Laboratory experiments on Convective and Rayleigh-Taylor Instabilities P. Gonzalez-Nieto,C. Yague, J.L. Cano y J.M. Redondo* Fac. FisicasUniv. Complutense de Madrid, Madrid, Spain *Univ. Politecnica de Catalunya, Barcelona Spain e-mail:email@example.com Abstract A simple laboratory model of turbulent mixing between two miscible fluids under an initial situation of top heavy stratification in a gravitational field has been performed. The mixing processes are generated by the evolution of a discrete set of unstable forced turbulent plumes. We describe the corresponding turbulent mixing processes measuring the density profiles and the heights of the fluid layers by means of flow visualization. We characterize the partial mixing process and the role of a viscoelastic gel that hampers mixing and controls in a random fashion the initial conditions. The mixing efficiency and the Atwood number (ranging between 9.9 x 10-3 to 0.13)are related showing an increase up to a level of 10-20% .The evolution of the convective structures produced by bubbles is also discussed comparing the mixing efficiencies in both stable and unstable initial conditions. Other convective mixing situations induced by two phase convective stirring are also compared in order to evaluate the mixing efficiency in different parameter ranges. 1. Introduction The quantification of mixing efficiencies produced by convective body forces is an interesting problem to investigate the mixing process in many environmental situations such as top cloud entrainment or deep convection in the ocean, or the behaviour of mixing in the ocean surface layers when an injection of air or CO2 is used as an input of mechanical energy. Laboratory experiments will be described here comparing the mixing efficiency and the main flow descriptors, further LES numerical simulations of convective flows are presented in Yague and Redondo(1996), the model solves the Boussinesq set of equations in a two dimensional grid and is based on Rees(1987). Together with the momentum equations, the continuity equation for incompressible flow is used taking into account only x and z components. Initially the subgrid scaling assumed a constant turbulent viscosity defined in terms of dimensional scaling with a simple turbulent parametrization based on local mixing length scales as ν = l2 / ∆ t and taking the integral scale initially as a constant in terms of the mesh size as l = 0.23 (∆x ∆ z)1/2 Different aspect ratios of the convective flow, generated by buoyancy are considered where a complex convective pattern develops showing some asymetry as in the experiments of Kimura and Iga(1995). Figure 1. The experimental set up and apparatus used in a Plume array 2. Mixing efficiency From the turbulent kinetic energy equation, comparing buoyancy with the production term K p' 1 u g u' u' u' u' u' 2 ' ' u ' u ' t t Z We obtain the Flux Richardson number considered as a local mixing efficiency in stably stratified flows, in an unstably stratified flow the sign will be negative. Fernando (1999) g ' u' Rf u u' ' Z And the gradient and flux Richardson numbers are related by the ratio of the momentum and scalar turbulent diffusivities, given as a turbulent Prandtl or Schmidt number, Prturb so that Rf= η = Ri / Prturb is the mixing efficiency Turner(1973). The mixing efficiency may also be calculated per unit base area in a practical experimental manner by evaluating the ratio of the gain in potential energy divided by the amount of kinetic energy provided to the fluid during the mixing. It has to be stressed that mixing is a transient process and turbulent structure only occurs between the initial simple structure (two separate fluids) and the final mixed structure (homogeneous, well mixed flow) or linear profile. Most of the interface geometrical descriptors are calculated as averages over the centre region of the interfacial region leaving buffer regions to the sides of the experimental box to avoid lateral influences from the walls. The potential energy at a set time may be evaluated with the following integral (Linden & Redondo, 1991; Redondo, 1989; Yagüe, 1992). Ep (t ) S g. z, t .z.dz h 0 where S the base surface, h is the fluid layer height and (z,t) is the vertical density profile at time t. We will discuss here mostly results of experiments described fully in Gonzalez-Nieto(2004) and in Yague(1995) evaluating the overall mixing efficiency calculated from the initial (top heavy) and the final density profiles. The later may be Neutral (well mixed), Stable with a density step (no mixing) or linearly stratified. Figure 3 shows the range of Atwood numbers examined and the linear fits. Several other geometrical parameters defining the initial conditions, such as the viscoelastic gel viscosity and the separation between