UNSTEADY PHENOMENA IN HORIZONTAL GAS-LIQUID SLUG FLOW

Transcription

1 4th International Conference on Multi-Phase Flow, Nice, France. UNSTEADY PHENOMENA IN HORIZONTAL GAS-LIQUID SLUG FLOW B. Caussade, J. Fabre, C. Jean Institut de Mécanique des Fluides, UA CNRS 005 Avenue Camille Soula, Toulouse, France P. Ozon Elf Aquitaine, Boussens, Saint-Martory, France B. Théron Institut Français du Pétrole, 1 et 4 Avenue de Bois-Préau, Rueil-Malmaison, France ABSTRACT The purpose of this study is (i) to illustrate from a set of experiments, the transient response of a two-phase flow line and (ii) to test the capability of classical two-phase flow models to reproduce fast transient behaviour. The present work is limited to slug flow. The experiments have been performed, in pipe of 5 cm inside diameter and 90 m length, with air and water. The line response to flow rate variations produced at the pipe inlet was analyzed from pressure, void fraction and outlet liquid flow rate recordings. Slug flow being by itself unsteady, the deterministic variations resulting from sudden variations of gas or liquid flow rate were extracted from uncorrelated fluctuations linked to the slug structure, using conditional averaging. Fourteen runs corresponding to seven different flow conditions have been recorded. The pressure and void fraction disturbances are clearly displayed. The pressure disturbance is shown to travel at the highest velocity and is weakly damped. Void fraction waves appear to be conservative in shape. The void fraction wave celerity has been compared to the Taylor bubble velocity. For non bubbly slugs, they are identical; when the liquid slugs contain high gas volume, the Taylor bubbles travel at a higher velocity than the void fraction waves. The outlet liquid flow rate was shown to vary over a period starting at the arrival time of the pressure disturbance and ending at the arrival time of the void fraction wave. The transient drift flux model, -including two equations for the gas and liquid mass conservation and one equation for the mixture momentum conservation plus two closure laws, one for the wall shear stress and the other for the void fraction-, was tested. The explicit MacCormack method was used for the numerical model. The results agree satisfactorily with the experiments, especially for the prediction of the outlet liquid flow rate. However some details in the transient behavior are not correctly reproduced by the model: they are significative of inertia effects that the drift flux model is unable to take into account.

2 1. INTRODUCTION Transient phenomena in gas liquid flows pose some of the more important questions arising in two phase systems. A considerable task has to be achieved in addressing fundamental aspects, like the behaviour and interaction of the void fraction waves and pressure waves. Experimental results are expected to clarify the closure problem that appears more crucial for transient than for steady flows. Up to now, basic experimental studies concerning propagation phenomena have been essentially limited to bubbly flows. The very first studies concerned pressure propagation in two-phase media (see for example Campbell and Pitcher (1), Mori et al (2,3), Kuznetsov et al (4)). More recently many authors have focussed on the physical behaviour of the void fraction waves (Mercadier (5), Micaelli (6), Bouré & Mercadier (7), Pauchon & Banerjee (8)). One of the most significant results commonly accepted is that the transition from bubble to slug flow is associated with void fraction wave instabilities (see for example Matuszkiewicz et al (9), Tournaire (10), Bouré (11)). Incidentally, the same conclusion was given theoretically (Lin and Hanratty (12)) for the transition from stratified to slug flow: instability of surface waves is responsible for the change in the flow pattern. For many reasons, transient phenomena in slug flows have not been extensively studied though they are of primary importance in many practical situations. Experiments on void fraction waves in vertical flow have been mentioned by Pauchon & Banerjee (8). In the present study we have investigated some aspects of transient phenomena occurring in horizontal gas liquid slug flows, especially the downstream motion of finite disturbances of pressure and void fraction. The classical method which consists of analyzing the behaviour of harmonic disturbances of small amplitude (as used by Micaelli (6) and Tournaire(10)) has been rejected, with respect to the particular structure of the mixed flow pattern. Indeed the experimental study of wave dispersion may have a physical significance only for wave lengths that are greater than the characteristic length scale of random motion (twophase flows are always unsteady at some scale, even for steady boundary conditions). This length scale depends on the flow pattern, the pipe diameter for bubbly flow, the characteristic wave length for stratified flow and the characteristic cell length for slug flow. Since for slug flows the cell length may be of the order of several pipe diameters, the study of wave dispersion would be limited to large wave lengths. Moreover the analysis of slug flow from small amplitude waves leads to extracting weak deterministic signals from high level random noise. The problem concerns essentially the void fraction which takes high values in the Taylor bubbles and low values in the liquid slugs. In order to avoid the aforementioned drawbacks, we have chosen to study finite disturbances following sudden variation of liquid or gas inlet flow rate. The experimental results corresponding to 7 basic runs are presented and discussed. For the same experimental conditions, the transient drift flux model has been evaluated. The mathematical model was solved by using a finite-difference method (MacCormack (13)). 2. EXPERIMENTAL FACILITY The experiments have been carried out in the two-phase flow loop shown schematically in Fig. 1. The pipe of 53 mm inside diameter and 90 m length, made of transparent PVC, is supported on a steel frame and set up in horizontal position with an accurate level. The deviation of the pipe from horizontal is globally less than 0.003% and locally less than 0.1%. One of the most important problems concerns the pipe dilatation due to the external temperature variations (the maximum total dilatation was about 10 cm). The pipe was mounted free to glide on bolsters spaced at 1 m intervals.

