Image analysis of turbulent mixingafter a fractal Cantor grid

Richard E RileyMay 2009

Supervisor: Dr F. Nicolleau

Thesis submitted to the University of Sheffield in partialfulfilment of the requirements for the degree of Master of

Engineering

Abstract

The turbulent mixing of a marker species in a flow of water is investigated using

image analysis. A transient mixing regime is established comprising a homogeneous

upstream concentration, interrupted by a grid. The development of the downstream

concentration field is recorded in a sequence of digital images. Grid pairs are tested,

each consisting of a fractal grid in the form of extruded Cantor dust (generations

one to four), and a non-fractal control grid of equivalent flow area. The fractal

dimension of the species interface is calculated over the sequence using the box-

counting method. Statistical algorithms and a method of wavelet analysis are

developed and applied as measurements of mixing. A novel method of species

identification using pixel saturation is employed and is found to produce clearer

results, independant of shadow effects and non-uniform lighting. Flow through

higher generation grids is observed to be near-laminar. Where the flow is turbulent,

no special behaviour is observed in terms of the fractal dimensions of the species

interface. Wavelet analysis alludes to increased energy at the smaller scales of the

fractal grid concentration fields. Improvements to the method are discussed and an

experiment is proposed involving a stationary mixing regime for more appropriate

analysis.

Resume

On etudie le melange turbulent d’une teinture dans un courant d’eau par anal-

yse d’images. Le regime transitoire du melange se compose d’une concentration

homogene en amont d’une grille. Le developpement du champ de concentration

en aval de la grille est enregistre dans une sequence d’images numeriques. L’etude

concerne des grilles par paire; une grille fractale en forme extrudee de la poussiere

de Cantor, et une grille de controle avec une aire d’ecoulement equivalente. On

calcule la dimension fractale de l’interface d’espece a l’aide de la methode de box-

counting. On developpe des algorithmes statistiques et une methode d’analyse

d’ondelette que l’on applique comme mesure de melange. On utilise une nouvelle

methode d’identification, basee sur la saturation de pixel, qui fournit des resultats

i

plus clairs et independants des effets d’ombres ou d’eclairage non-uniforme. On ob-

serve l’ecoulement quasi-laminaire a travers les grilles de generations superieures.

Dans le cas ou l’ecoulement est turbulent, on n’observe pas de phenomene special en

termes de dimension fractale d’interface. L’analyse d’ondelette fait allusion a une

augmentation de l’energie aux echelles inferieures du champ de concentration des

grilles fractales. On propose des ameliorations a la methode ainsi qu’une experience

qui comprend un regime stationnaire de melange pour l’analyse plus competent.

ii

C(x, y) Marker species concentration field -

CoV Coefficient of variance -

D Dimension -

E Exposure -

G(λ) Global wavelet energy as a function of scale -

(m,n) Sampling point coordinates in discretised space -

N Number of covering objects -

r Eddy scale diameter m

Re Reynolds number -

S(m,n) Pixel saturation map -

(x, y) Rectangular coordinates in continuous 2-D space -

S ′ Threshold saturation -

Np Number of sampling points -

λ Scale/wavelet scale length m

iii

Contents

1 Introduction 1

2 Theory 3

2.1 The structure of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 The energy cascade . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Practical applications . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Mixing levels and mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Measuring mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 The wavelet transform . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 The global wavelet energy spectrum . . . . . . . . . . . . . . . . . 11

2.4 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.2 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.3 Box-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Procedure 17

3.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.2 Flow conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.3 Image capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.4 Dye injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.5 The grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Box-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Wavelet analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.3 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 26

iv

4 Results 27

4.1 Statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Global wavelet energy spectrum . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Discussion 32

5.1 Four stages of development . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 Wavelet analysis - the scales of segregation . . . . . . . . . . . . . . . . . 35

5.4 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.5 Grid comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.6 Method review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Conclusion and Recommendations 41

References 44

Appendices 46

A Image sequences 46

B Ancilliary information 55

C Algorithms 57

C.1 Time meta-data extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 57

C.2 Image conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

C.3 Box-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

C.4 Box-counting batch code . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

C.5 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C.6 Global wavelet energy spectrum . . . . . . . . . . . . . . . . . . . . . . . 73

v

Acknowledgements

The author would like to thank the following individuals:

Dr Franck Nicolleau and Dr Andrzej Nowakowski for their continued input and en-

thusiasm throughout the project.

Mr Wayne Oxley for his invaluable contribution to the construction of the apparatus.

Dr Mike Green and his colleagues at the Low Carbon Combustion Centre for the

extended loan of the digital camera.

Mr Malcolm Nettleship for the supply of a flow meter.

vi

1 Introduction

Turbulence is an extremely important phenomenon of fluid dynamics which plays a role in

many industrial processes. The topology of turbulence is of particular interest; the eddies

which characterise turbulent flow exist across a range of physical scales pertaining to the

transfer of energy from decaying larger scales to be dissipated viscously at the smallest

scales. The theory of fractal geometry also describes self-similar repeating scales. The

fractal nature of turbulence was first examined in detail by Sreenivasan and Meneveau

in 1986 [7]. Much subsequent work has been conducted by Catrakis and Bond, most

significantly, for this investigation at least, with regards to the numerical box-counting

method as a tool for calculating the fractal dimension of the turbulent/non-turbulent

interface.

Turbulence is highly dissipative and plays a large role in fluid mixing, which has a

number of important applications such as combustion, chemical synthesis, petro-chemical

production and water & sanitation, as well as environmental problems such as contami-

nant transfer in ground water [8].

This investigation examines the development of turbulence, and its effect on mixing,

by observation of a marker species in a flow of water through a grid. Analysis of the result-

ing concentration field is undertaken using digital image analysis. Box-counting, wavelet

analysis and statistical methods are employed to examine mixing and scale development

in the flow.

Mimicking the fractal nature of turbulence, fractal Cantor grids are compared to non-

fractal control grids of equivalent flow area. The investigation aims to examine the effect

of the Cantor grids on mixing and the development of turbulence. Particular attention

is paid to the fractal dimension of the dye interface, and the statistical properties and

development of scales in the concentration field.

The objectives of the investigation are as follows:

� To progress towards a thorough laboratory procedure which is fully controlled.

1

� Develop methods of analysis which are standardised and comparable across all

results.

� Investigate the effects of fractal and non-fractal grids on mixing and scale develop-

ment.

� Make recommendations for future work.

2

2 Theory

2.1 The structure of turbulence

A turbulent flow is defined by the dominance of inertial forces over viscous forces which

would otherwise tend to damp the response to perturbations in the flow field. The result

is instability of the flow and a sensitivity to small alterations of the boundary conditions.

Turbulence is characterised by a number of important properties.

Randomness in time and space Turbulence is a random process, comprising a large

number of degrees of freedom. The result is unpredictable on a detailed level; however,

the statistical properties are repeatable, allowing analysis on the macro scale [9].

A wide range of scales A turbulent flow may be viewed as a composition of eddies of

different sizes, or scales [12]. Energy cascades from the large driving scales to the smallest

scales, where it is dissipated viscously. These scales are superimposed in time and space,

accounting for the complexity of a turbulent field. A brief look at the non-linear term of

the Navier-Stokes equation indicates the origin of these scales in the mathematical model.

∂

∂tUi +

∂

∂xjUiUj

︸ ︷︷ ︸non-linear

= −1

ρ

∂

∂xiP + ν

∂2

∂xi∂xjUi (2.1)

Highly dissipative Turbulence is highly dissipative in terms of heat as well as energy,

and dispersive in terms of mass transfer, which is beneficial to fluid mixing [9].

Three-dimensional Turbulence is the creation of vortices, which interact in 3-D space

[9].

3

Figure 2.1: An illustration of the energy cascade.

2.1.1 The energy cascade

The flow of energy from large to small scales is known as the energy cascade. Kinetic

energy is introduced at the driving scales which, due to instability, decay into smaller

sub-scales. These sub-scales decay into yet smaller scales as illustrated in figure 2.1. This

process continues until viscous forces become dominant and energy can be dissipated

viscously. This theory was proposed by Richardson (1922) and it would be churlish not

to include his very effective poetic summary of the process:

Big whorls have little whorls,

which feed on their velocity;

and little whorls have lesser whorls,

and so on to viscosity.

The smallest scale present in a turbulent flow field - that at which energy is dissipated

viscously - is known as the Kolmogorov length scale, η, after Andrei Kolmogorov (1941).

Kolmogorov posed that this scale corresponds to a local Reynolds number Reη ≈ 1, based

on the characteristic dimension of the eddy. This signifies the point at which viscous forces

become locally dominant.

It is possible to relate the Kolmogorov scale to the driving scale, L, by considering

the transfer of energy through the cascade.

4

The driving energy of the flow is of the order of the square of its mean velocity.

E = O < u2 > (2.2)

A characteristic time, T , can be defined as follows.

T =L√

< u2 >(2.3)

Therefore, the energy dissipated over time T may be defined as the energy dissipation

rate, ε.

ε =< u2 >

T=< u2 >

32

L(2.4)

At a given scale, r, the energy flux, ε, may be defined.

