Surfacing and clustering of gyrotactic swimmers in free-surface turbulence

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Author’s e-mail:

bhatia.harshit@spes.uniud.it bhatia.harshit@tuwien.ac.at

Author’s address:

Dipartimento Politecnico di Ingegneria e Architettura Universit`a degli Studi di Udine

Via delle Scienze, 206 33100 Udine – Italia

Web: diegm-University of Udine

Cover:

Pattern of Nature: Clustering of plankton on surface of open channel turbulent flow correlated with Surface Divergence color map level (Red-upwelling event and Blue-downwelling event).

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Acknowledgments

The feeling of gratefulness fills me as I submit my work as a PhD thesis; I have many people to thank who made this possible, who have supported me and provided every possible help with good conscience.

First, I am thankful to God/Cosmos/Nature, as a hidden force within every soul; who kept me determined to stand through every situation of life, firmly.

This thesis is not only about what I have worked and learned in science but also about journey of life, which has been an amazing experience. Recent destination of this journey was Udine, Italy; where I spent beautiful three years and got an opportunity to work. I sincerely thank Prof Cristian Marchioli and Prof Alfredo Soldati, for inviting me to Italy and Austria; and providing me an opportunity to fulfil my childhood dream, to obtain a Doctorate degree. They have been very patient and kind to teach me on a regular basis which made me reach where I am today for which I am highly grateful to them. I would like to extend my sincere gratitude to Prof. Eric Climent who made my first step in Europe easy and made me realize that Science is not about big stuff, it is made by simple small basics which when combined in a systematic way, become big. He also set the path for my journey towards PhD. I thank Dr. Ugur Guven who has been my mentor since I entered the field of science and shaped me, what I am today. He installed a firm belief that if you have dream you have to chase it and protect it. I would also like to thank my colleagues: Francesco Zonta, Giovanni and Francesco Romano; for their support and useful brain storming sessions. Special thanks to Diego and Alessio who have always been ready for technical discussions with me and learning session together; Marco for helping me in every possible way, during all the official bureaucratic processes and made my arrival and stay as easy as possible. Arash, Mobin and Pageman thanks to you as well.

Special thanks to my Family: My evergreen inspiration father Jagdish, who made everything look so easy, even in the toughest times; My mother Raj, who has been my never-ending source of emotional support and love; My brothers Varshit and Himan-shu, who has been close friends since childhood. I dedicate my thesis to my family. Special mention for friends who have been there whenever needed and made life full of colors, Sabin, Hailu, Vasili, Paula, Shaima and Marcela.

I also want to thank my sources of inspiration, Leonardo da Vinci and Carl Sagan, work of these genius minds inspired me to study science at first place. A musical thanks to Linkin Park and Coldplay, songs with meaningful lyrics have kept me going through the journey of life.

Last and special, this is heartfelt thanks to that special girl who has been there for me and has provided all kind of support possible.

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Abstract

In this thesis, the behaviour of gyrotactic (both spherical and elongated) micro-swimmers in a turbulent open channel flow has been investigated. The surface of the turbulent open channel flow has been treated as a flat free-slip surface that encloses the three-dimensional volume containing turbulent flow. This configuration mimics the motion of active oceanic organisms (e.g. phytoplankton, drifters or floaters) when surface waves, ripples are smooth and stable stratification is absent.

The behaviour of turbulence is important for dynamics of micro-swimmers that swim and rise towards the upper layers of the flow. Surface turbulence and activity near the free-surface have been analysed and carefully studied via an extensive campaign of direct numerical simulations coupled with Lagrangian tracking of active inertia-less swimmers. The fluid governing equations are solved using a pseudo-spectral method, whereas the swimmers’ governing equations (for velocity and orientation) are solved using considering different values of the swimmer re-orientation time (defined as the time a swimmer will take to return to its stable condition, which is vertical direction upwards if the effect of disturbance (vorticity) is removed), which corresponds to different levels of vertical stability.

Surface is forced to be two-dimensional by means of boundary conditions that increase the surface compressibility: Because of this boundary condition, turbulent events that are essentially three dimensional appear to be two dimensional when approaching the free surface. The surface is continuously renewed with the events of upwellings that act as two-dimensional sources for the surface parallel fluid velocity and alternate to sinks that correspond to downdrafts of fluid from the surface to the bulk of the flow. Gyrotactic micro-swimmers reach to the surface (the air-water interface where light is enough to activate photosynthesis) mainly by two means: through the fluid upwellings but also exploiting their own ability to swim upwards. Once at the free surface, swim-mers leave the upwelling regions and shift to the neighbouring downwelling regions, where their vertical stability keeps them at the surface.

The results discussed in this thesis for the case of spherical swimmers show the impor-tance of vertical stability as, the fast to reorient swimmers are more vertically stable and those which are slow to reorient are weak in stability. Swimmers with high vertical stability are fast to swim and reach the free-surface while the swimmers with weak vertical stability stay dispersed in the bulk of flow. Study also suggest the correla-tion of swimmers’ ability to reach to free surface increases with increase of Reynolds number as the energy contained in the flow is higher and thus the swimmers can take advantage of turbulent events to their favour.

Results focusing on elongated swimmers show that shape plays an important role on surfacing. Especially when the vertical stability is not strong enough, more elongated

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ing, when the reorientation time is slow, elongation compensate and helps swimmers overcome the instability, contributing in the purpose of swimming towards the target.

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List of Figures

1.1 Food web and role of phytoplankton(Credit: NASA images) . . . 1

1.2 Plankton Bloom (Credit: NASA images by Robert Simmon and Jesse Allen, based on MODIS data.) . . . 2

1.3 Energy spectra for the case of (a) 3D turbulence and (b) 2D turbulence 5

2.1 Sketch of the computational domain . . . 12

2.2 Plot showing relative contribution of three components playing role in the swimmer orientation. Sample taken from simulation of elongated swimmers at ReLτ and aspect ratio λ = 10. . . 21

2.3 Gyrotactic micro-organisms swim with velocity vsin a direction given

by the orientation vector p set by a balance of torques. The torque due cell asymmetry (bottom heaviness: Tgrav ) tends to align the cell

to its preferential orientation along the vertical direction k whereas the torque due to flow (Tvisc) tends to rotate the cell. . . 22

2.4 Sketch of elongated swimmer modelled as prolate ellipsoids (showing swimmer with λ = 10). . . 24

3.1 Fluid statistics in fully developed turbulent channel flow for all three Reynolds number simulated, Panels: (a) Mean velocity Uxprofiles ; (b)

Mean RM S profiles at ReL

τ ; (c) Mean RM S profiles at ReIτ; and (d)

Mean RM S profiles at ReH

τ. . . 29

3.2 One-Dimensional (streamwise) energy spectra of the streamwise [(a,c,e) Ex(kx)] and spanwise [(b,d,f) Ey(kx)] surface parallel velocity

fluctua-tions. Center of channel is represented by square() and free surface is represented by filled triangles(N) . . . 31

3.3 One-dimensional dissipative spectra of the fluid, D(k) = k2_E(k),

stream-wise [(a,c,e)Ex(kx)] and spanwise [(b,d,f)Ey(kx)] surface parallel

veloc-ity fluctuations. Center of channel is represented by square() and free surface is represented by filled triangles(N) . . . 32

3.4 Components of turbulent kinetic energy(TKE) of the fluid, for (a) ReL τ,

(b) ReI

τ and (c)ReHτ. Note that x-axis is presented in log scale, to

visualize the activity near wall. . . 33

3.5 Q-criteria of open channel turbulent flow. Structures have been clipped at 1/4th_{of domain near free surface. Panel: (a)Re}L

τ, (b)ReIτ and (c)ReHτ 36

3.6 Instantaneous contour map of two-dimensional surface divergence_∇2D

computed at the free surface for (a)ReL

τ = 170 (b)ReIτ =509 (c)ReHτ =1018. 37

3.7 Correlation of swimmer clusters and surface divergence ∇2D at ReLτ:

(a) low gyrotaxis (b) intermediate gyrotaxis and (c) high gyrotaxis. Swimmers at free surface (z+< 1) are considered. Swimmers segregate in_∇2D< 0 (blue region) avoiding∇2D> 0 (red regions). . . 39

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3.8 Correlation of swimmer clusters and surface divergence _∇2D at ReIτ:

(a) low gyrotaxis (b) intermediate gyrotaxis and (c) high gyrotaxis. Swimmers at free surface (z+_{< 1) are considered. Swimmers segregate}

in∇2D< 0 (blue region) avoiding∇2D> 0 (red regions). . . 40

3.9 Correlation of swimmer clusters and surface divergence _∇2D at ReHτ :

(a) low gyrotaxis (b) intermediate gyrotaxis and (c) high gyrotaxis. Swimmers at free surface (z+_{< 1) are considered. Swimmers segregate}

in∇2D< 0 (blue region) avoiding∇2D> 0 (red regions). Note that we

are looking at million of swimmers at the free surface for high gyrotaxis. 41

3.10 Temporal evolution of correlation dimension for the three species of plankton: (a) ReL

τ (b) ReIτ and (c) ReHτ . . . 43

3.11 Relative contribution of swimmers reaching the free surface in the zone of upwelling and downwelling for (b) low gyrotaxis (c) intermediate gy-rotaxis and (d) high gygy-rotaxis at ReL

τ. Solid black line represent total

number of swimmers at the free surface, with the contribution of down-wellings represented by long-dashed line and contribution of updown-wellings by short-dashed line respectively. Also shown is the concentration pro-file (a), of swimmers in the channel for all values of gyrotaxis. . . 44

3.12 Relative contribution of swimmers reaching the free surface in the zone of upwelling and downwelling for (b) low gyrotaxis (c) intermediate gy-rotaxis and (d) high gygy-rotaxis at ReI

τ. Solid black line represent total

number of swimmers at the free surface, with the contribution of down-wellings represented by long-dashed line and contribution of updown-wellings by short-dashed line respectively. Also shown is the concentration pro-file (a), of the swimmers in the channel for all values of gyrotaxis. . . . 45

3.13 Relative contribution of swimmers reaching the free surface in the zone of upwelling and downwelling for (b) low gyrotaxis (c) intermediate gy-rotaxis and (d) high gygy-rotaxis at ReHτ. Solid black line represent total

number of swimmers at the free surface, with the contribution of down-wellings represented by long-dashed line and contribution of updown-wellings by short-dashed line respectively. Also shown is the concentration pro-file (a), of the swimmers in the channel for all values of gyrotaxis. . . . 46

3.14 Total number of spherical swimmers reaching the surface (z+ _{< 1) in}

time. Solid line represent the high gyrotaxis, dashed curve for inter-mediate gyrotaxis and dot-dashed curve for low gyrotaxis. Gray solid line showing the ideal curve the swimmers would follow in absence of turbulence. . . 48

3.15 Mean vertical orientation of cells in wall normal direction: (a) Low Reynolds number (b) intermediate Reynolds number and (c) high Reynolds number for all the gyrotaxis. In the second column show the orientation in streamwise direction for the three gyrotaxi. . . 49

3.16 Mean pdf profile of spherical swimmer cells in near free surface region (within 10 wall units): (a)ΨL(b) ΨI and (c) ΨHfor low Reynolds number. 51

3.17 Mean pdf profile of cells in near free surface region (0.01 < z+10): (a)ΨL(b) ΨI and (c) ΨH for intermediate Reynolds number. . . 52

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List of Figures v

3.18 Mean pdf profile of cells in near free surface region (0.01 < z+ _{< 10):}

(a)ΨL(b) ΨI and (c) ΨH for High Reynolds number. . . 53

3.19 Jpdf of swimmers in near free surface region (2 < z+ < 10): (a) High gyrotaxi (b) intermediate gyrotaxi and (c) low gyrotaxi for low Reynolds number. . . 54

3.20 Jpdf of swimmers in near free surface region (2 < z+_{< 10): (a) High}

gy-rotaxi (b) intermediate gygy-rotaxi and (c) low gygy-rotaxi for Inter Reynolds number. . . 55

3.21 Jpdf of swimmers in near free surface region (2 < z+_{< 10): (a) High}

gy-rotaxi (b) intermediate gygy-rotaxi and (c) low gygy-rotaxi for Hidh Reynolds number. . . 56

3.22 Correlation dimension profiles:(a),(c),(e) for ReL

τ and (b),(d),(f) ReIτ.

Panel: (a), (b) represent ΨL; (c), (d) for ΨI and (e), (f) for ΨH. . . . 58

3.23 Concentration profiles of elongated swimmers for all values of λ:(a),(c),(e) for ReL

τ and (b),(d),(f) ReIτ. Panel: (a), (b) represent ΨL; (c), (d) for

ΨI and (e), (f) for ΨH. . . 59

3.24 Mean vertical orientation of cells in wall normal direction(z) for all elongated swimmers: (a),(c) (e) at ReL

τ; (b),(d) and (e) at ReIτ, for all

values of elongated swimmers. (a),(b) ΨL, (c),(d) for ΨI and (e),(f) ΨH. 60

3.25 Mean horizontal orientation of cells in streamwise direction(x) for all elongated swimmers: (a),(c) and (e) at ReL

τ; (b),(d) and (e) at ReIτ, for

all values of elongated swimmers. (a),(b) ΨL, (c),(d) ΨI and (e),(f) ΨH. 61

3.26 Total number of elongated swimmers reaching the surface (z+ _{< 1) in}

time at ReL

τ, for all values of λ. Panel:(a) ΨL, (b)ΨI (c) ΨH. Gray

line showing the ideal curve the swimmers would follow in absence of turbulence. . . 62

3.27 Total number of elongated swimmers reaching the surface (z+ _{< 1) in}

time at ReI

τ, for all values of λ. Panel:(a) ΨL, (b)ΨI (c) ΨH. Gray

line showing the ideal curve the swimmers would follow in absence of turbulence. . . 63

3.28 Relative contribution of swimmers reaching the free surface for (a) ΨL

in the zone of downwelling (b) ΨH in the zone of downwelling, (c) ΨL

in the zone of upwelling and (d) ΨH in the zone of downwelling, at ReLτ. 64

3.29 Relative contribution of swimmers reaching the free surface for (a) ΨL

in the zone of downwelling (b) ΨL in the zone of downwelling, (c) ΨL

in the zone of upwelling and (d) ΨH in the zone of downwelling, at ReIτ. 65

3.30 Mean pdf profile of cells in near free surface region (within 10 wall units) of orientation for ReL

τ; top section (a),(b),(c) for ΨL, mid section

(d),(e),(f) for ΨI and lower section (g),(h),(i) for ΨH. x-axis represent

the orientation pi. . . 66

3.31 Mean pdf profile of cells in near free surface region (within 10 wall units) of orientation for ReI

τ; top section (a),(b),(c) for ΨL, mid section

(d),(e),(f) for ΨI and lower section (g),(h),(i) for ΨH. x-axis represent

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3.32 Joint pdf profile of cells in near free surface region ( 2 to 10 wall units from free surface) of orientation for ReL

τ; (a),(b) for ΨL,(c)(d) for ΨI

and ,(e),(f) for ΨH. Panel (a,c,e) represent λ = 2 and (b,d,f) λ = 10. . 68

3.33 Joint pdf profile of cells in near free surface region (2 to 10 wall units from free surface) of orientation for ReI

τ; (a),(b) for ΨL,(c)(d) for ΨI

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1

Introduction

In rivers, the water that you touch is the last of what has passed and the first of that which comes; so with present time.