3 Air and water were injected into the pipe through a mixer made of a Y junction. The system to control the flow rates of each phase consists of two parallel lines. The main line supplies the minimum flow rate set by a control valve, the secondary line is equipped with a control valve to adjust the rate that can be added when the electro-valve is open. Water was supplied from a centrifugal pump and air from a rotative compressor at the constant pressure of 7 bars. The minimum and maximum flow rates of each phase were carefully measured for steady conditions with two sets of orifice plates covering the range l/s for the liquid and kg/s for the gas. The pressure difference across the plate was recorded during transients. The flow was submitted to sudden change in gas or liquid flow rate by opening or closing one of the two electro-valves. In order to maintain constant gas flow rate during the transient period, the control valves were operated at choked condition by keeping the upstream pressure as constant as possible. To avoid compressibility effects in the line between the compressor and the control valves, a 50 liter tank was placed just upstream of the valves. The mixture at the outlet was separated by gravity in a free surface reservoir. The kinetic energy of the liquid slugs was damped by wood gratings floating at the surface. The instant flow rate at the outlet of the flow line was determined by measuring the water level in the reservoir with a pressure transducer placed at the bottom. Since the liquid flow rate supplied by the pump is known, the outlet flow rate has been deduced from the time variation of the water level at an estimated accuracy better than 5%. Moreover the holdup for steady conditions was obtained from the difference between the two levels corresponding to single-phase flow of liquid and two-phase flow respectively; the accuracy was about 2.5%. Local measurements have been taken at 5 cross sections, placed about 25 m apart. Instant pressure was measured with inductance transducers. For the liquid heights, resistance probes made of two stainless steel vertical wires of 0.7 mm diameter, 5 mm apart, were used. In order to compensate the variation of water conductance with the temperature, a similar probe was mounted between the pump and the mixer. The probes have been calibrated in standing liquid; the void fraction was obtained by calculation assuming the phases separated. Obviously the results for bubbly flows or bubbly slugs are qualitative; however their accuracy has been verified in steady flow by comparison with the global holdup. 3. EXPERIMENTAL PROCEDURE, DATA ACQUISITION AND PROCESSING Transient phenomena were obtained during short periods following a sudden change in gas or liquid flow rate. The steps were chosen (i) weak enough to maintain the same flow pattern, (ii) high enough to be able to measure significant variations of the void fraction. In each case the resulting changes of the mean void fraction was less than 10% for flow rate steps up to 30%. In steady slug flow, the instant void fraction fluctuations due to the succession of bubbles and liquid slugs may be greater than 50%. So some difficulties arise in determining the weak variations following the steps of flow rate. In order to extract the signal (coherent variations related to imposed flow rate) from noise (random fluctuations linked to the slug flow structure), ensemble averages were carried out over a large number of positive and negative steps (70 to 200 samples, depending on the run conditions). In practice, one of the two electro-valves was successively opened and closed. In order to avoid starting a new step before steady flow is reached, the half period of the opening-closing sequence was chosen 50% greater than the time of response of the line (estimated by L/U g where L is the pipe length and U g the average gas velocity corresponding to the final conditions). For each run the pressures and the liquid levels at the different measuring test sections, the pressure difference through the orifice plates and the liquid level at the outlet