ε =u3r

r(2.5)

By rearrangement, the velocity at r may be expressed as:

ur =√< u2 >

[ rL

] 13

(2.6)

When Reη ≈ 1 it can be shown that:

[ ηL

] 43

=ν

L√< u2 >

(2.7)

It is therefore possible to estimate the Kolmogorov length scale with some knowledge of

the driving scales. In general, this knowledge is obtained from direct velocity measure-

ments.

5

2.1.2 Practical applications

The dispersive effect of turbulence has implications in mixing. Flow mixing is important

in a number of engineering applications as well as natural processes. Many processes

require the mixing of two or more species. For example, combustion relies on the mixing

of fuel and air, while the production of a soft drink requires the mixing of the ingredients.

Understanding and improving mixing can improve the efficiency of these processes.

This investigation concerns a continuous mixing problem. Continuous mixing offers

some obvious advantages over batch processes. Continuous mixing systems require less

space, are more easily controlled for improved consistency, and require reduced labour

[11].

2.2 Mixing levels and mechanisms

In the most basis case, mixing is passive and does not affect the flow dynamics, which

may be considered separately. Dimotakis [2] labelled this as level 1 mixing.

Level 2 mixing describes a regime where the flow field properties are not independent

of the mixing process. Such mixing may occur between fluids of a different density,

temperature or salinity.

Level 3 mixing occurs when a change in the fluid results from chemical reaction. A

typical example is combustion.

Three basic processes of fluid mixing exist: stretching & folding, diffusion and break-

up [10]. The turbulent mixing of species can be broken down into 3 mechanisms, as

described by Eckart in 1948; entrainment, kinematic stirring (dispersion) and molecular

mixing.

Entrainment At the largest scales of turbulence, fluid in non-turbulent regions is en-

trained into the turbulent flow field.

6

Kinematic stirring Intermediate scales are responsible for stirring (folding and stretch-

ing), thus creating a large species interface and accelerating the mixing process.

Molecular mixing The final stage of mixing occurs at the small scales and is associated

with viscous dissipation and molecular diffusion [1].

Molecular and viscous diffusion rely on a concentration gradient. Fick’s first law states

that the mass flux per unit area associated with molecular diffusion is proportional to

the concentration gradient.

−→JA = −δAB · −−→gradCA (2.8)

The mass flux vector of fluid A per unit area is denoted−→JA, δAB is the coefficient of

diffusion of species A in fluid B, and CA is the concentration of species A. In level one

mixing, viscous diffusion is analogous to molecular diffusion, also relying on a gradient of

concentration [2]. Mass transfer occurs from regions of high concentration to regions of

low concentration, as indicated by the negative sign.

2.2.1 Measuring mixing

One may easily define a mixed and unmixed state, but intermediate states are more

problematic. Kukukova et al. [4] have proposed three dimensions to describe mixing.

During this investigation, the mixing of one marker species is considered. Concentration

of the marker species is defined as C.

Intensity of segregation The extent to which a species can be considered separate

is described by the intensity of segregation. A concentration field with an intensity of

segregation of unity contains only distinct homogeneous regions of a species. A value of

zero indicates that the entire field contains a homogeneous concentration. The intensity

of segregation can be defined as the coefficient of variance,

CoV =

√√√√ 1

Np

Np∑i=1

(Ci − CmeanCmean

)2

(2.9)

7

where Np is the number of sample points and Ci is the marker concentration at point i.

The coefficient of variance does not distinguish between a total absence of a species and

a homogeneous mix of a species.

Exposure Exposure may be considered a description of the length and sharpness of

the species interface. According to Kukukova et al. (2009) it is “analogous to a simplified

calculation of the rate of mass transfer across the interface ”[4], as implied by Fick’s law.

A simplified and normalised calculation of exposure, E, is used in this investigation,

E ∼= 1

Np

Np∑i=1

Na∑j=1

1

2|Ci − Cj| (2.10)

where Na is the number of adjacent sample points (2, 3 or 4 for a square sampling regime)

to the current sampling point i.

Scales of segregation Break-up can be described by the scales of segregation. The

scales of segregation are difficult to define for a large quasi-continuous field such as a

digital image. Wavelet analysis provides a useful tool and is discussed in more detail in

section 2.3.

A priori, in a state of homogeneous concentration (or a ‘perfect’mix), all three dimen-

sions tend to zero.

During this investigation, the concentration field, C(x, y), is represented by a quasi-

continuous saturation map, S(m,n), calculated from the digital images (see 3.2) - all

analysis is applied to this digital representation.

8

2.3 Wavelets

Wavelet analysis allows the decomposition of signal data into space and local scale infor-

mation, which is useful in the analysis of scales and provides a measure of the scales of

segregation in a concentration field. The one-dimensional wavelet transform is introduced

and the global wavelet energy spectrum is defined as a method of consolidating the scale

information into an easily comparable form.

2.3.1 The wavelet transform

The wavelet transform decomposes a function into a combination of wavelet packets which

are translated and scaled versions of a carefully chosen mother wavelet. The wavelet

transform is closely related to the Fourier transform, which decomposes a signal into

trigonometric functions. The translation and scale of the packets provides information

on a function’s scale composition in local space.

The mother wavelet is an oscillating function of finite length, subject to certain con-

ditions [3]. To be oscillatory, the mother wavelet must meet the following condition,

∫ ∞

−∞ψ(y)dy = 0 (2.11)

which implies that it is negative as often as it is positive. The finite energy content of

the function is generally normalised to unity and is verified by:

∫ ∞

−∞|ψ(y)|2dy = 1 (2.12)

The final condition is known as the admissibility condition. This is required to ensure

that the wavelet transform has an inverse and is expressed as,

∫ ∞

−∞

|Ψ(ω)|2|ω| dy = Kψ <∞ (2.13)

where Ψ(ω) is the Fourier transform of the mother wavelet.

9

In this study the popular and purely real Mexican-hat wavelet is used (equation 2.14,

figure 2.2).

ψ(y) =d2

dy2e−

y2

2 (2.14)

Figure 2.2: The mexican hat mother wavelet.

The wavelet transform of a 1-D function f(y) is defined as follows,

f(y, λ) =1

λ

∫f(y′)ψ∗

(y′ − y

λ

)dy′ (2.15)

where ψ∗ is the complex conjugate of the mother wavelet. The transform, f , is a function

of y, a coordinate in 1-D space, and λ, the wavelet scale in the y-dimension.

The wavelet transform is closely linked to the Fourier transform [5], which expresses a

function in terms of trigonometric components, providing information on frequency con-

tent. The oscillations of the trigonometric components continue across all space, causing

a delocalisation of the information contained in the original function. The advantage

of the wavelet transform is that it maintains the locality of the information by using

localised mother wavelets.

Figure 2.3 shows an example one-dimensional function and its wavelet transform.

The wavelet transform is equally applicable in higher dimensions using higher dimen-

sion mother wavelets.

10

(a) (b)

Figure 2.3: a) An example function and b) plot of the wavelet transform coefficients as afunction of space and scale.

2.3.2 The global wavelet energy spectrum

The energy associated with a given scale is described by the global wavelet energy spec-

trum, which is defined as follows.

G(λ) =1

λ

∫f(y, λ)2dy (2.16)

The spectrum expresses the distribution of a function’s energy across scale space,

providing a description of the scale composition of the function.

The wavelet transform contains a large quantity of information. The advantage of the

global wavelet energy spectrum is that it condenses the information into a form that is

often more appropriate for direct comparison.

11

2.4 Fractals

2.4.1 Introduction

The term fractal refers to a geometric shape that cannot be described by traditional

Euclidean concepts. The statistics of a standard geometry may be easily quantified; for

example, the perimeter of a circle is a finite quantity and can be described as a function

of its radius. Fractal geometries do not display such properties. The Koch flake, shown

in figure 2.4, is an example of a fractal.

Figure 2.4: The Koch flake.

The Koch flake is a mathematical fractal comprising infinite repeating scales. Every

edge is interrupted by further repeating detail. The length of the perimeter of the Koch

flake is thus infinite. The area, however, converges to a finite value.

Fractals occur frequently in nature. Unlike mathematical fractals such as the Koch

flake, naturally occurring fractals contain a finite number of scales, eventually becoming

smooth at small scales.

Turbulence exhibits notable fractal behaviour pertaining to the dissipation of energy

through the scales of the energy cascade. The range of scales of the turbulent eddies is

analogous to the repeating scales which characterise fractal geometry. In the study of

turbulence, fractals are useful as standard Euclidean concepts are often inadequate [7].

12

(a) (b) (c)

Figure 2.5Fractals in nature - a) a fern leaf, b) the coastline of Norway, c) the human lung.1

2.4.2 Fractal dimension

Dimensionality is a well known concept of Euclidean geometry and may be extrapolated

to fractals. Fractal dimension provides a measure of fragmentation which is useful in the

study of flow interfaces.

Standard geometric shapes fall into four categories.

Category Dimension, D

point 0

line 1

area 2

volume 3

From these categories, it follows that an object of unit size and dimension, D, may

be covered by N similar objects scaled by factor λ, according to relationship 2.17.

N = λ−D (2.17)

For example, a 2-D square is covered by 4 similar squares scaled by a half, while a

3-D cube is filled by 8 similar cubes scaled by a half (see figure 2.6).

