Effect of radial inner cylinder vibration on Taylor ...

21ème

Congrès Français de Mécanique Bordeaux, 26 au 30 août 2013

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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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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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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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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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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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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

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