Thesis Final Julio 15 Emil

8/10/2019 Thesis Final Julio 15 Emil

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UNIVERSITAT POLITCNICA DE CATALUNYA

Departament de Fsica Aplicada

THE STRUCTURE OF TURBULENT JETS:

APPLICATION OF EXPERIMENTAL AND

ENVIRONMENTAL METHODS

EMIL SEKULA

Barcelona, July 2010

Director: Prof. Jos Manuel Redondo Apraiz

Codirector: Prof. Ana Mara Tarquis Alfonso


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Dedicated to my girlfriend, Amaia and to my parents, Zbigniew and Lucyna.


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Gratitude

To Professor Jos Manuel Redondo Apraiz for his constant help and ideas in the process of

redaction of this study. His impressive scientific experience in such difficult field as

turbulence had important influence on the presented work. His friendliness helps me during

these years not only in the research life.

To Professor Ana Mara Tarquis Alfonso being the coordinator of this thesis.

To Dr. Joan Grau for his help in the field of programming language and for offering

facilities of the Ima_Calc program.

To Dr. Alexei Platonov for the collection of Synthetic Aperture Radar (SAR) images and

his commentaries related to the satellite images analysis.

To Dr. Raffaele Marino and Dr. Luca Sorriso-Valvo for their cooperation.

To Prof. Allen Bateman for make accessible some useful data in present work and for his

cooperation.

To Departament de Fsica Aplicada de la Universidad Politcnica de Catalunya for facilities

founded during my academic period.

I would like to thank to all persons not mentioned personally here but having some

influence on this thesis.

Finally, but not meaning with less importance, I would like to thank to my girlfriend Amaia

for her invaluable support in both, research and personal life and to my parents, Zbigniew

and Lucyna for priceless help during all these years of my youth.

Thank you!


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INDEX

GRATITUDE

CHAPTER 1. INTRODUCTION 1

1.1 State of the art 6

1.2 Thesis aims 9

1.3

Thesis structure 11

CHAPTER 2. BACKGROUND AND THEORETICAL CONSIDERATIONS 13

2.1 Turbulence and turbulent flows 13

2.2 Homogenous turbulence 15

2.3 Non-homogenous turbulence 23

2.4 Extended Self Similarity (ESS) 24

2.5 2D Turbulence 25

2.6 Turbulent jets 28

2.7 Buoyant plumes 32

2.8 Plane turbulent free jet 38

2.9 Circular turbulent jet 48

2.10 Plane turbulent wall jets 54

2.11 Other jet and plume configurations 62

2.12

The turbulent boundary layers 66

2.13

Geophysical turbulence 70

2.14 Mixing efficiency 71

CHAPTER 3. THEORY SAR AND FRACTALS 74

3.1 SAR (Synthetic Aperture Radar) 74


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3.1.1 SAR applications 77

3.2 Fractal analysis 80

3.3

Self-affine fractals. Relationship between turbulence spectra and

fractal dimension 82

3.4 The spectralmethod 84

3.5 Fractal characteristics of intermittent turbulence 85

3.6 Box-Counting Method 87

3.7 Multifractal characterization 88

CHAPTER 4. METHODOLOGY AND EXPERIMENTAL SETUP 92

4.1 Introduction 92

4.2 Acoustic Doppler Velocimeter (ADV) 93

4.3 WinADV 98

4.4 Laser Induced Fluorescence (LIF) and Planar Laser Induced

Fluorescence (PLIF) 99

4.5 Particle Image Velocimetry (PIV) 102

4.6 DigiFlow 104

4.7 Particle Tracking Velocimetry 106

4.8 ImaCalc 106

4.9 Experimental setup 109

CHAPTER 5. EXPERIMENTAL RESULTS 117

5.1

Lower Reynolds number jet experiments 117

5.1.1 Velocity profiles 117

5.1.2 Jet velocities 118

5.1.3 Reynolds number 119

5.1.4 Standard deviation (r.m.s. turbulence) 121

5.1.5 Turbulence intensity 124

5.2 High Reynolds number wall jets experiments 126


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5.2.1 Velocities 126

5.2.2 Turbulence and mean velocity parameters 127

5.2.3 Reynolds number 129

5.2.4 Standard deviation 129

5.2.5 Skewness 131

5.2.6 Kurtosis 132

5.2.7 Correlation 133

5.2.8 Covariance 134


5.2.10 Other results 1365.3 High Reynolds number, two-phase Bubble jet 137

5.3.1 Velocities 137

5.3.2 Standard deviation 138

5.3.3 Skewness 139

5.3.4 Kurtosis 140

5.3.5 Correlation and Covariance 140


5.3.7 Other results 141

5.4 Data stability 142

5.5 Instrumental error 144

5.6 Comparison of the three cases of the jet configurations 144

CHAPTER 6. SPECTRAL MEASUREMENTS, STRUCTURE FUNCTION

TECHNIQUES AND SCALES ANALYSIS 147

6.1 Energy spectrum Fast Fourier Transform (FFT) 147

6.1.1 Normal jet, y = 4 cm case 149



6.1.4 Bubble jet, y = 4 cm case 150



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6.1.7 Energy spectrum results discussion 152

6.2 Structure function analysis 159







6.2.7 Extended Self Similarity (ESS) 1636.3 Correlation and scales 168

CHAPTER 7. SAR AND EXPERIMENTAL IMAGES FRACTAL RESULTS 175

7.1 Introduction 175

7.2 River jets SAR images 175

7.3 Sea vortex analysis 179

7.4 Experimental jet results 182

7.4 Fractal dimension analysis conclusions 190

CHAPTER 8. CONCLUSIONS 192

8.1 Future works 197

REFERENCES 199

APPENDIX 1 207

APPENDIX 2 225


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1

CHAPTER1

INTRODUCTION

Scientific work on turbulence is difficult but fascinating, this field is still not totally

discovered and exist many unsolved and open problems. From the philosophic point of

view, a goal of every study should be based on the most possible conclusions drawn from

assumptions even they finally result to be erroneous. This falsity give us some idea and

serve as additional information for future works, it is like successive step to discover the

totality of existing problem. Before the detailed study is presented in this thesis we

familiarise with existing hypotheses and assume and test our own ones: then the next step is

to prove their correctness remembering that is not in the scientific spirit to think that our

final ideas are absolutely correctly and irrefutable. So to present this thesis we should ask

some basic questions: what is an object of our problem, why do we want to study it and

how we will conduct our work. In this chapter we will answer these questions and finally

specify the detailed objectives and general purposes of this study on the structure of

turbulent jets and plumes.

The basic and overwhelming, research problem of this study is based on the

understanding of real turbulence and its structure. Turbulence is a phenomenon that can be

found anywhere, in every field of life, for example, from the stirring of a coffee cup to the

wind in the atmosphere. Most practical flows occurring in nature and in engineering

applications involve non-homogeneous turbulent flows that are affected by boundaries or

body forces such as: currents below the surface of the oceans, the gulf stream as a turbulent

wall-jet kind of flow, the boundary layer in the earths atmosphere, jet streams in the upper

troposphere, strong cumulus clouds and deep turbulent convection in the ocean, boundary

layers growing on aircraft wings and propellers, combustion processes and mixing in

turbulent chambers, wakes of ships, cars, submarines and aircrafts. There are many more

existing examples. In our definition we can observe closely to one of the first feature of the

real turbulent flows, which has not been studied extensively: non-homogeneity. In physics,

homogeneity is the quality of having all properties independent of the position, i.e.


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translational invariance. Obviously in non-homogeneity these properties depend on the

position.

The study of turbulence clearly is an interdisciplinary activity. It is important to know

that turbulence is not a feature of fluids but of fluid flows. If the Reynolds number is large

enough, the major characteristics of turbulent flows are not controlled by the molecular

properties of the fluid in which the turbulence occurred. Great progress has been made in

the last century (since Kolmogorovs work K41 and K62 theories (1941, 1962)) on the

structure and theory of homogeneous and isotropic turbulence, but non-homogeneous or

boundary affected flows still lack a comprehensive theory. Andrey Kolmogorov was aSoviet mathematician who made major advances in different scientific fields (among them,

the probability theory, topology, intuitionistic logic, turbulence, classical mechanics and

computational complexity). Kolmogorov is widely considered one of the prominent

mathematicians of the 20th century. On the later part of scientific life he switched his

research interests to the area of turbulence, where his 1941 (K41) works had significant

influence on the field. In classical mechanics he is best known for the KAM theory

(KolmogorovArnoldMoser theorem).