1Images are re-used under the GNU Free Documentation License.

13

Figure 2.6: A demonstration of self-similar object coverage in 2- and 3-D.

By rearrangement, the dimension of the object may be expressed as follows.

D = − logN

log λ(2.18)

Application of this relationship to fractal objects demonstrates an interesting property.

Cantor dust is one of the simplest fractal examples. Starting with a one-dimensional

line of unit length, with each successive generation the middle third of every contiguous

line is removed resulting in lines of one third the length as demonstrated in figure 2.7.

Figure 2.7: Cantor dust.

For Cantor dust, N and λ can be be expressed as a function of the generation n as in

2.19.

Generation, n N λ

0 1 1

1 2 13

2 4 19

N = 2n, λ = 3−n (2.19)

A priori, the fractal dimension of Cantor dust is less than one as subtractions are per-

formed to a line of initially one dimension. Through application of relationship 2.18, the

14

fractal dimension of Cantor dust is found to be 0.63.

DCantor = − n log 2

−n log 3=

log 2

log 3' 0.63 (2.20)

This dimension indicates the extent to which the Cantor dust fills one-dimensional space.

The relationship between the dimension of a fractal object embedded in q-dimensional

space to the dimension calculated from a p-dimensional cross-section can be shown to

follow relationship 2.21 if the section is “independent of the fractal itself ”[7], holding for

any section of an isotropic fractal.

Dq = Dp + (q − p) (2.21)

For example, the true dimension of a 3-D isotropic flow field can be related to the dimen-

sion of a 2-D section, just as the dimension of a square can be related to the dimension

of a cube; by the relationship D3 = D2 + 1.

The fractal dimension of the mixing interface provides a measurement of folding - how

much the interface fills space. A dimension close to 2 of an object embedded in 2-D space

indicates that the object nearly fills that space. If a dimension of 2 is calculated for a

2-D slice of an object embedded in 3-D space, for an iso-tropic fractal, the object has a

dimension of 3; indicating that it also fills 3-D space. Applied the species interface in a

mixing field, it has relevance to measuring mixing.

15

Figure 2.8: A box-counting example.

2.4.3 Box-counting

Box-counting is a numerical method used to calculate the fractal dimension of an object

by exploiting equation 2.18. The object of interest is covered with an array of boxes and

those which contain a portion of the object are counted. With each successive iteration the

size of the boxes is reduced. The number of occupied boxes, N , is plotted as a function of

the reciprocal of the box edge length, 1λ. On a log-log scale, a constant gradient indicates

a continuous distribution of scales and is equal to the fractal dimension.

D =logN

log 1λ

= − logN

log λ(2.22)

Figure 2.8 is an illustration of the box-counting method on the black/white interface

of a binary map.

16

3 Procedure

A flow of marker fluid through fractal and non-fractal grids is observed using digital

photography. The resulting flow patterns are analysed using the box-counting, wavelet

and statistical methods discussed, to examine the effect of the grids on the flow in terms

of fractal behaviour, scale development and mixing.

3.1 Experimental procedure

3.1.1 Apparatus

An experimental apparatus has been constructed comprising a perspex channel, into

which a grid can be fixed. A coloured marker fluid is introduced into a main-flow of

water at the entrance to the channel. Immediately downstream of the grid, the flow is

photographed in a viewing section using a tripod mounted digital camera (see figure 3.2).

The flow channel has a rectangular internal cross-section measuring 162 mm × 20

mm. The upstream section is 1.25 m in length.

Figure 3.1 shows a schematic of the apparatus.

3.1.2 Flow conditions

Flow through the apparatus is supplied by the reservoir and is driven by gravity. The

flow rate must be limited to ensure an established flow before the grid.

For a channel flow, the Reynolds number is expressed as,

Re =ρudhµ

(3.1)

where dh is the hydraulic diameter and is defined as,

dh =4A

P(3.2)

where A is the flow area and P is the perimeter of the flow area.

17

Fig

ure

3.1:

The

exper

imen

talse

t-up.

18

For the purposes of calculation the flow is considered laminar, thus the relationship

between dh and the development length, Le, required for flow to establish is as follows.

Ledh≈ 0.06Re (3.3)

For the available length in the upstream channel, Le = 1.25 m, a maximum Reynolds

number of Remax ≈ 585 is found. This is the around the upper limit for laminar flow and

corresponds to a flow rate of 3.2 l min−1 (∼= 1.6 cms−1 mean velocity in the channel).

To meet the criterion of developed flow at the grid, a flow rate of 3 l min−1 is main-

tained. While the upstream flow is laminar, turbulence is generated by the grid.

3.1.3 Image capture

Compressed JPEG images are taken at the viewing section using a tripod mounted digital

SLR2 camera as shown in figure 3.2. The digital camera specifications are included in

appendix B.1.

Images are captured for a period of around 200 s at the fastest rate obtainable with

the camera. The capture rate slows from an initial 3 frames per second to one frame

every 4 seconds due to time-to-disk limitations. A remote trigger is employed to avoid

movement during image capture.

All images are taken with a shutter time of 1/49 s, an ISO speed of 100 and an

aperture value of 4.97 EV (f/5.6).

Lighting is provided from below using a white laminated sheet as a diffuse background.

The white balance is calibrated using an image taken with no marker species in the flow.

Certain artefacts must be avoided to reduce interference with the image analysis.

Following each experiment, the viewing section is cleaned to remove debris and dye

deposits. As far as possible bubbles are removed from the viewing section prior to the

experiment. A polarising filter and photographic umbrella are used to reduce reflections

2Single Lens Reflex

19

from surrounding light sources.

During analysis the images are assumed to represent a two-dimensional slice of the

concentration field. Furthermore, the assumption is made that the image represents an

orthographic view.

One sequence of images is captured per grid (approximately 70 images).

3.1.4 Dye injection

Water soluble paint is used in the marker fluid. The paint is pre-mixed with water with

a volume ratio of 1:150. The marker fluid is introduced into the main flow by direct

injection using a syphon feed.

The driving head of marker fluid, with respect to the level of the main reservoir, is

fixed across all experiments. The resulting flow rate of dye is close to constant at 1.25 l

min−1. The entire dye tank capacity of 1.5 l is released during an experiment.

Blue dye is used as it is easily distinguishable from the yellow tinted indoor reflections

that are most prominent in the laboratory. This is to allow the option of employing a

hue filter to the image to remove associated artefacts.

A number of injection methods have been considered. Previous work by Brishni

(2008) [15] was carried out using a submersible pump. This method presents a number

of disadvantages: In order to submerge the pump, a large quantity of marker fluid is

required and a significant quantity cannot be delivered. Furthermore, with the pump

deactivated, a syphoning effect leads either to continued dye delivery or draw-back of the

main flow into the dye tank.

A syringe pump system offers accurate flow control and minimal wastage, but limits

the volume of fluid that can be delivered. An expensive compromise is offered by a

peristaltic pump.

The syphon feed is inexpensive and delivers a near-steady flow, flexibility in the quan-

tity of dye to be delivered, and minimal wastage.

20

Figure 3.2: The image capture set-up.

Figure 3.3: Control grid 1 in place with the viewing section detached.

21

3.1.5 The grids

A total of eight perspex grids are tested in four pairs; the fractal Cantor grid and its

corresponding control grid, which has the same flow area but a non-fractal geometry.

The four pairs correspond to extruded Cantor dust generations one to four (see section

2.4.2). Figure 3.4 shows control grids one to four and the fourth generation Cantor grid

with the relevant dimensions. Figure 3.3 shows a grid in place, with the viewing section

detached.

Due to the constraints of the material properties and cutting tools, the minimum

distance between flow areas is 2 mm. This limits the fractal grids to the fourth generation

and constrains the shape of the control grids. The control grid geometries employed are

arbitrary and are not unique solutions.

Grid anatomy comprises flow areas, through which fluid may pass, and blockage areas

which are impenetrable. The blockage areas lead to blockage regions in the immediate

downstream flow.

3.2 Numerical analysis

The image sequences are prepared using Linux/Unix Bash scripts and Matlab and all

image analysis is undertaken using Matlab. The University of Sheffield Iceberg server

and Sun Grid Engine are employed where more intensive computing power is required.

Prior to analysis, the relevant data is extracted from the digital images. Figure 3.6

shows the definition of the x and y axes used for the analysis of the digital images.

In order to identify the coloured dye against the white background, the image encoding

is converted from standard RGB (Red, Green, Blue) colour space to HSV (Hue, Saturation

and Value) colour space (see [14, 13]). A map of saturation, S(m,n) ∈ [0, 1], is created,

where m and n are the pixel coordinates. S(m,n) is a digital representation of the

concentration field, C(x, y). The marker fluid shows high saturation values while the

white background exhibits a near zero value. The saturation value is assumed to be

22

Figure 3.4: Dimensioned drawings of control grids 1, 2, 3 & 4, and the fourth generationCantor grid.

23

indicative of dye concentration and the relationship is assumed linear.

The conversion of digital images to saturation maps provides a clear representation of

the

concentration field, independent of shading effects. Figure 3.5 shows a comparison of

the value map and saturation map methods and clearly shows the improved clarity and

independence of lighting effects obtained using the saturation method.

