2. Experimental apparatus and electrodiffusion technique

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1 TOPICAL PROBLEMS OF FLUID MECHANICS 221 DOI: EXPERIMENTAL INVESTIGATIONS OF THE SPHERICAL TAYLOR- COUETTE FLOW USING ELECTRODIFFUSION TECHNIQUE M. Mahloul 1, 2, A. Mahamdia 2 and M. Kristiawan 3 1 Departement of Physics, Preparatory School Science and Technology of Algiers, B.P 474, Place des Martyrs, Algiers, Algeria. 2 Departement of Energetics and fluid mechanics, Faculty of Physics, University of Sciences and Technology Houari Boumediene, BP N 32 El Alia Algiers Algeria. 3 INRA, UR 1268 Biopolymers Interactions and Assemblies, 44316, Nantes, France. Abstract In this paper, we study the hydrodynamic instabilities between two concentric spheres, the inner rotating while the outer is at rest, through visualization and electrodiffusion technique. The exploration of the flow regimes is carried out for different values of the Taylor number Ta and the aspect ratio Γ, but with one dimensionless gap width δ = The bifurcation diagram of the flow is determined by classical visualization. On the other hand, by means of electrodiffusion technique, we measured the friction factor at the inner wall of the outer sphere. Time series obtained by the electrodiffusion technique, using FFT, permitted the identification of the fundamental frequencies and confirmed part of the bifurcation diagram obtained by the classical visualization. Keywords: Spherical Taylor-Couette, visualisation, electrodiffusion, instability. 1 Introduction The understanding of dynamic systems manifested in spherical geometry flows is still increasing concern of researchers. Particularly considerable progress has been realize in recent years to study the various instabilities and the transition to the turbulence of an incompressible viscous fluid between two concentric spheres in rotations. (Wimmer, [1] ; Bühler, [2] ; Mamun and Tuckerman, [3] ; Nakabayashi et al, [4] ; Yuan, [5] ; Tigrine et al, [6] ; Loukopoulos, [7] ; Feudel et al, [8] ; Lalaoua and Bouabdallah, [9] ; Mahloul et al, [10, 11]. The observation of the spherical flow and the description of the successive structures during the changes of the regimes is necessary for the understanding of the dynamics of the concerned phenomena. In this type of flow, the transition to turbulence is a repetition of bifurcation phenomena. Mahloul et al [11] studied experimentally the evolution of flow structures during the laminar turbulent transition in the spherical Couette system. The hydrodynamic instabilities were investigated by spectral analysis of time series recorded for different flow regimes. Different flow structures were also simulated numerically by Allaoua and Bouabdallah [9]. They presented the velocity time series and power spectral density corresponding to three type of instabilities, i.e. the spiral mode and wavy mode (SM+WM), the wavy mode (WVF) and the chaotic fluctuation (CF). The relaminarization phenomenon has been experimentally investigated in spherical Taylor Couette system by Nakabayashi et al [4, 12] and Mahloul et al [10]. These studies have shown the similarity of evolution of the fluctuation intensity in this flow by the use of two different methods. The aim of this work is experimental investigation of the spherical Taylor-Couette flow by classical visualization and electrodiffusion technique. The results obtained allow us to present the evolutions of the mean friction coefficient f * as a function of the Taylor number Ta. Spectral analysis of the time series permitted the identification of several bifurcation path obtained by the classical visualization during the laminar turbulent transition. 2. Experimental apparatus and electrodiffusion technique The experimental apparatus shown in Fig. 1 consists of two concentric spheres made of transparent Plexiglas. The inner one is rotated and the outer one remains at rest. The inner and outer spheres have a radius of R ih = 49.6mm and R oh =54.9 mm, respectively, resulting in the gap width d = R oh R ih = 5.3 mm