Turbulent flow is the motion of a fluid having local velocities and pressures that

fluctuate randomly. In this movement subcurrents in the fluid display turbulence, moving in

irregular patterns, while the overall flow is in one direction. Turbulent flow is common in

nonviscous fluids moving at high velocities. Almost all flows, natural and man-made, are

turbulent. Occurrence of turbulent flows in many situations forces to next works on this

subject.

Tennekes and Lumley (1972)proposed a list of some basic characteristics of turbulent

flows:

Irregularity (or randomness): the turbulent flow is unpredictable.

Diffusivity: which cause rapid mixing and increased rates of momentum, heat and

mass transfer.


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Mentioned before high Reynolds number

Three-dimensional vorticity fluctuations: turbulence is rotational and three-

dimensional.

Dissipation: turbulent flows are always dissipative.

Continuum: turbulence is a continuum phenomenon, governed by equations of fluid

mechanics.

Flows: turbulence is a feature of fluid flows and not of fluids as was commented

before.

Diffusivity is the diffusion coefficient. It is proportionality constant between the molar

flux due to molecular diffusion and the gradient in the concentration of the species (or the

driving force for diffusion). The higher the diffusivity (of one substance with respect to

another), the faster they diffuse into each other. This coefficient has the units of

(length/time).

Reynolds number [for Osborne Reynolds] is dimensionless quantity associated with the

smoothness of flow of a fluid. It is an important quantity used in aerodynamics and

hydraulics. At low velocities fluid flow is smooth, or laminar, and the fluid can be pictured

as a series of parallel layers, or lamina, moving at different velocities. The fluid friction

between these layers gives rise to viscosity. As the fluid flows more rapidly, it reaches a

velocity, known as the critical velocity, at which the motion changes from laminar to

turbulent with the formation of eddy currents and vortices that disturb the flow. The

Reynolds number for the flow of a fluid of density ; and dynamic viscosity through a

pipe of inside diameterDis given by

DU=Re (1.1)

where Uis the flow velocity.


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If the Reynolds number is not too large, the flow will be laminar. At higher Reynolds

number, the flow becomes chaotic in both space and time. The critical Reynolds number for

laminar flow in cylindrical pipes is about 1000.

It is not easy to solve turbulence problems because in fact it is mixed, random and not

regular process and it has not been possible to find a complete theory that describes the

phenomenon till now. There is not still a set of equations that could be used to efficiently

compute turbulent flows. The complexity of turbulence is also related to a large number of

scales of the flow playing an important role. The fundamental dynamical equations that

govern turbulent flow are the Navier-Stokes equations; their computational complexity

becomes hard task for large Reynolds numbers. It is a system of related non-linear partial

differential equations and must be supplemented by initial and boundary conditions and

always faced with the closure problem (a set on n-1 equations with at least n unknown

variables in it). Computational Fluid Dynamics tries to resolve this problem using one-

equation models; the two equation models (for example, k model) and the second-order

closure models. There are some models existing for some specific flows because computers

have recently become more and more powerful but they are not universal.

In 1922 Richardson proposed fully developed turbulence as a hierarchy of eddies of

different size. He assumed a cascade process of eddies breaking down. At eddies of sizeL

energy is injected, then energy is transmitted to smaller and smaller eddies and finally it is

dissipated in small eddies of scale where a viscosity plays a dominant role. A central role

in this scheme plays the mean rate of energy transfer per unit mass. The Navier-Stokes

equation, which describes the evolution of the velocity field vof the fluid, is:

fvpvvt

v rrrrr

++=+

21

(1.2)

where is the mass density, p is the pressure and the dynamic viscosity. The term

vv rr

is the nonlinear term and implies a breaking of bigger eddies into smaller eddies, the

term vr2

is the viscous term and represents a dissipation of kinetic energy as internal

energy of fluid, while the term fr

represents the external forcing acting on the fluid.


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Kolmogorov in 1941 studied (based on Richardsons cascade idea) fully developed

turbulence (turbulence which is free to develop without imposed constraints). Works of

Kolmogorov are mentioned in every standard textbook on turbulence or fluid mechanics.

Kolmogorov made assumption that turbulence should exhibit universal and isotropic

statistics for scales smaller than the integral scale L. Moreover, for scales larger than the

Kolmogorovs scale , the viscosity should play no dynamical role. There is a range of

length scales called the inertial range, in which the flow statistics are expected to be

universal, isotropic and independent of the viscosity. Since then extensive experimental and

numerical studies have attempted to describe the statistical behaviour of fully developedturbulence. Landau was the first to point out that the Kolmogorovs theory (K41) could not

be true because he did not take into account intermittency. Landau stated that the energy

dissipation displays important fluctuations about its mean value. A consequence is that

Kolmogorovs theory must certainly be corrected in order to contemplate this intermittency

character. Taking note of Landaus suggestions, Kolmogorov and Obukhov introduced a

refined similarity hypothesis called log-normal model. In this version (K62) they assumed

that the energy dissipation is log-normally distributed.

After the refined similarity hypothesis, different types of intermittency models were

proposed to describe the turbulence cascade and particularly the behaviour of scaling

exponents (Frish 1995). The success of these models can be evaluated especially on the

basis of experiments and this is next reason why is so important to do experiments in this

field. However, there are no models that agree with all experiments, although each model

works quite well within a limited group of available data. For that reason it is not possible

generally recommend one model over the others. Some of the most popular models for

fully developed turbulence are:

She-Leveque model

p-model

-model or

random-model


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Turbulent jet is a turbulent, coherent stream of material ejected from a nozzle into a

surrounding medium, or a nozzle designed to produce such a stream. Presented study refersto a fluid jet particularly. The subject is the concern of engineers in calculating pipe flows,

jets or wakes, for examples.

The mechanics of the turbulent jets, although studied during the last decades, still is a

paradigm of flow behaviour, together with wakes and boundary layers and it is great

interest of researchers. In recent years, thanks to improved remote sensing we can observe

concentration on the environment, for example dilution and mixing of pollutants in water

bodies at many scales. For these reasons we need continue an investigation work in three

scientific fields: experimental, environmental and numerical to obtain a greater knowledge

of the existing research problems.

The turbulent wall jet configuration occurs often in many environmental and industrial

processes. The most popular applications come from the fields of aeronautics design, heating,

cooling, ventilation and environmental fluid dynamics.

There are many books precisely describing problems of turbulence and turbulent

phenomenon. We mention in Appendix 1 some more complete and used during the

evolution of this study.

1.1 State of the art

Apart from the well-known and basic information in the major books about turbulence, a

large amount of work and specific publications refer to this subject and in particular to the

structure of wall jets, both in laboratory conditions and in environmental relevant flows.

We will discuss here the wide range of specific publications used in this thesis, either fromexperimental, technical aspects of measurement technique, in classical jet and plume theory

or in image processing or structure function and spectral techniques.

All the most important steps in turbulence of last century are summed up in the

publication of Lumley and Yaglom (2001). It is a brief, superficial survey of some very

personal points of view of the statistics and most important works of the last hundred years

in turbulence research, few clear conclusions are reported.


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The most general treatment of multi-scale and fractal analysis of fluid turbulence is done

by, a well-known author in this area, the director of the Trieste Physics Institute:

Sreenivasan (1999). A few aspects of turbulence research in the last century are briefly

reviewed and a partial assessment is made of the present directions. There are two possible

scenarios. Our computing abilities may improve so much that any conceivable turbulent

problem can be computed away with adequate accuracy, so the problem disappears in the

face of this formidable weaponry. The other scenario which is common in physics is that a

particular special problem that is sufficiently realistic and close enough to turbulence will

be solved in detail and understood fully. There is well-developed body of knowledge in

turbulence that is generally self-consistent and useful for problem solving. However, thereare lingering uncertainties at almost all levels.

The phenomenology of small-scale turbulence is done by Sreenivasan and Antonia

(1997).Small-scale turbulence has been an area of especially active research in the recent

past and several useful research directions have been pursued. The authors selectively

review this work.

There are many possibilities of jets configurations from the basic ones to more

complicated. Here and in Appendix 1 we comment most relevant publications used in the

preparation of this thesis in the different research lines, as examples for wall jet discussion,

the paper by Craft and Launder (2001) explores, using different levels of turbulence

closure, the computed behaviour of the three-dimensional turbulent wall jet in order to

determine the cause of the remarkably high lateral rates of spread observed in experiments.