It is important to note that the saturation map may only store data in the range [0, 1].

To the fullest possible extent, the camera settings have been selected such that the full

range of concentration values is captured within this range.

(a) (b)

Figure 3.5: The concentration field represented by a) an inverted map of pixel value, andb) a map of pixel saturation.

The saturation map, S(m,n), is analysed directly using the wavelet method. For

box-counting, a binary map π(m,n) is created using a chosen threshold value, S ′. The

pre-processing procedure is illustrated in figure B.1.

π(m,n) =

1 if S(m,n) ≥ S ′

0 if S(m,n) < S ′(3.4)

The turbulent field is non-isotropic and three-dimensional; however, all analysis is

carried out on a two-dimensional sample, which is assumed to be representative of the

full field.

To allow analysis of the development of the flow over time, the image capture time is

24

Figure 3.6: Axis definitions for the digital images

extracted from the image meta-data, saved to file and read by the analysis algorithms.

A Linux BASH script is employed to prepare the image sequences for Matlab. Using

the ImageMagick library the images are rotated to compensate for any skew in the image

capture angle and then cropped to a 15× 20 cm region in the centre of the viewing area,

using reference markings on the background. The image capture time is extracted from

the meta-data using the EXIFTOOL library. Time t = 0 is defined as the point at which

the instantaneous threshold, S ′ = 0.1.

Separate Matlab codes are employed to run batch processing of all images in a se-

quence.

Thresholding The choice of a threshold value is very important as it has a significant

effect on the results of the box-counting. Two methods of thresholding are employed.

For a given image, an optimum threshold, SO′, may be calculated. For this investigation,

the Otsu method is used (Otsu 1979 [6]). A fixed concentration iso-level is also tracked.

The two methods are as follows:

Instantaneous threshold method Each of the images in a sequence is converted to

a binary map πO(m,n) based on its Otsu threshold.

Fixed threshold method The image sequence is first analysed to obtain the instan-

taneous Otsu threshold as a function of time. Each image is converted to a binary map,

πm(m,n), based on the median threshold, SO′, of the entire sequence.

25

3.2.1 Box-counting

A box-counting algorithm has been developed to calculate the coordinates of the boundary

between the black and white regions of the binary map π(m,n).

An initial box edge length of lB10

, where lB is the largest edge of the image, is chosen

to ensure statistical validity. The covering box size, λ, decreases with each iteration, n,

according to the relationship λ = lBn1.2 . A minimum box size of 3 x 3 pixels is imposed to

avoid sampling beyond the image resolution.

The algorithm examines each boundary point in turn to find the box into which it

falls. The box is identified using a simple search algorithm.

Many existing box-counting codes divide the image into 2n boxes. The result is that

the boundary between two boxes always fall in the same place. The algorithm written

for this investigation divides the image into boxes of size lBn1.2 . The boundaries therefore

fall in different places with each iteration, avoiding any possible aliasing effects that may

occur.

3.2.2 Wavelet analysis

A wavelet method has been developed, based on the global wavelet energy spectrum

(see section 2.3). The spectrum of a one-dimensional section of the saturation map is

calculated at given distance downstream from the grid. This section is taken in the

cross-flow y-axis.

The global wavelet energy spectrum of S(m′, n) (where m′ is fixed) has a purely

qualitative significance, and no direct physical meaning.

3.2.3 Statistical analysis

A separate code is employed to calculate the coefficient of variance and the exposure of

each of the saturation maps in a sequence.

26

4 Results

Some example images are presented, showing the development of the flow through the first

generation control grid. Subsequently the statistical, wavelet and box-counting analyses

are presented for each equivalent pair of grids and are briefly described.

(a) (b) (c) (d)

Figure 4.1: The saturation map of the flow through the first generation control grid att = 2, 30, 70 and 120 s.

(a) (b) (c) (d)

Figure 4.2: The binary map of the flow through the first generation control grid att = 2, 30, 70 and 120 s, using the fixed value threshold method (SO

′ = 0.4941).

(a) (b) (c) (d)

Figure 4.3: The binary map of the flow through the first generation controlgrid at t = 2, 30, 70 and 120 s, for the instantaneous Otsu threshold (S ′O =0.0510, 0.5098, 0.8902, 0.7608).

27

4.1 Statistical properties

(a) Grid 1 (b) Grid 2

(c) Grid 3 (d) Grid 4

Figure 4.4: Exposure, E, Coefficient of Variance, CoV, and Otsu threshold as a functionof time for equivalent grid pairs.

At time t ' 0, the exposure is close to zero as the marker species is not present and

the saturation map indicates a uniform zero concentration field (neglecting the effect of

artefacts). Exposure rises steadily as marker enters the image, reaching a maximum as

the dye begins to dominate the flow. The exposure drops as coverage increases, reducing

the interface size and sharpness. As the marker injection ceases, the pattern repeats as

the colourless flow returns.

The coefficient of variance is initially high, as the species are well separated - dropping

as mixing occurs and the intensity of segregation reduces.

28

4.2 Global wavelet energy spectrum

(a) Grid 1 (b) Grid 2

(c) Grid 3 (d) Grid 4

Figure 4.5: Global wavelet energy spectrum of a slice perpendicular to the flow, at x ' 10cm from the downstream image edge for equivalent grid pairs.

Figure 4.5 shows the global wavelet energy spectrum of 1-D cross-flow slices taken 10

cm from the downstream image edge. The slices are taken from a single image from each

sequence. The time criterion for the single sample image is S ′O = S ′O. This occurs at

approximately t = 20 s for all sequences.

The peak energy generally occurs at a scale of 7 . λ . 8 cm. The first iteration

peaks at a lower value, reflecting the entrainment of the flow into a single central flow.

The fractal grids exhibit a step at lower scales.

29

4.3 Fractal dimension

(a) Grid 1 (b) Grid 2

(c) Grid 3 (d) Grid 4

Figure 4.6: Fractal dimension of the median threshold interface for equivalent grid pairs(see figure 4.2 for example images).

Fractal dimension, D, calculated at the median threshold value, is close to zero at

time t → 0. The fractal dimension rises sharply as the dye enters the image, reaching

a peak. As the above-threshold region consumes the image, the fixed interface moves to

the edges and the dimension tends towards one.

The presence of artefacts in the image cause some interference, appearing as small

above-threshold regions. This effect is negligible compared to the regions representing

the dye and only manifests itself for a few seconds.

30

(a) (b)

Figure 4.7: a) The correlation coefficient of the box-counting slope for control grid 1between t = 0 and 20 s, b) comparison of fractal dimension of the flow through the firstgeneration grids, calculated using the instantaneous threshold method.

The correlation coefficient of the box-counting slope, logNlog 1

λ

, indicates its linearity, where

1 is a perfect linear gradient and 0 shows no correlation. Figure 4.7 a) is an example

correlation taken from control grid 1. Linearity is high and from around 10 s maintains

a value of 1. This is an indication of the distribution of scales and the applicability the

box-counting method.

Figure 4.7 b) shows the fractal dimension of the flow through the first generation

control and fractal Cantor grid, calculated using the instantaneous optimum threshold as

discussed in section 3.2. The CPU time required to complete this calculation for higher

generations is significant (> 200 hours per grid) due to very large interfaces. As such

only the first generation grids have been analysed using this method.

31

5 Discussion

Before the results are discussed and interpreted, it is useful to establish some definitions

regarding the development of the concentration field over the sampling time window and

within the sampling frame (the image). These definitions are constructed from general

observation and related to the analytical results. The significance of the measurement

data is considered and a comparison of the grids is made. Finally, the assumptions and

limitations of the method are reviewed. Where these limitations are relevant to the

understanding of the results, they will be mentioned as appropriate.

5.1 Four stages of development

For the purposes of description, the term flow describes the velocity vector field and

concentration field is a scalar field describing species distribution.

The marker fluid is inert and, being composed of water soluble paint in water, exhibits

properties close to that of the main flow; therefore, the problem is classed as level 1 mixing

- the fluid mechanics are independent of the mixing process. Although the mixing problem

is unsteady and anisotropic, the main flow is stationary. Marker fluid is introduced into

the downstream main flow and quickly mixes in a turbulent regime, becoming laminar

before reaching the grid.

Figure 5.1: Image of the flow regime for the first generation fractal grid (manipulated forclarity). 1) Homogeneous incident concentration, 2) grid flow area, 3) grid blockage area,4) blockage region, 5) flow region.

Figure 5.1 shows the flow regime around a grid. The incident flow (immediately

32

upstream of the grid) is by calculation and observation, laminar. The fully established

incident concentration field is close to homogeneous. This homogeneous state is inter-

rupted by the grid. An unsteady mixing regime is established whereby the entrained flow

develops towards re-establishing a homogeneous concentration.

The entrained flow (immediately downstream of the grid) is highly dependant on the

grid geometry and conditions range from weakly transitional to turbulent. The number

of discrete flow areas in a grid determines the number of individual flows that comprise

the entire entrained flow. These discretised flows shall be labelled subsidiary flows.

The development of the concentration field may be described in four chronological

stages, during which different mechanisms may be observed.