2 222 Prague, February 15-17, 2017 and the non-dimensional gap width δ = d/r ih equal to The aspect ratio is defined as Γ = H / d where H is the height of fluid in the spherical gap. The inner sphere is driven by a DC motor (MC 100) with a power of P = 160 W with a double winding offering a very stable range of speeds. 2.1 The solution used: For the implementation of electrochemical measurements, an electrolytic solution containing an electrochemical couple (equimolar concentrations of 2 mol/m 3 of potassium ferricyanide and ferrocyanide) and a supporting electrolyte (2% by mass of potassium sulphate) was used. The high concentration of supporting electrolyte is necessary to avoid the migration of the active species in the electrical field. 2.2 Flow visualization: To ensure a good visualization of the flow, we added to the fluid 2% Kalliroscope AQ 1000 which is a product in liquid form of organic origin. This additive, which is chemically inert, does not contaminate the electrochemical solution and does not modify the physical properties of the fluid. Figure 1: Experimental apparatus and notations (dimensions in mm) The electrodiffusion method requires the installation of small electrodes flush with an inert surface. It is often used to measure local shear rate values (Hanratty and Campbell [13]). The principle of this method is based on the rapid electrochemical reduction of a reagent in solution, generally ionic, under particular conditions. The reduction of ferricyanide ions on a platinum cathode is generally used for the study of parietal velocity gradient. The equation of the redox reaction involved is: Fe( CN ) + e Fe( CN ) (1) The electrodiffusion method works in the regime of limiting diffusion current, recognized by a plateau on voltage current diagram. The working electrodes (probes) are platinum wires with a diameter of 0.5 mm flush mounted in the inner wall of the outer sphere. The counter electrode (anode) is a square sheet (50 20mm) of platinum attached to the lower part of the flow system. The counter electrode is large in size so as not to limit the current flowing in probes (see Fig. 2). The platinum is chosen for its polarizable character and rapid reaction.

3 TOPICAL PROBLEMS OF FLUID MECHANICS 223 2, R 2 Anode Figure 2: The electrodes location in the outer sphere. 3. Electrochemical electrode calibration 3.1. Polarograms To check correct function of the measuring electrodes, we plotted polarograms in a stable laminar hydrodynamic regime. The polarogram were obtained by varying the voltage (E A - E C ) between measuring electrode and anode from 0V to 0.7 V in steps of 20 mv and by measuring the corresponding current, see Fig.3. Plateau corresponding to the limiting diffusion current was found in the interval (0.2; 0.4 V). Small differences in the currents are caused by different area of the electrodes. 3,0 2,5 Courant of diffusion I(µA) 2,0 1,5 1,0 0,5 0,0 Electrode 1 Electrode 4 Electrode 5 Electrode 6 Electrode 7 Electrode 8 Electrode E A -E C (mv) Figure 3: Polarograms I = f(u) for different electrodes

4 224 Prague, February 15-17, The electrode calibration After verification of the correct function of the electrodes, they were calibrated in stable laminar regime for which the velocity field is perfectly known. Calibration consists of determining the transfer coefficient K as a function of the mean velocity gradient S and of verifying the law of proportionality in power 1/3. The expression for azimuthal velocity v in the laminar regime was given by Bühler (1990): Ω R ( r R ) v( r, θ) = sinθ r R R ih oh ( ih oh) Where θ is the angle relative to the sphere axis. The corresponding value of at inner wall of the outer sphere (r = R oh ) is expressed as: (2) S v 3Ω R = = r R R 3 1 ih 3 3 r = R oh ih oh sinθ (3) Under the following operating conditions: T θ = 23 C ; ν = 1, m 2 /s and Co = 2mol/m 3, the results of K. d calibration are shown in Fig. 4 as a function of the non-dimensional mass transfer coefficient K + = r D S. d 2 and Peclet number S + = r, where d r denotes diameter of the electrode and D denotes diffusion D coefficient. K Electrode 4 Electrode 5 Electrode 7 Electrode 8 Electrode 9 Electrode 11 Electrode 0 Courbe théorique: K + = 0,807.(S + ) 1/3 1 0, S + Figure 4: Calibration curves for electrodiffusion electrodes.

