Three-dimensionality of shallow flows
Citation for published version (APA):Cieslik, A. R. (2009). Three-dimensionality of shallow flows. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR640133
DOI:
10.6100/IR640133
Document status and date: Published: 01/01/2009 Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
providing details and we will investigate your claim.
Three-dimensionality
of
Printed by Universiteitsdrukkerij TU Eindhoven, Eindhoven, The Nether-lands
A catalogue record is available from the Eindhoven University of Technology Library: http://library.tue.nl/catalog/LinkToVubis.csp?DataBib=6:640133
Cie´slik, Andrzej Ryszard
Three-dimensionality of shallow flows / by Andrzej Ryszard Cie´slik. - Eindhoven: Technische Universiteit Eindhoven, 2009. - Proefschrift.
ISBN 978-90-386-1516-5 NUR 926
Trefwoorden: ondiepe stromingen, dipool, vloeistof wervels, quasi-twee-dimensionale turbulentie
Subject headings: shallow flows, dipole, vortex dynamics, quasi-two-dimensional turbulence
Three-dimensionality
of
shallow flows
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 28 januari 2009 om 16.00 uur
door
Andrzej Ryszard Cie´slik
prof.dr.ir. G.J.F. van Heijst en
prof.dr. H.J.H. Clercx
Copromotor: dr.ir. L.P.J. Kamp
for Fundamental Research on Matter (FOM).
CONTENTS
Outline and Motivation . . . 1
1. Introduction . . . 3
1.1 Two-dimensional and quasi-two-dimensional turbulence . . . 3
1.1.1 Turbulent flows . . . 3
1.1.2 The theory of homogeneous 2D turbulence . . . 4
1.1.3 2D versus 3D turbulence . . . 7
1.1.4 Geophysical flows . . . 8
1.2 Experiments on quasi-two-dimensional flows . . . 9
1.2.1 General overview . . . 9
1.2.2 Experiments in shallow fluid layers . . . 10
1.3 Concluding remarks . . . 14
2. Experimental and numerical techniques . . . 15
2.1 Chapter outline . . . 15
2.2 Experimental setup . . . 15
2.2.1 Electrolysis . . . 18
2.2.2 Electromagnetic forcing . . . 19
2.3 Experimental techniques . . . 20
2.3.1 Velocimetry: the principle and settings . . . 20
2.3.2 Particle Image Velocimetry . . . 21
2.3.3 Stereoscopic Particle Image Velocimetry . . . 22
2.3.4 Estimation of differential quantities . . . 26
2.3.5 Measurements of free surface deformations . . . 28
2.4 Three-dimensional numerical simulations . . . 29
3. Dipole-wall collision in a shallow layer . . . 33
3.1 Chapter outline . . . 33
3.2 Introduction . . . 33
3.3 Details of experiments and simulations . . . 35
3.4 Experimental and numerical results for H = 11.4 mm layer . . 38
3.6 Conclusions . . . 56
4. A dipole on a sloping bottom . . . 59
4.1 Chapter outline . . . 59
4.2 Introduction . . . 60
4.3 Experimental and numerical techniques . . . 61
4.4 Comparison between experimental and numerical results . . . 62
4.5 Results of 3D numerical simulations . . . 67
4.6 Experiments and simulations with the angle α = 4.8◦ . . . 74
4.7 Conclusions . . . 75
5. Decaying turbulence in shallow fluid layer . . . 77
5.1 Chapter outline . . . 77
5.2 Introduction . . . 78
5.3 Experimental and numerical details . . . 79
5.4 Experimental results from H = 11.4 mm layer . . . 80
5.5 Comparison between experiment and simulation I . . . 88
5.6 Comparison between 3D and 2D numerical simulations . . . . 91
5.7 Different boundary conditions: 3D simulation with a stress-free bottom . . . 94
5.8 Results of experiment and simulation II . . . 97
5.9 Free surface deformations . . . 97
5.10 Chapter summary . . . 99
6. Conclusions . . . 101
A. Derivation of the forcing term in the Navier-Stokes equation . . . 105
Summary . . . 115
Acknowledgements . . . 117
OUTLINE AND MOTIVATION
A large body of scientific effort has been devoted to study the dynamics of homogeneous, non-rotating shallow fluid layers in the laboratory [36]. These experiments are believed to be relevant in order to gain insight into some classes of geophysical flows like coastal and river flows, where the density stratification and rotation of the Earth does not play an important role. Expe-riments on shallow flows are also performed searching for similarities be-tween real turbulent laboratory flows and idealized two-dimensional (2D) turbulence [81]. Finally, another type of research involves studies of vortex dynamics in shallow layers, where coherent structures like a dipolar vortex are created and measured [75]. These coherent vortices are of a special in-terest to geophysical flows since they are very frequently observed in such large-scale flow systems.
