OCCURRENCE OF TAYLOR-DEAN INSTABILITY IN FLOW BETWEEN HORIZONTAL COAXIAL ROTATING CONES

Y. Hamnoune 1, F. Yahi 1,2† and A. Bouabdallah 1 1 Thermodynamics and Energetic Systems Laboratory, Faculty of Physics., University of Sciences and Technology Houari Boumediene, B.P.32 El Alia 16111 Bab Ezzouar. Algiers, Algeria. 2 Genie Physical of hydrocarbons Laboratory Faculty of hydrocarbons and chimestry, University M’Hamed Bougara–35000 Boumerdes-Al

Abstract The present work is concerned with an experimental study of free surface effect on the onset of the Taylor-Dean instability. The experimental device consists of two horizontal coaxial truncated cones, the inner cone is rotating and the outer one is at rest. The cones have the same apex angle giving a constant radial gap. The rotational speed variation occurs in a quasi-static mode. This is to highlight the occurrence conditions of the Taylor-Dean instability in this kind of flow system by using two photometry techniques (transmission and reflexion of natural light). Therefore, by varying the aspect ratio Γ=H/d in the range 6.71 ≤ Γ ≤ 13.95 we examine the interaction of free surface on the appearance of Taylor-Dean instability in laminar-turbulent transition. In each case, one measures the features of the associated structures in vicinity of the critical points of their appearance. The obtained results are compared with the classical cylindrical Taylor-Dean flow system. The main result is to find that the occurrence of the instabilities is only limited to the primary mode in the conical flow system. The triplet mode is not observed in our case.

Keywords: horizontal rotating coaxial cones, aspect ratio, free surface, laminar-turbulent transition regime, primary mode.

1 Introduction Taylor-Dean flow is a combination of flow between concentric rotating circular cylinders Taylor-Couette flow [12] and Dean Flow with application of a pressure gradient in the azimuthal direction [6].

The Taylor-Dean flow is largely studied theoretically and experimentally due to the simplicity of the geometry and the symmetry properties. The so called Flow has a various applications in the industrial sector (manufacture of paper pulp and the galvanizing line in steel mill industry).

The Taylor-Dean instability was considered theoretically by Diprima [7], then Hughes and Reid [10] they showed that the stationary modes do not exist. Thereafter, Raney and Chang (1971) showed that the stationary modes are replaced axially by the nonsymmetrical oscillating stationary mode. Gibson and Cook [11] studied the same problem but with small gap configuration. Joo and Shaqfeh [9] studied the stability of the flow for a viscoelastic fluid (Oldroyd-B) confined between two contra-rotating concentric cylinders with two different speeds Ω1 and Ω2. That flow is subjected to an azimuthal pressure gradient. Eagles et al. [8] modified the traditional Taylor-Dean flow system, where the inner circular cylinder is rotating and the noncircular outer cylinder is fixed. In order to determine the stability characteristics they have interested to the small gap configuration. In the case of a constant gap or believes in the direction of the basic flow, the flow becomes unstable. Conversely, when the gap decrease the flow becomes stable.

Since then, Brewster, Grosberg and Nissan [2] treated experimentally the problem with presence of the rotation force and azimuth pressure gradient. They were interested in the critical value and the formation conditions of Taylor vortices and they tried to find a common parameter for Dean and Taylor-Couette flows. Other experimental work relating to the Taylor-Dean flow includes those of Mutabazi et al. [14]. In the case of a primary instability in the model of the rollers inclined in displacement which is simply periodic in space and time. Slightly above this instability, it appears the release of the second instability moved in a periodic modulation of rolls with a high axial and a small wavelength frequency. The rollers inclined in triplet mode, is shown at very low frequency. Ait Aider [1] made an experimental study on the visualization flow of Taylor-Dean in a system with two coaxial cylinders open transversely.

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Laghouati and Bouabdallah [13] performed an experimental study of the first modes of instabilities in Taylor-Dean flow in the case of a viscoelastic fluid on the appearance of the two structure modes. They concluded that the wavelength of the triplet mode corresponds to almost three times the wavelength of the primary mode. Using a visualization technique Daimallah and Bouabdallah [5] determined the form of vortices as well as the critical parameters according to various rates of particles concentration and the aspect ratio Γ of the flow system.

The aim of this study is to highlight the conditions of appearances of the Taylor-Dean instability in a conical geometry and to compare them with the traditional case (cylindrical geometry). By varying the rate of filling Γ one evaluated the effect of the variation of the free face on this type of flow.

2 Experimental device The experimental device consists of two coaxial cones made of insulating and transparent material (Plexiglas) in order to allow a good visualization of flow regime. Both cones have the same apex angle Ф = 12° giving a constant annular gap δ = d/R1max where d = (9.68± 0.2) mm. The inner cone can rotate and the outer cone is maintained at rest [15].

