BUOYANT MIXING PROCESSES AND FRACTAL STRUCTURE IN TURBULENT PLUMES

1
BUOYANT MIXING PROCESSES AND FRACTAL STRUCTURE IN
TURBULENT PLUMES
INTERNATIONAL SUMMER COURSE ON NON-HOMOGENEOUS
TURBULENCE
2

1. INTRODUCTION AND AIMS
2. EXPERIMENAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 ?L and 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 hD and 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 dont 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 A0.134
• 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 1.01 s
t 1.61 s

• PARTIAL MIXING EVENT WITH THE MOST VISCOUS GEL
  ?Gel 1.03 g/cm3 and A0.130
• 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 ?Gel1.02
  g/cm3, µGel 16000 cps and A0.019 .
• 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. GLOBAL MIXING RESULTS

1. FRACTAL ANALYSIS
2. FRACTAL AND MULTIFRACTAL RESULTS
3. CONCLUSIONS

• These figures show a turbulent mixing process
  with the most viscous CMC gel (µGel44000 cps)
  and the Atwood number is A 0.130. And another
  mixing process with the less viscous gel
  (µGel16000 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. GLOBAL MIXING RESULTS

1. FRACTAL ANALYSIS
2. FRACTAL AND MULTIFRACTAL RESULTS
3. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. GLOBAL MIXING RESULTS

1. FRACTAL ANALYSIS
2. FRACTAL AND MULTIFRACTAL RESULTS
3. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 dont 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. GLOBAL MIXING RESULTS

1. FRACTAL ANALYSIS
2. FRACTAL AND MULTIFRACTAL RESULTS
3. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 hM was measured
  experimentally and it is 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 hM increases
  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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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/hL increases 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 VNON-MIXING has 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 VNON-MIXING, the
number of plumes nP and 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 np increases. Therefore, it might exist a
relation between the mixing efficiency and the
number of plumes.
BUOYANT MIXING PROCESSES AND STRUCTURE IN
TURBULENT JETS
28

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 related to the mixing
volume VMIXING and is also related to the
inverse of the non-mixing volume VNON-MIXING
which 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 function and 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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) and for
  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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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
  A0.03.
• 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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 A0.03. 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 A0.03 We can
observe there is a similar behavior at all
selected times.
BUOYANT MIXING PROCESSES AND STRUCTURE IN
TURBULENT JETS
34

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. GLOBAL MIXING RESULTS

1. FRACTAL ANALYSIS
2. FRACTAL AND MULTIFRACTAL RESULTS
3. 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 A0.03.
• 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MIXING PROCESSES
4. 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

1. INTRODUCTION AND AIMS
2. EXPERIMENTAL SETUP
3. PLUME ARRAY MXING PROCESSES
4. 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_at_bio.ucm.es azufre2_at_hotmail.com ?
913945072
BUOYANT MIXING PROCESSES AND STRUCTURE IN
TURBULENT JETS