Their computations confirm that the strong lateral spreading arises from the creation of

streamwise vorticity, rather than from anisotropic diffusion. The driving vorticity source is

created by the anisotropy of the Reynolds stresses in the plane perpendicular to the jet axis

rather than to the bending of mean vortex lines. It may be observed too, since the three-

dimensional wall jet is an acutely sensitive flow for assessing turbulence models; it would

be desirable to establish definitive experimental or, possibly, les results for the fully

developed limit.

A morespecificreport on turbulence and mixing in geophysical flows was presented in

the book by Redondo and Linden (2000) Turbulent mixing in geophysical flows and in

Redondo and Linden (1998). In this report, the papers of workshop on Mixing in


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Geophysical Flows in 1997 are summarized giving a state-of-the-art overview of present

research in geophysical turbulent mixing. The main topics discussed are stratified flows,

rotating stratified flows, gravity waves, instabilities and mixing, convection, experiments

and numerical simulations of geophysical flows and turbulent mixing. The mixing

processes are the key mechanisms for mass and momentum transfer in the oceans and

atmosphere and play a crucial role in determining the environmental conditions in which

mankind lives and operates. The approach almost universally adopted is to isolate one or

two particular processes and study these in detail using a combination of theoretical,

numerical, experimental and observational techniques.

More advanced data techniques and more sophisticated analysis leading to a betterunderstanding of the scale to scale transfer of energy may be obtained through the study of

the higher order structure functions and the intermittency. The problem of non-

homogeneous turbulence was investigated in this department (UPC), among others by

Mahjoub (2001). In this work a classification is proposed to determine the intermittency

and mixing ability. The variation of the structure functions and the scaling exponent in

decaying non-homogeneous turbulence produced by a grid and by a jet is measured with

sonic velocimetry, hot film and wire probes. In Mahjoub et al. (1998) the investigation

shows the advantage of using Extended Self Similarity (ESS) and also that in most cases

that 3 (absolute scaling exponent) is not one in complex non-equilibrium flows and

depends strongly on the separation distance, except near the source of turbulence at high

Re. The BDF model, after Babiano, Dubrulle and Frick (1995, 1997) was used to check

their experimental results and relate the values of the relative scaling exponents. The

different flows studied show different distributions of intermittency, which was defined in a

more general way. Mahjoub et al. (2001) presented in an environmental coastal flow a

statistical analysis of velocity structure functions for turbulent flows at 1m above the

bottom in a shallow (2m) bay in Denmark. High frequency (25 Hz) time series were

collected in different days. For the two times series studied, there was not clear inertial

range when the absolute scaling component had a scale-independent behaviour. The authors

also used the ESS concept to measure the relative scaling exponents. ADV measurements

of waves and turbulence captured the peaks in wind and current motions in the bay. The

difficulty in calculation of higher order structure functions in short time series lead us to


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design specially quite long stable measurement periods of about 10 minutes (in some cases

up to 4 x 104data points).

Intermittency was investigated by Anselmet et al (2001)in laboratory experiments with

turbulent flows, also Jou (1997) and Schertzer investigate multifractal cascade and

turbulence intermittency. The paper of Jou provides prelude to the consequences of

intermittency on the statistical properties of fully developed turbulence, mainly on scaling

laws for the different moments of velocity, energy distribution and diffusion behaviour. The

description of intermittency is carried out in the fractal model and in a more general

multifractal perspective. He emphasized that the energy transfer between eddies of different

scales strongly fluctuates from place to place, yielding intermittent bursts of turbulence.The active regions occupied by the eddies do not fill the whole volume but only a subregion

of it, which in the simplest model may be characterized by a fractal dimension D 2.87.

The intermittency modifies the energy spectrum by attributing less energy to small scales in

comparison with that predicted by the standard Kolmogorov distribution; and it enhances

diffusion with respect to the usual Richardson law of non-intermittent turbulence. More

detailed analyses of intermittency must focus on the exponents of the scaling laws for the

different moments of the velocity. We strived to improve on the diagnostic methods in

order to study the structure of turbulence, in Appendix 1 a list of specific papers that show

the state of art of the present line of research is discussed, a recent update on environmental

turbulence may be found in the 31st issue of Nuovo Cimento C.

1.2 Thesis aims

Studies of the behaviour of relatively basic kind of jets (the free turbulent jet, wall jet,

buoyant jets and plumes) are an important basis for understanding more complex

configurations. Although the investigation of the above mentioned kinds of flow has been

done for many years aiming at the mean flow predictions, there are still some important

problems unsolved in the behaviour of the turbulent cascades and their structure because of

the past limits of measurements methods, Launder and Rodi (1983) or List (1982) .


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We aim to understand the behaviour of turbulent jets in a deeper and more detailed way

incorporating the recent advances in non-homogeneous turbulence, structure function

analysis, multifractal techniques and extended self-similarity. Part of this thesis is based on

extending the measurements on turbulent structure on flows with similar wall-jet

configurations, but main aim is to get new results on the detail structure of the non-

homogeneous turbulent cascade processes and thus complement previous experiments,

which were mostly concern with mean structure and global fluxes.

One of the parts of this work includes multi-fractal analysis of SAR (Synthetic Aperture

Radar) and experimental jet (and plume) images. Scaling analysis allows us to investigatethe structure of ocean surface detected jets (SAR), to compare coastal and boundary effects

on the structure of river jets and to investigate the turbulent and fractal structure of non

homogeneous jets affected by different levels of turbulence upstream and downstream. An

innovative technique used to investigate the turbulent interactions within the inverse and

direct cascades near jets is to measure the spectral and fractal structure of the non

homogeneous jets and develop multi-fractal techniques which we believe will be useful for

environmental and industrial monitoring.

The present work is based on experimental and environmental jets configurations

showing similarity of the occurring phenomenon at different scales. Presented thesis has

been performed with experimental technique available at this time in the laboratory of Fluid

Dynamics of the UPC.

Experimental techniques develop very fast so we can use the new technology that will

increase our knowledge, even repeating some classical experiments under new light and

improved techniques. Moreover application of different experimental techniques affirm or

not a usefulness showing advantages and faults of each one and allows us to confirm the

previous results. Using on the same experiments more than one method of diagnostic

permits us to improve the understanding of the laboratory techniques; this is also usually an

important argument for research group. This thesis was performed in the laboratory

equipped, among others with Laser Induced Fluorescence (LIF), Particle Image

Velocimetry (PIV), Particle Tracking Velocimetry (PTV) and Acoustic Doppler


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Velocimeter (ADV), these are very useful velocimetry methods measuring velocity of

fluids and used to solve fluid dynamics problems or to study fluid networks. Industrial and

process control applications or the creation of new kinds of fluid flow sensors are

advantages of these methods also.

It was also very useful for the development of this thesis the research period spent at the

Czech Technical University, being able to work at the Department of Technical

Mathematics.

Here we present the general and specific objectives of this thesis:

Understanding of the dynamics of non-homogenous turbulent motions (jets, plumes,

vortices)

Comparison of the experimental end environmental features of the jets (different

Reynolds numbers and scales)

Application and evaluation of the Acoustic Doppler Velocimeter in laboratory

experiments

Use of multifractal analysis of the experimental and environmental images of jets

and plumes

Statistical analysis of the experimental results in the turbulence phenomenon point

of view

Effect of the boundary layer in wall jet configuration

1.3 Thesis structure

The present thesis is divided into 8 chapters and 2 appendices. We start with a general

introduction to turbulence and describe the basic information about the thesis subject,

discussing the background of the thesis and mentioning previous works leading to

Appendix 1 on a description of the state of art and the most relevant bibliography.

Chapter 2 refers to all theory in order to familiarize a reader with the work subject. This

chapter is divided into different parts, explaining turbulence and turbulent flows theoretical

considerations, focusing special attention on non-homogenous turbulence, extended self


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similarity, 2D turbulence, completing with information about various configurations of

turbulent jets and plumes.

Chapter 3 explains theory about Synthetic Aperture Radar (SAR), fractal analysis,

relationship between turbulence spectra and fractal dimension and description of the

different fractal methods.

Chapter 4 shows used experimental configurations with description of the applied

methods and techniques. Different laboratory configurations are explained.

In Chapter 5 experimental ADV results are presented for different configurations

including data stability and instrumental errors. Some conclusions are drawn.

Separate part (Chapter 6) of this thesis is related to spectral measurements, structurefunction techniques and scale analysis results including final conclusions.

Chapter 7 contains SAR and experimental images fractal results for river jets, ocean

vortices and other sea surface structures.

In Chapter 8 we present overall discussion and conclusions and possible future works.

Additionally, the used bibliography is quoted in alphabetical order.