Leonardo da Vinci

The macroscopic phenomena such as patchiness and clustering of marine landscape are influenced by the continues interaction between flow and motility of micro-swimmers. What makes this phenomenon special is the size, there is a pattern and striking beauty at all scales, large and small. To understand details of these phenomena, role of tur-bulence on these micro-organisms must be understood better and is a key research question. Among various types of micro-swimmers, phytoplankton are the most dom-inant one. In the water bodies, the plankton begin the marine food chain (Fig 1.1). Cell like structures use the sun’s energy to utilize carbon dioxide and water to create

Figure 1.1_{– Food web and role of phytoplankton(Credit: NASA images)}

oxygen and energy for themselves in the process known as photosynthesis. Despite being micro-size they are so numerous in environment that they account for about half of all the photosynthesis on this planet. Phytoplankton play two vital roles for survival of human beings. First they are the base of food chain in marine environ-ment, providing food to other plankton and fish; second their role in carbon cycle of the earth. Through photosynthesis these organisms transform inorganic carbon in the atmosphere into organic compounds, making them an essential part of Earth’s

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carbon cycle. They take up carbon dioxide from the atmosphere, when they die, they sink and they carry this atmospheric carbon to the deep sea, making phytoplankton an important factor in the climate system. In recent years this cause has become even more important as carbon dioxide, being a green house gas, is contributing to the global warming of the Earth. It is becoming a part of bigger discussion, how the invisible forest of ocean plays important role in reduction of green house gases in air. Worldwide, these plankton which act as biological carbon pump transfers huge amount (ranging in giga tonnes) of carbon from the atmosphere to the deep ocean each year. Even small changes in the growth of phytoplankton may affect atmospheric car-bon dioxide concentrations, which would feed back to global surface temperatures[47]. Their dynamics have effects on marine life and on the environment. Plankton species are known to inhibit the upper sun-lit layer of almost all oceans and bodies of fresh water. They obtain energy through the process of photosynthesis and must therefore live in the well-lit surface layer of an ocean, sea, lake or any other body of water. When observed from satellite we can see these phytoplankton in the upper layer of water bodies, forming interesting clusters which also informs that they respond to the turbulence activity of water body. The figure 1.2 shows the plankton bloom off the coast of New Zealand. It is very clear that the patterns that appear in the image suggest they are responding to vorticity and gradients of flow. These surface clusters are large scale and range upto few kilometres.

Figure 1.2_{– Plankton Bloom (Credit: NASA images by Robert Simmon and Jesse Allen, based} on MODIS data.)

There were significant attempts to capture the size and vastness of these clusters. Attempts that were made by observatory ships and tow boats to sample the concen-tration of plankton blooms. With the advances in the satellite observation and imagery these plankton blooms were captured and studied with high precision measurements. Study by Gowler et al.(1980) gives insight about the phytoplankton patchiness and oceanic structures, providing details used in satellite observation models used. They presented first output of spectral analysis from satellite imagery (data collected over 1976) of phytoplankton patchiness water motions at mesoscale level.They commented that bloom tend to occur earlier in shallow water on the continental shelf than in deeper water offshore and also mention that the relationship between phytoplankton

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Introduction 3

concentration and the amount of radiance upwelling(slight back scatter as observed from satellite) from beneath the sea surface is not in general linear [20].

Recent studies have addressed the natural perspective of oceanic patchiness by study-ing phytoplankton dynamics by followstudy-ing up the trajectories of water parcels in which the organisms are embedded [33]. With the rise in usage of satellite oceanography use of Lagrangian perspective has provided many valuable information on different aspects of phytoplankton dynamics, distribution patterns, and more recently bottom up mechanisms.

While we know surfacing and clustering of phytoplankton occur in nature frequently the role of turbulence and its contribution to surfacing and clustering and is not very well understood. How are these clusters formed? What are the factors that contribute (or counteract) to this natural phenomenon? To the best of our knowledge, the interconnected role of turbulence, and its effect on surfacing and clustering is not fully examined in detail yet. This is the motivation of present work. In this thesis, we will try to understand and answer these questions posed, by study of free-surface turbulence flow and its effect on the swimming behaviour and surface clustering of micro-swimmers like phytoplankton. We will also answer how these micro-swimmers interact with the turbulence and form these clusters at the surface.

We study this phenomenon using a simplified model of open channel water bodies, with turbulent flow where the surface waves are smooth and stratification due to temperature absent. Model used for study is capable to mimic the configuration and provide clear understanding of clustering and surfacing of inertia-less micro-swimmers. Visual example of plankton patchiness presented in 1.2 is definitely on scales larger than that addressed in this thesis, but, the work has objective to present the point that the patchiness that occur at large scales is not limited to one specific order of magnitude, patchiness is a multi-scale phenomena and occurs at scales of few kilome-tres, few meters and at cellular level as well. We in this thesis are not targeting to mimic the patchiness at natural scales but trying to contribute to the understanding of role played by turbulence in the clustering and patchiness phenomena. Dimensional size of study is discussed in chapter of methodology(2).

Main ingredients of this work are two fold, study of turbulence in open channel water bodies (focusing on activity near free-surface) and understanding the phenomenon of micro-swimmers, rising to surface to form clusters.

Turbulence is one of the most complex, unresolved issue of classical physics. It is non-linear multi-scale problem that exists all around us and plays a role in our daily life. which leaves lots of possibilities to understand and answer questions that are still unanswered related to nature.

Things happening in nature like movement and circulations of the atmosphere and oceans can help us in improving weather prediction and controlling pollution (sus-pended particles in air and water). Applications of turbulent flow also exists in large range of industrial flows where we want to take advantage of things nature has to offer us to improve overall efficiency of system and cost effectiveness. Understanding how mechanisms of transport and mixing processes work it is a challenge and needs to be addressed from both theoretical point of view (the study of chaos and diffusion in turbulence) and practical view (plankton dynamics). This is a challenging problem both from the theoretical point of view (the study of diffusion and chaos in geophysical

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systems) and for practical issues (plankton dynamics or the fate of pollutant spills) Reference[31],[15],[17],[66].

In the next sections we will provide introduction to some of the well-known character-istics of turbulent flows will be reviewed from a phenomenological point of view. It is interesting to know what happens close to the free surface will be discussed (surface turbulence). In next section general consideration about the clustering of swimmers in turbulence will be given.

Phenomenology of free-surface turbulence

Predicting and understanding of transfer of fluxes of momentum, heat and nutrient components (chemical species in general) at ocean-atmosphere interface is of impor-tance to deal with present climate change scenarios and other environmental issues. For non-breaking interfaces (absence of waves) energy transfer is controlled mainly by the dynamics of free surface turbulence. Even if it is restricted by two-dimensional space the actual behaviour of flow configuration is different from that of a simplified two-dimensional flow modelling. Direct Numerical Simulation of three-dimensional free surface channel flow allows us to demonstrate that energy transfer near surface is characterized by an inverse cascade from smaller to larger flow scales. Surface divergence allows us to demonstrate that even if the flow under the surface is three-dimensional activity at the interface is two three-dimensional and is direct result of local flow structures.

As mentioned in the introduction, turbulence is a multi-scale non-linear phenomenon. The way in which the different scales interact plays a key role in determining how the energy flows. Few studies of turbulence have been more important than the phe-nomenology of Kolmogorov-Richardson direct cascade.

To briefly explain this phenomenological picture, it is necessary to start from the Navier-Stokes equation,

∂tv + v.∇v = −1

ρ∇p + ν∆v + f (1.1)

where f is an external driving force, ν and ρ are the viscosity and density of the fluid respectively, p is the pressure. The forcing term is acting on a characteristic scale L and injects energy at an average rate of < f_{· v >= ǫ, where the brackets indicate} the average over space and time. The non-linear terms (v_{· ∇v and ∇p) preserve} the total energy and thus simply redistribute it among the modes, i.e. the different scales. Finally, the viscous term dissipates energy at an average rate νP

i,j(∂jvi)2.

The dissipation term is proportional to (∂jvi)2which, in Fourier space, means a term

proportional to the square of wave number (k2_{). This term becomes important at}

large wave numbers and thus at very small scales. At statistically stationary states, the rate of energy dissipation balances the input rate νP

i,j(∂jvi)2 ∼ ǫ.

According to the Kolmogorov-Richardson cascade, the system is forced at scale L (the injection scale) and due to nonlinear terms, large scale structures break into smaller and smaller eddies. This cascade ends at scales lDwhich are dominated by dissipation.

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1.1. Phenomenology of free-surface turbulence 5

comes from the non-linear (inertial) terms. A first way to look at the distribution of energy among the scales is by looking at the kinetic energy of the Fourier modes in an infinitesimal shell of wave-numbers (E(k)dk). It is known from experimental and numerical simulations that the energy spectrum exhibits a universal behavior which closely follows a power law E(k)∝ k−5/3 _{over the inertial range (}_1.3_).

Figure 1.3_{– Energy spectra for the case of (a) 3D turbulence and (b) 2D turbulence}

This power law seems to be independent of the fluid and the detailed geometry of forcing (with a correction due to intermittency introduced in the multifractal model by Parisi and Frisch in 1985). The two crossovers in figure 1.3 (a), are respectively the scale at which the energy is forced (L_{∼ k}_L−1)and the scale at which the energy is dissipated (lD ∼ kD−1). For k < kL the spectrum depends on the forcing or boundary

condition, for k > kL it shows a power law decay.