Entry Marker fluid begins to enter the frame, entrained by the grid. During this stage

the entrainment mechanism is easily observed. Entry is characterised by a peak in the

intensity of segregation (as described by the coefficient of variance) and a steady increase

in the exposure. During entry a sharp rise in the fractal dimension of the fixed value

interface occurs, from zero, indicating no interface (neglecting artefacts in the saturation

map) to a value between 1 and 2.

Mixing Subsequently dye bridges the entire frame. The kinematic stirring mechanism

can be observed as the intermediate turbulent scales distribute the dye. The dye interface,

however, remains well defined as the visible effects of molecular mixing are thus far

negligible. The result of a well defined interface is a peak in exposure. The intensity of

segregation begins to fall as the marker region grows; the mean saturation value increases

and thus the variance of the saturation map reduces. As the mixing stage begins, the

fractal dimension of the fixed value interface peaks.

Uniformity As the molecular mixing mechanisms act, the mixing interface becomes

less defined and the concentration field approaches a homogeneous state. In the cases

where a fully homogeneous state is achieved, it can be considered that full mixing has

33

occurred. Uniformity is characterised by a plateau in the exposure and intensity of seg-

regation. In a perfectly uniform saturation map, the exposure and coefficient of variance

are equal to zero. As the fixed value interface migrates to the edges of the frame, its

fractal dimension, D → 1.

Clear-down A finite quantity of dye is released and eventually the incident concen-

tration diminishes and regions of low concentration enter the frame. This stage can be

considered a similar repetition of the earlier stages.

5.2 Statistical properties

The significance of the development of exposure and intensity of segregation is discussed.

The scales of segregation are discussed separately in section 5.3.

As expected, the intensity of segregation provides a good indication of mixing with ref-

erence to uniformity. Where a homogeneous concentration field is reached, the coefficient

of variance reaches a value very close to zero.

It is important to mention the discrepancy between the coefficient of variance and

the intensity of segregation that it describes. At time t = 0, there exists a homogeneous

concentration field containing no marker species, thus IoSt=0 = 0; however, in practice,

the saturation map is sensitive to artefacts present on the channel walls, such as scratches,

bubbles or debris, which may appear as distinct regions of a second species and CoVt=0 >

0.

Exposure responds to two specific factors which serve to increase the length of the

dye interface. The discretisation of the flow by entrainment at the grid leads directly to a

larger interface. For example, the first generation Cantor grid entrains flow into a single

subsidiary flow with two interface origins (one at either extremity of the flow area). The

equivalent control grid entrains three subsidiary flows with a total of six interface origins.

The kinematic stirring mechanism, induced by recirculation in the blocked regions, also

increases the complexity of the interface, increasing its length.

34

Molecular and viscous diffusion serve to reduce the sharpness of the interface. This

manifests a reduction in exposure.

High exposure coincides with clarity of the kinematic stirring mechanism; however, it

is an indication of the concentration gradient and according to Fick’s law must therefore

be linked to the mass flux attributed to the diffusion processes.

5.3 Wavelet analysis - the scales of segregation

The one-dimensional global wavelet energy spectra are intended to provide an indication

of the scale composition of the saturation map and thus an insight into the scales of

segregation of the concentration field. In order to reduce the information sufficiently for

comparison, the spectra are fixed in time and space. Difficulty arises in specifying an

equivalent time location for each of the spectra. As the problem is unsteady and involves

a large number of variables, it is difficult to define an equivalent time for each grid. An

attempt is made to select equivalent time locations according to the saturation threshold;

however, this method does not necessarily produce comparable results. The results do not

show any particular patterns that suggest comparability between data sets. The wavelet

method may be more appropriately applied to a stationary mixing problem.

Nevertheless, the results allude to some interesting phenomena which may be con-

firmed by a stationary experiment. This is discussed as part of the grid comparison

(section 5.5).

The possibility of estimating the length of the Kolmogorov scale is shown in section

2.1.1; however, it is necessary to know the size of the integral scales. It may be useful to

compare these calculated scales with the global wavelet energy spectra. In general, flow

velocity measurements are required to measure the integral scales, although they can be

observed in some, but not all, of the images. This is an unfortunate limitation of the

non-invasive techniques used and the acquisition of only scalar data.

35

5.4 Fractal dimension

The two thresholding methods discussed in section 3.2 lead to very different box-counting

results.

Instantaneous threshold method The instantaneous threshold interface is very large

for a near homogeneous saturation map. The CPU time required to compute the devel-

opment of the fractal dimension of such a large interface exceeds 200 hours per grid.

The first generation grid flows do not reach a homogeneous state and can be analysed

using the instantaneous threshold method, with a CPU time of around 30 hours. The

absence of results for higher generation grids using this method prevents comparison and

interpretation.

Fixed threshold method The development of the fixed interface dimension illustrates

well the four stages of mixing and displays similar attributes to the exposure value. The

fixed value interface method imposes a binary definition of mixing; all sub-threshold re-

gions are defined as unmixed while all above-threshold regions are mixed. This simplistic

view limits the significance of fractal dimension as a measurement of mixing.

One significant result is that the fractal dimension of the fixed value interface peaks

consistently at 1.35 . D . 1.4. During their investigations into the fractal nature of

turbulence, Sreenivasan and Meneveau (1986) consistently found the dimension of the

turbulent/non-turbulent interface to be in the region of 1.3 to 1.4 for a two-dimensional

slice of a fully established isotropic turbulent flow field. It is also worth noting that

this peak closely coincides with the median sequence threshold value - the peak occurs

when the fixed threshold is equal to the instantaneous optimum threshold. This observa-

tion implies that the optimum threshold value indicates well the turbulent/non-turbulent

interface.

It is possible that some information may be acquired from the slope of the fixed value

interface dimension as it rises from an initial value of zero. However, low time resolution

36

makes comparison difficult. Furthermore, although no detailed study has been conducted,

the linearity of the box-counting calculations appears to be relatively low during the entry

stage, implying that a fractal dimension may not be applicable. A more in-depth study

of the correlation coefficients of the box-counting results would be required to assess the

applicability of a fractal dimension at this stage.

5.5 Grid comparison

The comparison of the statistical properties of the grids shows that higher generation

grids behave similarly to their equivalent control counterparts. Furthermore, the results

show that faster mixing occurs in higher generation grids; however, this result is biased

by the nature of the problem. The incident concentration field is homogeneous and since

the higher generation grids have a minimal effect on the flow, the entrained flow returns

very quickly to this state.

The first generation grids have large blockage areas at the extremities. The effect

of this geometry is large blockage regions in the entrained flow which inhibits the de-

velopment of a fully homogeneous downstream concentration. This exacerbates the bias

towards apparently faster mixing in higher generation grids.

With reference to the various measurements taken, some marked effects can be ob-

served, especially in the first generation grids.

The exposure of the first generation control grid reaches a significantly higher value

(E ' 0.024) than the other grids. This implies improved kinematic stirring and there-

fore improved molecular diffusion. The grid is defined by flow discretisation and large

blockage areas which induce strong turbulent recirculation. The first generation fractal

grid, containing blockage regions but no flow discretisation, reaches a significantly lower

peak exposure of around E = 0.017. Higher generation grids with high discretisation

but very small turbulent induction due to small blockage areas, peak at up to E = 0.02.

Observation of the saturation maps indicate that the flow around the blockage areas of

the third and fourth generation grids is close to laminar. This observation suggests that

37

there may be balance to achieve between discretisation and blockage area size. Further

investigation would be required to test this hypothesis.

Further observation of the flow field suggests that the most important factor in the

induction of turbulence is the geometry of the blockage areas rather than flow areas. The

Cantor grid model used in this investigation is such that, while the flow areas are of

different sizes the blockage areas are of the same size. A more significant effect may be

imposed on the flow by choosing an inverted model whereby the blockage areas are varied

and the flow areas constant. This may effect a greater impact on the flow development.

An inverted Cantor model would lead to very small flow areas. Different fractals could

be explored to increase the flow area.

Although comparison of the global wavelet energy spectra is difficult, there are some

apparent differences in the scale distributions in the saturation maps of the fractal and

control grids. The spectra show a small step at the small scales which appears to be exag-

gerated in the fractal results. A stationary experiment would allow this to be investigated

more thouroughly by eliminating variables for better comparison.

Interestingly, while the flow through the fourth generation grids are, by observation,

laminar, the fractal dimension nevertheless peaks at D ' 1.35, as would be expected for

a fully turbulent interface. This tends to contradict the idea that turbulence is a special

case in terms of fractal analysis of the interface.

5.6 Method review

It is important to reflect upon the assumptions made during the investigation and the

possible effects on the results. These assumptions are discussed, followed by a review of

the limitations of the experimental procedure.

There is a disparity between the method of image capture and the subsequent image

analysis which arises from the assumption that the saturation maps generated from the

digital images represent an orthogonal view of a two-dimensional slice of the concentration

field. In reality they represent a perspective projection - they can be considered as many

38

slices superimposed upon one-another - the detail in a slice of fluid is masked by the layers

around it. Box-counting calculations are affected by the masking of internal boundaries

and possible smoothing effects on the external boundaries. Exposure results are affected

by the cumulative effect of the projection which may cause the flow to appear more

homogeneous - exposure may be under-calculated. A closer approximation to a two-

dimensional slice could be achieved using a laser light sheet, which would also reduce the

effect of artefacts present on the channel walls.

