Reynolds number Ro= b Ω d / ν was 
fixed to –50 and the inner Reynolds number Ri= a Ωd / ν was 
varied between 106 and 120 (a is the inner cylinder radius and b 
is the outer cylinder radius, Ω the rotational speed, d the gap and 
ν the kinematic viscosity). 
A good agreement was found by determining the bifurcating 
critical value of Ri=106.1 with an accuracy of 0.6% compared to 
the results of Antonijoan et al. [4]. The study of the energy 
distribution at the different modes showed that the predominant 
mode in the spiral regime is the mode energy with k=1 as can be 
seen in Fig. 1. 
In the study of laminar spirals between conical cylinders, the first 
computations started with initial conditions that were a rest state 
to which was added a random perturbation of order 10-6 in the 
mode k=1. In the geometry of conical cylinders, both conical 
cylinders had the same apex angle α of 24, the inner conical 
cylinder rotating and the outer one stationary. Fixed endplates 
were imposed at the top and bottom of the flow system and the 
Reynolds number defined by:  Ri = a Ωd / ν was fixed to 200. 
When the solution reached steady state as ascertained by constant 
time series of the energies in all the azimuthal Fourier modes, the 
apex angle was decreased to 16 degrees and the simulations 
started again from the previous obtained solution. The procedure 
was repeated for an apex angle of 4 degrees until steady state was 
also reached. Figure 2 shows the flow structure obtained for an 
apex angle of 16 degrees and a Reynolds number of 200. At the 
final stage, for the same latter Reynolds number the apex angle 
was set to zero, which yields a cylindrical annulus configuration 
and the same procedure was adopted. The spiral structure 
obtained in the circular Couette case is presented in Fig. 3 for a 
Reynolds number Ri=300. 
 
Fig.1.  Distribution of the average azimuthal Fourier mode energy in the 
case of infinite counter-rotating cylinders with η=0.8, Ro=-50 and 
Ri=110. 
 
Calculations conducted with the outer cylinder fixed have 
revealed the persistence of laminar spirals in the infinite cylinder 
geometry. Figure 4 presents the spiral regime calculated at 
Ri=110 and Ro=0 for a gap ratio η=0.8.  
Time series of the velocity component revealed that the axial 
periodic drift of the spiral structure proceeds with a frequency fs 
that has a ratio to the rotational frequency fr of the inner cylinder, 
which decreases as Reynolds number increases. The ratio fs/fr has 
a value of 0.35 at Ri=220 and decreases to 0.32 at Ri=300 in the 
infinite cylinder geometry. 
 
 
Fig.2. Laminar spirals in the system of conical cylinders. Ri=200, α=16 
degrees, η=0.84 at the top of the system, Γ=H/d=12.50.  (H is the height 
of the fluid column, d the gap and α the total (apex) opening angle of the 
conical cylinders).  Isosurfaces show +/- radial velocity 
 
Fig.3. Laminar spirals in the system of concentric cylinders with 
endplates. Re=300, η=0.75, Γ=12.98, fs / fr = 0.35.  
Isosurfaces show +/- radial velocity 
 
 
With the presence of the top and bottom endplates, Ekman 
vortices are present near the fixed plates due to the pumping fluid 
layers. The spiral vortices become confined between the Ekman 
vortices and are separated at each end from the Ekman layer by a 
vortex dislocation. These dislocations observed also between 
conical cylinders, can be seen in Figs. 2 and 3 presenting 
isosurfaces of the axial velocity component for conical cylinders 
and concentric cylinders respectively. 
 
The analysis of the flow properties at the dislocation showed 
some interesting results. Figure 5 presents a contour plot of the 
radial component of the vorticity taken at an axial cross section 
of the cylinders. The section is located at the middle of the spiral 
vortex where the dislocation occurs.  
 
It can be noticed from Fig.5 that at the dislocation the vorticity of 
the spiral vortex is higher than the vorticity of the remaining 
vortex, which belongs to the Ekman layer. The spiral vortex 
occupies 3/8 of the cross section and is located near the inner 
cylinder.  
 
 
 
 
412
References 
 
 
 
[1] Taylor, G.I., Stability of a viscous fluid contained between 
rotating cylinders, Philos. Trans. R. Soc. London, Ser. A 
223, 1965, 289-343.  
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Taylor-Couette flow, Phys. Fluids 10, 1998, 3233-3235. 
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modulated Taylor-Couette flows, Fluid Dyn. Res.,  36, 2005, 
61-73. 
[4] Antonijoan, J., Marques, F. & Sanchez, J., Non-linear spirals 
in the Taylor-Couette problem, Phys. Fluids, 10, 1998, 829-
838.  
[5] Andereck, C.D., Liu, S.S. & Swinney, H.L., Flow regimes in 
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cylinders, J. Fluid Mech., 164, 1986, 155-183. 
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(Springer, Berlin, 1994). Fig.4. Laminar spirals in the system of infinite counter-rotating cylinders.  
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 [8] Edwards, W.S., Tagg, R.P., Dornblaser, B.C. & Swinney, 
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[11] Coles, D., Transition in circular Couette flow, J. Fluid   
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[12] Krueger, E.R., Gross, A. & Diprima, R.C., On the relative 
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[13] Snyder, H.A., Stability of Rotating Couette Flow. I. 
Asymmetric Waveforms, Phys. Fluids, 11, 1968, 728-734.   
Fig.5. Representation of vorticity contour lines at an axial cross section in 
finite cylinders geometry. The section is taken at the location of the 
vortex dislocation separating the  top Ekman vortex from the spiral 
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[16] Noui-Mehidi, M.N., Ohmura, N. & Kataoka, K., Dynamics 
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Conclusions 
The present results have shown that spiral vortices between 
concentric cylinders in a finite geometry have properties different 
from spiral vortices between infinite cylinders when the outer 
cylinder is stationary. There is need to pursue furthermore both 
numerical and experimental investigations to better understand 
the mechanisms behind the formation of spiral vortices and 
highlight the characteristic properties associated with this kind of 
flow regimes.   
[17] Blackburn, H.M.. & Sherwin, S.J., Formulation of a 
Galerkin spectral element-Fourier method for three-
dimensional incompressible flows in cylindrical 
geometries,  J. Comp. Physics, 197, 2004, 759-778. 
 
 
 
 
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Laminar Spirals in the Outer Stationary Cylinder Couette-Taylor System

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Laminar Spirals in the Outer Stationary Cylinder Couette-Taylor System

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