Change of ow patterns in thermocapillary convection in liquid bridge

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1 Available online at Acta Astronautica 54 (2004) Change of ow patterns in thermocapillary convection in liquid bridge V.M. Shevtsova, D.E. Melnikov, J.C. Legros MRC, Universite Libre de Bruxelles, CP-165, 50 Ave. F.D.Roosevelt, Brussels 1050, Belgium Received 2 April 2002; received in revised form 19 December 2002; accepted 23 May 2003 Abstract A parametric investigation of the onset of chaotic ows in a non-isothermal liquid bridge is numerically carried out for a liquid with medium Prandtl number, Pr = 4, and unit aspect ratio. Spatiotemporal patterns of thermocapillary ow are successively studied beginning from onset of instability up to the appearance of the non-periodic ow. Well tested numerical code is used for solving the three-dimensional, time-dependent Navier Stokes equations in cylindrical coordinate system. The dependence of viscosity upon temperature is taken into account. The two-dimensional steady-state ow becomes oscillatory with azimuthal wave number m cr = 2 resulting from a Hopf bifurcation at Re 0 cr = 630. Between this critical Reynolds number and up to Re 810 convective ow evolves being uniquely determined by the set of characteristic parameters and independent of initial conditions. A secondary ow appears with an independent frequency and independent wave number m = 3 at Re 1 cr =810. Two solutions with m=2 and 3, each with a dierent pattern of symmetry, co-exist for Re Re 1 cr. Dierent modes reveal dierent behaviors in the supercritical area. Moreover, the traveling wave belonging to the critical mode m cr =2 always remains periodic, but the mode excited later on, with azimuthal wave number m = 3, exhibits chaotic features at Re c 2003 Elsevier Ltd. All rights reserved. 1. Introduction A large number of numerical and experimental studies in a half-zone model have been devoted to the transition from steady to oscillatory ow. The real crystal growth processes as a rule deal with a melt far above from the critical temperature dierence, T T cr. Little information has been published regarding the temperature and ow elds with increase T above the threshold of instability up to chaotic regime. Corresponding author. Tel.: ; fax: address: vshev@ulb.ac.be (V.M. Shevtsova). From theoretical point of view the liquid bridge model is a good example of dissipative dynamical system. Moreover, up to now the transition to chaos has been extensively studied for convective ows in closed system. Finite size systems with an open interface on which Marangoni forces act have been out of focus. As for the liquid bridges, to the best of our knowledge there is only one publication by Frank and Schwabe [1] in which the transition from steady ow to the chaos has been experimentally studied in liquid bridges for liquids with Pr =7; 49; 65. Petrov et al. [2] have considered liquid bridge as a non-linear dynamic system to control an isolated unstable state far away from critical point for Pr = 35. In both cases experiments have been carried out in terrestrial conditions. Due to Marangoni forces the steady-state motion sets in the /$ - see front matter c 2003 Elsevier Ltd. All rights reserved. doi: /s (03)

2 494 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) Nomenclature d k m P Pr r R R Re S SW t T TW Ṽ z height of bridge thermal conductivity azimuthal wave number pressure Prandtl number radial coordinate radius of endwalls relative variation of viscosity Reynolds number strain rate tensor standing wave state time variable temperature traveling wave state velocity vector axial coordinate Greek letters coecient of thermal expansion aspect ratio T temperature dierence between rods distance from critical point 0 T T deviation from linear dimensionless temperature prole dimensionless temperature with respect to the cold endwall density of uid surface tension coecient in the linearized, surface tension equation of state dimensionless kinematic viscosity kinematic viscosity coecient in the linearized, kinematic-viscosity equation of state azimuthal coordinate! 0 fundamental frequency! 1 subfrequency Subscripts 0 estimated at the reference uid temperature c cold endwall cr critical value h hot endwall z projection on z-axis liquid bridge as soon as a small temperature gradient is applied along the free surface. In the phase space, a single point corresponds to the image of the steady state. When the temperature gradient along the free surface achieves the critical value, T cr, this steady-state ow bifurcates to a time-dependent three-dimensional ow. The transition from steady state to oscillatory ow is rather well understood [3 5]. As a rule two hydrothermal waves, propagating in opposite directions, bifurcate from two-dimensional state at critical point. Standing (SW) or traveling (TW) wave may result from the interaction of these two waves. The close line corresponds to the image of this oscillatory ow in the phase plane. In the case of TW near the critical point this curve has a shape of the circle, the radius of which is proportional to the amplitude of the oscillations and the frequency of rotation along this circle is proportional to the frequency of the oscillatory ow! 0. The possible existence of further time-dependent instabilities was not yet numerically investigated. Probably, it is due to the mathematical complexity. It is known, that at some temperature dierence T =T 1 cr the periodic state becomes unstable and a second bifurcation takes place. The type of this second bifurcation depends upon the route to the chaos, chosen by particular dynamic system. The goal of this paper is for the rst time to identify numerically the route to the chaos for pure thermocapillary convection in a liquid bridge. 2. Mathematical formulation of the problem A liquid bridge is considered, consisting of a uid volume held between two dierentially heated horizontal at concentric disks of radius R, separated by a distance d. The temperatures T h and T c (T h T c ) are