Two appendices are attached as supplementary information to give more detailed list of

existing works on the subject (Appendix 1) and utilized during the elaboration of this study

and description of BDF method (Appendix 2) to calculate intermittency.


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CHAPTER2

BACKGROUND AND THEORETICAL CONSIDERATIONS

2.1 Turbulence and turbulent flows

A general introduction on turbulence and turbulent flows was explained in Chapter 1.

Here we complete the specific and necessary theory about this thesis, more detailed theory

is available in any of the mentioned bibliography about turbulent jets and mixing.

As we mentioned before, it is very difficult to give a precise definition of turbulence.

There are many efforts to define it considering different aspects, as an example, proposed

Tennekes and Lumley (1972).

Some of the characteristics are: turbulent flows are always dissipative. Viscous shear

stresses perform work, which increases the internal energy of the fluid at the expense of

kinetic energy of the turbulence. Turbulence needs a continuous supply of energy to make

up for viscous loss. If energy is not supplied, turbulence decays rapidly. The majordifference between random waves and turbulence is that waves are non-dissipative (though

they are often dispersive), while turbulence is essentially dissipative.

Figure 2.1 shows some examples of turbulent flows occurring in real life.


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Figure 2.1 Some examples of turbulent flows. Turbulence in the tip vortex from an airplane

wing (top-left) (source: Langley Research Center of the United States NASA).Turbulent

flow around an obstacle (top-right) (courtesy Wikipedia) and laminar and turbulent water

flow over the hull of a submarine (below) (source: Wikipedia).

Despite the importance and abundance of turbulent flows, the community of scientists

as Reynolds (1883) or Richardson (1922, 1929) has encountered many difficulties in

developing a satisfactory scientific general theory. One of the most important steps in our

understanding came in 1941 when Kolmogorov developed his theory (K41) (Kolmogorov

1941) about how the energy that is put into large turbulent motions cascades down to very

small scales where it is converted into heat by viscosity.


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The cascade theory of energy proposed Kolmogorov in 1941. This energy cascade we

can observe in Figure 2.2 where, kis the wave number equal to 2/l, lis length scale,Lis

the largest scale and U is velocity.

a b

Figure 2.2 Energy spectra (a) and cascade of energy by Richardson and Kolmogorov (b).

The energy comes from big to small whirls and without any source of energy,

turbulence decay quickly.

Real turbulent flows such as geophysical flows are non-homogenous and non-isotropic.

In this work, we present the theories and assumptions related to both (with homogenous

turbulence) but more emphasis will lay on true-life cases.

2.2 Homogenous turbulence

Kolmogorov described his model for fully developed, locally homogenous and isotropic

turbulence using the concept of stationary and continuous energy cascade process. There

are many intermittency models prominent in such a simple turbulent regime based on

characterization of the random but homogenous nature of the energy dissipation field.

These models have limitations in real life, environmental and industrially relevant flows

where the turbulence is usually non-homogenous, non-isotropic and non-stationary.


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In 1942 Kolmogorov introduced his theory (K41) for locally, homogenous, isotropic

and stationary turbulence using velocity structure functions. The velocity structure

functions of orderpare defined in terms of the moments of velocity differences as:

( ) ( ) ( )( ) ( )qlq

q uxulxulS =+= r

r

r

(2.1)

where stands for ensemble average and uis the velocity component parallel to lr

.

Kolmogorovs theory is based on the following similarity hypothesis:

For all distances llr

= small compared with integral scaleL, l


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In this model, the energy is transferred to small scales in steps. At eddies of size L

(scales are of the order of the flow width, contain most of energy and dominate the

transport of momentum, mass and heat) energy is injected, then energy is transmitted to

smaller and smaller eddies, until it is dissipated into heat at smallest eddies of size (small

scales responsible for most of the energy dissipation).

The essential hypothesis of this model is that for high Reynolds numbers Re and l

smaller than the integral scale Land larger than the Kolmogorov scale , the longitudinal

velocity structure functions satisfy the relation:

( ) ( ) 3/qpq lClS = (2.2)

where Cp is universal constant.

There is an exact dynamical relation for the third order longitudinal velocity structure

function, which can be derived from the Navier-Stokes equations for homogenous and

isotropic turbulence. For incompressible turbulence, when 0= u , then we have the

famous 4/5th

law of Kolmogorov:

lul5

43 = (2.3)

The scaling relationship for structure functions in range between and L plays an

important role in experiments, for example in fixing the extent of the inertial range and in

estimating energy dissipation rate per unit mass , in turbulent flows:

2

15

=

x

uisotropic

(2.4)

valid only for locally isotropic flows Hinze (1975).

It can be shown that the structure functions in homogenous, stationary and isotropic

turbulence which is in local equilibrium have a scaling behaviour:


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18

( ) qllSq

(2.5)

where the scaling power qis usually called the scaling exponent of the structure function

of order q. For the Kolmogorov theory (K41):

3

qq = (2.6)

indicating that the scaling exponents of structure functions of order qare scale-independent

and universal quantities. It was then assumed that there is a dynamically determinate scale

which can be constructed just from the average rate of energy dissipation and the

kinematic viscosity as:

4/13

=

(2.7)

This length-scale which would represent the smallest eddy size not damped by

dissipation is called the Kolmogorov length-scale. Similarly we can rewrite Kolmogorovs

time and velocity scales defined as:

( ) 2/1 =k (2.8)

( ) 4/1=kv (2.9)

Hence, the local Reynolds number with reference to the two scales and vk is equal to one:

1=

kv (2.10)

Many experimental studies have been done to verify the relation for scaling exponent of

the structure function of orderppredicted by Kolmogorov (1941), specially for the density

of turbulent energy per unit of mass at scalel forq= 2


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19

( ) ( ) 3/21' lClE = (2.11)

and for its spectral equivalent

( ) 3/53/2

2' = kCkE (2.12)

Here, C1 and C2 in principle, are universal constants if the full Kolmogorov K41

hypothesis is met, lmust be within the inertial subrange and k=2/l is the corresponding

wave number. Many works shows the approximately value of these constants, named also

after Kolmogorov.

This Kolmogorov relation basically predicts that (if Re is large enough) the energy

spectrum of fully developed homogenous turbulence is divided into three distinct wave

number regions (Figure 2.4):

1. The region of energy injection at largest scales

2. Inertial range where the energy is transmitted from large to small scales

3. Dissipative range, where the energy is dissipated by viscosity into heat from the

small scales, which are compared to the Kolmogorov length scale .

Scheme of energy spectrum as a function of the wave number scale is shown below

(Figure 2.4).

Figure 2.4 The shape of energy spectrum as a function of wave

number (modification of Redondo notes image).


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(Probability Distribution Functions) of the velocity increments ul to the moments of l.

Then, the corresponding prediction for the qth-order moment of the velocity incrementul

as function of the scale separation l is usually formulated as,

q

lu ~3/3/ qq

l l ~ ql

(2.13)

Considering that there is a scale dependence of the dissipation as:

3/q

l ~

3/ql

(2.14)

now the scaling exponents are:

3/3/ qq q += (2.15)

where l is the locally defined energy dissipation per unit mass over a volume of size

llr

= centred at space-position xr

, 3/q is the scaling exponent of3/q

l , refers to

averaging over all positions vectors xr

and q= 1,2 is the order of the statistical moment.

In a self-similar situation corresponding to a homogenous and isotropic turbulence

characterized by a scale uniform random dissipation field, 1 = 0 and the correction 3/q for

3q in relation (2.15) is only induced by the intermittency correction. Then, this relation

guarantees also the basic results 3 = 1 for locally homogenous and isotropic turbulence.

Many experimental studies have been done to verify these relations predicted by

Kolmogorov. Here we present homogenous intermittency model called model

introduced by Frisch (1995). The idea behind this model comes from the Richardson

cascade, in which at each level of the cascade the energy of the large eddies Lis uniformly

distributed over the other, eddies of size las

n

n Lll = (2.16)


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where n= 0,1,2.. and 0 < l < 1.

We can define the energy per unit mass on scales lnas

D

nnnnn

L

lvvPE

=

3

22 (2.17)

The parameterDis called a fractal dimension.

In this formPnis the fraction of decreasing of the eddies, which has factor(0


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3

3

3

1 Dh

= (2.22)

and the structure function of orderpis written as

( )q

L

lvvPlS n

q

L

q

nnnq

= (2.23)

with

( )

+=

313

3qDqq (2.24)

The energy spectrum is given as

( )

+

33

3

5 D

kkE (2.25)

which is derived as a correction to the k-5/3law of the Kolmogorov theory (K41).