the dense layer and the gel were varied. The hole distribution was also modified but will be reported elsewhere The expression for the mixing efficiency of the top heavy initial profile is based in (Linden & Redondo, 1991; Linden, Redondo & Youngs, 1994) applicable to cases with complete or parcial mixing: Wusefull EpPARTIAL 1 1 Wavailable EpNMix EpNoMix W PARTIAL W PARTIAL 1 EpNoMix EpNoMix Where W indicates Work, Ep PARTIAL the increase in potential energy for the incomplete mixing situation and Ep NoMix the potential energy change in case of no mixing. After describing some of the random situations forced by the viscoelastic layer separating initially the dense and light fluids, the evolution of the density front thickness and the overall mixing efficiencies will be presented and discussed. The mixing efficiency of the overall convective process is particularily simple when the two layers containing the dense and light fluids are of the same height H/2, then if the heavy fluid, initially on top has density ρ1 = ρ2 + Δρ, with ρ2 the lighter fluid, there are two limiting cases that will give the maximum and minimum mixing efficiencies, i.e. when the final profile is constant in height with density ρ2 + Δρ/2, and when there is no molecular mixing and the top and bottom layers just exchange positions. There are two possible ways to calculate the mixing efficiency assuming that all the kinetic energy used for mixing the flow comes from the available potential energy, so calculating from the integral expression for the potential energy per unit area, the initial potential energy is: Ep(t=0)=3/8 g H2 Δρ And the final potential energy if complete mixing has occurred is EpMix(t=tf)=1/4 g H2 Δρ While if no mixing has taken place, the final potential energy is EpNoMix(t=tf)=3/8 g H2 Δρ. Similar expressions may be found for other configurations and for unequal depths of the light and heavy fluid. If the mixing efficiency is defined as Ep (t ) Ep NoMix Ep Mix Ep NoMix Then, the maximum mixing efficiency is ½ as discussed in Linden and Redondo (1991), but if the definition is made considering the actual potential energy used in the process, then the range of mixing efficiency is 0-1 using Ep (t ) Ep NoMix Ep (to) Ep (t ) A simple relationship may be found between these two alternative definitions as η=ζ / 2 – ζ as seen in figure 2, with a practical maximum mixing efficiency of 0.33 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Figure 2. Description of the mixing efficiency relationship, 3. Description of the experiments 2.1. Measurement system Turbulent mixing is generated experimentally under an unstable density distribution in a fluid system. In our case, the fluids that constitute the unstable density stratification are miscible and the turbulence will produce molecular mixing. Local unstable density stratifications occur when there is a density overturning in a stratified fluid. These unstable density stratifications correspond theoretically to a pure Rayleigh-Taylor instability which appears in fluid systems placed in a gravitational field when the density of the system decreases in the direction of the gravity. The hydrodynamic situation occurring in the experiment described here is an unstable density distribution and, although it does not exactly correspond to a Rayleigh-Taylor situation, one may still use several of the theoretical concepts of the Rayleigh-Taylor instability as a reference. The turbulent mixing processes under an unstable density distribution in the fluid system is generated by a discrete number of forced turbulent plumes whose behaviour and interaction result in the mixing. In this experiment our principal aim is the study of the properties of the mixed fluid that is produced after the turbulent mixing process. Figure 3 shows a diagram of the initial and final set ups of the three layer system including the Gel layer that provides random initial structure as it breaks. Figure 3. Description of the mixing process as the dense fluid percolates the Gel layer. tests were always performed with the same number of orifices in the box system. The orifice diameter affected the initial properties of the turbulent plumes16 increasing their moment and bulk flow. Additionally, the system of boxes is located at a height Ho from the CMC gel stratum and produces an increase in the overall initial potential energy of our system. This increase in energy will be analyzed in section 4. From the description of our setup we can conclude that the two essential characteristics of this experimental procedure are the presence of the CMC gel stratum, and the initial appearance of a set of turbulent plumes.16 The experiment begins when the orifices are in the open position and the denser fluid impinges on the CMC gel stratum. This