3 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) Fig. 1. Geometry of the problem. prescribed at the upper and lower solid liquid interfaces, respectively, yielding a temperature dierence T =T h T c. The free surface is considered cylindrical and non-deformable. The geometry of the problem is shown in Fig. 1. The surface tension and kinematic viscosity are taken as linear functions of temperature (T )=(T c ) T (T T c ); (T )= (T c )+ T (T T c ); where T T = const; T T = const. The governing Navier Stokes, energy and continuity equations are written in non-dimensional primitivevariable formulation in a cylindrical co-ordinate + V V = P + R S ( + +(1 + R ( + z))v; V + V = V z + Pr ; where velocity is dened as V=(V r ;V ;V z ), 0 =(T T c )=T is the dimensionless temperature and is the deviation from the linear temperature prole = 0 z, S i =@x k k =@x i is the strain rate tensor. At the rigid walls the no-slip conditions are used Ṽ(r; ; z =0;t)=0, Ṽ(r; ; z =1;t) = 0 and constant temperatures are imposed (r; ; z =0;t)=0, (r; ; z =1;t)=0. On the cylindrical free surface (r =1,06 6 2, 0 6 z 6 1), the stress balances are V r =0 and ( ) 1 (1+R (+z))s e r +Re e z +e (+z)=0: The free surface is assumed thermally r (r =1; ;z;t)=0: The Prandtl, Reynolds numbers and aspect ratio are dened as Pr = o k ; Re= T Td 0 2 ; = d 0 R : The distance from critical point is measured by the parameter = Re Re cr : Re cr A new parameter R describes the relative variation of viscosity. R = T T ; =1+R ( + z): 0 Here, is dimensionless viscosity scaled by 0 = (T c ). Throughout this parametric study the Prandtl number, the variation of viscosity and the aspect ratio are kept constant, Pr =4,R = 0:5 and =1. The description of the numerical code and tests of validation can be found in [4,6]. 3. Backgrounds Unlike some other non-linear dynamic system in liquid bridges there is only one stable solution corresponding to the time-dependent ow near the threshold of instability for a xed set of parameters. Respectively, the only one fundamental frequency,! =! 0, exists in spectrum, the amplitudes of harmonics are negligibly small. (The rare case of mixed modes at the onset of instability is not considered here). The close circle corresponding to the image of this oscillatory ow in the phase plane is a

