BUBBLE DISTRIBUTION IN A TURBULENT PIPE FLOW Catherine Colin, Dominique Legene, Jean Fabre Institut de Mécanique des Fluides de Toulouse, UMR 55 CNRS-INP/UPS Avenue du professeur Camille Soula 314 Toulouse, France Tel : (33) 5 61 8 58 5 Fax : (33) 5 61 8 58 1 Email: colin@imft.fr ABSTRACT In the gas-liquid turbulent bubbly pipe flows, the prediction of the spatial distribution of the phases is crucial for the design of the thermohyaulic loops. This prediction remains difficult because of the coupled effects of the bubble ift velocity (due to gravity), the turbulence of the liquid phase, the dynamics of the bubbles and the vicinity of a wall. Up to now, no satisfactory model does exist, especially in micro gravity conditions. In order to analyse the role of the gravity and the turbulence of the liquid phase upon the bubbles distribution, experiments and numerical simulations are performed under normal gravity conditions and in micro gravity. They consist of a lagrangian tracking of single bubbles in a turbulent pipe flow. The experimental and numerical results show that, on earth, under the action of the lift force, the bubbles move radially towards the pipe wall in vertical upward flow, towards the pipe centre in downward flow. In micro gravity, in absence the ift velocity, the lift force vanishes. The action of the large turbulent eddies is dominant on the bubble dispersion and the radial bubble distribution is more homogeneous. INTRODUCTION In the gas liquid bubbly pipe flows, the spatial distribution of the phases controls the pressure op and the wall heat transfer, therefore its prediction is crucial for the design of the thermohyaulic loops. A research programme concerning the gas liquid pipe flows on earth and under micro gravity conditions is carried out at the Institut de Mécanique des Fluides de Toulouse (IMFT) with the support of the Centre National d Etudes Spatiales. One of the objectives of this programme concerns the prediction of the radial distribution of the bubbles in a gas-liquid turbulent bubbly pipe flow. Previous investigation (Colin et al., 1993 ; Kamp et al., 1995) of the local structure of a turbulent bubbly flow in a pipe of 4 cm diameter and 4 m long was carried out. Water was axially injected in the tube and the air bubbles were injected through 4 hypodermic needles of.34mm diameter. The superficial velocity of the liquid ranged between.3 and 1 m/s, the superficial velocity of gas up to.5 m/s and the bubble diameters between 1 and 4 mm. At.8 m from the bubble injection, the tube was equipped with local probes, which can be moved in a radial direction: - a single hot film probe for the measurements of the axial velocity of the liquid phase, - a double optical probe for the measurements of void fraction, bubble diameter and velocity. High-speed video pictures of the flow were also taken to determine the bubble size after image processing. Experiments were performed in the laboratory in vertical upward and downward flow and under micro gravity conditions during parabolic flights aboard the Caravelle and Airbus A3 Zero-G aircrafts. The radial distributions of the void fraction α, the axial mean velocities of the liquid U L and gas U G and the RMS velocities of the phases have been measured (Kamp, 1996). The main result is the strong influence of the gravity upon the void fraction distributions. On earth, the classical peak of void fraction near the wall in upward flow and the void coring effect in downward flow are observed. In micro gravity conditions, the radial distribution of the bubbles is rather flat with a maximum near the pipe centre (figure 1). The radial distribution of the bubbles is attributed to different effects: the lift force due to the ift velocity of the bubbles and the vorticity of the liquid flow, the action of the turbulence of the liquid phase and the bubble dynamics. The shape of the void fraction profiles can be analysed from the radial momentum balance equations of the liquid and the gas phases, written for a steady, quasi parallel flow. After elimination of the pressure gradient in the radial direction and considering that the turbulence of the liquid is homogeneous in the pipe section, the following equation is obtained (Kamp, 1996): d"! L v' L = M Gr +! L (1 # ") dv' L M Gr = 6 d F r [1] where v' L is the radial contribution to the turbulent kinetic energy and M Gr the density of interfacial momentum transfer in the radial direction, related to the total averaged force F r acting on the bubbles of diameter d in the radial direction. These different forces are detailed in the r.h.s. of the bubble dynamics equation [3]. In most of the eulerien models, the mean forces acting in the bubbles and the interfacial momentum transfer are expressed only versus the mean velocities of liquid and gas. Thus, in the expression of F r, only the lift force remains and equation [1] becomes:
d! v' L = (1"!) dv' L " 1 C L ( U G " U L ) du L [] where U L and U G are the axial components of the mean liquid and gas velocities plotted in figure. Equation [] can be used to predict the sign of dα/ and the qualitative shape of the void fraction distribution. The gradient of the mean liquid velocity being negative, the sign of the lift force is the same as this of the ift velocity U G - U L. It is positive in upward flow, negative in downward flow (figure ) and it follows that, for nearly spherical bubbles, the lift force pushes the bubbles towards the pipe wall in upward flow and towards the pipe centre in downward flow. The gradient of v' L in the radial direction is always positive, except very close to the wall, but its effect is smaller than this of the lift force. In micro gravity condition, the ift velocity is very close to zero and the mean lift force vanishes. Thus equation [] predict that the gradient of the void fraction is positive, which is not in agreement with the experimental results. The micro gravity experiments clearly pointed out the lacunae of the classical eulerien two-fluid models. In the laboratory experiments, the ift velocity is significant and the role of the turbulence on the bubble motion is often hidden by the forces due to the mean flow. On the other hand, in micro gravity, all the forces due to the mean flow vanish after averaging. Therefore, it is crucial to take into account the effect of the turbulence in the bubble dynamics equation..8 1.6!.7.6.5.4.3..1 U L 1.4 U G 1. (m / s) 1.8...4.6.8 1 r/d.6..4.6.8 1 r/d Figure 1: Void fraction distribution Figure : Radial distributions of U L, U G vertical upward flow, downward flow, microgravity flow, U G : open symbols, U L : closed symbols In order to analyse the role of the mean flow and the turbulence of the liquid phase on the bubble distribution, basic experiments and numerical simulations are performed on earth and also in micro gravity conditions. Both physical experiments and numerical simulations consist of the determination of the trajectory of single bubbles in a single-phase turbulent shear flow. In this basic situation, the single bubbles do not influence the turbulence of the liquid phase. EXPERIMENTAL DEVICE Experiments are carried out in a pipe of 4 mm diameter and 4 m long with the two-phase flow loop EDIA (figure 3). Water is circulated in a tube of 4 cm diameter and 4 m long by a centrifugal pump with a superficial velocity of 1 m/s. Spherical bubbles with diameter smaller than 1 mm are injected one at a time, through a small tube at the pipe centre, in the water flow. The bubbles are injected m downstream a turbulence grid located at the pipe inlet. A bubble injection device has been developed to inject bubbles with a constant size. The bubbles are created inside a box, at the outlet of an hypodermic needle of.15 mm dia. Water is injected in this box with a very low flow rate by a secondary pump and it carries the bubbles through a small tube to the centre of the pipe of 4 cm diameter. In that way, the bubble created in the box have a constant diameter of.9mm. Two synchronised high-speed video cameras located in two perpendicular plans take pictures of the bubbles downstream from the injection. After image processing, the three-dimensional trajectory of each bubble is rebuilt. The probability density function of the radial position of the bubbles can be obtained in different pipe sections downstream the bubble injection in order to be compared to the statistical results of the numerical simulations. Experiments are carried out in laboratory in vertical upward and downward flow. Micro gravity experiments have also been performed during a parabolic flights campaign aboard the Airbus A3 Zero G aircraft. Two different zones are investigated. Just downstream the injector, the bubbles are still in the central part of the pipe, where the mean velocity gradients are weak and the turbulence of the liquid phase nearly homogeneous. In this region a dispersion coefficient of the bubbles is estimated. A second zone at m from the bubble injector is also investigated. The radial distribution of the bubbles doesn t evolve anymore in the axial direction. The results are compared to the previous results of Kamp (1996) obtained with the injection of several bubbles.
Water circuit Visualising test section Light Cameras 4 cm dia. pipe tube Orifice Bubble extractor Pump Bubble injection device Valve Flow meter Box air Grid Motor EXPERIMENTAL RESULTS Figure 3: Experimental set-up A first set of experiments has been carried out in laboratory in vertical upward flow for a liquid superficial velocity of 1 m/s. Some measurements have been performed close to the injection at a distance z smaller than 15 R (R being the pipe radius). The radial distributions of the bubbles obtained after image processing, at different axial positions z, are plotted in figure 4. The bubbles are injected at the pipe centre, but quickly move toward the pipe wall under the effect of the turbulence and the lift force. The shape of the bubble distribution becomes larger as z increases due to the bubble dispersion by the turbulence. At z=1r, some bubbles have already reached the pipe wall.