4 separator, were digitized on a LMS-DIFA analog/digital converter at a frequency of 10 Hz. The conditional averages of the output liquid flow rate, the local pressure and the void fraction were computed in real time. The digital data were stored and processed on a HP 1000 computer. Each run might extend over 8 hours during which the flow rates corresponding to the final steady conditions had to be kept constant. Then for that run, the response to both positive and negative steps of flow rate was obtained. A typical example of averaging is given in Fig. 2. In the initial void fraction signal, the fluctuations due to the flow intermittence conceal the change of the void fraction resulting from the variation of the inlet flow rate; a similar conclusion holds for outlet liquid rate. After the averaging process, the perturbation of the void fraction is clearly seen. For the pressure however, the signal is not too noisy and the results are less spectacular. The true mean value µ x (t) of a physical variable x(t) falls, with a confidence of 95%, within the interval (Bendat et Piersol, (14)): [ x (t) - 2! x (t) "##N, x (t) + 2! x (t) "##N ] where N is the number of samples from which the average x (t) is obtained, and! x (t) is the variance. For the case plotted in Fig. 2, the variance of the void fraction was equal to 0.21; the average was determined from 100 samples so that the mean void fraction has been obtained with an uncertainty of ± Seven runs have been carried out. Their flow conditions are given on the flow pattern map of Fig. 3, as well as the transitions predicted from the model of Taitel & Dukler (15). The flow conditions were chosen so that (i) the slug length was small compared to the length of the pipe, (ii) the flow pattern did not change after the steps. 4. EXPERIMENTAL RESULTS The experimental and numerical results corresponding to the runs 21 and 25 are summarized in Fig. 4, 5, 6 and 7. From the top to the bottom of each figure, the void fractions, the pressures and the outlet liquid flow rate are plotted. The left part of each graph corresponds to the positive step of flow rate starting at t=0, the right part, to the negative step starting at the time specified by the dotted line. In order to distinguish the void fraction corresponding to different measuring sections, the curves have been shifted vertically. Moreover, a zoom has been added for the pressure, in order to amplify the details of its time variation, close to the disturbance starting time (the pressure difference P' between the instantaneous pressure P and the initial pressure P 0 has been plotted for clearness). In Fig. 4 and 6, the flow was submitted to changes of gas flow rate. In Fig. 5 and 7, the liquid flow rate was varied. First let us focus on void fraction results. In the upper graph, one can identify more or less clearly a void fraction wave, that travels downstream at constant velocity. Its shape remains remarkably conservative, even in the details like in Fig. 6 and Fig. 7. It moves without significant amplification or damping: this is a phenomenon with a "large memory", like the conveying slug flow structure. Besides some details are indicative of the formation of a very long Taylor bubble; examples are given in Fig. 6 (left side corresponding to positive step of gas flow rate) and Fig. 7 (right side corresponding to negative step of liquid flow rate) where the void fraction increases during a few seconds. This phenomenon agrees with experimental observations and is amplified for conditions close to the transition (like for the run 27). It is interesting to compare the celerity of the void fraction wave and the gas velocity. Indeed the drift flux model investigated in the present study predicts the following results (see appendix): the gas velocity does not change across the void fraction wave; this wave moves at the same velocity as the gas.

5 The celerity of the void fraction wave has to be compared to the velocity of the gas corresponding to the final steady conditions. The comparison is shown in Fig. 8. For convenience, the gas velocity has been replaced by the Taylor bubble velocity, obtained by the cross correlation method. The results agree for low velocities corresponding to non bubbly slugs. For velocities greater than 4 m/s the slugs contain bubbles and the gas velocity is probably smaller than the Taylor bubble velocity. The discrepancies are qualitatively in agreement with the theory. The records of pressure contain some interesting features. As expected, a pressure disturbance following the initial variation of flow rate travels downstream at a larger velocity than the void fraction wave. In each cross section, the pressure varies with time according to the following sequence. When the disturbance reaches the section, the pressure changes suddenly (pressure disturbance); then it remains practically constant until the void fraction wave reaches this point; in a third stage, the pressure slowly evolves toward the final steady conditions that are obtained when the void fraction wave has reached the pipe outlet. A simplified scheme of the transient phenomenon is given in Fig. 9. In fact the pressure is controlled by both the inertia effect giving the fast variations and the wall friction responsible for the slower ones. One interesting problem is the possible coupling between void fraction and pressure. From experiments it is hard to say whether the instantaneous void fraction varies when the pressure disturbance occurs. However the time average of R g was determined before and after the passage of the pressure disturbance. The results showed a difference of a few percent, indicating that the pressure disturbance induces a small change of void fraction. Again the influence of void fraction wave on pressure is seen more easily in some cases (in Fig. 7, the pressure P 3 displays a fluctuation related to the void fraction wave at t=110 s). These features will be discussed further. The outlet liquid flow rate changes when the pressure disturbance reaches the outlet. Next it evolves slowly until the void fraction wave arrives and changes again to approach its final value. It is worth noting that the accurate prediction of the instantaneous outlet flow rates will result from the correct determination of wave velocity. 5. TRANSIENT MODEL AND NUMERICAL RESULTS Although the drift-flux model is not beyond criticism, it was tried for evaluation in the present study. Pertinent equations for transient flow have been proposed elsewhere (Fabre et al. (16)). The present numerical results have been obtained with a more simplified model including 2 equations for the mass conservation of each phase and 1 equation for the momentum conservation of the mixture. $ $t $ $t (% l R l ) + (% g R g ) + $ $x (% l R l U l ) = 0 ( 1 ) $ $x (% g R g U g ) = 0 ( 2 ) $ $t (% l R l U l + % g R g U g ) + $ $x (% l R l U 2 2 l + %g R g U g ) + $ $x P = 4 D & - (R l % l + R g % g ) g sin ' (3) R l + R g = 1 (4)