4 496 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) limit cycle which has some region of attraction. It means that introducing dierent disturbances to the two-dimensional state, the system will move by different trajectories, but nally it will achieve the cycle. This limit cycle plays the role of an attractor and in the present case of the liquid bridge there is only one stable attractor. Going further to the supercritical area higher order harmonics are excited, and the circle in phase plane changes the shape remaining a close curve. When a broadband noise is present in the power spectrum, even if it is small in comparison with the amplitudes of the main frequencies and their linear combinations, it is a non-periodicity sign. Dierent routes to non-periodic motion have been identied for the convective ows in closed systems [7]: (a) The most classical way begins from appearance of secondary ow with an independent frequency! 1. As a result a quasi-periodical motion is established in liquid bridge, characterized by two incommensurate frequencies, the ratio! 0 =! 1 is irrational number. Third, fourth, etc., incommensurate frequencies will appear in the system. The distance between following and previous frequency is diminished, and it will lead to non-periodic motion. (b) The onset of non-periodicity is associated with the loss of entrainment as Re is increased. (c) Another route to the chaotic (turbulent) ow involves successive subharmonics (or period doubling) bifurcations of a periodic ow. There are several ways to recognize chaotic behavior features in the oscillatory system, such as analyzing their return map, phase space trajectory and power spectrum. One of the fundamental characteristic of a chaotic physical system is its sensitivity to the initial state. It means that if the same system is started at two dierent but close initial conditions (initial guesses in our numerics) their dynamical state will diverge from each other very quickly in phase space. As for the liquid bridge problem, the latter denition of chaos may be rather confusing as dierent oscillatory modes may co-exist at the same temperature dierence, and there are experimental proves of it. In numerics they may be excited by appropriate initial guesses, that can be treated as perturbations in the real system. For example, if the initial perturbations introduced into the system are of the form sin(m 1 ) the solution will be described by the same sin(m 1 ) function if this oscillatory solution is stable. Otherwise, upon some time it will bifurcate from initially taken azimuthal wave number m 1 to another solution with a dierent m 2 that is stable for a given set of the parameters [4]. Thus despite the symmetry of disturbances the system will arrive to the solution in the form of waves with m 2 wave number. Indeed, for our particular system with parameters Pr =4, R = 0:5 and =1, in the region of Reynolds numbers 630 Re 810 the only oscillatory mode, m =2; is stable. The calculations have been done for Re = 700 choosing the initial guess with a symmetry m = 3. The results shown in Fig. 2 demonstrate that after decaying the mode m=3 the stable solution with a mode m=2 is established. To record the oscillations four equidistant numerical thermocouples were placed inside a liquid bridge at a transversal section. For the TW with a mode m = 3 the temperature signals from dierent thermocouples have constant phase shift (see left insertion in Fig. 2). For the mode m = 2 the oscillations at the opposite thermocouples are in phase and due to this instead four temperature proles only two are visible at the right insertion in Fig. 2. The higher modes are excited above the threshold of instability. In that region the system can admit two stable periodic solutions at the same time with two dierent wave numbers m 1 and m 2. The nal solution which will be found depends on initially chosen wave number guess. In present case, the state of the system is considered to be chaotic (or weakly turbulent) if a spectrum has broadband noise and broad peaks. 4. Processing of the results The rst series of calculations have been done on the mesh ( ), where the numbers show the amount of the intervals in radial, azimuthal and axial directions, respectively. This mesh was utilized for the calculations from the rst bifurcation point where the oscillatory convection is set in and up to Re The traveling wave with azimuthal wave number m = 2 represents the only stable solution near the onset of instability, Re cr = 630. For integration the governing equations at a supercritical Reynolds number, solution for the previously investigated Re was taken as initial guess, i.e. the initial location of the system in phase space has always symmetry m=2. The oscillations have always been periodic with clear spectra up to Re = Neither broadband noise nor

5 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) Fig. 2. Ascertainment of stable oscillatory solution with wave number m = 2. Initial guess is a ow eld with a symmetry m = 3. The temperature proles correspond to Re = 700, Pr =4, =1. frequencies dierent from the fundamental frequency and its harmonics in spectra were observed throughout the range of investigated Reynolds numbers. One of the rst explanation of this scenario was that the resolution is not sucient to treat correctly the problem near the cold corner (the region of the largest gradients and velocities). It can happen that the thin region adjacent to the cold, and perhaps the hot, corner is a source of disturbances giving birth to the turbulence. Therefore, the calculations have been repeated on non-uniform grid ( ) with a smaller space intervals near the hot and, especially, near the cold corner. It was obtained that the characteristics of oscillations such as amplitude of temperature oscillations, net azimuthal ow and power spectra, corresponding to the two dierent grids, coincide with very good accuracy. Below one will nd the reason of such the non-chaotic behavior of the liquid bridge. 5. Results and discussion 5.1. Spatiotemporal properties of TW m =2 The supercritical ow in liquid bridge far away from the threshold of instability admits the existence of at least two stable oscillatory solutions at the same time with two dierent wave numbers. Staying on the m = 2 solution, the critical mode at the onset of instability, the calculations have never led to chaos. The periodic oscillations of the temperature and the axial velocity with time are shown in Fig. 3 far from the threshold of instability, Re = 5000, =6:94. Varying the Reynolds number in the wide range 630 Re 5000 a weak secondary instability occurred only once at Re At this point oscillations of axial velocity switched from one-maximum prole to two-maxima ones as shown in Fig. 3b. But the oscillations of temperature are left without changing, see Fig. 3a. At higher values of Reynolds number dierence between the maxima of axial velocity remains with good accuracy constant, see Fig. 4. Further increasing of the temperature dierence did not lead to transition to turbulence. Up to Re = 5000 no signs indicating pre-chaotic state of the system, such as frequency skip, period doubling or quasi-periodicity were observed. The image of this thermoconvective ow in phase plane is a closed curve, shown in Fig. 5. There were always only one fundamental frequency and its harmonics presented in spectra.