1.5,5 1 z/r=8,5 1 z/r=1 z/r=1 1 1 1 1.5 1 r/r z/r=,5 z/r=,5 z/r=4,5 Figure 4 : Radial distributions of the bubbles of.9 mm diameter, at different distance z from the injection for a superficial liquid velocity j L =1m/s. The axial evolution of the mean radial position of the bubbles is plotted in Figure 5. It is compared to the trajectory of a single bubble of same diameter calculated from the dynamics equation for a spherical bubble:! G " b dv dt = (! G #! L )" b g + C D (Re b ) $d 8! L (u L # v) u L # v +! L " b (1+ C M ) Du [3] % L dv( # C &' Dt M dt )* +! L " b C L (u L # v) + (, + u L ) The r.h.s. of equation [3] are the different forces acting on a spherical bubble: buoyancy, ag, added mass and lift forces. u L and v are the instantaneous velocities of the liquid phase and the bubble, ρ G, ρ L are the densities of gas and liquid,! b is the bubble volume, d is the bubble diameter, C M is the virtual mass coefficient taken equal to 1/, C D is the ag coefficient function of the bubble Reynolds number Re b = u L! v d / " L (Mei et al., 1994), ν L the cinematic viscosity of the liquid. C L is a lift coefficient depending on both Re b and the shear rate (Legene and Magnaudet, 1998). This equation is valid for a bubble with a diameter smaller than the length scales of the flow inhomogeneity. By considering only the effect of the mean velocity of the liquid, the bubble motion in a steady, parallel, vertical shear flow can be calculated by the projections of equation [3] on the vertical (axial) and horizontal (radial) axis: C M dv r dt C M dv z dt = C L ( U L! V z ) du L = C L V r du L + 3C D 4d! 3C D 4d V r v ( U L! V z )v + g [4] V r and V z are the components of the mean bubble velocity in the horizontal and vertical directions. The trajectory of a bubble injected at 1 mm from the pipe axis is calculated from [4] and plotted in figure 5. The comparison with the measurements of the mean radial position of the bubbles, clearly points out that the mean radial motion of the bubbles is not only due to the mean velocity of the liquid. Then the turbulent velocities plays also an important role in the mean motion of the bubbles and have to be taken into account. The effect of the turbulence on the bubble dispersion has also been investigated. Although, the flow near the pipe centre of a tube of 4 cm for a Reynolds number of 4, is not very homogeneous and isotropic, we tried to estimate a dispersion coefficient of the bubbles in the radial direction and to compare it to the results of Spelt and Biesheuvel (1997) obtained by direct numerical simulations of bubble motions in homogenous isotropic turbulence. By analogy with the turbulent diffusion of fluid particles (Hinze, 1975), the dispersion coefficient of the bubbles in the radial direction D r can be calculated versus the mean square displacement of the bubbles in the radial directions r :
D r = 1 d dt r [6] For the long times, greater than the lagrangian integral time scale of the particle motion τ r the dispersion coefficient reaches a constant value, corresponding to a linear evolution of r with time (Hinze, 1975). For bubbles in an homogeneous, isotropic turbulence, Spelt (1996) and Spelt and Biesheuvel (1997) give an expression of the lagrangian integral time scale τ r and of the dispersion coefficient of the bubbles in the direction perpendicular to the bubble ift:! r = 1 " L # 1 + 4(! rel /µ) & % u $ 1+ 3(! rel / µ) ( D r = 1 ' "u L + 3 8 "u L )! rel, +. * µ - for " = u / V T << 1 and! rel µ with! rel = V T / g µ = / /V T V T = gd / L 361 L [7] u is the velocity scale of the turbulence, L is the eulerian integral length scale of turbulence, λ is the Taylor micro scale (Hinze, 1975), V T is the terminal velocity of the bubble, τ rel is the bubble relaxation time and µ is the interaction time between the bubble and a turbulent eddy of size λ. u, λ and L are estimated at the pipe centre from the measurement of the liquid velocity by hot film anemometry, for j L =1m/s. The different time, length and velocity scales are estimated for our experiment (Marino, ): u = 4cm / s V T = 7cm /s L = 4mm! = 1.8mm " rel = 14ms µ = 7ms " r # 1ms and D r # 9 $1 %6 m / s [8] The lagrangian integral time scale of the bubble motion in the radial direction τ r is about equal to 1 ms, then the bubble dispersion for the long time scales can be studied for a distance to the injection z greater than V z τ r = 1mm =.6R, V z being the vertical velocity of the bubble near the pipe centre. Experimentally, the dispersion coefficient is calculated from equation [6] using the data of the figure 6. 