5 TOPICAL PROBLEMS OF FLUID MECHANICS Results and discussions 4.1 Classical visualization The different flow regimes during the laminar-turbulent transition are identified by Taylor number [10, 6, 9] with Ta = ( R Ω d ν ) d R. Mahloul et al [11] gave a diagram of the appearance of ih ih ih instability modes for different aspect ratios Г. Table 1 summarizes the different regimes of Taylor- Couette flow between two concentric spheres in rotation. Fig. 5 shows spherical flow at different Taylor numbers with a partially filled annular space. Table 1: Taylor critical numbers defining the apparition of different regimes [10]. The critical Taylor number Tc 1 Tc 2 Tc 3 Tc 4 Tc 5 Tc 6 Tc 7 Instability mode TVF Taylor SM Spiral SM + SWM Spiral WM Wavy Apparition of fluctuations Chaos Vortex Mode WM Wavy Mode Flow Mode SM, Ta = 50 SWM, Ta = 100 Chaos, Ta= 650 Figure 5: Visualization of some modes of instability during the transition for a partially filled annular space. Figure 6: Diagram of path to chaos for Γ = 19.

6 226 Prague, February 15-17, 2017 Fig. 6 shows the bifurcation diagram of the flow during the transition to chaos in the wave plane (n, m), related to the various modes of hydrodynamic instabilities. The quantities n and m are respectively associated with the numbers of Taylor cells (axial wave number) and circumferential (azimuthal) waves. 4.2 Evolution of friction coefficient After comparing four formulas taken from the literature, Mahamdia et al [14, 15] chose the equation for definition of the mean friction coefficient f *. _ τ (R + R ) * ih oh f = (4) 2ρΩ 2 R 2 d.r 1 ih ih Where τ = µ S, is the shear stress of the flow exerted on the wall inner surface of the outer sphere (r = R oh ) The proposed expression of f * is analogous to that of Wendt [16], after introducing a correction term, such as: f * τ R = 2 (5) f w τ. dr i ih where R is the mean radius of the annular space and R = ( R + R ) / 2 ih oh x Γ=18 Friction factor f * Mahamdia (Γ=10) Tc 4 Tc 7 Ta Figure 7: Evolution of friction factor f * versus Taylor Number for Γ= 18 The evolution of the mean friction coefficient f * as a function of the Taylor number Ta (Fig. 7) shows that the friction level is higher in the case of spherical flow than in that between cylinders. The evolution is similar in the range of Taylor number between 10 and 200 and are inversed beyond Ta=200. Moreover, the local minima corresponding to the critical values of Taylor number confirm the intervention of the azimuthal waves on the spiral movement at Tc 4 (SWM) and the appearance of chaos at Tc 7 obtained by the classical visualization.

7 TOPICAL PROBLEMS OF FLUID MECHANICS Power spectral density In order to deepen our quantitative investigations, we carried out a more detailed analysis to clarify the evolution and properties of the transition flow. For this purpose, a spectral analysis was applied in order to determine the conditions of successive transitions. The equation of the normalized fluctuating component of the recorded time series is given by the relation (Noui-Mehidi et al [17]): S( t) Smean s( t) = SD where S(t) is the total signal recorded, S mean is the mean component of the signal, and SD is the conventional standard deviation. The results of Fast Fourier Transformation applied on s(t) are shown in Fig. 8. (6) Log power-spectal density F 1 =n 0 f 0 Ta= Frequency Hz Log power-spectal density ,4 0,2 F 1 =n 0 f 0 Ta= ,5 1 1,5 2 2,5 3 3,5 4 Frequency (Hz) Figure 8: Power spectral density of regime Wavy Mode obtained with the electrode 5 for Γ = 19. The two spectra show fairly similar characteristics and we can note a frequency jump, f 1 = 0.48 at Ta = 180 and f 1 = 0.60 at Ta = 195. The ratio of these fundamental frequencies is 0.6 / 0.48 = 1.25 = 5/4. This result corresponds to a particular point of the bifurcation. It is the passage from the number of waves n of 5 to the value 4. We see that the spectra confirm the bifurcation path obtained by the classical visualization (Fig. 6).