In two-dimensional flows the situation is rather different due to the simultaneous conservation of kinetic energy and enstrophy. As consequence there is an inverse energy cascade from small to larger scales. The interest in 2D turbulence is supported by the fact that the reversal of the energy flux has been observed in geophysical flows subjected to the Earth’s rotation [59], [39].

In 1967 Kraichnan [29] posed the basis for a theory in 2D flows. The basic idea is that the energy and the enstrophy are injected at a scale LI (figure 1.3(b)) at a

rate < f· v >= ¯ǫ and < ∇ ∧ f >= ¯η respectively. Then a double cascade estab-lishes due to non-linear transfer of energy and enstrophy among the scales: energy flows toward the larger scales (l > LI) (inverse cascade) while enstrophy towards the smaller scales (l < LI) (direct cascade). In figure 1.3(b) the energy of spectra for the 2D case is shown. From dimensional analysis and theoretical derivations, it is known that for 1/L(t) ≪ k ≪ kI, the power spectrum behaves as in 3D turbulence:

E(k) _{≈ k}−5/3_¯_ǫ−2/3 _{while for k}

I < k < kD it is obtained: E(k) ≈ k−3η¯−2/3. It is

clear from previous discussion that three- and two-dimensional flows are driven by different phenomenologies. Moreover, three-dimensional turbulence is characterized by anomalous scaling and small-scale intermittency [19], whereas the inverse cascade is apparently self-similar [4]. The transition between the two behaviours has been mainly investigated in models of turbulence where the dimension was introduced as formal parameter [7].

In Chap.3 how the confinement induced by the presence of the free surface, causes a direct or inverse cascade, will be discussed. From experiments of free-surface Rashidi

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and Banerjee [54] as well as from numerical simulation of Pan and Banerjee [45], it was observed that the turbulence near the surface is dominated by upwellings: blobs of fluid impinging on the surface originated from the hairpin vortex in the bottom boundary layer. These regions are separated by downwellings and spiral vortices attached at the surface in a process known as vortex connection. As discussed in a review article by Sarpakaya [57], the phenomenology of vortex connection is linked to the hairpin vortex ejected from the turbulent bottom boundary layer and approaching the surface. These structures break into sections which remain attached to the surface. The dynamics of upwellings and downwellings guides the energy transfer close to the free-surface as observed by Perot and Moin [50], [51] and Nagaosa [42]. Upwellings are associated to stagnation points at surface with high pressure and a negative gradient of the normal velocity. In a stagnation point, the pressure has a maximum. This leads to a negative vertical component of the pressure-strain correlation. Momentum is transferred to surface-parallel fluctuations. In downwelling, the situation is reversed, and energy is transferred from horizontal to vertical fluctuations. Magnaudet [36] showed that these phenomena depend on anisotropy of the turbulence below the free surface underlying that this behaviour could not be reproduced completely by isotropic decaying turbulence. It remains to understand if the turbulence at the surface is mainly three-dimensional or two-dimensional. Pan and Banerjee [45] noted that the one-dimensional velocity spectra shows a scaling region∼ k−3_{which is consistent with the}

prediction of two-dimensional turbulence by Kraichnan. However, Walker [64] showed that the contribution to the production of normal surface vorticity by vortex-stretching has its maximum near the free-surface. This process could not develop, by definition, in two-dimensional flows and it shows a three-dimensional nature of surface.

Phenomenology of swimmer clustering in turbulence

Many phytoplankton species exhibit motile nature and they migrate towards the re-gions rich in the nutrients and other favourable areas. Even though their swimming speeds are typically smaller than ambient flow speeds, there is well documented evi-dence that the interplay between motility and turbulence can result in complex and ecologically important phenomena. One such complex phenomena arising from motil-ity is clustering of swimmers at the surface of water bodies. Motilmotil-ity can lead to a striking focusing effect known as gyrotaxis when coupled with shear in the form of vertical gradients in horizontal fluid velocity. Gyrotaxis is the directed motility of cells arising from the combination of gravitaxis (which stabilizes cell orientation in the vertical direction, typically through bottom heaviness) and destabilization by the ambient fluid shear. This results in a balance between the gravitational torque due to the uneven density distribution within the cell, which tends to keep the center of mass below the center of buoyancy, and the hydrodynamic torque exerted by the fluid that surrounds the cell. In recent years it was discovered that plankton accumulation due to gyrotaxis can occur also in synthetic or homogeneous isotropic turbulence (HIT here after)[9],[12],[13].

To advance current understanding of how gyrotactic swimmers propel themselves near and below a free-shear air-water interface, in this thesis we investigate their dynam-ics for the reference case of turbulent open channel flow. We want to quantify the

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1.2. Phenomenology of swimmer clustering in turbulence 7

effect of the flow Reynolds number and of the cell shape on the vertical migration of swimmers at varying gyrotaxis (covering a wide range of re-orientation times) and at self-propelling speeds that are typical of the most common phytoplankton species. In recent years, a whole class of biological and physical systems which may be referred to as active matter or active particles, has been studied theoretically and experimen-tally [56], [60]. Examples of such systems are the self-propelled swimmers as phyto-plankton cells, whose interaction with the leading fluid is not trivial. A wide variety of external factors including nutrient concentration, gravity, and the rate of strain of the fluid affect the orientation of the swimming velocity of the cells [14],[13]. The resulting distribution is far from uniform.

Micro-organisms concentrate in the turbulent regions close to the surface and to the sea bed where the level of turbulence is also affected by external factors. The motion of an individual micro-organism is determined by its swimming and by the advection of the fluid, where vorticity and rate of strain re-orient it, and by the response to external stimuli and biases such as nutrient concentration, gravity and light. Phytoplanktons can respond to these stimuli and have been categorized based on these behaviours, such as geotaxis (Adams et al. 1999 [1]), phototaxis (Martin 1983 [38]), gyrotaxis (Kessler 1985 [14]) and chemotaxis (Adler et al. 1974 [2]). Gradients in concentration of plankton span a wide range of length scales ranging from regions of persistent upwelling at the equator with length scales of thousands of kilometres to micro-scale patchiness that occurs at the scale of centimetres. To deal with aquatic flows, motile micro-organisms have developed many ways of adaptation [14]. One of those tends to orient the cell’s swimming direction upward against gravity. The resulting balance between gravitational torque due to the asymmetric distribution of density and the hydrodynamic torque is known as gyrotaxis and will be examined in this thesis. Prants et al. (2017) [53] while commented that complex patterns arising from the coupling between swimmer motion and temporal variability of the velocity field, have roots in dynamical systems and chaos theory and play role in the patchiness observed. Lagrangian method based study takes this factor into account and that the velocity field is continuously changing while swimmers are being advected.

It is interesting to note that clustering that is particularly observed in inertial particles can also occur in the case of micro-swimmers which are specially inertia-less. When the swimmers are small and their densities are comparable with the carrying fluid they usually follow the local fluid motion like floaters [34],[35]. If any of these conditions are not fulfilled, the dynamics of the particles deviate from those of the fluid. The deviation can be a result of property of inertia (these particles being generically called ”inertial particles”) or due to fact that these floaters are active swimmers which possess their own ability to swim and choose direction to swim. The situation in this case becomes more complicated if the particles/swimmers are ”active”, e.g. if they can self-propel, such as microorganisms in the ocean (which will be discussed in Chap. 3

3.2), or if the particles exchange mass, momentum or energy with the carrying fluid, such as water droplets in clouds. Here in this thesis the case of inertia-less floaters which are active swimmers will be considered.

The clustering of particles/swimmers can occur in various cases. The authors in [52], [24] find that inertial particles sample preferentially flow regions in which their local concentration is higher with respect to the initial conditions. On the other hand,

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clustering could occur also in the case in which particles (even tracers) are advected by a compressible flow and follow the streamline (Boffetta et al [5]). The picture which arises is that, clustering is a general consequence of compressibility. Clusters that form at free surface due to various bias are known to be time persistent, which means that these cluster of swimmers stay at free surface even if the fluid event beneath them has been changed long ago in time [34],[35].

