effect of radial inner cylinder vibration on taylor
TRANSCRIPT
21ème
Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013
1
Effect of radial inner cylinder vibration on Taylor-Couette
flow with free surface
A. ABDELALIa, H. OUALLI
a , S. HANCHI
a and A. BOUABDALLAH
b
a. LMF, EMP Bordj El-Bahri 16111, Algiers, Algeria
b. LTSE, USTHB, Bab Ezzouar 16032, Algiers, Algeria
Résumé:
Le présent travail consiste à étudier numériquement et expérimentalement l’écoulement dans un système
de Taylor-Couette à surface libre. La stratégie du contrôle est interprétée par l’application d’une
déformation radiale au niveau du cylindre intérieur tournant. L’objectif principal est d’observer l’influence
de cette déformation, en présence d’une surface libre, sur l’évolution de la structure des vortex de Taylor
relatifs à l’apparition de la première instabilité d’une part, et sur la destruction de la couche d’Ekman
d’autre part. Les résultats numériques sont obtenus à partir du logiciel FLUENT pour les fluides
incompressibles. Le system de base se caractérise par une hauteur de 170mm, un rapport des rayons ɳ=0.9,
un taux de remplissage Γ=28.5 et un jeu radiale δ=0.1. Les résultats montrent que l’application de ce
contrôle actif retarde considérablement l’apparition de la première instabilité de Tac1=44.8 à Tac1=69.4
correspondant à ε=1.5%. Il est établi aussi que la fréquence de destruction de la couche d’Ekman (f>20Hz)
est nettement supérieure à celle de destruction des vortex de Taylor (f<3Hz).
Abstract:
A numerical and experimental study of the Taylor-Couette flow control with a free surface is presented in
this work. It is devoted to investigate the effect of the free surface oscillation associated with the inner
cylinder cross section variation on the Taylor vortex destruction. The numerical results are obtained using
FLUENT software package for a three dimensional and incompressible flows. The basic system geometry is
characterised by a height H=170mm, a ratio of the inner to the outer cylinders radii ɳ=0.9, an aspect ratio
Γ=28.5 and a gap ratio to the radius of the inner δ=0.1. It comes out for the obtain results that the first
instability mode of transition is delayed, and the vortices destroying process can be applied for all the flow
regimes encountered in the Taylor-Couette flow in route to turbulence. The Taylor vortices show a
particular sensitivity and can be easily destroyed using low deforming frequencies (f<3Hz), while the Ekman
layer exhibits larger resistance to actuation and relatively higher deforming frequencies (f>20Hz) are
required for this vertical layer to vanish after strong interface excitation.
Keywords: Taylor-Couette flow, free surface, active control.
1 Introduction – aim and motivation
Hydrodynamic instabilities define the incompressible fluid process to the turbulence. Our study aim is how
to manipulate or voluntarily change the flow velocity field in a Taylor-Couette system with a free surface, in
order to investigate the advance or the delay of the first instability establishment. However, this control is
characterized by an application of a radial inner cylinder deformation according to a sinusoidal law (*). We
aim to delay the first instability onset of Taylor-Couette, and to destroy the Taylor vortices by creating the
mixing phenomenon with the free surface (respectively corresponding to delay (**) or to advance (***) the
turbulence birth).
The interest of the turbulence prediction in the Taylor-Couette system has a great importance for both
academic and industrial sectors: **) the stratified fluids (oceanography), accretion disc (astrophysics), filtering
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Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013
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oils and waste water, shaft-bearing connection, ***): increasing heat transfer (cooling, heating), fluid mixture,
combustion.
The free surface effect on the flow in the Taylor-Couette system was introduced by a several authors. Orr
and Scriven [1] reported on two iterative methods for free surface flows in which the effects of surface
tension, viscous, gravitational and inertial forces are all important. Mahamadia and Bouabdellah [2] showed
the experimental results of the effect of the Reynolds number and the aspect ratio on the stability of the flow.
Djeridi et al [3] and Atkhen et al [4] studied experimentally the motion of air bubbles captured by the
ventilation near the free surface. Watanabe et al [5, 6] showed by numerical and experimental approaches,
the mode exchanges of flow between concentric and rotating cylinders with vertical axes.