More different intermittency models are explained in Mahjoub (2000).

2.3 Non-homogenous turbulence

In the previous sections we explained that in recent years many efforts have been made

to explain the intermittency phenomenon, particularly in homogenous flows. Some models

are mentioned in Chapter 1 of this thesis. In non-homogenous flows which are more

complex and have more practical interest, less attention has been given to the study of non-

local dynamics which seems separated from intermittency and also seems to play an

important role in non-homogenous turbulence.

There are many models concerning the energy cascade scale to scale processes and the

dissipation rate, see Frisch (1995) for an account.


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Kolmogorov and Obukhov in 1962 introduced the refined version of Kolmogorovs

similarity hypothesis taking into account intermittency. They assumed that for locally

homogenous and isotropic turbulence, the energy dissipation field strongly fluctuates in

both space and time. On the other hand, the average amplitude of the dissipation random

field scales quite uniformly in both space and time. In contrast, when the turbulence is non-

homogenous and non-isotropic, the dissipation random field is non-uniform in scale. This

means that the fluctuations and the amplitude of the variance of the energy transfer are

scale dependent quantities. In this case, the correction q/3 of relation (2.15) is associated

with both the intermittency phenomenon linked to the rarest events and the anomalous

dependence as a function of the length scale of transfer properties in the energy cascadescales. Therefore, this relation is not strictly valid, because q are anomalous and scale

dependent.

2.4 Extended Self Similarity (ESS)

The Extended Self Similarity (ESS) is technique which is a key way to analyze

homogenous and non- homogenous flows. It is a property of velocity structure functions ofhomogenous and non-homogenous turbulence. Instead of obtaining scaling exponents in

the usual way by plotting structure functions of the absolute velocity incrementsq

lu

against l,we plot them against the third-order structure function of the absolute velocity

increment3

lu and then we have:

q

lu ~3/3

q

lu (2.26)

where q/3is a relative scaling exponent and q is defined by

q

lu ql

(2.27)


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where, q is now the absolute scaling exponent and may be different from q for odd values

of qbecause absolute values of velocity increments are used.

Some limitations of ESS were mentioned by different authors. They pointed out that the

ESS does not seem to work when the shear is strong, such as in the shear behind a cylinder

and in boundary layer turbulence. In contrast, some authors found that the ESS also works

well in these situations even. This suggests that the ESS may be also a specific and

convenient tool to analyze non-homogenous turbulence.

The use of3

l

u instead of 3l

u in ESS may be physically explained because it

refers to the scale by scale absolute balance of transferred energy at a given scale l and

includes both energy transfers from larger to smaller scales (normal cascade) and the

anomalous energy transfers from smaller to larger scales (inverse cascade). This fact

suggests that the ESS relation must be expressed in term of3

lu .

Some intermittency models for non-homogenous case are explained in Mahjoub (2000)

special attention focus on BDF model.

2.5 2D Turbulence

In the non-homogenous case, the energy spectrum can be steeper than k-5/3

and saturates

to k-3

. This behaviour is illustrated in Figure 2.6,which shows clearly the transition from

homogenous and local dynamicsk-5/3

to non-local and non-homogenous dynamics k-3

. For

example, Babiano demonstrated in his numerical study for 2D non-homogenous turbulence,

that the spectral slope increases with degree of non-homogeneity and can reach up to k-3

.


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Figure 2.6 The evolution of energy spectrumE(k) from non-local to local dynamics

(Mahjoub 2000).

On the other hand, the scaling exponent of the second order velocity structure function

2shows an important deviation from Kolmogorovs 2/3 prediction.

Kraichnan (1975) set forth, called the energy-enstrophy model, which was based on

spectral truncation of the underlying fluid dynamics equations, leading to the two-

dimensional turbulent energy-enstrophy cascades as an extension of Kolmogorovs K41

and K62 theories. He predicted an inverse cascade of energy (Figure 2.7) in two-

dimensional Navier Stokes fluid turbulence and proposed an inertial range with a k-5/3

power-law energy spectrum, just as in three-dimensional turbulence, but with a flux of

energy from small to large scales rather than the reverse. It is one of the most important

phenomena in fluid dynamics occurring, for example, in the atmosphere or ocean

In agreement with Kraichnan (1967) the energy and enstrophy conservation make that

energy actually flow to larger scales and this is a basic difference between 2D and 3D

turbulence (where energy flows toward small scales in a direct cascade). The enstrophy can

be interpreted as the quantity directly related to the kinetic energy in the flow model that

corresponds to dissipation effects in the fluid. It is defined as:


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2

2

1r

=Z (2.28)

where urr

= is vorticity.

Figure 2.7 The inverse energy cascade characterized by ( )35k in the power spectrum.

It is impossible to find in nature strictly 2D case. It exists only for numerical

simulations or for theoretical considerations.

Theoretically it is possible to resolve The Navier-Stokes equations from the 2D point of

view, where the flow velocity has only two components. The equation of the vorticity has

form,

+

=

+

+

2

2

2

2

yxyv

xu

t (2.29)

where, simplified z= .

The relation between mean enstrophy and energy spectrum is

( ) ( ) ( )

==0 0

2 ,, dktkdktkEktZ (2.30)

where (k,t) is the enstrophy spectrum.


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We consider a turbulent flow stationary, where the energy is injected by force f(k)

around ki, then the kinetic energy injection

( )

=0

dkkf (2.31)

and the injection of enstrophy

( )

=0

22 ikdkkfk (2.32)

Kraichnan propose that if the Reynolds number is sufficiently high, there is a wave

number range ki< k


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Turbulent jets are fluid flows produced by a pressure drop through an orifice. Their

mechanics, although studied for over fifty years, has recently received research attention that

has resulted in a much-improved understanding of the process by which they entrain

surrounding fluid.

Examples of turbulent jets are presented in Figure 2.8.

Figure 2.8 Different examples of turbulent jets. Vorticity magnitude at Reynolds number

3960 (source: Laboratory for Aero & Hydrodynamics Delft University of Technology)

(left). A simulation of a fully turbulent jet flow by David Glase (Purdue University)(right).

The greyscale contours represent mixture fraction. The lower half of the plot shows

velocity vectors, colored and scaled by their relative magnitude.

There is now overwhelming evidence that the initial growth of turbulence is a direct

consequence of large-scale motions generated at the jet boundaries. These large-scale motions

are primarily responsible for jet noise production and the initial entrainment of ambient fluid.The basic sequence for axisymmetric jets seems to be as follows: in the immediate

neighbourhood of the orifice, the high-speed jet flow causes a laminar shear layer to be

produced. The shear layer is unstable and grows very rapidly; forming ring vortices that carry

turbulent jet fluid into the irrotational ambient fluid and irrotational ambient fluid into the jet.

It is clear that the production of vortices is the key element in initial jet dilution. Each vortex

wraps ambient fluid about itself, then, as the vortices pair, the fusion process mixes the


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ambient and jet fluid. Circumferential instability, and possible interaction with helical modes,

leads to the apparent eventual self-destruction of the large-scale structures and this, in turn,

generates the subsequent small-scale turbulent mixing.

Different configurations of turbulent jets are shown below (apart from the further, more

detailed cases).

The radial turbulent jet is shown in Figure 2.9.

Figure 2.9 Definition sketch of radial turbulent jet (Rajaratnam 1976).

The compound jet is presented in Figure 2.10.

Figure 2.10 Definition sketch for compound jet (Rajaratnam 1976).

The confined jet is demonstrated in Figure 2.11.


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Figure 2.11 Definition sketch for confined jet axisymmetric case (Rajaratnam 1976).

The jet in cross-flow is shown in Figure 2.12

Figure 2.12 Definition sketch of circular jet in cross-flow (Rajaratnam 1976).

The radial wall jet is presented in Figure 2.13.

Figure 2.13 Radial wall jet (Rajaratnam 1976).

The plane compound wall jet is demonstrated in Figure 2.14.


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Figure 2.14 Definition sketch of a plane compound wall jet (Rajaratnam 1976).

The average equations of motion of several cases will be discussed in next sections.

2.7 Buoyant plumes

A plume (in hydrodynamics) is a column of one fluid moving through another.

Turbulent plumes are fluid motions whose primary source of kinetic energy and momentum

flux is body forces derived from density inhomogeneities. Several effects control the motion

of the fluid, including momentum, diffusion, and buoyancy (for density-driven flows).