Our system is characterized by an outer cone with largest radius R2max = (45 ± 0.2) mm and lowest radius R2min = (12± 0.2) mm. The largest radius of the inner cone is R1max = (35. 31 ± 0.2) mm, while the lowest radius is R1min = (2.31 ± 0.2) mm. The length of the fluid column is fixed at H = (155 ± 0.2) mm. A DC motor connected to the rotating axis by a flexible in order to avoid the adverse effects of vibration (Fig. 1) drives the inner cone.

R1max R2max

H

Figure 1: Experimental device

The working fluid is a solution of 20% of Vaseline oil CHALLALA, favoring a better suspension of the particles in the fluid visualization, which is added to 80% of a petroleum product SIMILI to reduce the viscosity of the oil, with a concentration of 2 g/l of aluminum flakes. Such a mixture constitutes a Newtonian fluid characterized by its kinematic viscosity ν = 4.8·10-6 m²/s and its density ρ = 777.23 kg/m3 with an accuracy of 1%.

In order to characterize the onset of the hydrodynamic instabilities, it is necessary to introduce dimensionless numbers involving viscous forces, which play a stabilizing role, and centrifugal forces, which have a destabilizing effect. These dimensionless numbers, which serve as control parameters of the flow, are the Reynolds number Re, the Taylor number Ta and the Froude number Fr defined in Table 1.

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Table1: control parameters

Geometric Dynamic

Radial Gap

∆ = d/R1max = 0.27

Aspect ratio Г = h/d

6.7 ≤ Г ≤ 13.9

Reynolds Number Re = (R 1max Ω d) / υ

Taylor Number Ta = Re (δ )1/2

Froude Number Fr = (R1maxΩ)/(ghcosΦ)1/2

Reflection of natural light

This method is based on the light beam reflection on the seeding particles. The light is supplied by an external source located in front of the experimental device. The intensity of the reflected light will depend on the orientation of the aluminum flakes, which are aligned with the local velocity vector. As a consequence, if the velocity vector is axial, the flakes will reflect light strongly and bright zones will appear on the images. On the contrary, if the velocity has a significant radial component, the flakes will be oriented parallel to the light rays and will let the light pass without reflection, giving dark zones on the images (Figure 2).

Figure 2: Visualization by reflection Γ = 11.36 at Ta = 422

Figure 3: Visualization by transmission Γ = 10.84 at Ta = 402.4

Transmission of natural light

This method of visualization is based on the optical transmission of the light source, which is placed behind the experimental device. The light rays pass through the flow and provide in-depth structure of the flow. In contrast with the previous technique, dark zones correspond to axial velocity vectors and bright zones to radial orientations, and all the dark and bright zones are well visible (figure 3).

3 Results and discussions To study the Taylor-Dean instability, the experimental device must be in a horizontal position with an inclination angle of the system α = 90° with the presence of a free surface. In this investigation, we study the effect of free surface on the appearance of this instability, why we are conducted to vary the filling ratio 6.71 ≤ Γ ≤ 13.95. To proceed, we change quasi-statically the rotational speed of the inner cone until the appearance of the Taylor - Dean Cells.

The initial instability is manifested by the presence of two states; a laminar state and a second state in which the inclined rollers are present (see Figure 4). Increasing speed, the flow is dependent on time and at a certain speed the flow becomes turbulent.

For a fixed filling rate Γ = 13.95 Ta = 218.9, we observe near the top of the cone the appearance of vortex the rest of the flow is laminar. At Ta = 365.8, it appears n = 7 cells inclined. Cells where the rollers become dependent in time, we notice that the lower part of the flow system still laminar.

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Figure 4: Visualization of Taylor-Dean instability Γ = 13.95 , Ta = 294.2

At Ta = 377 another set of rollers is formed in the upper part of the cone, inclined rollers displaced in two different directions separated by source of instability. Increasing speed of rotation at inner cone we observe the appearance of a disturbance zone in the upper part of the cone, the lower part is invade by the Taylor- Dean rollers (see Figure 5), increase in advantage of speed disturbance is spreading over the entire cone.

Figure 5: Visualization of Taylor- Dean instability

Γ = 12.90, a) Ta = 372.1,b) Ta = 393.1 c) Ta = 544.5,d) Ta = 748

In the case of the conical flow system, we observe the occurrence of the primary mode and the absence of the triple mode, but it appears in the classical system: the cylindrical system (figure 6).

Figure 6: Visualization of Taylor- Dean instability between two cylinders a) primary mode Re = 263 b) tripled mode Re = 303 by Mutabazi (1995)

(d)

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The Taylor number correspondence with appearance of the Taylor-Dean instability into function of the filling rate evolves according to increasing law polynomial of 2nd order.