Gyrotaxis was discovered in 1985 by John Kessler [49],[48] by showing that phyto-plankton cells tend to collect along the centerline of a laminar Poiseuille flow. Recent experimental observation shows that gyrotactic algae cells could be trapped in hori-zontal layers in laminar vertical shear. The phenomenon called ‘gyrotactic trapping’ [14], [13] occurs when vertically migrating cells accumulate where vertical gradients in horizontal velocity exceeds a critical shear threshold, causing cells to tumble end over end. Only recently it was demonstrated that phytoplankton clustering could occur also in turbulence [12], [9], [10], [68], [11]. Numerical simulations have shown that gyrotactic algae generate small-scale clusters with fractal distribution [12], [68]. One way to classify different phytoplankton species is by the shape factor. While some types of plankton are spherical in shape (referred to as spherical swimmers hereinafter) and other types of plankton exist in various forms of elongated shapes (referred to as elongated swimmers hereinafter). Assigning realistic and robust values to the many associated parameters in global circulation models is an active area of research. More-over, since global warming has effects on the carbon cycle of the environment, it is fundamental to investigate how plankton dynamics contributes to it, which is still an open question.

Large section of marine life is not spherical, so it is interesting to have a look at how elongated species of planktons have advantage over their spheroidal shaped counter-parts.

In the available literature Zhan et al [68] presented the effect of elongation in HIT. They observed that elongated micro-orgamisms show some level of clustering in the case of swimmers without any preferential alignment whereas spherical swimmers re-main uniformly distributed. Micro-organisms with one preferential swimming direction (e.g. gyrotaxis) still show significant clustering if spherical in shape, whereas prolate swimmers remain more uniformly distributed. They suggested that due to their large sensitivity to the local shear, these elongated swimmers react slower to the action of vorticity and gravity and therefore do not have time to accumulate in a turbulent flow. These results show how purely hydrodynamic effects can alter the ecology of microorganisms that can vary their shape and their preferential orientation.

While spherical swimmers preferentially sample local downwelling flow, for elongated swimmers Borgnino et al.[6] and other similar studies observe that in homogeneous isotropic turbulence spherical swimmers cluster more at small stability and speed numbers, while for large values of the parameters elongated cells concentrate more. Study by Parsa et al.(2012) [46] show that the rotation rate for elongated shaped pas-sive swimmers is influenced directly by alignment, particle orientations become well correlated with the velocity gradient tensor, and this alignment depends strongly on the shape of swimmer. This study explains how settling larvae changing the offset of the centres of buoyancy and of gravity can preferentially accumulate in updrafts, favourable for dispersal, or downdrafts, favourable for settlement, thus exploiting the

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1.2. Phenomenology of swimmer clustering in turbulence 9

hydrodynamics of the vorticity near the sea bed (Grunbaum & Strathmann 2003 [21]). Studies [68] also suggest that micro-organisms like the dinoflagellate Ceratoco-rys horrida, are able to reversibly change its morphology in response to variations of the ambient flow (Zirbel, Veron & Latz 2002 [69]), can exploit hydrodynamics effects to increase or decrease encounter rates in an active way. In general, the clustering pattern is more pronounced for prolate swimmers in chaotic flow and have clear ad-vantage over spherical particles which cannot enter the clusters if initially outside of the chaotic zone[63]. Spherical swimmers can be trapped for long time while the elon-gated counterparts are observed to have ability to enter and leave the cluster zones faster.

In this study, we have considered both spherical and elongated swimmers (e.g., the Chlamydomonas augustae), yet, many motile organisms are characterized by a non-spherical shape and changes in morphology can dramatically affect the function of these organisms in fluid flows: For example, elongation changes flow-induced cell rota-tion, thus affecting nutrient uptake [22]. Non-spherical particles are known to respond not only to the turbulent vorticity but also to turbulent strain: As a result, the orien-tation of non-spherical swimmers become correlated with the velocity gradient tensor, and the resulting alignment depends strongly on the shape.[23], Zhan et al. [68] have shown that, even in the absence of both stratification and gyrotaxis, elongated micro-swimmers in homogeneous isotropic turbulence exhibit some level of clustering whereas spherical swimmers remain uniformly distributed. An opposite behaviour is observed for gyrotactic micro-swimmers: elongated ones remain more uniformly distributed, whereas spherical ones show significant clustering.

This finding was explained considering the shape-dependent sensitivity to the local shear: Elongated swimmers react more slowly to the combined action of vorticity and gravity and therefore need longer time to accumulate. Clearly, the coupling with stratification is expected to further modify swimmer dynamics, with possible effects on their capability to form sub-surface clusters and/or thin layers near the thermocline. However, this effect of elongation on micro-swimmers has have not been studied before in open channel turbulent flow configuration. Last section in results chapter 3.3 is dedicated to discussion on the effect of elongation in addition to reorientation time and Reynolds number effect.

The present work in this thesis is structured as follows:

• Chapter 2 Methodology: the governing equations and the numerical method are introduced and discussed in detail; in the first section, the geometry of problem and the equation for the open channel configuration for turbulence, are reported. In the second and third sections the numerical approach and the solution procedure are provided; in the third and fourth sections the spectral approximation of the solution is shown and the discretized set of equations is reported. In the last section information about lagrangian tracking for active floaters is provided.

• Chapter 3 Results and Discussion: Bias in preferential clustering of inertia-less swimmers at the free-surface is studied. We start with a discussion about turbu-lence and its behaviour near free-surface. In the first section once the flow field

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statistics for the neutrally buoyant turbulent open channel flow are provided, we discuss about the main contributing components of energy budget and then introduce the energy spectra and dissipative spectra at the free surface, which will be compared with the centreline of channel to make the effect of free-surface clear. In the second section, the case of spherical self-propelled inertia-less par-ticles and their surfacing phenomena and clustering is studied and the statistics with respect to temporal and spatial behaviour are presented and discussed. Fi-nal and the third section of results the elongated swimmers in turbulent open channel flow will be discussed, linked to the dynamics of turbulence.

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2

Methodology

Deep in the human unconscious is a pervasive need for a logical universe that makes sense but the real universe is always one step beyond logic. Zbigniew Herbert

This chapter shall give a brief overview of the governing equations of motion and the observed phenomena in fully developed turbulence. In this chapter we provide overview of the equations used to describe the dynamic behaviour of an incompressible viscous fluid flow, in the first part. In the second part of the chapter the numerical method employed to solve the balance equations is presented in detail. In the last and third section of this chapter the Lagrangian Particle Tracking algorithm is introduced.

Governing equations

The numerical analysis of dispersed turbulent multiphase flows requires the proper definition and solution of a set of governing equations that describe the behaviour of each constituent of the system. Depending on the physical properties of the con-stituents and different regimes considered, their governing equations can be based on diverse approaches: i.e. the continuous fluid phases are usually described by a set of Eulerian continuity and momentum balance equations (Navier-Stokes equations), while different approaches can be adopted for the dispersed phases.

In the continuum approach the dynamics of fluids are governed by the balance equa-tions of mass, momentum and energy. These equaequa-tions, together with the constitutive laws for the stress tensor can describe the motion of any kind of fluid flow. With reference to the schematic of figure2.1, we consider an incompressible and Newtonian turbulent flow of water in a plane channel.

The reference geometry consists of two infinite flat parallel walls; the origin of the coordinate system is located at the center of the channel and the x, y and z axis point in the streamwise, spanwise and wall-normal directions respectively. Indicating with 2h the channel height, the size of the channel is 2πh× πh × h, in x−, y− and z− axis respectively.

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Figure 2.1_{– Sketch of the computational domain}

For the fluid velocity, we enforce no-slip boundary conditions at the bottom wall and free slip boundary condition at the top wall (Free Surface). In the X and Y direction periodic boundary conditions are applied.

The governing equations are non-dimensionalised with the channel height h, the fric-tion velocity uτ=

q _τ

w

ρref, where subscript w indicates wall. The shear stress τw used

to define the friction velocity is the horizontally averaged value of τ at the wall which must balance the vertically integrated pressure gradient for the steady state.

With the previous choices the governing balance equations in dimensionless form read as: ∂ui ∂xi = 0, (2.1) ∂ui ∂t = Si+ 1 Reτ ∂2_u i ∂x2 j ! −_∂x∂p i (2.2) where ui is the ithcomponent of the velocity vector.

The S-terms contain the non-linear convective terms, the dimensionless mean pressure gradient and the buoyancy term:

Si=−uj

∂ui

∂xj

+ δi,1 (2.3)

In the above equations, δi,1 is the mean pressure gradient that drives the flow in the

streamwise direction.

Eqs. 2.1and2.2are subject to the following boundary conditions:

@W all : ux= uy= uz= 0 (2.4)

@F ree_{− surface :} ∂ux ∂z =

∂uy

∂z = uz= 0 (2.5)

The dimensionless Reynolds number is defined as Reτ =uτh

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2.2. Numerical approach 13

where ν is the kinematic viscosity of the fluid, h is channel height and u is the shear velocity.