Bubbles in the viewing section are also a common source of artefacts as they refract

light. Where possible most bubbles are removed from the viewing section but this is not

possible in all cases. Bubbles close to the grid on the upstream side may also affect the

flow conditions. A vertically orientated flow channel could alleviate such problems.

The application of single statistical scalars (dimension, coefficient of variance and

exposure) to the entire frame is incompatible with the nature of the problem, which is

anisotropic and inhomogeneous. It may be more appropriate to divide the frame into

zones, or take sample slices normal to the mean flow. Application of two-dimensional

wavelet analysis with a zoned system may also be a useful method of analysis. The

application of a 2-D sampling regime is also in conflict with the 3-D nature of turbulence.

Certain difficulties arise in the definition of a time origin. An approximation is made

with relation to the threshold value; time t = 0 is defined as the point at which the

instantaneous threshold, SO′ = 0.1. The sampling time resolution is also limited by

the image capture equipment and inhibits the study of transient effects. Furthermore,

the large sampling time required leads to a limitation of time resolution due to the

practicalities of analysing large amounts of data. The time variable would be negated

altogether by stationary flow conditions. Marker injection at the grid would create a

stationary problem and allow better comparison of the grid effects.

The relationship between the saturation map and the concentration field is assumed

to be linear, although this has not been proven. Calibration could be performed on known

homogeneous concentrations under the same photographic conditions in order to quantify

39

this link.

A major limitation of the experiment is the restricted sampling frame. As the energy

cascade develops in the direction of flow, small scale turbulence and molecular mixing ef-

fects occur downstream of the viewing section and are not present in the results. Analysis

should be carried out on these downstream effects.

It has also been observed that the flow is not consistently turbulent. Higher order

grids display largely laminar properties. This is restricted by the development length

imposed by the length of the upstream channel. An increased flow rate (with a longer

upstream channel) or altered grid patterns could improve turbulent induction.

A single set of control grids is used and the designs are not unique. Further designs

should be tested.

40

6 Conclusion and Recommendations

An investigation has been carried out into turbulent mixing of a marker fluid in a flow

through fractal Cantor grids using image analysis. Following previous work, the exper-

imental procedure has been improved, particularly with regard to image capture tech-

niques and the inclusion of control experiments using non-fractal grids. A set of analyt-

ical tools has been developed, improving existing box-counting methods and developing

wavelet and statistical methods.

Fractal cantor grids have been compared to non-fractal control grids of equivalent flow

area. The problem consists of an incident flow of homogeneous concentration, entrained

by a grid into a transient mixing field. The flow conditions in the mixing field range from

weakly transitional/laminar to turbulent. Image sequences of the entrained flow have

been converted to saturation maps which are assumed to represent an orthoganol 2-D

slice of the marker species concentration field.

The saturation map sequences have been analysed using image processing techniques.

The intensity of segregation (represented by the coefficient of variance) and exposure

have been calculated. An attempt has been made to examine the scales of segregation

using global wavelet energy spectra of 1-D slices of the saturation map at a fixed time

location. The fractal dimension of a fixed value saturation threshold has been calculated

using the box-counting method.

The following conclusions have emerged from the analyses and observation of the

concentration field.

� The results suggest that exposure shows a strong positive response to turbulent

kinematic stirring and discretisation of the entrained flow. Both serve to increase

the length of the interface, contributing to stretching, folding and break-up of the

marker species.

� Exposure is linked to the concentration gradient and by Fick’s law is shown to be

a measure of the mass flux associated with molecular mixing.

41

� Wavelet analysis alludes to increased energy at the small scales in the flow through

the fractal grids.

� The iso-value interface dimension peaks at 1.35 . D . 1.4, which is in agreement

with the findings of Sreenivasan and Meneveau (1986). This result is equally ob-

served in the cases where the flow regime is observed to be near-laminar. Any

special fractal properties for turbulent cases is not observed.

� Mixing may benefit from a balance between flow discretisation and kinematic stir-

ring induced by larger blockage areas at the grid.

Further conclusions may be drawn regarding the improvements made to the experi-

mental procedure.

� The use of a saturation map in lieu of a value map reduces the effects of non-

uniform light and shadow and more effectively identifies the marker fluid against

the background.

� A polarised filter coupled with a photographic umbrella serves to render the effect

of reflections negligible.

The results obtained from this investigation successfully highlight necessary improve-

ments to the method.

� The application of single scalar measurements to the sampling frame is inappropri-

ate. A zoning scheme should be considered.

� A more comprehensive study of the applicability of the box-counting algorithm

should be conducted, examining the linearity of the box-counting results.

� A stationary problem could be created by injecting the marker species at the grid,

eliminating the time variable and allowing better comparison. Time-averaging could

be employed.

42

� A quantified relationship between the saturation map and concentration field should

be obtained.

� The sampling frame should be extended to capture the downstream flow.

� A longer upstream section should be used to allow higher flow rates.

� A wider set of fractal grids should be investigated including different fractal models,

inverted fractal models and varied designs of control grid.

� The effect on mixing of the size of the blockage areas and discretisation of the flow

areas should be investigated further.

� A laser light sheet, or similar method, should be employed to achieve a closer

approximation to a 2-D slice.

� Implementation of a vertical flow channel should be considered to mitigate bubble

accumulation.

43

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[3] Marie Farge. “Wavelet transforms and their applications to turbulence”. In: Annual

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[4] Alena Kukukova. “A new definition of mixing and segregation: Three dimensions

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[5] F. Nicolleau and J. C. Vassilicos. “Wavelet analysis of wave motion”. In: Personal

communication, F. Nicolleau, University of Sheffield (2000).

[6] Nobuyuki Otsu. “A threshold selection method from gray-level histograms”. In:

IEEE Transactions on systems, man, and cybernetics 9.1 (Jan. 1979), pp. 62–66.

[7] K. R. Sreenivasan and C. Meneveau. “The fractal facets of turbulence”. In: Journal

of Fluid Mechanics 173 (1986), pp. 357–386.

[B] Books

[8] C. W. Fetter. Contaminant Hydrogeology. 2nd. Prentice Hall, 1999. Chap. 3.

[9] Jean Mathieu and Julian Scott. An Introduction to Turbulence. Cambridge Univer-

sity Press, 2000.

[10] J. M. Ottino. The Kinematics of Mixing: Stretching, Chaos, and Transport. Cam-

bridge University Press, 1987.

44

[11] E. L. Paul, V. Atiemo-Obeng and S. M. Kresta. Handbook of Industrial Mixing:

Science and Practice. John Wiley & Sons Inc., 2003.

[12] Stephen B. Pope. Turbulent Flow. Cambridge University Press, 2000.

[13] Milan Sonka, Vaclav Hlavac and Roger Boyle. Image Processing, Analysis, and

Machine Vision. Third. Thomson, 2008.

[C] Online

[14] Richard E Riley. A comparison of methods for identifying a dyed fluid in an image.

www.reriley.co.uk/idfluid.php. 2008. url: www.reriley.co.uk/idfluid.php.

[D] Masters thesis

[15] Brishni Mukhopadhyay. “Image analysis of flow through a Cantor grid”. MA thesis.

The University of Sheffield, 2008.

45

Appendices

A Image sequences

The following pages contain the saturation map sequences for each grid.

46

Figure A.1: Saturation map sequence for control grid 1.

47

Figure A.2: Saturation map sequence for fractal grid 1.

48

Figure A.3: Saturation map sequence for control grid 2.

49

Figure A.4: Saturation map sequence for fractal grid 2.

50

Figure A.5: Saturation map sequence for control grid 3.

51

Figure A.6: Saturation map sequence for fractal grid 3.

52

Figure A.7: Saturation map sequence for control grid 4.

53

Figure A.8: Saturation map sequence for fractal grid 4.

54

B Ancilliary information

Table B.1: Digital camera specifications

Manufacturer: CanonModel: EOS 300D

Resolution: 6.2 Mega pixelsLens: Canon Zoom EF-S 58 mm Ø

18-55 mm1:3.5 - 5.6

55

Figure B.1: The image processing data flow.

56

C Algorithms

This appendix contains the algorithms used for image processing. Slightly modified ver-

sions of these codes are used on the University of Sheffield Iceberg Server where augmented

computing power is required.

All code is the work of the author.

C.1 Time meta-data extraction

The following Linux BASH code reads the meta data from each image in a sequence and

writes to file.

# Dialogue to choose file

srcdir=$(zenity --file-selection \

--directory --title "Choose containing directory...")