results in the generation of several forced plumes which are gravitationally unstable (figure 2). The development of these plumes makes the turbulent mixing process possible and the details of this interaction will be described in the following section. There exist different procedures to obtain experimentally unstable density interfaces13-20. We propose a new experimental method to obtain an unstable density distribution. The fluid system consists of three homogeneous fluids with different densities that are initially at rest. The fluids are inside a cubic glass container. At the bottom of the container there is a fluid with lower density L making a layer designated as the “light layer” with a height hL. On top of this layer, a sodiumcarboximethyl celulose gel stratum, or CMC gel,21,22 is placed with density G and a height of hG. Finally, a system made of two metacrylic boxes, one fitting inside the other is placed at a height Ho from the CMC gel layer. The bottoms of the boxes are pierced with orifices that have apertures that can be regulated. These boxes contain the fluid of greater density D which constitutes the “dense layer”. The dense fluid reaches a height hD inside the boxes and is colored with sodium fluorescein which acts as a passive tracer23. The CMC gel is a non-newtonian time-dependent fluid and it presents a thyxotropic behaviour. One purpose of the presence of the CMC gel in the laboratory model is to represent in a faithful way real situations in which the unstable layers are separated by stable zones such as in the atmosphere. Another reason is the practical advantage of slightly delaying the mixing process so that it may better be observed experimentally. The presence of the gel stratum influences the energy balance of the fluid system which has been analyzed quantitatively. The experiment begins when the orifices of the box system are switched to the open position, then the dense fluid flows from system of boxes and it is injected through the orifices pierced in the bottoms of the boxes because there is a pressure change. The dense fluid flows as jets through the air layer and these dense fluid jets come into the gel layer. Therefore the denser fluid impinges on the CMC gel layer and goes through the gel locally arriving in the shape of jets which break down the surface tension of the gel. The high viscosity of the gel and the small width of the gel layer make that the dense fluid flows in the laminar regime through the Gel layer, two different gel viscosities were used. As a consequence, the dense fluid flows through the gel as jets again and it comes into the lighter fluid layer. The result is the generation of several forced plumes/jets which are gravitationally unstable. The development of these plumes makes the turbulent mixing process possible. There is no mixing between the denser fluid and the CMC gel. As the turbulent plumes develop, the denser fluid comes into contact with the lighter fluid layer and the mixing process between them begins. The behaviour of this fluid system is governed by the turbulent energy source generated by the buoyancy. Specifically, it depends on three factors: one, the CMC gel used which has viscosity G; second, the ratio of densities of the fluid system or Atwood number A and the third factor that influences overall mixing produced by the convective situation is the direct geometrical and topological effects that the initial conditions have over the volume where mixing can take place23,24. 2.2 Data Analysis Four suction conductivity probes which are located at different heights within the fluid system are connected to this instrument.17,18 The conductivimeter record is controlled by an analogous-digital converter Unidata Starlogger Model 6004B and the output data stored in a computer. The proper function of the conductivity probes requires the use of a suction system composed of a flow-regulation beaker, a venturi pipe and a multiple hydrodynamic connector. The flow-regulation beaker is a device which controls the input mass flow in the venturimeter, and venturi pipe acts as a small suction pump. The hydrodynamic connector permits the simultaneous connection of the four conductivity probes to the same venturi pipe so that their suction mass flow will be approximately equal. The evolution of the fluid flow is visualized by the shadowgraph technique or directly by fluorescence induced by light. Finally, all the experiments are recorded by a video camera for their subsequent digitalization with DigImage and Adobe Première 5.1. This software permits the grabbing of independent frames such as those shown in figures 2, 3 and 4. These A Venturi pipe acting as a small suction pump was used to eliminate the fouling of the four conductivity suction probes used to measure the