When momentum effects are more important than density differences and buoyancy effects,

the plume is usually described as a jet. Usually, as a plume moves away from its source, it

widens because of entrainment of the surrounding fluid at its edges. Plume shapes can be

influenced by flow in the ambient fluid (for example, if local wind blowing in the same

direction as the plume results in a co-flowing jet). This usually causes a plume which has

initially been 'buoyancy-dominated' to become 'momentum-dominated'. This transition is

usually predicted by a dimensionless number called the Richardson number. The

Richardson number is the dimensionless number that expresses the ratio of potential to

kinetic energy

2u

ghRi= (2.35)

wheregis the acceleration due to gravity, ha representative vertical length scale and ua

representative speed.

A further phenomenon of importance is whether a plume has laminar flow or turbulent

flow. Usually there is a transition from laminar to turbulent as the plume moves away from


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its source. This phenomenon can be clearly seen in the rising column of smoke from a

cigarette.

When high accuracy is required, computational fluid dynamics (CFD) can be employed

to simulate plumes, but the results can be sensitive to the turbulence model chosen.

Plumes have not been studied in the same detail as jets but nevertheless there have been

some recent gains in the understanding of their mechanics because they are of considerable

importance in the dispersion of air pollution. Some examples of different plumes are shown

on Figure 2.15

Figure 2.15 Different examples of plumes. Plume of the Space Shuttle Atlantis after launch

(left) (source: NASA APOD), industrial air pollution plumes (centre) (source: Wikipedia)

and large fire induced convection plume (right) (source: Wikipedia).

Basic to the understanding of all free turbulent flows is the process of entrainment

(Turner 1973) or mixing of outside fluid into the plume. It is observed that (like jets)

turbulent plumes have a sharp boundary separating nearly uniform turbulent buoyant fluid

from the surroundings. This boundary is indented by large eddies and the mixing process

takes place in two stages, the engulfing of external fluid by the large eddies, followed by

rapid smaller scale mixing across the central core. The vertical velocity and the turbulence

measured at a fixed point of the axis have an intermittent character. Though an

instantaneous profile across a plume is sharp-edged, the time averaged profile of velocity is

mother and can be well fitted by Gaussian curve. A detailed theory of the mechanism of

entrainment has been given by Townsend (1970).


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As Batchelor (1954) pointed out, the increasing vertical flow in a plume also implies

that there is a mean inflow velocity across the boundary which varies as z-1/3

. That is, the

linear spread of radius with height implies that the mean inflow velocity across the edge of

the plume is proportional to the local mean upward velocity.

=

b

rfzFw 1

3

1

3

1

0 (2.36)

=

b

rfzFg 2

3

5

3

2

0' (2.37)

zb = (2.38)

where wis the vertical velocity,

=0

0 '2 rdrwgF (2.39)

0/'' gg= (2.40)

g is the acceleration due to gravity, the difference of density, 0 constant

environment density, r is the radial distance from a vertical line above the source, z

height, f1 and f2 are functions explicated later, b is the radial length scale and is a

constant to be defined for particular profiles.

The actual form of functions f1 and f2 must be obtained using either more detailed

theories (with some questionable assumptions) or directly by experiment. The existing

solutions show that there is a flow from the environment into the turbulent plume, since the

mass flux is clearly increasing with height.

When the simplest entrainment assumption is made, so that the inflow velocity is taken

to be some fraction of the upward velocity, the equations of conservation of mass,

momentum and buoyancy can be reduced to the form

( )wb

dz

wbd2

2

= (2.41)


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( )'2

22

gbdz

wbd= (2.42)

( ) ( )zNwbdz

gwbd 222

' = (2.43)

The velocity w and width bare defined by integrating the mass and momentum fluxes

across the plume:

( )

=0

2 2 rdrrwbw , ( )

=0

222 2 rdrrwbw (2.44)

N2is the square of the local buoyancy frequency and equal to:

( )( )dzdgN // 012 = (2.45)

1 is some standard density in the environment and0 environment density.

When the environment is of uniform density, N= 0, so the third equation is a formal

statement of the fact that the buoyancy flux is constant. The multiplying constants such as

are now in terms of the entrainment constant :

zb 5

6= (2.46)

3

13

1

10

9

6

5

= zFw

(2.47)

3

53

1

10

9

6

5'

= zF

Fg

(2.48)

Other, more general kinds of plumes can be treated by the same method.

The numerical value to be chosen for cannot be obtained theoretically and it must be

taken from laboratory experiments.


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Lets calculate, as an example, two typical length scales for thermal. To gain insight in

the determining parameters, a dimensional analysis is executed, following the approach of

Fischer (1979).

First the volume flux Q, the momentum fluxMand the buoyancy fluxBare defined as:

wDQ 2

4

= (2.49)

QwM = (2.50)

'0QgB= (2.51)

in which wis the initial velocity andg0 the reduced gravity, defined as gg='0 .

Here, =

From these expressions, typical length scales can be derived:

2/1

QLQ = (2.52)

2/1

4/3

B

MLMB= (2.53)

in which LQ represents the length over which the geometry of the injection nozzle

influences the propagation of the thermal and for lengths greater than LMB, the flow is

characterized by the buoyant forcing. For a thermal source buoyancy conservation

depends upon the temperature distribution in the environment and the temperature

dependence of the volumetric coefficient of thermal expansion. For a mass source, the

ambient fluid density must be constant. Thermals are discussed in detail by Turner (1973).

We will look now at the simple dimensional description of plumes using empirical

parameters to scale the volume (or mass) flux Q, the momentum fluxMand the buoyancy

fluxB. In a 3D flow the respective dimensions are:

[Q] =L3T

-1, [M] =L

4T

-2and [B] =L

4T

-3


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but in a two-dimensional flow we have

[Q] =L2T-1, [M] =L3T-2and [B] =L3T-3

We can scale the advance of a plume due to momentum only

3/23/1 tMCz m= (2.54)

or due to buoyancy

tBCz b 3/1= (2.55)

It is easy to see that for distances greater than the jet length

3/2MBCL lm = (2.56)

the plume will behave as a pure buoyancy driven plume. In this exampleMis kept low. Cm,

Cband Clare constants.

Time-dependent plumes and jets with decreasing source strengths are investigated by

Scase et al. (2006) as an example and more examples are mentioned in Appendix 1.

As we told before, some part of this work is related to the multi-phase jet/plumes. As an

example, two-phase plume is shown on Figure 2.16


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Figure 2.16 Definition sketch for a two-phase plume in a crossflow. Socolofsky and Adams

(2002).

It illustrates two effects of separation for a bubble plume in a crossflow. First, at some

height above the source, the crossflow separates the entrained fluid from the rising bubbles.

This occurs as the rise velocity of the entrained fluid decreases with height allowing the

crossflow to have an increasing effect.

Second, as observed, crossflows transport bubbles having different slip velocities

(terminal rise velocities) differentially downstream; this is called fractionation.

Fractionation distributes the buoyancy over an increasing horizontal area with height. We

show here that the crossflow separates the entrained fluid from the bubbles at a discrete

height, hS, below which the bubbles and entrained fluid behave like a mixed, coherent

plume and fractionation is negligible, and above which the separated fluid may be treated as

a buoyant momentum jet.

2.8 Plane turbulent free jet

It is for example a jet of water coming from a plane nozzle of large length into a large

body of water or a jet of air into a large expanse of air. If we use suitable flow visualisation

techniques, we will find that the jet mixes violently with the surrounding fluid creating

turbulence and the jet itself grows thicker. Figure 2.17 shows a schematic representation of the

jet configuration discussed above, which is known as the plane turbulent free jet.


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Figure 2.17 Definition sketch of plane turbulent free jets.

Experimental observations on the mean turbulent velocity field indicate that in the axial

direction of the jet, one could divide the jet flow into two distinct regions.

In the first region, close to the nozzle, known commonly as the flow development region,

as the turbulence penetrates inwards towards the axis or centreline of the jet, there is a wedge-

like region of undiminished mean velocity, equal to U0. This wedge is known as the potential

core and is surrounded by a mixing layer on top and bottom. In the second region, known as

the fully developed flow region, the turbulence has penetrated to the axis and as a result, the

potential core has disappeared.

In the fully developed flow region, the transverse distribution of the mean velocity in the

x-direction, i.e. the variation of uwithyat different sections, has the same geometrical shape.

At every section, udecreases continuously from a maximum value of umon the axis to a zero

value at some distance from the axis. Let us now plot u/umagainsty/b(Figure 2.18). The free

jets have top-hat velocity distributions in the potential core.


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Figure 2.18 Velocity distribution for plane turbulent free jets.

Because a free jet entrains fluid from both sides, it spreads faster, and, therefore, it

centreline velocity decays faster than that for the wall jet in the flow development region near

the nozzle exit.