In this paper, the study is devoted to scrutinize the free surface effect coupled with the inner cylinder
diameter variation, on the flow behaviour. The numerical results are obtained using FLUENT software
package for the three dimensional and incompressible flows. The basic system has a height H=170mm, a
ratio of the inner to outer radii 9.0 , an aspect ratio, corresponding to the cylinder height reported to the
gap length Γ=28.5 and a ratio of the gap to the radius of the inner cylinder is δ = 0.1.
In the first step, the study is carried out on the nominal case without control. The results show that the
first instability mode of transition is delayed when the aspect ratio decreases. In the second part of this study,
we apply a sinusoidal radial vibration on the inner rotating cylinder using a variable amplitude ε (from 0 to
1.5%) of the inner radius, and a fixed frequency equal to f=48Hz. It’s also reported that the radial
deformation of the inner cylinder associated to the free surface dynamic increase both the axial wave number
and the free surface elevation, and delay the appearance of the first bifurcation of the flow. At the end, the
visualisations show that the free surface oscillation strongly contributes in the vortices destruction.
2 Formulation
The radii of the stationary outer cylinder and the rotating inner cylinder are respectively R1 and R2. The
axes of the cylinders are vertical and parallel to the direction of gravitational acceleration. The bottom of the
system of Taylor-Couette is a solid stationary wall while the top is the free surface of the working fluid with
air. The height of the evolving fluid is Hf. The radial gap between the two radii is d= R2-R1. The sinusoidal
vibration of the rotating inner cylinder is executed according to the law: with: r is
the variable radii,
is the deforming amplitude and f is the deforming frequency.
The numerical process is based on the resolution of the Cartesian Navier-Stocks equations using the
Volume Of Fluid (VOF) scheme. The solution is carried out using the FLUENT computational fluid
dynamics package. This computer program applies a finite volume method to integrate the equations of
motion. An explicit scheme is used to discretize time and a third-order MUSCL scheme is used to discretize
the convective terms in the momentum equations. The time step is fixed equal to Δt=0.0002 s. Modulation
of the inner cylinder diameter is carried out using the “dynamic mesh” FLUENT program.
3 Main results
3.1 Nominal case: first instability for different aspect ratios
A comparative study between experimental and numerical results is developed. The figure 1 shows that
when the aspect ratio increases, the establishment of the first bifurcation is advanced (from Γ=2.5 according
to Tac1=64 to Γ=28.5 according to Tac1=44.8, where Tac1 is the critical Taylor number). However, we note
that for both aspect ratios, the same number of the vortices is established. Globally, the deviation between
the numerical and the experimental results in terms of the critical Taylor number Tac1 is around 3%.
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Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013
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3.2 Controlled case: vortices development at the free surface
The figure 2 depicts the evolution of the Taylor vortices at the free surface in terms of the velocity field.
When the amplitude is minimal (ε = 0), the Taylor vortices are located at the nominal free surface without
elevation. The displacement of the inner wall from dmax = 3.5mm at ε = 0 to dmin = 3.02mm for ε = 1.5% leads
to increase the fluid elevation and the vortices number.
Γ = 2.85
(Tc= 64)
(Tc=65.66)
2 vortices
Free surface
Γ = 5.71
(Tc= 54.1) (Tc= 55.88)
3 vortices
Free
surface
Γ = 17.14
(Tc= 51.4) (Tc= 52.61)
9 vortices
Free
surface
Γ = 11.42
(Tc= 54.1) (Tc= 55.88)
6 vortices
Free surface
9 vortices
Free surface
FIG. 1- First instability establishment for different aspects ratio / numerical and experimental
comparative results
Γ = 22.85 (Tc= 47.1) (Tc= 48.53)
12 vortices
Free
surface
(Tc=44.8) Γ = 28.57
(Tc= 46.28)
13 vortices
Free
surface
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Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013
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The vortices are established when the deformation rate is maximum for each amplitude. The size of the
free surface vortex is smaller than that of Taylor vortices. The width of the Taylor vortices extends on the
entire gap for all the considered amplitudes. When increasing progressively the deforming amplitude, the
number of vortices increases. Note also that the axial wavelength remains almost constant for the amplitudes
0 - 0.2 - 0.5 - 0.7% and decreases slightly for amplitudes 1.25 and 1.5% indicating that the vortices number
increases with the elevation of the fluid.