15 15 z /R 1 z /R 1 5 5 4 6 8 1 5 1 15 r (mm) r (mm ) Figure 5: Mean radial positions of the bubbles ( ) Figure 6: Mean square of the bubble radial positions trajectory of a single bubble calculated from [5] For z/r between 1.5 and 5, the mean square of the bubble radial positions evolves quasi linearly with z. For greater values of z (>8R), the distribution of the radial positions of the bubbles becomes strongly asymmetrical (figure 4), the effect of the lift is dominant and the values of r increase a lot. Then the dispersion coefficient will be estimated from the data obtained for z/r < 5. The bubble vertical velocity V z being about equal to 1. m/s for 1.5 < z/r < 5, the dispersion coefficient can be estimated from the slope of the curve plotted in the figure 6 (Hinata et al., 198): D r = 1 V z d dz r! 6"1 #6 m /s [9] The value of the dispersion coefficient of the bubbles determined from the experimental data is very close to the value [8] estimated from the expression of Spelt and Biesheuvel (equation [7]). Consequently, close to the injection, where
the bubbles are present in the central part of the pipe and where the effect of the lift force is not dominant, their radial dispersion by the turbulence is well predicted by the theory developed for the homogeneous, isotropic turbulence. In vertical upward flow, the radial repartition of the bubbles has also been determined from the statistics of 4 bubbles trajectories, at a distance z=1r from the injection. From this distance the bubble distribution is established. In the figure 8, the bubble distribution (dashed line) is compared to the void fraction distribution (white squares) measured at the same velocity j L =1m/s and for a mean void fraction of %. The shapes of these two distributions are very similar in spite of the different the bubble sizes. They display a sharp peak near the wall and an absence of bubbles near the pipe centre. This comparison is very promising, and it seems reasonable to study the mechanism of the void fraction distribution, at low void fraction, by analysing the dynamics of isolated bubbles. NUMERICAL SIMULATIONS The numerical simulations are performed with the code JADIM, allowing Large Eddy Simulations of the turbulence (Calmet & Magnaudet, 1997) and a lagrangian tracking of particles (Climent & Magnaudet, 1997). The local instantaneous characteristics of a wall turbulent shear flow are obtained by LES with a great accuracy (close to that obtained with Direct Numerical Simulations) for Reynolds number up to 4,. The trajectory of isolated spherical bubbles can be computed by using equation [3]. The instantaneous velocity u L of a turbulent liquid flow is computed for a D geometry by the code JADIM, the bubble velocity v is calculated from equation [3] and the instantaneous trajectories of single bubbles are obtained after integration of the velocity v. The probability density function of the radial repartition of the bubbles can be deduced from a statistical analysis of the instantaneous trajectories and compared to the experimental data. The first results of the numerical simulations performed for bubbles of.5 mm diameter (Legene et al., 1999) confirm the tendencies experimentally observed (figure 7). Under the action of the lift force, the bubbles move radially towards the pipe wall in vertical upward flow (-1g), towards the pipe centre in downward flow (1g). In micro gravity (g), in absence of a ift velocity, the lift force vanishes. The action of the large turbulent eddies is dominant on the bubble dispersion and the radial bubble distribution is more homogeneous. The trajectories of the bubbles between z=6 R and z=1r are used to calculate the probability density function of the radial positions of the bubbles. In figure 8 these dimensionless radial distributions ( for bubbles of.5 mm diameter) computed for upward, downward and micro gravity flows are compared to the dimensionless void fraction distributions measured by Kamp (1996) with the same superficial velocity (j L =1m/s), but with simultaneous injection of several bubbles at low void fraction α=%. The divergence between the numerical simulations and the experiments can be attributed to different factors. Only 1 bubble trajectories have been computed and the convergence of the results is not certain. Furthermore, the bubble sizes are different:.5 mm in the numerical simulations and 3 to 4 mm in the upward and downward flow experiments, then the bubble ift velocity is smaller in the numerical simulations and the effect of the lift force weaker. The interactions between the bubbles and the turbulent eddies are also different for bubbles of different sizes. On the other hand, in micro gravity, the numerical simulations are qualitatively in good agreement with the experiments of Kamp performed with bubbles of 1. mm diameter. The bubble motion is controlled by the instantaneous added mass force (third term of the r.h.s. of equation [3]), taking into account the temporal variations of the turbulent structures.