6 498 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) (a) (b) Fig. 3. Oscillations of temperature and axial velocity with time in supercritical area, = 6:62, for the traveling wave with mode m = 2, Re = 4800, Pr =4. Fig. 4. Axial velocity maxima vs. Reynolds number for m = 2 oscillatory mode Spatiotemporal properties of TW m =3 Fig. 5. Phase plane: m =2, Re = Another independent mode, m = 3, appears with an independent frequency at Re 1 cr 810. Two solutions with m = 2 and 3, each with dierent pattern of symmetry, co-exist for Re Re 1 cr. Spatiotemporal patterns depend on the azimuthal wave number. The temperature disturbance eld structure is organized so that after the axisymmetric component is subtracted from the 3-D temperature eld there are m hot and m cold spots observed in a transversal section and on the free surface. The snapshots of the temperature disturbances eld in a horizontal cross-section (z =0:5) and on the free surface are shown in Fig. 6 at the same time moment. The temperature eld with two hot and two cold spots in Fig. 6 represents TW with a mode m = 2, and the snapshots with three

7 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) (a) (b) Fig. 6. Temperature disturbances in z=0:5 transversal section (above) and on the free surface (bellow) for Pr=4, Re=1500, =1;R = 0:5. The averaged in azimuthal direction temperature eld is subtracted from the total distribution. (a) m = 2 mode and (b) m = 3 mode. hot and three cold spots in Fig. 6b conrm the existence of periodic solution with m = 3. Both patterns, in Figs. 6a and b correspond to the same set of parameters, Re = It is worth mentioning that in both cases the waves propagates practically azimuthally, an oblique angle with respect to z-axis in the temperature distribution on the free surface in Fig. 6 is close to zero. A frequency skip always occurs, when the system changes the mode of the oscillatory regime. The frequency skip is a quick change of the dominant frequency caused by small increase of T. Indeed, the fundamental frequency for m=3 wave is always larger than for m=2 wave number pattern. For example, the fundamental frequency of the m=2 at Re=1500 is equal to! 2 =34:36 versus! 3 =35:59 for the mode m=3. The dierence!, which is about 3.6%, cannot be related with an error, as the same amount of points have been used for calculations and for Fourier analysis. Even in logarithmic scale the power spectrum for the mode m = 2 shown in Fig. 7a is similar to that one for mode m = 3 shown in b, just the fundamental frequencies have dierent values:! 2 =48:74 versus! 2 =54:61. It indicates that up to certain value of the Re number the both mode, m = 2 and 3 progress in similar way. The splitting of the maxima of the axial velocity, shown in Fig. 4 for the mode m = 2, is also observed for mode m =3atRe 3000.