density profiles (Redondo 1990) The evolution of the fluid flow was visualized by the shadowgraph technique or directly by light induced fluorescence. All the experiments are recorded at 50 Hz with video camera for their subsequent digitalization. This process is carried out by the use of two image softwares: DigImage and Adobe Première 5.1. This software permits the grabbing of independent frames such as those shown in figures 2, 3 and 4. These correspond to the time evolution of different turbulent mixing processes. The experiments were done at room The present work is based on more than 150 experimental tests. The experimental conditions were modified through the use of two CMC gels with different viscosities and 10 different values of Atwood number for each gel. The number of repetitions of each experiment varied between a minimum of 5 and a maximum of 10. The values of the direct and indirect physical magnitudes corresponding to the same initial experimental conditions were obtained by means of an ensemble average. The experiments were done at room temperature. Two CMC gels with different viscosity, but with a similar thyxotropic behavior, were used. The most viscous gel, or gel 1, had a viscosity of 1.9x105 cps and a density of 1.030 g/cm3. The less viscous gel, or gel 2, had a density of 1.025 g/cm3 and a viscosity of 1.7x105 cps. The density of the lighter fluid was pre-determined as a function of the CMC gel because of the necessity to create a stable density interface between them. Another parameter investigated systematicly in the experiment was the density of the denser fluid. As this density changed, the Atwood number and the initial buoyancy of the fluid system changed. We used a total of 20 different saline solutions to constitute the denser fluid layer. The density ratio between the lighter fluid and the denser one provided a wide range of Atwood numbers which extended from 9.8x10-3 to 1.34x10-1, where rD/rL is nearly unity. 4 Mixing by a convective unstable plume array The behaviour of the mixing process is shown in figures 4 and 5 which show frame sequences corresponding to the time evolution of several turbulent mixing processes visualized with the shadowgraph technique. Figure 4 shows the time evolution of a turbulent mixing process where the CMC gel used is the less viscous one and the Atwood number is A=0.134. Figure 5 shows the time evolution of another mixing process where the CMC gel used is the more viscous one and the Atwood number values A= 0.092. Figures 4a, 4b and 4c correspond approximately to the same time interval as, respectively, figures 5a, 5b and 5c. If we compare these figures we observe clearly the initial absence of turbulent plumes and the appearance of gel protuberances in figure 5 because the gel viscosity has increased. As the gel viscosity is reduced, the probability of initial generation of some gel protuberances reduces, so that the probability of the formation of turbulent plumes increases as we can see in figure 4. These two phenomena, protuberances and plumes, are not mutually excluding23. (a) (b) (c) (d) Figure 4. Time evolution of a partial mixing process through its frames sequence. Experiment made with the less viscous CMC gel (Gel=1.02 g/cm3) and A= 0.134. (a) Starting of turbulent plumes (t=0.24 s). (b) Development and lateral interaction of turbulent plumes (t=0.44 s). (c) Non uniform evolution of the mixing convective front (t=0.72 s). (d) Intense turbulent microstructure (t=6.68 s). (a) (b) (c) (d) (e) Figure 5. Time evolution of a partial mixing process corresponding to an experiment made with the most viscous CMC gel (Gel=1.03 g/cm3) and Atwood number of 0.092. (a) Slight protuberance in the CMC gel layer and an incipient formation of some turbulent plumes (t= 0.52 s). (b) Growth of the plume which breaks up the only protuberance made in the CMC gel layer (t= 1.72 s). (c) General interaction between plumes while the protuberance breakup continues through other plumes (t= 2.12 s). (d) Interaction of the fluid system with the physical contours of the container (t= 5.12 s). (e) Final state after the partial mixing process. We can observe the mixing layer limited by the stable density interface (t= 81.52 s). The other factor having an effect on the fluid flow is the Atwood number A. This adimensional parameter represents the strength of the buoyancy effect as a consequence of the density difference between the fluids. The greater the value of the Atwood number, the more the available potential energy for the mixing process. This implies a greater quantity of mixed fluid, and therefore, a greater height of the mixed layer and the mixing efficiency. Comparing figures 4 and 5 we observe that the rate of the turbulent plumes are greater in figure 4 than in figure 5. This is