EQUATIONS OF MOTION

In this section we will show the equations of motion for the plane turbulent free jet. The

Reynolds equations in the Cartesian system are written as:

+

+

+

+

+

=

+

+

+

z

wu

y

vu

x

u

z

u

y

u

x

uv

x

p

z

uw

y

uv

x

uu

t

u '''''1 2

2

2

2

2

2

2

(2.57)

+

+

+

+

+

=

+

+

+

z

wv

y

v

x

vu

z

v

y

v

x

vv

y

p

z

vw

y

vv

x

vu

t

v '''''1 2

2

2

2

2

2

2

(2.58)

and:

+

+

+

+

+

=

+

+

+

z

w

y

wv

x

wu

z

w

y

w

x

wv

z

p

z

ww

y

wv

x

wu

t

w 2

2

2

2

2

2

2 '''''1

(2.59)

The continuity equation is written as:

-3 -2 -1 0 1 2 3y/b

0

0.2

0.4

0.6

0.8

1

U/U

m


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0=

+

+

z

w

y

v

x

u (2.60)

where thex-axis defines the axial direction of the jet, they-axis is perpendicular to thex-axis

and is in the direction of the height of the nozzle and the z-axis is third axis of the co-ordinate

system; u, vand wand u, vand ware the turbulent mean and fluctuating velocities in thex-

,y- andz-coordinatedirections,pis the mean pressure at any point, is the kinematic viscosity, is

the mass density of the fluid and tis the time variable.

Because the mean flow is two-dimensional, w= 0, z / of any mean quantity is zero;

0'' =wu ; 0'' =wv and since the mean flow is steady 0/ = tu and 0/ = tv . Further,since the transverse extent of the flow is small, u is generally much larger than v in a large

portion of the jet and velocity and stress gradients in they-direction are much larger than those

in the x-direction. With these considerations, the equations of motion could be shown to

reduce to the form:

x

u

y

vu

y

uv

x

p

y

uv

x

uu

+

=

+

2

2

2 '''1

(2.61)

yv

yp

=

2

'10

(2.62)

0=

+

y

v

x

u (2.63)

Integrating with respect toyfrom yto a point located outside the jet, differentiating and

substituting, we get:

( )222

2

''''1

vuxy

vu

y

uv

dx

dp

y

uv

x

uu

+=

+

(2.64)

where p is pressure outside the jet.

The last term in the above equation is smaller than the other terms and could be dropped.

Hence we obtain the reduced equations of motion as:


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y

vu

y

uv

dx

dp

y

uv

x

uu

+=

+

''12

2

(2.65)

and:

0=

+

y

v

x

u (2.66)

where p is simply written aspfor convenience. We could rewrite the last two terms as:

( ) ( )t

y

vu

yy

u

y

+

=

+

1

1''

11 (2.67)

where1 and t are, respectively, the laminar and turbulent shear stresses and is the

coefficient of dynamic viscosity. In free turbulent flows, due to the absence of solid

boundaries,t is much larger than 1 and hence it is reason able to neglect 1 and rewrite:

ydx

dp

y

uv

x

uu t

+=

+

11 (2.68)

Further, because in a large number of practical problems the pressure gradient in the axial

direction is negligibly small and also to study the jet under relatively simpler conditions, let usset dp/dx=0. Then:

yy

uv

x

uu t

=

+

1 (2.69)

0=

+

y

v

x

u (2.70)

which are the well-known equations of motion for the plane turbulent free jet with a zeropressure gradient in the axial direction.

The integral momentum equation may be deduced by multiplying now byand

integrating fromy= 0 toy= :

=

+

0 0 0

dyy

dyy

uvdy

x

uu

(2.71)

Let us now consider the different terms of the above equation:


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( )

=

=

0

2

0 0

2

2

1

2

1dyu

dx

ddyu

xdy

x

uu (2.72)

=

=

00 0

0 dy

x

uudy

y

vuuvdy

y

uv (2.73)

since fory= 0; u= um, v= 0 and for y ; u= 0, v= ve

where veis a finite quantity known as the entrainment velocity.

Then:

=0

2

0dyudx

d

(2.74)

The equation tells us that the rate of change of the momentum flux in the x-direction is

zero; that is the moment flux in thex-direction is conserved (or preserved).

If the plane jet is issuing from an orifice of height 2b0with a uniform velocity of U0, for

every unit length of the orifice, the momentum flux2

000 2 UbM = . If we imagine that this

momentum flux is emanating from a (fictitious) line source, located at the so-called virtual

origin from whichxis measured, we have:

=0

0

22 Mdyu (2.75)

The momentum fluxM0is an important physical quantity controlling the behaviour of the

plane jet. It effectively replaces individual values of b0and U0.

Using the integral momentum equation, we will now develop a method of predicting the

variation of the velocity and length scales. For the plane turbulent jet, we have seen that the

velocity distribution in the fully developed region is similar. That is:

( )fuum

=/ (2.76)

where by /=

Let us assume simple forms for umand bas:


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p

m xu (2.77)

qxb (2.78)

wherepand qare the unknown exponents to be evaluated (do not confuse with pressure and

structure function order). Substituting last equations, we get:

02

0

2=

dbfudx

dm (2.79)

wheref2stands for ( )2f . Rewriting:

=0

220 dfbu

dx

dm (2.80)

where

0

2 df is a constant, then:

( ) 02

=mbudx

d

(2.81)

We can say that bum2is independent ofx. That is:

02 xbum (2.82)

THE INTEGRAL ENERGY EQUATION

Let us multiply the first equation of motion by u and integrate it with respect toyfromy

= 0 to =y . We get:

=

+

0 0 0

2 dyy

udyy

uuvdy

x

uu

(2.83)


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Let 2/2uE = , the kinetic energy per unit volume.

dyDt

DEdy

y

Ev

x

Eudy

y

uuvdy

x

uu

dyy

Evdy

y

uuv

dyx

Eudy

x

uu

=

+

=

+

=

=

0 0 0 0

2

0 0

0 0

2

(2.84)

where D/Dt stands for the particle derivative and DE/Dt is the total rate of change of thekinetic energy.

=

=

0 0 0

0 dy

y

udy

y

uudy

yu

(2.85)

yu / is the rate of production of turbulence, by the Reynolds shear stress working on the

mean velocity gradient. We have:

=

0 0

dyy

udy

Dt

DE (2.86)

which says that the rate of decrease of the kinetic energy is equal to the rate at which

turbulence is produced.

For our present purposes, we will rewrite the above equation in a slightly different form.

=

=

=

=

0 0

2

0

2

0

2

0 0

222

222

222

dyx

uuuvdy

u

yvdy

y

Ev

dyx

uuu

u

xdy

u

xudy

x

Eu

(2.87)

Adding the above two expressions:


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=

=

+

0 0 0 0

22

22 udy

u

dx

d

dyu

u

xdyy

E

vdxx

E

u

(2.88)

We could now write:

=

0 0

2

2dy

y

uudy

u

dx

d

(2.89)

We see that the rate of decrease of the kinetic energy flux is equal to the rate at which

turbulence is produced. Using our earlier assumptions we could rewrite:

=0 0

233 '2

dfguudfbudx

dmmm (2.90)

=0 0

333 '2

1 dgfudfbu

dx

dmm (2.91)

Letting:

=0

1

3

2

1Fdf and

=0

2' Fdgf

WhereF1andF2are constants, we could rewrite:

( )

1

2

3

3

F

F

u

budx

d

m

m

= (2.92)

As before, the integral moment of momentum equation may be deduced as follows.

Let us multiply the equations of motion byyand integrate fromy= 0 to = . We obtain

then:

=

+

0 00

1dy

yydy

y

uvydy

x

uuy

(2.93)


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=

=

00

2

0

0 0

2

2

1

2

1

uvdydyyudx

ddy

y

uvy

dyyudx

ddy

x

uuy

(2.94)

Adding:

=

+

0 0 0 0

2 uvdydyyudx

ddy

y

uvydy

x

uuy (2.95)

=

00

11dydy

yy

(2.96)

The integral moment of momentum equation becomes:

=000

2 dyuvdyydyudx

d (2.97)

Substituting

=

00

'' fdbufdfbuv mm (2.98)

where b=db/dx, ( ) muuf /= , dxduu mm /' =

Now:

( ) ( )[ ] bdbJuJbufuuvdy mmm

=0

21

0

'' (2.99)

where ( ) =

01 fdfJ and ( ) =

02 fdJ

We may write:

( ) ( )

=0

2

0

2

2

0

1

2

0

222 '' gdbudJbuudJbbudfbudx

dmmmmm (2.100)

Letting:


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( ) ( ) ,,, 50

24

0

13

0

2 FdJFdJFdf ===

and

=0

6Fgd

whereF3,F4,F5andF6are constants, we have:

( ) 0'' 27262

5

22

4 =++ buFbuuFbbuFbudx

dF mmmmm (2.101)

2.9 Circular turbulent jet

Most jets are generated by flow outlet from a pipe as a point source, like the plume from a

volcano in geophysics, this configuration is described as a circular jet of diameter demerging

from a nozzle with a uniform velocity of U0into a large stagnant mass of the same fluid. If we

observe the jet, we would find that the size of the jet increases steadily as it travels away from

the nozzle as shown in Figure 2.19.