Numerical approach

Eqs. 2.1-2.2are discretized using a pseudo-spectral method where the field variables are transformed in wavenumber space, through Fourier representation for homogeneous (periodic) directions x and y and Chebychev representation for non-homogeneous (wall normal) direction z. A two-level explicit Adams-Bashfort scheme for the non-linear terms and an implicit Crank-Nicolson method for the viscous terms are employed for the time advancement.

As it is quite commonly done in pseudo-spectral method, the convective non-linear terms are first computed in physical space and then transformed in the wavenumber space using a de-aliasing procedure based on the 2/3-rule; derivatives are evaluated directly in the wavenumber space to maintain spectral accuracy. More details will be provided in the next section.

Throughout this thesis we will analyze different type of swimmers in the free surface turbulence (spherical/elongated). However, the structure of the governing equations for fluid does not change from one situation to another, that we have observed in Sec. 2.1. As a consequence, a unique numerical scheme can be adopted to analyze a different set of swimmers properties. Without loss of generality, we will take the simulations with free surface turbulence as reference. Indeed, the numerical approach developed for the solution of Eqs.2.1-2.2 is discussed. In this section it is preferred to refer to x,y,z (streamwise, spanwise and wall normal direction respectively) to as x1,x2, x3 and the same is true for ux, uy, uz which will be referred to as u1, u2, u3.

Solution procedure

The present scheme solves for the balance equations of motion (Eqs. 2.1-2.2) through the elimination of pressure. The pressure field can be removed upon taking the curl of Eq. (2.2), to give: ∂ωk ∂t = ǫijk ∂Sj ∂xi + 1 Re∇ 2_ω k (2.7)

where ωk = ǫijk∂u_∂xj_i is the kthcomponent of the vorticity vector. Taking the curl twice,

of Eq. (2.2) and using Eq. (2.1) together with the vectorial identity,

∇ × (∇ × v) = ∇(∇ · v) − ∇2_v _(2.8)

which is a 4th_{order equation in u}

i, we can obtain: ∂(∇2_u i) ∂t =∇ 2_S i− ∂ ∂xj ∂Sj ∂xj + 1 Re∇ui (2.9)

Eqs.(2.1)-(2.2) can be written with respect to the normal components, i.e. for ω3 and

u3: ∂ω3 ∂t = ∂S2 ∂x1 − ∂S1 ∂x2 + 1 Re∇ 2_ω 3 (2.10)

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∂(∇2_u 3) ∂t =∇ 2_S 3− ∂ ∂x3 ∂S2 ∂x2 + 1 Re∇ 4_u 3 (2.11)

These two equations are numerically solved for ω3and u3. With ω3and u3 known, u1

and u2can be obtained by solving the following equations simultaneously,

∂u1 ∂x1 +∂u2 ∂x2 =₋∂u3 ∂x3 , (2.12) ∂u2 ∂x1 + ∂u1 ∂x2 =−ω3. (2.13)

Here, Eqs. (2.12) and (2.13) are derived respectively, from continuity and from the definition of vorticity. Although not needed for the time advancement of the solu-tions, pressure can be obtained by solving a Poisson-type equation after all velocity components have been found:

∇2p = ∂Sj ∂xj

(2.14)

Spectral representation of solutions

To represent the solution in space, finite Fourier expansion in the homogeneous (x1

and x2) directions is used:

f (x1, x2, x3) = N1 2 X |n1| N2 2 X |n2| ˆ f (k1, k2, x3)ei(k1x1+k2x2) (2.15)

where ˆf represents the Fourier coefficients of a general dependent function, i =√_−1, N1and N2are the number of Fourier modes retained in the series, and the summation

indices n1 and n2 are chosen so that−N₂1 + 1≤ n1 ≤ N₂1 and −N₂1 + 1≤ n1≤ N₂1.

The wavenumbers k1 and k2 are given by:

k1= 2πn1 L1 (2.16) k2= 2πn2 L2 (2.17) with L1and L2being the periodicity lengths in the streamwise and spanwise directions.

Because of the orthogonality of the Fourier functions, the Fourier transform ˆf can be obtained as: ˆ f (k1, k2, x3) = 1 N1N2 N1 2 X |n1| N2 2 X |n2| f (x1, x2, x3)e−i(k1x1+k2x2) (2.18)

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2.4. Spectral representation of solutions 15

with x1and x2 are chosen to be the transform locations

x1= n1 N1 L1 (2.19) x2= n2 N2 L2 (2.20)

In the cross-stream (wall-normal) direction x3, Chebyshev polynomials are used to

represent the solution,

ˆ f (k1, k2, x3) = N′ 3 X n3=0 a(k1, k2, n3)Tn(x3), (2.21)

where the prime denotes that the first term is halved. The Chebyshev polynomial of order n3 in x3is defined as

Tn3(x3) = cos(n3arccos(x3)), (2.22)

with −1 ≤ x3≤ 1. Orthogonality also exist for Chebyshev polynomials, which leads

to the following inverse transformation:

ˆ a(k1, k2, n3) = 2 N3 N′ 3 X n3=0 ˆ a(k1, k2, x3) Tn3(x3). (2.23)

In physical space the collocation points along the cross-stream direction are related to Chebyshev indexes in the following way:

x3= cos

n3π

N3

. (2.24)

The advantage of using Chebyshev polynomials to represent the solution in the cross-stream direction is that such a representation gives very good resolution in the regions close to the boundaries, because the collocation points bunch up there. In wall bounded flows, resolution close to the wall is very important, because this is the region where large gradients occur. For in-depth discussion on Chebyshev polynomials and their applications in numerical analysis, see Fox and Parker [18]. Therefore the spectral representation (in all three directions) of a generic dependent variable takes the final form f (k1, k2, x3) = N1 2 X |n1| N2 2 X |n2| N′ 3 X n3=0 ˆ a(k1, k2, n3) ei(k1x1+k2x2)Tn3(x3). (2.25)

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Discretization and solution of the equations

Momentum equations

With the spectral representation given by Eq.(2.4.1), Eq.(2.3.4) can be written as: ∂ ∂t ∂2 ∂x2 3 − k 2 ˆ u3= ∂2 ∂x2 3 − k 2 ˆ S3 −_∂x∂ 3 ik1Sˆ1+ ik2Sˆ2+ ∂ ∂x3 ˆ S3 + 1 Re ∂2 ∂x2 3 − k 2 ∂2 ∂x2 3 − k 2_u_ˆ 3, (2.26) where k2_{= k}2

1+ k22, Time advancement of Eq. (2.26) is done using a two level explicit

Adams-Bashfort scheme for the convective terms and an implicit Crank-Nicholson method for the diffusion terms. The time-differenced form of Eq. (2.24), based on the above schemes, is ∂2 ∂x2 3 − k 2 (ûn+13 − ûn3) ∆t = 3 2 ∂2 ∂x2 3 − k 2_S_ˆn 3 − 1 2 ∂2 ∂x2 3− k 2_S_ˆn−1 3 −_∂x∂ 3 ik1 3 2Sˆ n 1 − 1 2 ˆ S1n−1 −_∂x∂ 3 ik2 3 2Sˆ n 2 − 1 2 ˆ S2n−1 − ∂ 2 ∂x2 3 3 2Sˆ n 3 − 1 2 ˆ S₃n−1 + 1 Re ∂2 ∂x2 3 − k 2 ∂2 ∂x2 3 − k 2 (ûn+13 − ûn3) 2 (2.27)

where superscripts n_{− 1, n and n + 1 indicate three successive time levels. By defining} γ = ∆t

2Re we can rearrange Eq. (2.27):

1− γ ∂ 2 ∂x2 3 − k 2 ∂2 ∂x2 3 − k 2 ˆ un+13 = k2 3 2Sˆ n 3 − 1 2 ˆ S3n−1 ∆t −_∂x∂ 3 ik1 3 2Sˆ n 1 − 1 2Sˆ n−1 1 ∆t −_∂x∂ 3 ik2 3 2Sˆ n 2 − 1 2Sˆ n−1 2 ∆t + γ ∂ 2 ∂x2 3 + (1− γk2₎ ∂2 ∂x2 3 − k 2 ˆ un3. (2.28)