because the Atwood number is greater as explained before and the gel viscosity is greater than in figure 4.23 The final result of the mixing process is a mixed layer located at the bottom of the experimental container which is separated from the unmixed lighter fluid by means of the stable density interface (figure 5(e)). Since not all of the lighter fluid participates in the mixing, we qualify it as a partial mixing process. On the other hand, the mixed layer obtained experimentally is not homogeneous, but has a density stratification. Summing up, we have axysimetric turbulent plumes at an initial stage which develop at the intermediate stage. This development is caused by the lateral interaction between the plumes. At the final stage the turbulent mixing decays, leading to the appearance of a mixed layer that has a stable density stratification. The partial mixing process is characterized by two global properties: the mixing efficiency and the mixed layer height hM which exclusively depend on the characteristics of the fluid system in the initial and final states. As the mixing process is influenced by the initial buoyancy, these two global properties are analyzed versus the Atwood number A which is proportional to the buoyancy driven forcing. In addition, the mixing efficiency and the mixed layer height are influenced by the height of the denser fluid Ho and the presence of the CMC gel. The height Ho is due to the experimental setup because we place the denser fluid on the CMC gel layer by means of two boxes. Therefore, Ho is the height of the boxes relative to the gel layer and, finally, it is the initial height of the denser fluid relative to this gel layer. The presence of the height Ho is directly related with the role of initial conditions on the mixing efficiency. All the experiments were always performed with the same fixed height Ho which has a value of 1.5 cm. The effect of the height Ho is manifested by an increase in the available initial potential energy of the fluid system. We compare the available initial potential energy of the fluid system with and without the height Ho. As Ho is constant, we could think that the increase of the initial potential energy of the fluid system is constant. But this increase of the potential energy changes potentially to the Atwood number changed mainly by density difference of the denser fluid. The amount of increase in the initial potential energy depends on the Atwood number and is on the order of 1.5%.23 The final state of the fluid system is characterized by the presence of a stable density stratification which appears because the mixed layer is separated from the unmixed lighter fluid by a stable density interface. The mixed layer height hM represents the final height of this stable density interface and it was measured experimentally. Of course, the height hM is directly proportional to the final quantity, or volume, of the mixed fluid. The mixed layer height increase as the Atwood number grows because hM represents the volume of the mixed fluid; in other words, as the buoyancy effect increases so does the convective turbulent mixing. This behaviour is shown in figure 6(a) which represents the adimensional mixed layer height hM versus the Atwood number. The effect of the CMC gel viscosity may also be observed in this figure: the mixed layer height hM increases if the gel used is the less viscous one because the greater number of turbulent plumes that appears when the gel viscosity is reduced. The denser fluid and the lighter one do not mix completely which implies that the mixing process is only partial. Here, the mixing efficiency of this process is analyzed, where is defined as the fraction of the available energy that is used to mix the fluids. The values of the mixing efficiency are obtained theoretically from the potential energy change that occurs during the mixing process. The definition of the mixing efficiency used was that of Linden and Redondo(1991) Ep PARTIAL 1 (9) Ep NOMIX where (Ep)PARTIAL is the actual variation of potential energy associated to the partial mixing process. (Ep)NOMIX is the potential energy change of a process without mixing in which the denser fluid and the lighter one only exchange positions. In figure 6(b) we plot the average mixing efficiency as a function of the Atwood number of the fluid system. This mixing efficiency corresponds to the case of the stratified mixed layer. In our experiment, the mixed layer is stratified because mixing is not complete. From observations of the final density profiles that showed a strong density step, we assume that the stratification is made up by two layers. This has been found to be due to the interpenetration of the unstable plumes only through a fraction of the area at the top, precisely because once the dense fluid looses its potential energy it may not mix with lighter fluid above. 