# Make a directory for processed data files if one does not already exist

mkdir -p "$srcdir"/prep/

# Extract creation date and time and read into a temporary file

exiftool -p ’$filename $CreateDate’ -q -f \

"$srcdir" >> "$srcdir"/prep/timedata.tmp

# Sort the data and write to file, delete tmp file

sort -d "$srcdir"/prep/timedata.tmp >> "$srcdir"/prep/timedata.dat

rm "$srcdir"/prep/timedata.tmp

# Rotate and crop...

# Take parameters

rotangle=$(zenity --entry --text "Enter CW rotation angle in degrees")

57

cropstr=$(zenity --entry --text "Enter crop string in the form \

wxh+x+y (i.e. size then offset) e.g. 1208x1312+403+198")

rsiz=$(zenity --entry --text "Enter output size (%)")

# Run mogrify commands an feed output to a progress dialogue box

(

echo "# Copying images..." ; sleep 1

cp "$srcdir"/*.JPG "$srcdir"/prep/

echo "25" ; sleep 1

if [ "$rotangle" -ne "0" ]

then

echo "# Rotating..." ; sleep 1

mogrify -distort SRT "$rotangle" \

"$srcdir"/prep/*.JPG

echo "50" ; sleep 1

fi

echo "# Cropping..." ; sleep 1

mogrify -crop "$cropstr" "$srcdir"/prep/*.JPG

echo "75" ; sleep 1

echo "# Resizing..." ; sleep 1

mogrify -resize "$rsiz"% "$srcdir"/prep/*.JPG

echo "100" ; sleep 1

echo "# Preparation complete. Click ’OK’." ; sleep 1

) |

58

zenity --progress \

--title="Image preparation" \

--text="Initialising..." \

--percentage=0

59

C.2 Image conversion

The following Matlab code converts the digital image to a saturation map or binary map.

1 % Image to concentratuion map.

2 % Author: Richard E Riley 20/11/2008

3

4 % Converts a colour image to an intesity scale using saturatuion value to

5 % determine the concetration of a coloured dye.

6

7 % Call as in the following example:

8 % im2conc(image,conversion type,level) where:

9

10 % <image> is the image to analyse

11

12 % <conversion> is the type of conversion to perform. Possible options are:

13 % 'bin' = binary conversion. This option should be followed by <level>; the

14 % greyscale threshold. a value of 0 uses the Otsu Method

15 % 'scale' = greyscale conversion where pixel intensity indicates

16 % dye concentration

17

18 function [image,olev] = im2conc(rgb image,conversion,level)

19

20 % Convert to HSV, to greyscale based on saturation value.

21 hsv image = rgb2hsv(rgb image);

22 s = hsv image(:,:,2);

23 if strcmp(conversion,'bin') % Compare instruction string

24 % Convert to binary using Otsu's method to find the threshold. if

25 % no threshold is specified.

26 if level == 0

27 level=graythresh(s);

28 disp(['Otsu Level = ' num2str(level)]);

29 end

60

30 image = im2bw(s,level);

31 olev = level;

32 elseif strcmp(conversion,'scale')

33 % Simply report greyscale image

34 image=s;

35 else

36 disp('Please specify conversion type as bin or scale');

37 end

61

C.3 Box-counting

The following Matlab code peforms the box-counting calculation on a single image.

1 % A box−counting algorithm Author: Richard E Riley 26/02/2009

2

3 % Call as in the following example:

4 % boxcount(image,xdim,ydim,iterations,level) where: <image> is the image

5 % to analyse <xdim> & <ydim> are the dimensions of the image in cm

6 % <iterations> is the number of iterations to perform, this is limited to a

7 % 3x3 pixel box. <level> is the saturation threshold (between 0 and 1). a

8 % value of 0 uses the Otsu's Method

9

10 % The box counting size decreases according to the number of iterations N

11 % with the following relationship: 1/ Nˆ−1.212

13 % Function returns [dimension,correlation] where dimension is the final

14 % dimension and correlation the correlation coefficient of the box−count15 % result.

16

17 function [dimension,correlation] = boxcount mk2(rgb image,x real,y real,

Nmax,level)

18

19 % Convert to HSV and to greyscale based on saturation value, and then

to

20 % binary.

21 [bw,olev] = im2conc(rgb image,'bin',level);

22

23 % Combine and plot all boundaries

24 B = bwboundaries(bw); % Detect boundaries

25

26

27 % Plot image and boundaries

62

28 figure(1); clf; imshow(rgb image); hold on;

29 for k=1:length(B)

30 boundary = B{k};31 plot(boundary(:,2), boundary(:,1), 'g','LineWidth',2)

32 end

33

34 % Convert from cell data into a matrix of x−y coordinates describing

35 % all distinct boundaries.

36 B=cell2mat(B);

37 if B

38

39 % Scale coordinates to physical dimensions.

40 [b,a]=size(bw);

41 sfx=double(x real/a);

42 sfy=double(y real/b);

43 B real = [];

44 B real(:,2)=B(:,2).*sfx;

45 B real(:,1)=y real−(B(:,1).*sfy); % Image origin is the top−leftwhereas plot origin is bottom−left

46

47 % Here the bounding box is define as the size of the image. In fact,

this is a

48 % legacy to older code.

49 xmin=0;

50 xmax=a*sfx;

51 ymin=0;

52 ymax=b*sfy;

53

54 % Plot boundary points on a real scale and display the bounding box and

55 % gives some information

56 figure(2); clf; plot(B real(:,2), B real(:,1),'b.','markersize',3);

57 axis([0 x real 0 y real]);

58 gbox(1) = rectangle('Position',[(xmin),(ymin),(xmax−xmin),(ymax−ymin)],'EdgeColor','r');

63

59 hold on;

60

61 % Box−counting...62

63 % Count boundary points

64 np=size(B);

65 np=np(1);

66 disp([num2str(np) 'boundary points']);

67

68 % Redefine origin to that of the bounding box

69 % Padded with the equivalent length of one pixel (sf−).70 B real(:,2) = B real(:,2) − xmin + sfx;

71 B real(:,1) = B real(:,1) − ymin + sfy;

72

73 % Define the inital variables for box counting

74 BBx=xmax − xmin + sfx; % Length of bounding box in x

75 BBy=ymax − ymin + sfy; % Length of bounding box in y

76 BB=max([BBx,BBy]); % Choose largest edge (thus all boxes will be

square)

77 fullbox=[]; % Matrix to hold the grid references of all full boxes

78 bcount=[]; % Matrix to hold box size vs. the number of full boxes

79

80 % Start at box size smaller than bounding box so that enough boxes

81 % exist to give a good statistical average.

82 lbs=BB/10;

83 sf=1.2; % Box shrink factor

84

85 % Limit max iterations to give a minimum box size of 3x3 pixels.

86 if (lbs/Nmaxˆsf)<((max([sfx,sfy]))*3)

87 Nmax=floor((lbs/(3*max([sfx,sfy])))ˆ(1/sf));

88 disp(['Iterations limited to ' num2str(Nmax)]);

89 end

90

91 % Calculate number of calcs needed and create a progress dialogue. This

64

92 % provides only an estimation of progess. The main purpose is to

93 % reassure the user that the program is advancing.

94 calcs=np*Nmax;

95 prog = waitbar(0,'Please wait...');

96 itprog = waitbar(0,'Iteration progress...');

97

98 % For each box size step

99 for N=1:Nmax

100 % Define length of each box edge. Power increases reduction steps

101 % with each iteration.

102 lb=lbs/Nˆsf;

103

104 % For each boundary point, find which box it falls into.

105 for k=1:np

106

107 bx=1;

108 while B real(k,2) > (bx*lb)

109 bx=bx+1;

110 end

111 by=1;

112 while B real(k,1) > (by*lb)

113 by=by+1;

114 end

115

116 fullbox(k,:)=[bx,by]; % Read grid reference into array

117 waitbar((k/np),itprog,['Point ' num2str(k) ' of ' num2str(

np)]);

118 end

119 fullbox=unique(fullbox, 'rows'); % Allow each full cell to

appear only once

120 [f,g]=size(fullbox); % f is now equal to the number of full

boxes

121 bcount(N,:)=[(1/lb),f];

65

122 waitbar((N*np)/calcs,prog,['Iteration ' num2str(N) ' of '

num2str(Nmax)]); % Update progress dialogue

123

124 % Plot the boxes which contain the boundary giving some

125 % reassurance of what the script is actually doing.

126 delete(gbox(:)); % Delete previous boxes

127 gbox=[];

128 for k=1:f

129 gbx=((fullbox(k,1)−1)*lb) + xmin;

130 gby=((fullbox(k,2)−1)*lb) + ymin;

131 gbox(k) = rectangle('Position',[gbx,gby,lb,lb],'EdgeColor',

'r');

132 end

133

134 % Use a single order poly fit and take first order coefficient

135 % to obtain fractal dimension

136 if N > 1

137 p = polyfit(log(bcount(1:N,1)),log(bcount(1:N,2)),1);

138 dimension = p(1);

139 dimlog(N,:)=[N,p(1)];

140 else

141 p(1)=0;

142 dimension = p(1);

143 end

144

145 end

146 close(itprog);

147 close(prog);

148

149 % Final data processing

150

151 % Plot log−log of dimension as a function of box size

152 figure(3); clf; loglog(bcount(:,1),bcount(:,2));

66

153 grid on; xlabel('1/\epsilon (cmˆ−ˆ1)','fontsize',12); ylabel('Fractal

dimension','fontsize',12)

154

155 % Return final dimension (for ease of use as opposed to accessing the

156 % dimlog matrix.

157 dimension = p(1)

158

159 % Calculate the correlation coefficient of the box−count to indicate

160 % linearity and appicability of method to given image.