6 R, U, %, are the fraction, the velocity, the density of each phase respectively (g: gas; l: liquid). P is the pressure, D the pipe diameter, & the wall shear stress, g the gravitational acceleration and ' the pipe inclination. Three closure equations are added to express the drift velocity, the wall shear stress and the relationship between the pressure and the gas density. In order to avoid solving the equation of energy, an isothermal process was assumed (a complete drift flux model including the conservation of energy of the mixture as well as the thermal equilibrium between phases would lead to the same conclusion, as far as the external temperature is constant). We must recognize that the assumption is wrong for fast transients: however the results are not too sensitive to this choice except for the pressure disturbances: % g = P r T (5) Where r is the gas constant and T the temperature. The gas velocity was assumed equal to the Taylor bubble velocity which is expressed by (Bendiksen (17)): U g = C 0 (R g U g + R l U l ) + C ( "### gd (6) C 0 = sin 2 ' C ( = 0.35 sin ' cos ' for U M "### gd < 3.5 ( 7 ) C 0 = 1.2 C ( = 0.35 sin ' for For the wall shear stress the following closure law was considered: U M "### gd > 3.5 ( 8 ) & = - f % M U M U M 2 Where % M, U M are the density and the velocity of the mixture. The finite difference explicit method, developed by MacCormack (13) to solve the Navier Stokes equations for compressible flows, has been used here. The details of the method, extensively discussed by Anderson et al. (18), are outside the scope of the present paper. The numerical stability is obtained by applying the following CFL condition between the time step )t and the mesh size )x: )t * )x / max { +, 1 +, +, 2 +, +, 3 +} (9) The results are presented in Fig. 4-7 (bold curves). They agree qualitatively with the experiments: the propagation of the void fraction wave is well predicted. It must be pointed out that the shape of the disturbance remains constant as expected. The pressure and, as a consequence, the liquid flow rate at the pipe outlet present the general features of the experimental results. However some discrepancies appear, especially for the run 25. Some fluctuations are not correctly calculated (see for example the void fraction and the liquid flow rate at the outlet, Fig. 6 & 7).

