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BUOYANT MIXING PROCESSES AND FRACTAL STRUCTURE IN TURBULENT PLUMES

INTERNATIONAL SUMMER COURSE ON NON-HOMOGENEOUS TURBULENCE. BUOYANT MIXING PROCESSES AND FRACTAL STRUCTURE IN TURBULENT PLUMES. INTRODUCTION AND AIMS EXPERIMENAL SETUP PLUME ARRAY MIXING PROCESSES GLOBAL MIXING RESULTS. FRACTAL ANALYSIS FRACTAL AND MULTIFRACTAL RESULTS CONCLUSIONS.

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BUOYANT MIXING PROCESSES AND FRACTAL STRUCTURE IN TURBULENT PLUMES

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  1. INTERNATIONAL SUMMER COURSE ON NON-HOMOGENEOUS TURBULENCE BUOYANT MIXING PROCESSES AND FRACTAL STRUCTURE IN TURBULENT PLUMES

  2. INTRODUCTION AND AIMS • EXPERIMENAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS • INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  3. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS INTRODUCTION. AIMS • Turbulent mixing is a very important issue in the study of geophysical phenomena because most fluxes arising in geophysics fluids are turbulent. • We study turbulent mixing due to convection using a laboratory experimental model with two miscible fluids of different density with an initial top heavy density distribution. • The conclusions of this experimental model relate the mixing efficiency and the volume of the final mixed layer to the Atwood number, ranging from 0.010 to 0.134. • Mixing produced in convective flows is investigated comparing different experiments: • experiments with and without an intermediate gel layer; • experiments with a line plume array and a bidimensional plume array. • We also study the fractal structure of non homogeneous plumes affected by different levels of buoyancy and initial potential energy. • We analyze the time evolution of the fractal dimension as plumes develop and we make a multifractal analysis. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  4. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS DIFFERENT KINDS OF EXPERIMENTS DENSE LAYER DENSITY: D= ( , ) g/cm3 LIGHT LAYER DENSITY: L=(1.03, 1.04) g/cm3 CMC GEL LAYER DENSITY: Gel= 1.02 g/cm3 Gel= 1.03 g/cm3 EXPERIMENTS WITH A CMC GEL LAYER EXPERIMENTS WITHOUT A CMC GEL LAYER 1 PLUME EXPERIMENT 9 PLUME EXPERIMENT EXPERIMENTS WITH A LINE PLUME ARRAY (1D) EXPERIMENTS WITH A BIDIMENSIONAL PLUME ARRAY (2D): 54 PLUMES NUMBER EXPERIMENTS: 200 ATWOOD NUMBER: (0.01, 0.14) NUMBER EXPERIMENTS: 20 ATWOOD NUMBER: 0.03 and 0.07 BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  5. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS EXPERIMENTAL SETUP FOR EXPERIMENTS WITH A CMC GEL LAYER BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  6. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS EXPERIMENTAL SETUP FOR EXPERIMENTS WITH A CMC GEL LAYER • Turbulent mixing is generated experimentally under an unstable density distribution in a fluid system. The fluids that form the initial unstable stratification are miscible and the turbulence will produce molecular mixing. • Our experiment consists of three homogeneous fluids with different densities that are initially at rest. • The fluids are placed inside a cubic glass container. At the bottom of the container, it is the fluid with lower density Land with a height hL. On top of this light fluid layer, a sodiumcarboximethyl celulose gel stratum, or CMC gel, is placed with density G and depth of hG. The gel generates a random initial structure. Finally, the fluid of greater density D(brine), which constitutes the dense layer, reaches a height h’Dand is coloured with sodium fluorescent (at low concentration as a passive tracer). • The experiment begins when the dense fluid flows forming jets and it impinges on the CMC gel layer, breaks down its surface tension and goes through the gel locally. The high gel viscosity (from 16000 cps to 44000 cps) and the small width of the gel layer make that the dense fluid flows in the laminar regime. There is no mixing between the dense fluid and the CMC gel. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  7. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS EXPERIMENTAL SETUP FOR EXPERIMENTS WITH A CMC GEL LAYER • Finally, the dense fluid comes into the light fluid layer and it generates several forced plumes which are gravitationally unstable. This development is caused by the lateral interaction between these plumes at the complex fractal surface between the dense and light fluids. • As the turbulent plumes develop, the dense fluid comes into contact with the light fluid layer and the mixing process grows. • The final result of the mixing process is a heavier mixed layer located at the bottom of the container. • As the turbulence decays, a stable situation with internal waves takes place. • The mixed layer is separated from the non mixed light fluid by a stable and sharp density interface which final height is the mixed layer height hM BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  8. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS EXPERIMENTAL SETUP FOR EXPERIMENTS WITH A CMC GEL LAYER MAIN CHARACTERISTIC OF EXPERIMENTS TWO METACRYLIC BOXES WHOSE BOTTOMS ARE ALTERNATIVELY PIERCED WITH ORIFICES THE CMC GEL LAYER The CMC gel is a non-newtonian time dependent fluid and presents thyxotropic behaviour. (a) (b) Thyxotropic behaviour of the sodiumcarboximethyl gel. (a) Time evolution of the gel viscosity for the more viscous gel (curve 1, ) and for the less viscous one (curve 2, ) with a rotation speed of 0.6 rpm. (b) Evolution of the gel viscosity with the shear rate for the less viscous gel. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  9. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS EXPERIMENTAL SETUP FOR EXPERIMENTS WITH A GEL LAYER • If there is a gel layer, there is a random initial distribution of 2D-plume array. Therefore, we don’t control the number of plumes nP and their location. • If there is a gel layer, the mixing efficiency and the mixed layer height hNM are smaller, as some graphical results will show later. • We use two metacrylic boxes whose bottoms are alternatively pierced to locate the denser fluid layer. These pierced bottoms can control the number and geometry of the plumes, but the CMC gel layer will randomize the initial distribution in a way in which the initial conditions (viscosity of gel) are seen to modify the overall mixing efficiency. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  10. 4 3 2 1  Vp • INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS THE TWO METACRYLIC BOXES WHOSE BOTTOMS ARE ALTERNATIVELY PIERCED WITH ORIFICES WHOSE POSITION CAN BE REGULATED. These boxes contain the fluid of greater density D which constitutes the “dense layer” and is coloured with sodium fluorescein (a passive tracer). PUSH WITHDRAWAL VELOCITY OF THE PLASTIC, Vp THE BOTTOM HOLES OF THE METACRYLIC BOXES ARE NOT SUPERIMPOSED: CLOSED POSITION THE BOTTOM HOLES OF THE METACRYLIC BOXES ARE SUPERIMPOSED: OPENED POSITION BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  11. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS PARTIAL MIXING PROCESS WITH A BIDIMENSIONAL PLUME ARRAY AND A GEL LAYER • PARTIAL MIXING EVENT WITH THE LESS VISCOUS GEL: Gel= 1.02 g/cm3 and A=0.134 • Initial experimental state (0 s). • Starting of turbulent plumes (0.24 s). • Development of the turbulent plumes (0.32 s). • Lateral and front interactions between the turbulent plumes (0.64 s). • Interaction of the fluid system with the physical contours of the container (1.00 s). • Final state after the partial mixing process (96.70 s). t= 0.32 s t= 0.60 s t= 1.61 s t= 1.01 s • PARTIAL MIXING EVENT WITH THE MOST VISCOUS GEL: Gel= 1.03 g/cm3 and A=0.130 • Small protuberance in the CMC gel layer (0.32 s). • Appearance of two gel protuberances which fill up with the denser fluid (0.60 s). • Breakup of one of the protuberances through a turbulent plume (1.01 s). • Simultaneous growth of the plume and the protuberance which is emptying and distorting the CMC gel layer at the same time. New turbulent plumes begin (1.61 s). t= 0 s t= 0.24 s t= 0.32 s t= 0.64 s t= 1.00 s t= 96.70 s BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  12. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS PARTIAL MIXING PROCESS WITH A BIDIMENSIONAL PLUME ARRAY AND A GEL LAYER • Frame sequence of an experimental mixing process whose experimental characteristics are: Gel=1.02 g/cm3, μGel= 16000 cps and A=0.019 . • Start of the time evolution of the mixing process with several turbulent plumes which are clearly separated (t= 0.28 s). • Vertical development of the plumes. There is no protuberance in the CMC gel layer (t= 0.40 s). • The lateral interaction of turbulent plumes starts while they are growing (t= 0.56 s). • The lateral interaction between turbulent plumes is greater than in (c) (t= 0.80 s). • The mixing convective front evolves through the light fluid layer. General interaction between plumes (t= 1.44 s). • Non uniform evolution of the mixing convective front. The interaction of the fluid system with the contours of the container starts (t= 2.36 s). • A gravity current develops through the light fluid layer. This gravity current reaches the front of the experimental container (t= 4.76 s). • The mixing process fills the volume of the experimental container. Incipient formation of the mixed layer (t= 11.96 s). • Final state after the partial mixing process. We can observe the mixing layer limited by the stable density interface (t= 94 s). BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  13. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS • These figures show a turbulent mixing process with the most viscous CMC gel (μGel=44000 cps) and the Atwood number is A= 0.130. And another mixing process with the less viscous gel (μGel=16000 cps) and the Atwood number is A= 0.134. Both Atwood numbers are almost equal because we want to just describe the gel effect. ATWOOD NUMBER INFLUENCE GEL INFLUENCE As the gel viscosity is reduced, the probability of initial generation of gel protuberances is reduced and the formation of turbulent plumes increases. We observe that the number of turbulent plumes is greater if the Atwood number grows which implies that there is a greater quantity of mixed fluid. THE BEHAVIOUR OF THE FLUID SYSTEM IS INFLUENCED BY SEVERAL FACTORS ATWOOD NUMBER, A INITIAL POTENTIAL ENERGY GEL VISCOSTY, G NUMBER OF PLUMES, nP BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  14. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS THE BEHAVIOUR OF THE FLUID SYSTEM IS INFLUENCED BY SEVERAL FACTORS ATWOOD NUMBER, A INITIAL POTENTIAL ENERGY GEL VISCOSTY, G NUMBER OF PLUMES, nP INFLUENCE ON FRACTAL STRUCTURE ? INFLUENCE ON THE OVERALL MIXING MIXED LAYER HEIGHT, hM RELATION? MIXING EFFICENCY,  BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  15. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS TO STUDY BETTER THE FOLLOWING RELATIONSHIPS: ATWOOD NUMBER, A MIXING EFFICENCY,  INITIAL POTENTIAL ENERGY MIXED LAYER HEIGHT, hM NUMBER OF PLUMES, nP INFLUENCE ON THE FRACTAL STRUCTURE ? NEW EXPERIMENTS WITH A LINE PLUME ARRAY AND WITHOUT A GEL LAYER BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  16. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS EXPERIMENTAL SETUP FOR EXPERIMENTS WITHOUT A GEL LAYER BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  17. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS EXPERIMENTAL SETUP FOR EXPERIMENTS WITHOUT A GEL LAYER • If there is not a gel layer, there is not a random initial distribution of plumes. • We don’t use the viscoelastic gel because we want to control the number of plumes and their geometric configuration into a line array: • one plume; • two plumes; • three plumes; • ...... • ...... • and nine plumes. • We can control the geometric setup of the plumes by means of the orifices located at the bottoms of the two metacrylic boxes. • If there is not a gel layer, the mixing efficiency and the mixed layer height hNM are greater. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  18. 4 3 2 1  Vp • INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS THE TWO METACRYLIC BOXES WHOSE ARE ALTERNATIVELY PIERCED WITH ORIFICES WHOSE POSITION CAN BE REGULATED. These boxes contain the fluid of greater density D which constitutes the “dense layer” and is coloured with sodium fluorescein (a passive tracer). PUSH WITHDRAWAL VELOCITY OF THE PLASTIC, Vp THE BOTTOM HOLES OF THE METACRYLIC BOXES ARE NOT SUPERIMPOSED: CLOSED POSITION THE BOTTOM HOLES OF THE METACRYLIC BOXES ARE SUPERIMPOSED: OPENED POSITION BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  19. 5 6 4 3 2 1 5 6 4 3 2 1 • INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS DIFFERENT HOLE GEOMETRIES DIFFERENTS PLUME GEOMETRIES 1 PLUME GEOMETRY 9 PLUME GEOMETRY ONE OPEN HOLE (IN GREEN) NINE OPEN HOLES (IN GREEN) BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  20. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS GLOBAL MIXING RESULTS: MIXED LAYER HEIGHT Behaviour of the non dimensional height of the mixed layer with the Atwood number for experiments made with the most viscous CMC gel (Curve 1, ●), and with the less viscous one (Curve 2, ). The figure shows the linear fits done. • The mixed layer height hMwas measured experimentally and itis directly proportional to the final quantity, or volume, of the mixed fluid. The volume of the mixed fluid increases as the Atwood number grows and, consequently, the height of the mixed layer, hM, is greater. In other words, as the buoyancy effect increases so does the convective turbulent mixing and the mixed layer height. • The effect of the gel viscosity may also be observed in this figure. The height hMincreases if the gel used is the less viscous one because the number of turbulent plumes is greater when the gel viscosity is reduced. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  21. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS GLOBAL MIXING RESULTS: MIXING EFFICENCY Mean mixing efficiency versus the Atwood number for experiments made with the most viscous CMC gel (Curve 1, ●), and with the less viscous one (Curve 2, ). The corresponding empirical fits are shown. • The mixing process is only partial and we can analyze the mixing efficiency , which is defined as the fraction of the available energy used to mix fluids: • We observe that the efficiency increases as the Atwood number A does which, physically, implies that the buoyancy effect grows and it produces a greater mixing process with a greater efficiency. • Besides,Curve 2 shows that mixing efficiency is greater forh the less viscous gel if we compare it to experiments made with the most viscous gel (curve 1). BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  22. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS GLOBAL MIXING RESULTS: MIXING EFFICENCY Behaviour of the mean mixing efficiency versus the Atwood number considering the mixed layer homogeneous (Curve A,) and stratified with two layers (Curve B, ) corresponding to experiments made with the most viscous CMC gel. • The final mixed layer is stratified because the final density profiles show a strong density step, and, therefore we assume that the stratification of this mixed layer is made up by two layers. • If the final profile is stratified, then the mixing efficiency is about 0.17 and it has an upper limit of 0.18. • 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. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  23. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS GLOBAL MIXING RESULTS: MIXING EFFICENCY OUR MIXING EFFICIENCY IS ABOUT 20% OF THE MAXIMUM MIXING EFFICIENCY IN SIMILAR EXPERIMENTS THE EFFECT OF THE TURBULENT PLUME ARRAY THE EFFECT OF THE GEL VISCOSTY, G THE GEL LAYER REDUCES MIXING EFFICIENCY ABOUT 40% IF WE COMPARE IT TO EXPERIMENTS WITHOUT GEL. DYNAMICS OF THE TURBULENT PLUMES THE NUMBER OF THE PLUMES BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  24. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS GLOBAL MIXING RESULTS: MIXING EFFICENCY Mean mixing efficiency  versus the Atwood number A for experiments made with the most viscous CMC gel (μGel= 44000 cps, Curve 1, ), with the less viscous one (μGel= 16000 cps, Curve 2, ●) and without gel (Curve 3, ). • This figure shows that the gel layer reduces mixing efficiency about 40% if we compare it to experiments without gel. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  25. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS THE EFFECT OF THE TURBULENT PLUME ARRAY DYNAMICS OF THE TURBULENT PLUMES The two-dimensional plume array makes a conical volume without mixing as the figure shows because once the dense fluid looses its potential energy it may not mix with the lighter fluid above. There is an interpenetration of the unstable plumes only through a fraction of the area at the top. Therefore, the denser fluid and the lighter one do not mix completely. This non-mixing volume makes the mixing efficiency decrease. All turbulent plumes feed on the ambient light fluid. As a consequence there is a height h which represents the start of plume lateral interaction which determines the non-mixing volume. The start time of lateral interaction ranges from 1.042 s for a two plumes experiment to 0.336 s for a nine plumes experiment. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  26. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS THE EFFECT OF THE TURBULENT PLUME ARRAY Vertical development of the turbulent plumes with a superimposed scheme which represents their initial growth. Every plume is represented by a cone which radius is the plume radius, R. The lateral interaction between plumes starts at a depth h. The non-dimensional mixing volume • Represents the decrease of the mixing volume if the height ratios h/hLincreases because plumes reach a larger depth without interacting. The greater mixing volume appears as soon as the plumes interact. The non-dimensional mixing volume varies from 86% if the height ratios is (h/hL)=1/5 to 66% if (h/hL)=1/2. • The non-mixing volume V*NON-MIXINGhas the opposite behaviour which influences the mixing efficiency. • We demonstrate that the dynamical behaviour of plumes reduces the mixing efficiency because they generate a smaller volume useful to mix. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  27. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS THE EFFECT OF THE TURBULENT PLUME ARRAY THE NUMBER OF TURBULENT PLUMES The second hypothesis to understand our mixing efficiency values is that the mixing efficiency increases if the number of turbulent plumes is greater. To verify this hypothesis, we perform new experiments with a line of plumes –from one to nine plumes- as described before. The new experiments are performed without using the viscoeleastic gel because we want to control the number of plumes and their geometric configuration into a line array. We want to investigate more in depth the relation between the non-mixing volume V*NON-MIXING, the number of plumes nPand the mixing efficiency and to evaluate the result: the lower the gel viscosity, the higher the mixing efficiency is.If the gel viscosity is reduced, the number of plumes npincreases. Therefore, it might exist a relation between the mixing efficiency and the number of plumes. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  28. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS THE EFFECT OF THE TURBULENT PLUME ARRAY THE NUMBER OF TURBULENT PLUMES The mixing efficiency  is relatedto the mixing volume V*MIXING and is also related to the inverse of the non-mixing volume V*NON-MIXINGwhich can be represented by the non-mixing height, hNM. For these reasons, we measure, directly from the digitalisations of the experiments, the nonmixing height and we relate it to the number of plumes. The new results are shown in this figure. Graphic behaviour of the non-mixing height, hNM, versus the number of plumes, np. If the number of plumes, np, is greater, the non-mixing height decreases and, therefore, the non-mixing volume also decreases. Then the mixing efficiency is greater which agrees with the results deduced before. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  29. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL ANALYSIS • Fractal studies provide a natural method for analyzing turbulent fields like plumes and their turbulent cascade processes. • If there is a subrange where production and dissipation are at equilibrium, it is possible a functional relation between the exponent  of the spectral density functionand the fractal dimension D of the scalar field represented in the images: • The last aim is to investigate the intermittency of the mixing plumes (measuring the maximum fractal dimension and using results of another researchers relating to the sixth and third order structure function scaling exponents). BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  30. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL ANALYSIS • We investigate the fractal structure of non homogeneous plumes affected by different levels of buoyancy (different values of the Atwood number A), initial potential energy (several initial heights Ho of the source) andfor different number of plumes, np (from one to nine). • Fractal characterization of dispersing plumes like scalar concentration fields is imperfect but is a preliminary step toward a general multifractal description.Fractal dimensions between 1.3 and 1.35 are obtained from box counting methods for free convection and neutral boundary layers. Other results have been published which use the box counting method to analyze images of jet sections –produced from LIF techniques- and determined that the fractal dimension of jet boundaries was 1.36 • The fractal and multifractal analysis of the turbulent convective plumes was performed with the box counting algorithm for different intensities of evolving plume images using the special software Ima_Calc. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  31. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL RESULTS: 1 PLUME EXPERIMENT t= 0.168 s t = 0.168 s t= 0.262 s t= 0.420 s t= 0.748 s t= 1.0 s BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  32. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL RESULTS: 1 PLUME EXPERIMENT • Time evolution of a plume through its frame sequence corresponding to experiments made with A=0.03. • The first column of the figure shows the time evolution of the one plume experiment with its time frame. • The second column is the selected region of interest. • The third column represents the corresponding histogram which allows us to define the intensity or grey level range to study. • The fourth column is the fractal dimension or the plot of N(d) versus d. • Finally, the fifth column shows the multifractal results, i.e, the change of the fractal dimension related to the grey level. • The relation: • is used to determine the fractal dimension D (box-counting dimension) of the plume boundary by a regression line fit through the box-counting results. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  33. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL RESULTS: 1 PLUME EXPERIMENT This curve represents the time evolution of the fractal dimension D corresponding to one plume experiment with Atwood number A=0.03. At early stages, the fractal dimension has large changes. Later, it tends towards a value between 1.2 and 1.3. The study indicates a mean fractal dimension of 1.23 for the one plume experiment. As the turbulent plume is evolving, we can do a multifractal analysis. This figure shows the time evolution of the multifractal results for the same plume with Atwood number A=0.03 We can observe there is a similar behavior at all selected times. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  34. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL RESULTS: 9 PLUME EXPERIMENT • To verify if the number of plumes np affects the global mixing effiency  and the fractal dimension,we perform experiments with a plume geometric setup into a line –from one to nine plumes-. • Time evolution of nine plume experiment through its frame sequence corresponding to experiments made with A=0.03. • There is a lateral interaction between plumes as they evolve. As a consequence, it appears a joined convective front which time evolution is showed in the first column of the following figure. • The second column represents the selected region of interest of the front. • The third column is the corresponding histogram which allow us to define the intensity or grey level range to study. • The fourth column shows the results of the multifractal analysis of the interest region which shows the behavior of the fractal dimension versus the intensity level. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  35. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL RESULTS: 9 PLUME EXPERIMENT t = 0.336 s t = 0.504 s t = 0.630 s t = 0.706 s t = 0.850 s BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  36. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS FRACTAL RESULTS: 9 PLUME EXPERIMENT This figure shows the time evolution of the fractal dimension D associated to the convective front. Before 0.336 s there is no fractal dimension because there are individual turbulent plumes and not a convective front. The front fractal dimension has great changes and it is not clear it tends towards a limit. Non Convective Front The study indicate a mean fractal dimension of 1.082 for the nine plume experiment. As the front grows, this figure represents the time evolution of the front multifractal results. The behavior is similar at all times with a nearly plane region from lower intensities to 180 grey level. Afterwards, there is a decrease of the fractal dimension at (180, 220) grey range and, finally, it increases towards a maximum at higher intensities. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  37. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS CONCLUSIONS • To properly understand atmospheric and oceanic turbulence, for example, a deep understanding of the mixing processes is first required. • The global conclusions of this experiment are related to the mixing efficiency and the volume of the final mixed layer as functions of the Atwood number, the gel viscosity and the number of plumes. • We have verified that the initial conditions modify the overall mixing efficiency, i. e., the number of plumes affects the mixing efficiency because if the number of plumes decreases, the mixing effiency also diminishes because the non-mixing height increases. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  38. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MIXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS CONCLUSIONS • We compare the fractal results corresponding to one plume and nine plume experiments with the same Atwood number A= 0.03. First, the mean value of the fractal dimenson for the convective front of the nine plume experiment (1.082) is lower than the mean value of the one plume setup (1.23) which it is closer to the results of other researchers. • We also can compare the multifractal results. There is a clear difference at lower intensities (below 180) because the fractal dimension of the one plume experiment has not a nearly plane region. Later, the multifractal behaviour is more similar because it increases in both experiments. • Finally, we can compare the time evolution of the fractal dimension D. As mentioned before, the fractal dimension corresponding to the one plume experiment tends towards a limit value. This behaviour is not the same for the convective front of the nine plume experiment which fractal dimension changes. BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

  39. INTRODUCTION AND AIMS • EXPERIMENTAL SETUP • PLUME ARRAY MXING PROCESSES • GLOBAL MIXING RESULTS • FRACTAL ANALYSIS • FRACTAL AND MULTIFRACTAL RESULTS • CONCLUSIONS IF YOU WANT MORE INFORMATION: P. López González-Nieto Dpto. Física de la Tierra, Astronomía y Astrofísica II Avda. Ciudad Universitaria s/n. 28040 Madrid maplopez@bio.ucm.es – azufre2@hotmail.com  913945072 BUOYANT MIXING PROCESSES AND STRUCTURE IN TURBULENT JETS

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