0.14 0.85 0.80 0.12 0.75 2 2 0.10 0.70 h /(h +h ') 0.65 d 0.08 0.60 1 l 0.06 0.55 m 0.50 1 0.04 0.45 0.02 0.40 0.35 0.00 -0.020.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 A A (a) (b) (c) Figure 6. (a) Behaviour of the non dimensional height of the mixed layer with the Atwood number for experiments made with the more viscous CMC gel (Curve 1, ), and with the less viscous one (Curve 2, ). The figure shows the linear fits done. (b) Mean mixing efficiency versus the Atwood number for experiments made with the more viscous CMC gel (Curve 1, ), and with the less viscous one (Curve 2, ). The corresponding empirical fits are shown. (c) Time evolution of the vertical mean density profiles for experiments made with the less viscous CMC gel and Atwood number A=0.134. In general, we observe that the efficiency increases as the Atwood number A grows. Physically, the increase in Atwood number implies that the buoyancy effect grows, as does the convective acceleration. Therefore it produces a greater mixing process with a higher efficiency associated with it. Curve 2 of figure 6(b) shows that the experiments done with the less viscous gel have a mixing efficiency greater than the one corresponding to the experiments with the more viscous gel shown in curve 1. Again, the reason is the greater number of turbulent plumes created when the gel viscosity is reduced. Other scientific works state that the maximum mixing efficiency is reached when the final profile is totally mixed and homogeneous. This value is 0.5; if the final profile is stratified, as it is in our case, then the mixing efficiency is calculated about 0.17.25 We agree with other researchers that one must concentrate more on the overall mixing efficiency behavior in relation to the Atwood number than on the exact numeric values25,28. The mixing efficiency is a global quantity, so we do not need to use the intermediate density profiles measured experimentally. Although they are not used we show their time evolution in figure 6(c) because it is an important part of the time evolution studies. This figure 6(c) shows the time evolution of vertical mean density profiles for experiments made with the less viscous CMC gel and for Atwood number A=0.134. This figure shows the vertical density profiles by means of several linear fits made to the experimental data in spite that the density profiles may not be linear. These linear fits characterize the time evolution of density profiles globally by means of slope change in time, which aids in the understanding of the global process. 3. Results Figure 3. Evolution of the density contours as convection developes 4. Discussion Using the variance of the signal (t) defined from: V (T ) ( (t T ) (t )) 2 where < > denotes the average over the entire periode T and the dependence for fractal time series V(T) T2H (Voss 1985 a,b 1988).Using T = 1/f and the description of the spectral density function, S(f), we have equivalently S(T) T and T S T 2 (t ) e ift dt T V 0 so we can relate S ( f ) T V T 2 H 1 T 2 E 12 D so that considering both molecular and turbulent diffusivities we have dc d 2 c d dc D 2 k dt dx dx dx In the case when a Bounday, such as a wave breaker transkers momentum to the coastal flow through horizontal shear. The same hypothesis used by Von Karman for the Atmospheric Bounday Layer may be used, defining a relevant scale of velocity, that associated to the turbulent sidewise friction produced by the boundary as r τ=*u2 then assuming that eddies proportional to the distance from the breaker are most efficient we model the lateral shear as dU/dz = u*/kz being κ the so called Von Karman constant with value for the atmosphere of 0,41. Integrating the diferential equation we have a logarithmic current profile such as U(z) = u* ln(z/zo) / κ 5 Conclusions Another experimental and numerical observation is that while the anisotropy of the Reynolds stresses is obviously linked with the non-homogeneity taking the vertical axis (in stratified flows) and the rotation axis (in rotating flows); Scalar behaviour in such flows has non-linear mixing properties Redondo (2002). There are similar effects that depart from Kolmogorov’s K41 and also for K62 theories, not just in second order structure functions (and related spectra ) for spatial non-homogeneity, for anisotropy and for spatial and temporal intermittency. In this experimental method there is no other acceleration apart from gravity, and there are no obvious additional experimental mechanisms which would enhance the mixing efficiency. At the physical level, the turbulent mixing process depends mainly on the initial buoyancy—represented by Atwood number—and on the viscosity of the CMC gel used. Moreover, the