Inner cyl. Outer cyl. Free surface vortices
Taylor vortices
ε=0 ε=0.2% ε=0.5%
FIG. 2 - Vortices development
at the displaced free surface / Velocity field
Free surface vortices
ε=1.25% ε=0.7% ε=1.5%
FIG. 3 - Vortices number variation with the deforming amplitude
increase
Deforming amplitude %
Vo
rtic
es
nu
mb
er
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Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013
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From the figure 3, we note that the axial wave number increases with the deforming amplitude. The axial
waves number passes from 13 (26 vortices) for amplitude 0, to 17 (34 vortices) for the amplitude 1.5%,
corresponding to an increasing rate of 30.76%. This can be explained by the fact that the radial deformation
increases the height of the fluid in the gap inducing thus the vortices number increasing.
3.3 Taylor vortices destruction
The figure 5 illustrates in details, the process of the vortices structures destruction corresponding to the
amplitude ε=1.25%. This mechanism passes by several stages according to the applied deformation
frequency. Indeed, when the deforming frequency f=1.3Hz, we note a light azimuth undulation of the whole
vortices except Ekman layer which resists to actuation. When the deforming frequency is set equal to
f=1.6Hz, the figure shows the beginning of a dislocation localised on the deviated Taylor vortex in the (r-z)
plan. By increasing the frequency of deformation, a downward dislocation of the vortices takes place in a
privileged direction. This rupture does not reach the bottom of the system (Ekman layer) which remains
resistant even after the destruction of all the Taylor vortical structures (Taylor vortices) for f=3.1Hz. When
f=22Hz, the flow in gap is completely agitated, the free surface elevation is intensively disturbed with a so
strong oscillation movement to reach the system bottom and destroy the Ekman layer. At this stage, all the
structures are destroyed in the system. The free surface disturbing combined to the strongly agitated flow
lead to the total destruction of the vertical structures in the system of Taylor-Couette.
We note that the partial destruction frequency is much less than that of the total destruction respectively
3.1 and 22Hz. This result is illustrated by both the numerical and the experimental result, respectively,
figures 4 and 5.
FIG. 5 - Vortices destruction in Taylor-Couette system
with free surface- Experimental result
Strong free surface agitation
Total destruction of the vortex
Partial destruction Piled Taylor
vortices
Undulated vortices
Vortices Piled
f =0Hz f =1.6Hz
Beginning of dislocation
Ekman layer
Total destruction Partial
destruction Vortices rupture
f=1.9Hz f = 3.1Hz f=22Hz
FIG. 4 - Vortices destruction in Taylor-Couette
system with free surface- Numerical result
Ekman layer
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Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013
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4 Conclusion
The radial deformation of the inner cylinder is numerically and experimentally considered. This active
controlling strategy strongly affects the natural development of the flow in the Taylor-Couette system. The
birth of the Ekman layer and the first instability are largely delayed. The presence of the free face associated
with the deformation process induced an increase in the Taylor vortices number.
Vortices destruction (mixing phenomenon) in Taylor-Couette system with a free surface has also been
established. By applying a sinusoidal deformation on the rotating inner cylinder, we can destroy, totally (all
the Taylor vortices including the Ekman layer) or partially (vortical structures without the Ekman layer), the
vortices using a combined dynamic of the free surface oscillation and radial deformation of the inner rotating
cylinder.
(*) Physical oscillation of the inner cylinder using a camshaft (03 cams)
References
[1] Orr and Scriven “Rimming flow: Numerical simulation of steady, viscous, free surface flow with surface
tension” J. Fluid Mech. 1978, vol. 84, part one.
[2] Mahamadia and Bouabdellah “Ecoulement de Tylor-Couette en géométrie finie sur à surface libre’’ The
Canadian Journal Of Chemical Engineering, 2003.
[3] Djeridi et al. “Two phases Couette-Taylor flow: Arrangement of the dispersed phase and effets on the
flow tructures” Phy. Fluids16 (128) 2004.
[4] Atkhen et al. “Air bubbles in a Couette-Taylor flow” C.R. Acad. Sci. Paris. 327. Serie II.
[5] Watanabe et al. “Development of free surface flow between concentric cylindres with vertical axis’’ J,
Phys.Conf. series 14, pp 9-19, 14th INT CT, WORKSHOP, 2005.
[6] Watanabe et al. “Mode formation of free surface rotating flow between concentric vertical cylinders’’
Journal of Physics: Conference Series 137, 2008.
cam
Inner cyl. (pvc)
Shaft of
Inner cyl.
Physical variation of the inner cylinder R1