8 228 Prague, February 15-17, Conclusions The classical visualization technique made it possible to obtain the path of the bifurcation passing through different regimes of motion of a spherical Taylor-Couette flow. The study of the evolution of friction factor f* as a function of the Taylor number, made it possible to demonstrate the appearance of the spiral wavy mode and chaos. The spectral analysis revealed interesting indications on successive transitions of different flow regimes. The experimental results obtained by the electrodifussion technique and the classical visualization are in good agreement. References [1] M. Wimmer: Viscous flows and instabilities near rotating bodies, Progress in Aerospace Science, vol. 25 (1988) pp [2] K. Bühler: Symmetric and asymmetric Taylor vortex flow in spherical gaps, Acta Mech, vol. 81 (1990) pp [3] C. K. Mamun, L. S. Tuckerman: Asymmetry and Hopf bifurcation in spherical Couette flow, Phys Fluids, vol. 7 (1995) pp [4] K. Nakabayashi, W. Sha, Y. Tsuchida: Relaminarization phenomena and external-disturbance effects in spherical Couette Flow, Journal of Fluid Mech, vol.534 (2005) pp [5] L. Yuan: Numerical investigation of wavy and spiral Taylor-Gortler vortices in medium spherical gaps, Phys Fluids, vol. 24 (2012) [6] Z. Tigrine, F. Mokhtari, A. Bouabdallah, M. Mahloul: Experiments on two immiscible fluids in spherical Couette flow, Acta Mech, vol. 225 (2014) pp [7] V. C. Loukopoulos: Criteria and limits for flow modes of the spherical Taylor-Couette problem in medium and wide gaps, Journal Comp Method in Sci and Engng, vol. 15 (2015) pp [8] F. Feudel, L. S. Tuckerman, M. Gellert, N. Seehafer: Bifurcations of rotating waves in rotating spherical shell convection, (2015), Physical Review, E92, [9] A. Lalaoua, A. Bouabdallah: A Numerical Investigation on the Onset of the Various Flow Regimes in a Spherical Annulus, Journal of Fluids Engng, vol. 138 (2016) pp [10] M. Mahloul, A. Mahamdia, M. Kristiawan: The spherical Taylor-Couette flow, European Journal of Mech, vol. 59 (2016) pp.1-6. [11] M. Mahloul, A. Mahamdia, M. Kristiawan: Experimental Investigations of the Spherical Taylor- Couette Flow, Journal of Appl Fluid Mech, vol. 9 (2016) pp [12] K. Nakabayashi, W. Sha: Vortical structures and velocity fluctuations of spiral and wavy vortices in the spherical Couette Flow, Springer-Verlag Berlin Heidelberg LNP, vol. 549 (2000) pp [13] T. J. Hanratty, J. A. Campbell: Measurement of Wall Shear Stress, in Fluid Mechanics Measurements, (ed. R.J. Goldstein), Hemisphere Publishing Corporation, Washington, (1983) pp [14] A. Mahamdia, A. Dhaoui, A. Bouabdallah: Aspect ratio influence on the stability of Taylor-Couette flow, Journal of Physics: Conference Series, vol. 137 (2008) [15] A. Mahamdia, A. Bouabdallah, S. E. Skali: Effects of free surface and aspect ratio on the transition of the Taylor Couette flow, Comptes Rendus Mécanique, vol. 331 (2003) pp [16] F. Wedt: Turbulent Strömungen zwischen zwei rotierenden koaxialen zylindern, Ing Arch, vol. 4, (1933) pp [17] M. N. Noui-Mehidi, N. Ohmura, K. Kataoka: Dynamics of the helical flow between rotating conical cylinders, Journal of Fluids and Structures, vol. 20, (2005) pp