Figure 2.19 Definition sketch of circular turbulent jets.

If we use any time-mean velocity measuring device and measure the variation of the axial

velocity u with the radial distance r at different x-sections, we will obtain an interesting

description of the growth of the jet. We will find that up to section 1-1 (see Figure 2.19), there

is a core of flow with undiminished velocity equal to U0. At section 1-1, the turbulence

generated on the boundaries penetrates to the axis and the mean velocity on the axis begins to

decay withx. The core of fluid with the undiminished velocity is in the form of a cone and is

known as the potential cone or more familiarly as the potential core. This region from the


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nozzle to the end of the potential core is known as the flow development region whereas the

region away from the end of the potential core is known as the zone of fully established flow.

In the region of fully developed flow we find that, at any section, udecreases continuously

from a maximum value of umon the axis to zero for large values of r. On figure 2.20, u/umis

plotted against an undimensionless distance r/b, where bis the value of rand U= Um/2in the

case of the plane jet). This is done for Trupels data.

Figure 2.20 Velocity distribution in circular jets Trupels observations.

It is interesting to find that the velocity profiles are indeed similar.

In order to deduce the equations of motion, we start with the Reynolds equations in a

cylindrical system ( )zr ,, and apply the boundary-layer approximations since the jet

occupies only a small width in the transverse direction. The Reynolds equations in the

cylindrical system for steady axisymmetric flow may be written as:

+

+

+

+

+

=

+

r

v

r

vvv

zv

r

z

v

r

v

r

v

rr

vv

r

p

r

v

z

vv

r

vv

rzrr

rrrrrz

rr

222

2

2

22

22

'''''

11

(2.102)

0 0.4 0.8 1.2 1.6 2

=r/b

0

0.2

0.4

0.6

0.8

1

U/U

m


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+

+

+

+

=+

+

r

vvvv

zvv

r

z

v

r

v

r

v

rr

vv

r

vv

z

vv

r

vv

rzr

r

zr

''2''''

12

2

22

2

(2.103)

+

+

+

+

+

=

+

r

vvv

zvv

rz

v

r

v

rr

vv

z

p

z

vv

r

vv zrzzr

zzzzz

zr

'''''

11 22

2

2

2

(2.104)

and : 0=

+

zr rv

z

rv

r

wherezr vvv ,, are the time mean velocities in the r, andzdirections and ',' vvr and 'zv

are the respective velocity fluctuations. For the circular jets without swirl, 0=v and all the

terms containing v and its derivatives disappear from the above equations. Further, zv >>

rv ; gradients in the radial directions are much larger than those in the axial direction. Viscous

stresses could be assumed to be much smaller than the corresponding turbulent shear stresses

provided that the nozzle Reynolds number is greater than a few thousand. Further, turbulent

normal stresses are approximately equal in the radial and peripheral directions. With these

stipulations, the equations of motion become:

2'1

rvrr

p

=

(2.105)

++

=

+

2'''

''1

zzr

zrz

zz

r vzr

vvvv

rz

p

z

vv

r

vv

(2.106)

0=+

zr rvz

rvr

(2.107)

Integrating, substituting and simplifying in a manner similar to that of the plane jet, we have:

( )''11 zrzzzr vrvrrdz

dp

z

vv

r

vv

=

+

(2.108)


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wherepnow is the pressure outside the jet. For convenience, let us now call the axial distance

xand let the velocity components in the axial and radial directions be uand vrespectively and

let = '' zrvv . With these substitutions, the equations of motion become:

r

r

rdx

dp

r

uv

x

uu

+=

+

11 (2.109)

0=

+

rv

rru

x (2.110)

Since in most of the practical cases dp/dx is almost zero, let us consider zero-pressuregradient flows. As a result gets further simplified to:

r

r

rr

uv

x

uu

=

+

11 (2.111)

These equations are the simplified equations of motion for the circular jet.

In the same way as for other configurations, the integral momentum equation is

described for a circular jet diffusing into a stagnant environment of the same fluid with

zero-pressure gradient; it is easy to see that the momentum flux of the jet in the axial

direction is preserved. We will now develop this criterion in an elegant manner.

Let us multiply last point equations by r and integrate with respect to rfrom r= 0 to

r= . We get:

=

+

0 00

drr

rdr

r

uvrdr

x

uur

(2.112)

0

24

1

24

1

0

0

2

0

2

00

=

=

=

drr

r

urdrdx

ddr

r

uvr

urdrdx

ddr

x

uur

(2.113)


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Hence:

=0

2 02 urdrdx

d

The last equation states that the rate of change of the axial momentum flux in the axial

direction is zero or that the momentum flux in the axial direction is conserved.

Let us now develop equations for the velocity and length scales. Let:

( ) ( )fbrfuu m == // (2.114)

p

m xu (2.115)

and: qxb (do not confuse with pressure and structure function order).

With these substitutions, we have:

=0

222 02 dfbudx

dm (2.116)

and:

022 xbum (2.117)

We need one more equation to evaluate these exponents. We will develop this second

equation firstly by considering the similarity of the equations of motion, secondly from the

integral energy equation and thirdly using the entrainment hypothesis.

For a circular turbulent jet, the integral energy equation is derived as follows.

We have:

=

+

0 00

2 drr

rudr

r

uruvdr

x

uru

(2.118)


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=

=

0 0

2

2

0

2

0

222

1

22

2

1

rdrx

uudr

r

uruv

urdr

xudr

x

uur

(2.119)

Adding, we get:

=

+

0 0

2

0

2

22

2

1 udrur

dx

ddr

r

uruvdr

x

uru

(2.120)

The integrant in the above equation could be recognised as the kinetic-energy flux through

an element ring area. Let us now consider the remaining term:

=

0

22

1

r

urdrdr

r

ru

(2.121)

The integral represents the rate of production of turbulence so now we have the form:

=

0 0

2

22

2r

urdr

urdru

dx

d

(2.122)

This equation states that the rate at which the kinetic-energy flux decreases is equal to the

rate at which turbulence is produced. We have: ( )fuu m =/ and ( ) gum =2/

Substituting these relations and simplifying:

=

0

22

0

323 '21 dgf

bbuudfbu

dxd

mmm (2.123)

Letting:

=0

3

32

1 dfF and

=0

4 dgfF becomes:

( ) 53

4323 / FF

Fbubu

dx

dmm == (2.124)

43,FF and

5F are constants.


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Axisymmetric or axisymmetrical are adjectives which refer to an object having

cylindrical symmetry or axisymmetry. In this case we consider two dimensions (rand x).

For 2-D planar jets the coordinatesxandyare considered.

2.10Plane turbulent wall jets

The turbulent wall jet, even limiting attention to the topographically simple cases well

liked by academics, arguably provides more puzzles for those seeking an ordered set of

rules to describe turbulence than any other class of turbulent shear layer. Formally, we canregard a wall jet as a boundary layer in which, by virtue of the initially supplied

momentum, the velocity over some region in the shear layer exceeds that in the free stream.

Wall jets are of great and diverse engineering importance and engineering applications

often feature 2D or 3D wall jets. The best known and most challenging applications lie in the

field of advanced airfoil design and in problems of heating, cooling or ventilating-areas.

There are major industrial applications such as the film-cooling of the lining walls of gas-

turbine combustion chambers and of the leading stages of the turbine itself. In both cases the

aim is to introduce a cool layer of fluid adjacent to a solid surface in order to provide

protection from a hot external stream. In practice, for reasons of constructional strength, it is

not possible to provide a continuous cooling slot extending over the full lateral extent of the

region to be cooled. A series of short slots or holes is thus employed. One wants, therefore, a

rapid lateral spreading of coolant to fill in the gaps but a low mixing rate in the direction

normal to the surface in order that

Thesis Final Julio 15 Emil

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