8 500 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) (a) (b) Fig. 7. The power spectrum of the TW far from the threshold of instability for (a) m = 2 mode,! 0 =48:47 and (b) m = 3 mode,! 0 =54:61 when Re = Fig. 8. The dependence of the fundamental frequency on the Reynolds number for the mode m = 3 is shown by diamonds, for the mode m = 2 the solid line represent results of calculations on the mesh ( ) and asterisks on the mesh ( ). The dependence of the fundamental frequency upon Re number is shown in Fig. 8 for the all cases mentioned above. The solid line corresponds to the mode m = 2 obtained on coarse mesh ( ) while the asterisks represent the fundamental frequency for the same mode, calculated on ner grid ( ). One can say that the coarse mesh is enough ne to carry out such kind of parametric study. There is excellent qualitative agreement throughout the considered range of parameters. Below the Re 3000 the values of frequencies coincide, but even for Re = 4200, =5:67 the dierence is only about 1:35%. The diamonds shows the frequency for m = 3 mode, which approaches to the mode m =2atRe 810 having a small frequency skip. The frequency smoothly grows with the enlarging of the Reynolds number exposing in spectrum a new harmonics. As the Reynolds number is increased further, there is a continuous amplication of the amplitude of the certain harmonics. At Reynolds number Re 3500 the periodic solution with one frequency looses its stability and a second frequency appears in the spectrum. The second frequency! 1 appears as a subfrequency of the smaller amplitude which achieves rapidly more or less the similar value of the amplitude as the main frequency at Reynolds number Re At Re = 3950 period doubling takes place as! 1 =0:5! 0, which is accompanied by the splitting of the closed limit cycle in the phase plane. The phase plane for Re=3950 is shown in Fig. 9. At larger values of the Reynolds number, two subfrequencies appear in the spectra and return maps for axial velocity oscillations have rather complicated form.

9 V.M. Shevtsova et al. / Acta Astronautica 54 (2004) As the Reynolds number is increased further, the sharp spectral peaks are wiped out by a continuous amplication of dynamical noise. Acknowledgements Fig. 9. Phase plane indicating the period doubling: m=3, Re= Conclusions Spatiotemporal patterns of thermocapillary ow is successively studied beginning from onset of instability up to the appearance of the non-periodic ow for a liquid with Prandtl number, Pr =4, and unit aspect ratio. The dependence of viscosity upon temperature is taken into account. The two-dimensional steady-state ow becomes oscillatory with azimuthal wave number m cr =2atRe 0 cr = 630. The supercritical ow in liquid bridge far away from the threshold of instability admits the existence of two stable oscillatory solutions at the same time with two dierent wave numbers, m = 2 and 3. Staying on the m = 2 solution, the critical mode at the onset of instability, the calculations have never led to chaos. The mode with odd azimuthal wave number m = 3 exhibits chaotic behavior at Re For this Reynolds number two incommensurate frequencies have been found in spectrum. After two-frequency quasi-periodic regime, the three-frequency quasiperiodic state follows immediately without phase locking between them. This paper presents research results of the Belgian program on Interuniversity Pole of Attraction initiated by the Belgian state, Prime Minister s Oce, Science Policy Programming (PAI 04-6). The scientic responsibility is assumed by the authors. The authors wish to thank Prof. A. Nepomnyashchy for helpful discussions. References [1] S. Frank, D. Schwabe, Temporal and spatial elements of thermocapillary convection in a liquid zone, Experiments in Fluids [2] V. Petrov, A. Haaning, K.A. Muehlner, S.J. Van Hook, H.L. Swinney, Model-independent nonlinear control algorithm with application to a liquid bridge experiment, Physical Review E [3] M. Wanschura, V.M. Shevtsova, H.C. Kuhlmann, H.J. Rath, Convective instability mechanisms in thermocapillary liquid bridges, Physics of Fluids [4] V.M. Shevtsova, D.E. Melnikov, J.C. Legros, Threedimensional simulations of hydrodynamical instability in liquid bridges. Inuence of temperature-dependent viscosity, Physics of Fluids [5] M. Lappa, R. Savino, R. Monti, Three-dimensional numerical simulation of Marangoni instabilities in liquid bridges: inuence of geometrical aspect ratio, International Journal for Numerical Methods in Fluids [6] V.M. Shevtsova, D.E. Melnikov, J.C. Legros, Peculiarities of three-dimensional ow in liquid bridges at high Prandtl numbers, Computational Fluid Dynamics Journal [7] G.L. Baker, J.P. Gollub, Chaotic Dynamics: An Introduction, 2nd Edition, Cambridge University Press, Cambridge, 1996, p. 256.

THERMOCAPILLARY CONVECTION IN A LIQUID BRIDGE SUBJECTED TO INTERFACIAL COOLING

THERMOCAPILLARY CONVECTION IN A LIQUID BRIDGE SUBJECTED TO INTERFACIAL COOLING Melnikov D. E. and Shevtsova V. M. Abstract Influence of heat loss through interface on a supercritical three-dimensional