Figure 7: Evolution of the Taylor number and Froude number with varying aspect ratio

For lower Aspect factor (6.71 ≤ Г ≤ 12), a small quantity of fluid is located in the upper part of the flow system and induced by rotation of the inner cone. In this case, the flow becomes unstable.

At Γ ≥ 12, Taylor number decreases by increasing the aspect factor, which means that the free surface has a stabilizing effect.

The Froude number is a dimensionless number that characterizes fluid in the relative importance of the forces related to the speed and the force of gravity. This number is sensitive to the variation of the free surface; the evolution of this number is a polynomial law of decreasing order of 2.

We note that for high fill rate, the Froude number has minimum values but low fill rates, the Froude number increases. By decreasing the filling ratio, the free surface area becomes larger and gives the meaning of the Froude number, which is very sensitive to the filling ratio variations.

4 Evolution of wavelength for different flow regimes

The evolution of the axial length of dimensionless wave λ = λ*/(2d) for different filling rate Γ. It is found that the wavelength is substantially constant.

In all cases, we note that the wavelength λ* is almost constant as a function of the Taylor number for a filling rate Γ ≤ 12. However, it appears there is a change of behavior when for a filling rate Γ ≥ 12.9, axial wave length tends to decrease up to Ta = 330 then it becomes constant.

Figure 8: Evolution of the wavelength various to the Taylor number

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Therefore, this suggests that the dynamics associated with α = 90° is specifically different comparing with the other inclination angles describing the Taylor-Dean flow. For filling ratio Γ ≤ 10, there is a small linear decrease of λ*.

For all curves, we note that the dynamics of the Taylor-Dean flow is dominant for the angle α = 90 °.

5 Conclusion

This study has allowed us to highlight the effect of free surface flow between rotating coaxial cones In this case we observe the appearance of the primary mode, which corresponds to the

propagation of the inclined rolls, but we did not observe the triple mode as in the cylindrical system

For a filling rate Γ ≤ 12 the flow is unstable and a fill rate Γ ˃ 12 the flow is stable. The wavelength is substantially constant for different filling ratios

References

[1] Ait Aider A: Visualisation de l’écoulement de Taylor–Dean ouvert, C.R.Mecanique, 333197–203, 2005

[2] Brewster D.B., Grosberg P., Nissan A.H.: The stability of viscous flow between horizontal concentric cylinders, Proc. R. Soc. London, Ser .A 251, 76, 1959.

[3] A. Bouabdallah: Instabilités et Turbulence dans l’écoulement de Taylor-Couette, Thèse de Doctorat ès Sciences, INP Lorraine, 1980.

[4] Chen F., Chang M.H.: Stability of Taylor-Dean flow in a small gap between rotating cylinders, J.F.M., vol. 243, pp. 443455, 1992.

[5] Daimallah A., Bouabdallah A., Nsom B.: Onset of Instabilities in Taylor-Dean Flow of Yield-Stress Fluid, Applied Rheology, Volume 19, Issue 3, 2008.

[6] Dean W. R.: Fluid Motion in a Curved Channel, Mathematical and Physical Papers, Vol. 121, No. 787 , pp. 402-420, 1928.

[7] DiPrima R.C.: The stability of viscous flow between rotating cylinders with a pressure gradient acting round the cylinders, J.F.M, 6,462, 1959.

[8] Eagles P. M.: Taylor-Dean instability in channels with slowly varying curvature and gap-width, Fluid Dynamics Research, 10, pp 181-191, 1992.

[9] Joo Y. L., Shaqfeh E. S. G.: A purely elastic instability in Dean and Taylor-Dean flow, Phys. Fluids, A 4, 524, 1992.

[10] Hughes T. H. and W. H. Reid: The effect of a transverse pressure gradient on the stability of Couette flow, Z. Angew. Math. Phys., 15, 573-581., 1964.

[11] Gibson D. and A. E. Cook: The stability of curved channel flow, Quart. J.M.A.M., 27(2), 142-160, 1974.

[12] Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. London, Ser. A 233, 289 (1923)

[13] Laghouati1 Y., A.Bouabdallah, I. Mutabazi: Evolution des premiers modes d’instabilité dans le système de Taylor-Dean en solution viscoélastique, Rhéologie, Vol. 7, 55-60, 2005

[14] Mutabazi I., et al.: Spatiotemporal Pattern Modulations in the Taylor-Dean System, Physica Review letter S, vol E 64, N° r 15

[15] Yahi, F., Hamnoune., Y., Bouabdellah, H.: Experimental investigation of the free surface effect on the conical Taylor-Couette flow system, JAFM,vol 9, N 6 ,pp 2743-2751, 2016

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