turbulent mixing occurs between miscible fluids and therefore, there is a high degree of fine scale mixing. We analyze two global properties associated with this process, the mixing efficiency h and the mixed layer height hM. The mixing efficiency increases with the Atwood number but decreases as the viscosity of the CMC gel is increased. The values of the mixing efficiency are less than 0.20 and tend toward a steady value for higher Atwood numbers. We propose an empirical fit to represent this behaviour. The values of the mixing efficiency obtained with this experimental setup are somewhat lower than others reported in the scientific literature for mixing by RT fronts (Linden and Redondo 1991) but care has to be taken when different definitions are used as described in section 2. The direction of gravity plays a dominant roles in the two dimensionalization due to body forces so that dominance of the enstrophy cascade over the direct energy cascade, but it is important to realize that both direct and inverse cascades may not be in equilibrium at the same time. The intermittency coupled with the non-homogeneity and anisotropy act indistinguishably to modify the dispersion within the flow, the role of coherent structures is also relevant as described in Redondo et al (1998) Babiano (2002) and is expected to modify also the mixing efficiency. Acknowledgments The authors would like to thank professors Ruijy Kimura From Tokio University and Stuart Dalziel from Cambridge Universityy for help with image analisis and discussions comparing experiments and simulations, Peter Furmanek had a Socrates EU felowship. Support was provided by ERCOFTAC and grant 2001SGR00221 (2005) of the Generalitat de Catalunya. References 1 González-Nieto, P. L., Redondo, J. M., Cano, J. L. and Yagüe, C.: 2004, The role of initial conditions on Rayleigh-Taylor mixing efficiency, International Workshop on The Physics of Compressible Turbulent Mixing, Ed. Dalziel S., DAMTP, Cambridge University, U.K. 2 Yagüe, C., 1992. Estudio de la mezcla turbulenta a través de experimentos de laboratorio y datos micrometeorológicos. Ph. D. Thesis. Complutense University of Madrid. 3 Linden, P. F., Redondo, J. M. and Caulfield, C. P.: 1992, Advances in Compressible Turbulent Mixing, edited by W.P. Dannevik, A.C. Buckingham and C.E. Leith, Princenton University. 4 Linden, P. F.: 1979, Mixing in stratified fluids, Geophys. Astrophys. Fluid Dynamics. 13, 3-23. 5 Linden, P. F. and Redondo, J. M.: 1991, Molecular mixing in Rayleigh-Taylor instability. Part I: Global mixing, Phys. Fluids. A3 (5), 1269-1277. 6 Turner, S. ―Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows.‖ J. Fluid Mech. 173, 431-71 (1986). 7 Harindra J.S. Fernando, ―Turbulent mixing in stratified fluids.‖ Annu. Rev. Fluid Mech. 23, 455-93 (1991). 8 Ratafia, M.: 1973, Experimental investigation of Rayleigh-Taylor instability, Phys. Fluids 16(8), 1207- 1210. 9 Read, K. I.: 1984, Experimental investigation of turbulent mixing by Rayleigh-Taylor instability, Physica 12D, 45-58. 10 Popil, R. and Curzon, F. L.: 1979, Production of reproducible Rayleigh-Taylor instabilities, Rev. Sci. Instrum. 50(10), 1291-1295. 11 Böckmann, M. And Müller, S. C.: 2000, Growth rates of the buoyancy-driven instability of an autocatalytic reaction front in a narrow cell, Phys. Rev. Lett. 85 (12), 2506-2509. 12 Linden, P. F., Redondo, J. M. and Youngs, D. L.: 1994, Molecular mixing in Rayleigh-Taylor instability, J. Fluid Mech. 265, 97-124. 13 Sharp, D. H.: 1984, An overview of Rayleigh-Taylor instability, Physica 12D, 3. 14 Youngs, D. L.: 1991, Three-dimensional numerical simulation of turbulent mixing by Rayleigh-Taylor instability, Phys. Fluids A 3 (5), 1312-1320. 15 American Society for Testing and Materials.: 1989, Standard Test Methods for Sodium Carboxymethylcellulose, Designation D1439. 16 Mahjoub O.B., Redondo J.M. and Babiano A. (2000b) Hyerarchy flux in nonhomogeneous flows in Turbulent diffusion in the environment Eds. Redondo J.M. and Babiano A. 249-260. 17 Redondo J.M. (2002) Mixing efficiencies of different kinds of turbulent processes and instabilities, Applications to the environment in Turbulent mixing in geophysical flows. Eds. Linden P.F. and Redondo J.M. 131-157. 18 Kimura R. and Iga, I. 1995 in Mixing in Geophysical Flows Eds. Redondo J.M: and Metais, Ed CIMNE, Barcelona. 19 Kolmogorov, A.N., 1941. Local structure of turbulence in an incompressible fluid at very high Reynolds numbers. Dokl. Academia de Ciencias de la URSS, 30:299-303. 20 Redondo, J.M., 1990. The structure of density interfaces. Ph.D. Thesis. University of Cambridge. Cambridge. 21 Redondo J.M., M.A- Sanchez, I.R. Cantalapiedra and R. Castilla (1998) "Vortical structures in stratified turbulent flows". Annales Geophysicae. Abstract (16 ), 1133.