7 6. CONCLUSION Although the experimental method used in this study does not lead to the same level of information as the wave dispersion method, it provides useful information for the evaluation of two-phase transient models. The method involves ensemble averaging over several samples of the same experiment. The results are displayed through pressure and void fraction recordings. Two disturbances resulting from a sudden change in the inlet gas or liquid flow rate are clearly displayed in the experiments. The faster is essentially a pressure disturbance and possibly a velocity disturbance (see appendix) that controls the time at which the flow rate begins to change at the outlet. The slower is a void fraction disturbance travelling possibly at the gas velocity, conservative in shape, that controls the time at which the flow rates reach the final steady conditions at the pipe outlet. However we must keep in mind that the present experimental procedure is limited to the disturbances having a measurable amplitude and propagating downstream. A new procedure is now under development for studying the disturbances travelling upstream. The drift-flux model closed by analytical relations, is the simplest model to be tried: for stratified flows, it gives poor results; for bubbly flows it is probably unable to predict accurately the pressure and the void fraction. What is the issue for slug flows? The model that has been used here is certainly too simplified: it contains some assumptions regarding namely the inertia, and the closure relationships. However the main features of the transient flow are correctly predicted, in particular, the motion of the pressure and void fraction disturbances. The accuracy of the results is not yet satisfactory, although this work constitutes a good basis for future improvements and for comparing with the two fluid model. A complete evaluation will be possible after fulfillment of the experiments. ACKNOWLEDGEMENTS The authors are greatly indebted to the Elf Aquitaine and to the Institut Français du Pétrole for supporting this work. REFERENCES 1 Campbell, I. J. and Pitcher, A. S., "Shock Waves in a Liquid Containing Gas Bubbles", Proc. Roy. Soc. A, Vol. 243, pp , Mori, Y., Hijikata, K. and Komine, A., "Propagation of Pressure Waves in Two-phase Flow", Int. J. Multiphase Flow, Vol. 2, pp , Mori, Y., Hijikata, K. and Ohmori, T., "Propagation of a Pressure Waves in Two-phase Flow with Very High Void Fraction", Int. J. Multiphase Flow, Vol. 2, pp , Kuznetsov V. V., Nakoryakov V. E., Pokusaev B. G., Shreiber I. R., "Propagation of perturbations in a gas-liquid mixture", J. Fluid Mech. 85, 85-96, Mercadier, Y., "Contribution à l'étude des propagations de perturbations de taux de vides dans les écoulements diphasiques eau-air à bulles", Thèse, Université Scientifique et Médicale et Institut National Polytechnique de Grenoble, Micaelli, J. C., "Propagations d'ondes dans les écoulements diphasiques à bulles à deux constituants. Etude théorique et expérimentale", Thèse, Université Scientifique et Médicale et Institut National Polytechnique de Grenoble, 1982.

8 7 Bouré, J. A., Mercadier, Y., "Existence and properties of flow structure waves in twophase bubbly flows", Applied Scientific Research 38, , Pauchon, C., Banerjee, S., "Interphase Momentum Interaction Effects in the Averaged Multifield Model", Int. J. Multiphase Flow, Vol. 12, No. 4, pp , Matuszkiewicz, A., Flamand, J. C. and Bouré, J. A., "The bubble-slug flow pattern transition and instabilities of void fraction waves", Int. J. Multiphase Flow, Vol. 13, No. 2, pp , Tournaire, A., "Détection et étude des ondes de taux de vide en écoulement diphasique à bulles jusqu'à la transition bulle-bouchon", Thèse Institut National Polytechnique de Grenoble, Bouré, J. A., "Properties of Kinematic Waves in Two-Phase Pipe Flows. Consequence on the Modeling Strategy", European Two-Phase Flow Group Meeting", Brussels, May 30-June 1, Lin, P. Y., Hanratty, T. J., "Prediction of the Initiation of Slugs with a Linear Stability Theory", Int. J. Multiphase Flow, Vol. 12, No. 1, pp.79-98, MacCormack, R. W., "The Effect of Viscosity in Hypervelocity Impact Cratering", AIAA , Cincinnati, Ohio, Bendat, S. J, and Piersol, A. G., "Random Data: Analysis and Measurement Procedure, Chap. 10: Nonstationnary, Transient, and Multidimensional Data", John Wiley & Sons, Taitel Y., Dukler A. E., "A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow", AIChE Journal, 22, 1, 47-54, Fabre, J., Liné, A., Péresson, L., "Two Fluid/ Two Flow Pattern Model for Transient Gas Liquid Flow in Pipes", 4th Int. Conf. on Multi-Phase Flow, London, England, June, Bendiksen, K. H., "An Experimental Investigation of the Motion of the Long Bubbles in Inclined Tubes", Int. J. Multiphase Flow, Vol. 10, pp , Anderson, D. A., Tannehill, J. C., Pletcher, R. H., "Computational Fluid Mechanics and Heat Transfer", Hemisphere Publishing Corporation, McGraw-Hill Book Company, APPENDIX The response of the two-phase flow line to the sudden variation of the inlet flow rate can be given by transforming the hyperbolic PDE system (Eqs. 1-4) in an ODE system on the characteristic curves as follows: dp dt + W 2 dr g dt = 0 on C 1 (A.1) dp dt + W (U g - U l ) dr g dt - (U g - U l - W) R l du l dt + (U g - U l - W) G = 0 on C 2 (A.2)