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2.5. Discretization and solution of the equations 17

Introducing (β )2 ₌ 1+γk2

γ and recalling that ∂ ˆu3

∂x3 =−ik1uˆ1− ik2uˆ2 from continuity,

we can manipulate the last term on the rhs of Eq. (2.28): − γ ∂ 2 ∂x2 3 − β 2 ∂2 ∂x2 3 − k 2 ˆ un+13 = − k2 3₂Sˆ3n− 1 2 ˆ S₃n−1 ∆t_{− k}2 γ ∂ 2 ∂x2 3 + (1_{− γk}2) ˆ un 3 −_∂x∂ 3ik1 3 2Sˆ n 1 − 1 2 ˆ S1n−1 ∆t−_∂x∂ 3ik1 γ ∂ 2 ∂x2 3 + (1− γk2₎ ˆ un 1 −_∂x∂ 3 ik2 3 2Sˆ n 2 − 1 2 ˆ S2n−1 ∆t₋ ∂ ∂x3 ik2 γ ∂ 2 ∂x2 3 + (1_{− γk}2) ˆ un 2. (2.29)

By introducing the historical terms, which take into account the effect of previous time instance, on non linear convective terms:

ˆ Hn 1 = 3 2Sˆ n 1 − 1 2Sˆ n−1 1 ∆t + γ ∂ 2 ∂x2 3 + (1_{− γk}2₎ ˆ un 1, ˆ H2n = 3 2Sˆ n 2 − 1 2Sˆ n−1 2 ∆t + γ ∂ 2 ∂x2 3 + (1− γk2₎ ˆ un2, ˆ Hn 3 = 3 2Sˆ n 3 − 1 2Sˆ n−1 3 ∆t + γ ∂ 2 ∂x2 3 + (1_{− γk}2₎ ˆ un 3, (2.30) Eq. (2.29) becomes: ∂2 ∂x2 3 − β 2 + ∂ 2 ∂x2 3− k 2 ˆ un+1₃ = 1 γ(k 2_H_ˆn 3 + ∂ ∂x3 (ik1Hˆ1n+ ik2Hˆ2n) (2.31) If we put ˆHn= k2Hˆ3n+∂x∂3(ik1H n

1+ik2H2n) we come to the final form of the equation:

∂2 ∂x2 3 − β 2₊ ∂2 ∂x2 3 − k 2_u_ˆn+1 3 = ˆ Hn γ . (2.32) Defining ˆφ = _∂x∂22 3 − k 2_ˆ_un+1

3 the above fourth-order equation becomes a system of

two second-order equations:

∂2 ∂x2 3 − β 2_{φ =}_ˆ Hˆn γ (2.33) ∂2 ∂x2 3 − k 2 ˆ un+13 = ˆφ (2.34)

These equations are solved with the following four boundary conditions: ˆ

un+1₃ (_{±1) = 0} (2.35a)

∂ ˆun+1₃ ∂x3

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∂2_u_ˆn+1 3

∂x2 3

(_{−1) = 0} (2.35c)

where x3 = z = +1 indicates the wall and x3 = z = −1 indicates the surface. The

conditions (b) and (c) are obtained from applying the continuity equation at the wall (condition (b)) and its derivative in x3 at the free-slip surface (condition c). The lack

of real boundary conditions for ˆφ can be circumvented by decomposing it into three parts:

ˆ

φ = ˆφ1+ ˆAφ2+ ˆBφ3 (2.36)

where constants ˆA and ˆB are to be determined. These three individual components of ˆφ satisfy: ∂2 ∂x2 3 − β 2 ˆ φ1= ˆ Hn γ , φˆ1(1) = 0, φˆ1(−1) = 0; ∂2 ∂x2 3 − β 2 ˆ φ2= 0, φ2(1) = 0, φˆ2(−1) = 1; ∂2 ∂x2 3 − β 2 ˆ φ3= 0, φ3(1) = 1, φˆ3(−1) = 0; (2.37)

Likewise ˆun+13 can be splitted into:

ˆ

u3= ˆu3,1+ ˆAu3,2+ ˆBu3,3 (2.38)

Once the solution of Eqs. (2.37) has been carried out, we can solve: ∂2 ∂x2 3 − β 2 ˆ u3,1 = ˆφ1, uˆ3,1(1) = 0, ˆu3,1(−1) = 0; ∂2 ∂x2 3 − β 2 u3,2 = φ2, u3,2(1) = 0, u3,2(−1) = 0; ∂2 ∂x2 3 − β 2 u3,3 = φ3, u3,3(1) = 0, u3,3(−1) = 0; (2.39)

Finally the unknown constants ˆA and ˆB are determined by applying the boundary conditions of Eq.(2.35b) to ˆun+1₃ written in terms of its components:

∂ û3,1 ∂x3 (1) + Â∂u3,2 ∂x3 (1) + ˆB∂u3,3 ∂x3 (1) = 0 ∂ û3,1 ∂x3 (_{−1) + ˆ}A∂u3,2 ∂x3 (_{−1) + ˆ}B∂u3,3 ∂x3 (_{−1) = 0} (2.40)

With ˆA and ˆB determined, ˆun+13 is fully known. The above system of equations

are solved using a Chebyshev method so the solutions ˆun+13 will be represented by

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2.5. Discretization and solution of the equations 19

Therefore, the solution ˆun+13 will be function of k1, k2and n3:

ˆ

un+1₃ = un+1₃ (k1, k2, n3) (2.41)

where 0 < n3< N3, N3 being the number of coefficients and collocation points in the

wall-normal direction. Recalling Eq.(2.15), the solution in space will read as:

un+13 (x1, x2, x3) = N1 2 X |n1| N2 2 X |n2| N′ 3 X n3=0 ˆ un+13 (k1, k2, n3)ei(k1x1+k2x2)Tn3(x3) (2.42)

The other two velocity components will be determined through the normal vorticity component ˆω3. Following a discretization procedure similar to that of Eq. (2.8) we

can write: ∂2 ∂x2 3 − β 2_ω_ˆn+1 3 =− (ik1Hˆ2n− ik2Hˆ1n) γ (2.43)

with boundary conditions: ˆ ωn+13 = ik1uˆ2− ik2uˆ1= 0, x3= +1; (2.44) ∂ ˆωn+13 ∂x3 = ik1 ∂ ˆu2 ∂x3 − ik2 ∂ ˆu1 ∂x3 = 0, x3=−1; (2.45)

Once vorticity is known, ˆun+11 and ˆun+12 can be determined from solving:

−ik1uˆ1+ ik1uˆ2= ˆω3n+1, (2.46)

−ik1uˆn+11 − ik2uˆn+12 =−

∂ ˆω3n+1

∂x3

, (2.47)

that come from the definition of ˆω3 and from continuity equation, respectively.

Pres-sure can be calculated by the transformed Poisson equation Eq (2.14) ∂2 ∂x2 3 − β 2 ˆ pn+1= ik1Sˆn+11 + ik2Sˆ2n+1+ ∂ ˆS3n+1 ∂x3 . (2.48)

The above scheme is used to evaluate the solutions in Fourier-Chebyshev space for k2

6= 0. The case k2_{= 0 corresponds to the solution averaged over an x}

1 − x2 plane.

In this case the solution procedure is simpler: upon time discretization the x1 and x2

components of Eq. (2.2) in Fourier-Chebyshev space after time discretization give: ∂2 ∂x2 3− 1 γ ˆ un+11 =− ˆ H1 γ , (2.49) ∂2 ∂x2 3− 1 γ ˆ un+12 =− ˆ H2 γ , (2.50)

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that can be solved by applying the following boundary conditions: ˆ un+11 = ûn+12 = 0, x3= +1 (2.51) ∂ ûn+11 ∂x3 = ∂ û n+1 2 ∂x3 = 0, x3=−1 (2.52)

Using the continuity equation, Eq.(2.5.22), with k1 = k2 = 0 and the condition

ˆ

un+13 (±1) = 0 one can show that ˆun+13 = 0. To calculate ˆpn+1 it is necessary to

recall the transformed momentum equation, Eq.(2.3), in the x3 direction for k2 = 0

and ˆun+13 = 0: we have ˆpn+1=−(

\ ˆ

un+13 ˆun+13 ).