(-1g) (1g) (g) Figure 7: Computed trajectories of bubbles of.5 mm diameter in a turbulent channel flow (h= cm, Re=4,) in vertical upward flow (- 1g), downward flow (1g) and microgravity flow (g). 8 pdf 6 pdf pdf (-1g) (1g) (g) 1.5 1.5 4 1 1.5.5..4.6.8 1..4.6.8 1..4.6.8 1 r/r r/r r/r Figure 8: Comparison of the probability density function of the radial positions of the bubbles and the dimensionless void fraction profiles for Re=4,: numerical simulation (d=.5 mm)- - - experimental results (d=.9mm) experimental results of Kamp (α=%) : vertical upward flow, downward flow, microgravity flow
CONCLUSION In most of the eulerian models, used for the dispersed flow computation, the interfacial momentum transfer (including the forces acting on the bubbles) is modelised from the mean velocity fields, the effect of the turbulent velocities being not taken into account. Some discrepancies are then observed between the radial bubble distributions measured in a pipe flow and the predictions of these models, especially in micro gravity. To point out the effects of the mean velocity field and of the turbulence on the bubble motion, some experiments and numerical simulations on the dynamics of isolated bubbles have been carried out. The three dimensional trajectory of each bubble injected at the pipe centre, has been determined and the statistical distributions of the radial positions of the bubbles have been calculated. The first experiments performed in vertical upward flow, show that near the bubble injection point, the radial dispersion of the bubbles is similar to this observed in homogeneous isotropic turbulence. On the other hand, the mean motion of the bubbles is not well predicted by a bubble dynamics equation based on the mean velocity field. At a distance of 1R from the bubble injection, the statistical distribution of the bubbles is quite the same as the void fraction distribution measured in bubbly flow at low void fraction. Therefore, the statistical analysis of the dynamics of single bubbles is pertinent to study the void fraction distribution in bubbly flows at low void fraction. The first results of the numerical simulations are very promising. They point out the importance of the instantaneous characteristics of the turbulence upon the bubble dynamics. This aspect should be taken into account in the modelling of the interfacial momentum transfer (including the forces acting on the bubbles) in the eulerian models. Then, the prediction of the bubble radial distribution should be improved, especially in micro gravity where the bubbleturbulence interactions control this distribution. BIBLIOGRAPHY CALMET I., MAGNAUDET J., Large Eddy Simulation of high-schmidt number mass transfer in a turbulent channel flow, Phys. Fluids 9 (), 1997. CLIMENT E., MAGNAUDET J., Simulation d écoulements induits par des bulles dans un liquide initialement au repos, C.R. Acad. Sci. Paris, t. 34, Série II b, 1997. COLIN C., KAMP A. & FABRE J., Influence of gravity on void and velocity distribution in two-phase gas-liquid flow in pipe, Adv. Space Res., 13, 7, 141-145, 1993. HINATA S., KUGA O., KOBAYASI K., Diffusion of bubbles in two phase flow, second report : diffusion of a single bubble and eddy diffusivity of heat in single phase turbulent flow, bulletin of the JSME,, No.164, 1979. HINZE J.O., Turbulence, Mc Graw Hill, Second Edition, 1975. KAMP A., COLIN C. & FABRE J., The local structure of a turbulent bubbly pipe flow under different gravity conditions, nd International Conference on Multiphase Flow, Kyoto (Japan), April 3-7, 1995. KAMP A., Ecoulement turbulent à bulles dans une conduite en micropesanteur, Thèse INPT, 1996. LEGENDRE D., MAGNAUDET J., The lift force on a spherical bubble in a viscous linear shear flow, J. Fluid Mech., 368, pp 81-16, 1998. LEGENDRE D., COLIN C., FABRE J., MAGNAUDET J., Influence of gravity upon the bubble distribution in a turbulent pipe flow: comparison between numerical simulations and experimental data, Journal de Chimie Physique, 96, 951-957, 1999. MARINO D., Etude du mouvement de bulles isolées dans un écoulement turbulent en tube, DEA INP Toulouse,. MEI R., LAWRENCE C.J., ADRIAN R.J., Unsteady ag on a sphere at finite Reynolds number with small amplitude fluctuations in the free stream velocity, Phys. of Fluids, 6, pp 418-4, 1994. SPELT P.D.M., The motion of bubbles in a turbulent flow, PhD thesis University of Twente, 1996. SPELT P.D.M., BIESHEUVEL A., On the motion of gas bubbles in homogeneous isotropic turbulence, J. Fluid Mech, 336, pp.1-44, 1997.