9 dp dt - W (U g - U l ) dr g dt - (U g - U l + W) R l du l dt Where the equations of each characteristic are given by: + (U g - U l + W) G = 0 on C 3 (A.3) dx dt =, 1 = U g for C 1 (A.4) dx dt =, 2 = U l + W for C 2 (A.5) dx dt =, 3 = U l - W for C 3 (A.6) Eqs. A.1 to A.6 are written in dimensionless form by scaling, the velocities by U M, the coordinate by D, the time by D/U M and the pressure by % l U 2 M. G is the rhs of Eq. 3, scaled by % l U 2 M /D. W is a dimensionless velocity defined as follows: W 2 = % g % l rt U M 2 R g (1 - C 0 R g ) (A.7) We note in passing that W is much greater than unity (typically for the present conditions). For simplicity we shall assume that the relative variation of the inlet flow rate is small. A simplified description of the experiments is shown in Fig. 9. At t=0 the variation of the flow rate induces two disturbances that propagate downstream on the lines C 1 and C 2 at the velocities, 1 and, 2 respectively. These two characteristics split the (x,t) plane in three different domains (Fig. 9). We are interested in determining the jump of these variables across the lines C 1 and C 2. We define the jump of F across the line k by [F] and k start from the following assumptions: (i) friction is neglected in the neighborhood of the point where the jump is calculated; (ii) as a consequence the pressure, the void fraction and the phase velocities are nearly equal in two points of the same domain, close to each other (for ex. a & b); (iii) W is greater than unity by at least one order of magnitude; (iv) the dimensionless drift velocity U g -U l is much smaller than unity for cocurrent flow. Across C 2 the jumps are expressed from Eqs. 6, A.1 & A.3: [P] 2 + W 2 [R g ] 2-0 (A.8) [P] 2 - W R l [U l ] 2-0 (A.9) [P] 2 - W ( 1 C 0 - R g ) [U g ] 2-0 (A.10) The jump of void fraction is weak compared to the pressure jump: this conclusion agrees with the experimental observations. Across C 1 the jumps are expressed from Eqs. 6, A.2 & A.3:

10 [P] 1 + (U g - U l ) 2 [R g ] 1-0 (A.11) R l [U l ] 1 + (U g - U l ) [R g ] 1-0 (A.12) [U g ] 1-0 (A.13) The disturbance of the pressure is less important than the disturbance of void fraction: indeed it was shown (see Fig. 7) that the pressure exhibits small variations when large disturbance of void fraction occurs. Again the gas velocity does not change across the characteristic C 1 : this result is in agreement with the conservative shape observed experimentally (according to Eqs. A.4 & A.13 the wave is non dispersive: the same result could be obtained from the wave dispersion equation).

11 a. Experimental loop 89.5 m 64.8 m 44.8 m 24.8 m 0.7 m Separator Orifice plate Pump Orifice plate Compressor b. Separator Pipe c. Measuring test section Resistance wires Pressure transducer Reservoir Pressure tap Fig. 1: Experimental loop

12 1 Void fraction Void fraction 0 1 void fraction disturbance confidence interval Time (s) Fig. 2: Conditional averaging: void fraction a) sample of run 24; b) ensemble average over 100 samples Liquid superficial velocity (m/s) Intermittent flow Stratified flow Annular flow Gas superficial velocity (m/s) Fig. 3: Flow conditions

13 !!"#$% &'% ()*% +,'% -./".0"1*% 12% #.3% 2415% /.06!!! %'%*)76/"8.4%/63)403%9% ::: %'%6;<6/"76*03

14 !!"#$% &'% ()*% +,'% -./".0"1*% 12% 3"4)"5% 2316% /.07!!! %'%*)87/"9.3%/7:)30:%;% <<< %'%7=>7/"87*0:

15 !!"#$% &'% ()*% +,'% -./".0"1*% 12% #.3% 2415% /.06!!! %'%*)76/"8.4%/63)403%9% ::: %'%6;<6/"76*03

16 !"#$% &'% ()*% +,'% -./".0"1*% 12% 3"4)"5% 2316% /.07!!! %'%*)87/"9.3%/7:)30:%;% <<< %'%7=>7/"87*0:

17 8 Taylor bubble velocity (m/s) Velocity of void fraction disturbance (m/s) Fig. 8: Celerity of void fraction waves versus velocity of T-bubbles t domain influenced by final conditions C 1 t t transient domain C 2 domain influenced by initial conditions x R g P Fig. 9: Sketch of experiments in (x,t) plane