161 correlation=corrcoef(log(bcount))

162 corsiz = size(correlation)

163 if corsiz(1) == 1

164 correlation=0

165 else

166 correlation=correlation(1,2)

167 end

168

169

170 % What to do if no boundary points exist.

171 else

172 dimension=0;

173 correlation=NaN;

174 end

175

176 end

67

C.4 Box-counting batch code

The following Matlab code invokes the box-counting script for a sequence of images and

includes time data.

1 function dimensions = bcbatch(xdim,ydim,Nmax,level)

2 % BCBATCH Executes box−counting on multiple images and appends time data.

3 % See boxcount.m for variable meanings.

4

5 % Show dialogue to select the images in the sequence.

6 [fname,pname] = uigetfile('*.jpg;*.JPG;*.bmp','Select containing folder','/

home/rriley/Documents/Work/Year 4/Final Project/MatLab/Data/','

multiselect','on')

7 fname = sort(fname);

8

9 % Read the time data and sort into and array. Number of data lines should

10 % equal the number of images in the sequence.

11 [files, dates, times] = textread([pname 'timedata.dat'], '%s %s %s');

12 timedata=horzcat(files, times);

13 timedata = sortrows(timedata,1);

14

15

16 dimensions=[];% Create array to hold the time and calculated data.

17

18 % For each image invoke boxcounting algorithm

19 for k=1:length(fname);

20 f=cell2mat(fname(k))

21 p=([pname f]);

22 rgb image=imread([pname f]);

23 [dimensions(k,2),correlations(k)] = boxcount mk2(rgb image,xdim,ydim,

Nmax,level);

24 time=char(timedata(k,2));

25 dimensions(k,1)=datenum(time,'HH:MM:SS');

68

26 end

27

28 % Provide plots and save data to file.

29 dimensions(:,1)=(dimensions(:,1)−min(dimensions(:,1)))*24*60*60;30 plot(dimensions(:,1),dimensions(:,2),'o');

31 xlabel('Time (s)','fontsize',12); ylabel('Fractal dimension','fontsize',12)

;

32 fid = fopen([pname 'dimdev.dat'], 'w');

33 data=dimensions';

34 fprintf(fid, '%1.4f\t%1.4f\n', data);

35 fclose(fid);

36

37 plot(dimensions(:,1),correlations(:),'o');

38 xlabel('Time (s)','fontsize',12); ylabel('Correlation Coefficient','

fontsize',12);

39 fid = fopen([pname 'correlation.dat'], 'w');

40 dimensions(:,1)

41 correlations

42 data=[dimensions(:,1),correlations'];

43 fprintf(fid, '%1.4f\t%1.4f\n', data);

44 fclose(fid);

69

C.5 Statistical analysis

The following Matlab code calculates the coefficient of variance and exposure of all images

in a sequence and includes time data.

1 function [CoV,E] = mixdim(name)

2 % [CoV,E] = mixdim(name) calculates the Coefficient of Variance (CoV) and

3 % Exposure (E) of an image sequence. Data is output to tab delimited file

4 % as a function of time. Time stamps are read from the timedata.dat file

5 % included in the folder containing the image sequence. (timedata.dat is

6 % generated using a bash script). Output is saved to the same directory as

7 % the image files. Variable 'name' is the label to be attributed to the

8 % output files.

9

10 % Show dialogue to select the images in the sequence.

11 [fname,pname] = uigetfile('*.jpg;*.JPG;*.bmp','Select containing folder','/

home/rriley/Documents/Work/Year 4/Final Project/MatLab/Data/','

multiselect','on')

12 fname = sort(fname);

13

14 % Read the time data and sort into and array. Number of data lines should

15 % equal the number of images in the sequence.

16 [files, dates, times] = textread([pname 'timedata.dat'], '%s %s %s');

17 timedata=horzcat(files, times);

18 timedata = sortrows(timedata,1);

19

20 dimensions=[]; % Create array to hold the time and calculated data.

21

22 % For each image perform calculations.

23 for k=1:length(fname);

24 f=cell2mat(fname(k))

25 p=([pname f]);

26 rgb image=imread([pname f]);

70

27 conc = im2conc(rgb image,'scale',0); % Convert image to saturation map

28

29 % Intensity of Segregation calculation

30 mconc=mean(mean(conc));

31 I = ((mconc − conc).ˆ2)./(mconc*(1−mconc));32 dimensions(k,2) = sqrt(mean(mean(((conc−mconc)./mconc).ˆ2)));33

34 % Exposure calculation

35 [a,b] = size(conc);

36

37 % Begin by adding one pixel padding to the saturation map. Padding

38 % is a copy of the border pixels to achieve zero difference between

39 % image edges.

40 conc2 = conc;

41 conc2=cat(1,conc2(1,:),conc2);

42 conc2=cat(1,conc2(a+1,:),conc2);

43 conc2=circshift(conc2,[−1,0]);44

45 conc2=cat(2,conc2(:,1),conc2);

46 conc2=cat(2,conc2(:,b+1),conc2);

47 conc2=circshift(conc2,[0,−1]);48 [a,b] = size(conc2);

49

50 % Reset value and begin exposure calculation

51 d=0;

52 for m=2:a−153 for n=2:b−154 p=[];

55 p(1) = abs(conc2(m,n) − conc2(m−1,n));56 p(2) = abs(conc2(m,n) − conc2(m+1,n));

57 p(3) = abs(conc2(m,n) − conc2(m,n+1));

58 p(4) = abs(conc2(m,n) − conc2(m,n−1));59 d=d+0.5*sum(p);

60 end

71

61 end

62 dimensions(k,3)=d/((a−2)*(b−2)); % Normalise and store in matrix.

63

64 % Send time to matrix.

65 time=char(timedata(k,2));

66 dimensions(k,1)=datenum(time,'HH:MM:SS');

67 end

68

69 % Provide plots and save data to file.

70 dimensions(:,1)=(dimensions(:,1)−min(dimensions(:,1)))*24*60*60;71 subplot(2,1,1); plot(dimensions(:,1),dimensions(:,2),'o');

72 xlabel('Time (s)','fontsize',12); ylabel('CoV','fontsize',12); hold on;

73 subplot(2,1,2); plot(dimensions(:,1),dimensions(:,3),'x');

74 xlabel('Time (s)','fontsize',12); ylabel('Exposure','fontsize',12);

75 fid = fopen([pname name '−mixdim.dat'], 'w');

76 data=dimensions'

77 fprintf(fid, '%1.4f\t%1.4f\t%1.4f\n', data);

78 fclose(fid);

79

80

81 % Return answers

82 CoV = dimensions(:,2);

83 E = dimensions(:,3);

72

C.6 Global wavelet energy spectrum

The following Matlab code calculates the global wavelet energy spectrum of a slice of

each image in a set.

1 % Wavelet analysis script − Richard Riley, 5th Feb 2009

2

3 % GWES = waveline(x real, y real, x pos) calulates the global wavelet

4 % energy spectrum (GWES) of a slice of each selected image (via a dialogue)

. The

5 % slice is taken in the y direction (vertical) at position x=x pos where

6 % x pos is in chosen dimensions. x and y real are the real dimensions of

7 % the images in chosen units.

8 function spec = wavline(x real,y real,x pos)

9

10 % Show a dialogue to select the required files.

11 [fname,pname] = uigetfile('*.jpg;*.JPG;*.bmp','Select containing folder','/

home/rriley/Documents/Work/Year 4/Final Project/MatLab/Data/','

multiselect','on')

12 fname = sort(fname);

13

14 % For each image, perform calculations.

15 for k=1:length(fname);

16 f=cell2mat(fname(k))

17 p=([pname f]);

18 rgb image=imread([pname f]);

19

20 % Convert to HSV, to greyscale based on saturation value.

21 conc = im2conc(rgb image,'scale',0);

22 conc show = conc − min(min(conc));

23 conc show = conc show ./ max(max(conc show));

24 figure(1); imshow(conc show); hold on;

25

73

26 % Set the scale range

27 [a,b] = size(conc);

28 a=a−1;29 scales = 1:a;

30

31 % Define scale factors to give results in real dimensions

32 sfx=double(x real/b);

33 sfy=double(y real/a);

34

35 % Create scale array in real dimensions

36 s real=(1:a).*sfy;

37

38 % Define x pos in terms of pixels

39 x=ceil(x pos/sfx);

40 plot([x,x],[0,a]);

41

42 % Take slice and calculate continuous wavelet transform

43 slice = conc(:,x);

44 figure(2); plot(slice);

45 figure(3);

46 tslice=cwt(slice,scales,'mexh');% ,'plot');

47

48 % Calc GWES

49 for sc=1:a

50 h=tslice(sc,:).ˆ2;

51 spec(sc,2)=(intdump(h,length(h)))*(a*sfy)/(sc*sfy);

52 spec(sc,1)=sc*sfy;

53 end

54

55 % Plot and save to file in selected folder.

56 figure(4); plot(spec(:,1),spec(:,2));

57 fid = fopen([pname f '−wav.dat'], 'w');

58 data=spec';

59 fprintf(fid, '%1.4f\t%1.4f\n', data);

74

60 fclose(fid);

61 end

62 end

75