Lagrangian Tracking Model for swimmer

To track the swimmers we implement the Lagrangian tracking model for tracers as we consider the swimmers to be inertia-less. These swimmers are not passive tracers but are active tracers as they possess their own swimming velocity which allows them to escape the streamlines of fluid [49],[48]. The choice of modelling the swimmers to be inertia-less is due to the fact that these swimmers are very small in size and their densities don’t affect the fluid properties.

Swimmers used in this work are shown in visual form in the figure2.3 and2.4. Swimmers are treated with point-particle approach. They are injected into the flow at a concentration regime of dilute system conditions and therefore the effect of swim-mers on turbulence is neglected (one way coupling approach) as well as inter-swimmer interaction. Swimmers are invisible to each other in this scenario.

The dynamics of these swimmers is described by a set of ordinary differential equations for the position, xp, and the velocity, vp, of the swimmers. In vector form:

dxp

dt = uf(xp) + vsp, (2.53)

where uf(xp) is the fluid velocity at the position of the swimmer and p is the

orienta-tion of each swimmer which evolves in response to the biasing torques acting upon it. The fluid velocity and the fluid velocity gradients are interpolated at particle position using second order Lagrange polynomials. Computational code developed to solve the equations is capable of using higher order polynomials, but the choice was made to gain computational time when simulating a large number of swimmers. Swimmer position denoted by vector xp has components xp, yp and zp; swimmer velocity represented by

vector vp will have components up, vp and wp in stream-wise, span-wise and normal

direction respectively. In similar way the Swimmer position denoted by vector p will have components px, py and pz in respective directions. Same nomenclature will be

followed in the chapter of results and discussion3. dp dt = 1 2B[k− (k . p)p] + 1 2ωf(xp)∧ p + β(I − pp).p.E (2.54) We try to understand these three terms on right hand side of Eq.(2.54) separately. Lets call these three terms as A,B,C in this section.

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2.6. Lagrangian Tracking Model for swimmer 21 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 200 400 600 800 1000 1200 1400 T+ Gyrotaxis component Vorticity component Elongated component

Figure 2.2_{– Plot showing relative contribution of three components playing role in the swimmer} orientation. Sample taken from simulation of elongated swimmers at ReL

τ and aspect ratio λ = 10.

• Term A: The first term describes the tendency of a cell to remain aligned along the vertical direction due to bottom-heaviness. Here k = [0, 0, 1] is a unit vector in the vertical upwards direction, which is also the preferred direction of swim-ming.

B = µα⊥/(2~ρg) is the gyrotactic reorientation time (~ denotes the

center-of-mass offset relative to the center-of-buoyancy, α⊥ is the dimensionless resistance

coefficient for rotation about an axis perpendicular to p. B is a key parameter for the dynamic of the cells since it represents the characteristic time a perturbed cell takes to return to vertical if ω = 0.

• Term B: The second term on the right hand side of equation captures the ten-dency of vorticity to overturn a cell by imposing a viscous torque on it. Here where ω =_{∇ ∧ u is the fluid vorticity and p being the orientation vector.} • Term C: The third term (commonly called as Jeffery term (Jeffery 1922)[25]),

added takes care of morphological effect (shape factor). β = (λ2

− 1)/(λ2_{+ 1)}

is the cell eccentricity and λ is the aspect ratio. β = 0 for spherical swimmers. Here E is the symmetric part of the deformation tensor and I is the identity matrix.

To show the relative importance of the three terms in the equation we plotted the contribution in the normalized graph as shown in figure (2.2). This is real time plot from the simulation carried out to show the importance of all the three terms is comparable. Parameters of simulation used for this specific graph are Reτ = 170,

high gyrotaxis which are fast to reorient (φ = 1.113) and λ = 10 (highest value of the aspect ratio simulated).

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All three components contribute equally to the orientation evolution of swimmers in the turbulent field and hence forming the motivation to study the elongation effect of swimmers (details in chapter3). In this plot all three components fluctuate along the mean value and don’t change much with respect to initial values and with respect to each other as well. We don’t see any cross over in the graph. Another significant fact we would like to mention here is that the contribution curve of elongated component representing term C increases with the increase of aspect ratio or elongation but never crosses the curves of other components.

Periodic boundary conditions are imposed on the swimmers moving outside the compu-tational domain in the homogeneous directions x and y. In the wall-normal direction, the swimmers reaching the free surface still obey the force balance, and elastic rebound is enforced at the bottom wall. A 4th-order Runge-Kutta scheme is used to advance Eqs.(2.53)-(2.54) in time.

Cell positions and swimming direction were integrated using the non-dimensional form of equations2.55and2.56: dx∗ dt∗ = u ∗_(X∗ ) + Φp (2.55) dp dt∗ = Ψ[k− (k.p)p] + 1 2ω ∗ ∧ p + βp.E∗.(I− pp) (2.56) where time, lengths and velocity were non-dimensionalised by using the friction veloc-ity uτ and fluid viscosity ν. Dimensionless parameters are Φ = vs/uτ and Ψ = _2B1 _uν2

τ.

At each time step of the simulation, the local fluid velocity, vorticity and gradients at the particle positions, were calculated using Lagrangian interpolation.

T

visc

T

grav

y

z

v

s

u

_(z)

mg

x

Figure 2.3_{– Gyrotactic micro-organisms swim with velocity v}_s_{in a direction given by the} orien-tation vector p set by a balance of torques. The torque due cell asymmetry (bottom heaviness: Tgrav) tends to align the cell to its preferential orientation along the vertical direction k whereas

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2.7. Simulation parameters 23

Code Implementation

Computational code used to solve the governing equations was developed in-house to perform simulations with high range of machine independence. Eularian code was originally written in Fortran-2003 along with the implementation of parallel program-ming paradigm. Code uses the MPI standard and the parallelization is achieved by one-dimensional domain decomposition. Computations are split among the processes into sub-domains either in the z-direction (wall-normal) or in the y-direction (span-wise). First implementation of Eurlarian part of this code in the parallel configuration was performed in 2002 and implementation of the code in the open channel configu-ration was performed in 2013. Details can be found in [34],[37]. Lagrangian tracking code development and implementation was done as part of thesis using parallelization scheme for swimmers as well. It takes into account the non-homogeneity of swimmer concentration distributed in the flow field and the storage of fluid flow data which is distributed amongst the processors. Number of swimmers were distributed equally between the processors and as the simulation is performed each processor tracks the sub-set swimmers that were initially assigned to it. Such direct linkage between the local particle concentration and processor is important for numerical work load distri-bution over the calculation domain to perform parallel Lagrangian tracking efficiently. This choice of parallelization scheme is further justified as the number of swimmers are expected to rise with the simulation time and thus work load needs to be distributed effectively(and equally) which otherwise will result in heavy load on processor handling the swimmers at the free surface.

Simulation parameters

In this section the parameters presented are relative to three values of shear Reynolds number which are: ReL

τ = 170, ReIτ = 509 and ReHτ = 1018 respectively. In the

dimensional terms, they correspond to shear velocity of uL

τ = 0.00601ms−1, uIτ =

0.018ms−1 _{and u}H

τ = 0.036ms−1. These parameters can also be presented in the

terms of length size to make contact with ecological applications. If the shear velocity of the open channel is kept fixed the corresponding values of channel height are: 0.33m for ReH

τ , 0.17m for ReIτ and 0.056m for ReLτ. The size of computational domain in

wall units is L+

x × L+y × L+z = 2πReτ × πReτ × Reτ, discretized with grid points

corresponding to kx = i2π/Lx, ky = j2π/Ly and Tn(z) = cos[n.cos−1(z/h]) where

values of i, j and n are shown in the table [2.1]. Note that these values of i,j,n are before de-aliasing. kx ky n ∆x+ ∆y+ ∆z+M ∆z+m ReL τ 128 128 129 8.3448 4.172 2.066 0.025 ReI τ 256 256 257 12.492 6.246 3.1171 0.009 ReH τ 512 512 513 12.492 6.246 3.1171 0.009 Table 2.1_{– DNS parameters}

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2.7. Simulation parameters 25

developed turbulent field) to start simulation. Number of swimmers simulated kept same as well to make one on one comparison possible, N = 106 _{for Re}L

τ and ReIτ.

Boundary conditions in each direction remains the same. In wall normal direction, swimmers reaching the free-slip will align themselves to stream wise direction while no special boundary condition has been implemented for this cause. We carry out our study for elongated swimmers with values of β and λ as shown in the table below, at the Reτ = 170 and Reτ = 509 and for continuity we will represent them as ReLτ and

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