A large ratio between the typical horizontal length scale (e.g. the size of the flow domain) to the typical vertical length scale (e.g. the fluid layer depth) is generally believed to ensure quasi-two-dimensional (Q2D) flow properties of shallow fluid layers. [36, 81, 75, 76, 9, 27]. If this ratio is large enough one usually refers to “geometrical confinement” of the flow. In this case Q2D dynamics practically means that “geometrical confinement” is be-lieved to inhibit vertical velocities relative to horizontal motions and vertical profiles of the horizontal velocities become simply Poiseuille-like [25]. Fol-lowing this reasoning the majority of the work done in the past focused on the planar properties of motion, usually at the free surface of a shallow fluid layer. Yet relatively little is known about the vertical flow structures inside a shallow fluid layer and their interactions with the horizontal motion of the fluid, including the free surface flows.
As a response to that state-of-the-art, the research reported in this PhD thesis concerns the three-dimensional (3D) structures present in shallow flows. The main goal is to elucidate the importance or unimportance of 3D effects in a few shallow-flow configurations. Three typical flow structures are exa-mined: a dipolar vortex colliding with a no-slip vertical wall (Chapter 3), a dipole moving over an inclined bottom (Chapter 4) and decaying turbulence (Chapter 5) in a shallow fluid layer. A dipolar vortex is chosen since it is
a frequently occurring flow structure in real geophysical flows or in purely 2D turbulence [50]. Investigations on Q2D turbulence largely disregard the effects of lateral sidewalls and a simple and well-defined problem of dipole-wall collision is believed to shed some light on this more complicated turbu-lence problem. A dipole approaching a no-slip inclined bottom corresponds with the realistic and common flow phenomenon of a vortex in coastal areas. Finally, the 3D flow structures in decaying turbulence provide new insights into this classic experiment with assumed quasi-two-dimensionality.
The flows discussed in this thesis are driven by electromagnetic forcing. The flow phenomena are studied both by laboratory experiments and by 3D numerical simulations. Wherever the three components of the velocity vector in a single horizontal plane are measured in the laboratory the technique of Stereoscopic Particle Image Velocimetry (SPIV) was employed. In addition to these measurements, 3D numerical simulations have been performed with a realistic model for the Lorentz force.
These experimental and numerical aspects of the three-dimensionality in-side shallow fluid layers are summarized in the Chapter 6.
1. INTRODUCTION
1.1 Two-dimensional and quasi-two-dimensional turbulence
The experiments in shallow fluid layers, regardless whether with single vor-tical structures or with an array of interacting vortices, all are related to a particular theory of two-dimensional (2D) turbulence. This theory, followed by numerical and experimental attempts to confirm it in various circumstan-ces, is the subject of this introductory chapter.
1.1.1 Turbulent flows
The fluid motion around us is usually turbulent, meaning essentially unpre-dictable, with random fluctuations of velocity and pressure both in time and space and with enhanced mixing property. The strength of turbulence de-pends on quantities like a characteristic velocity of the fluid U , a typical length scale L, the kinematic viscosity ν of the fluid. These quantities to-gether form the Reynolds number Re = U L/ν. Rustling leaves in a windy autumn day, the flow of air around one’s head looking out of the window in a running train, the flow of water around a swimmer, are all examples of turbulent flows. These flows have a common characteristic: all the three components of the velocity vector are comparable with each other. On the other hand one can imagine an unrealistic motion confined only to two di-mensions in numerical simulations. This subclass of turbulent flow with suf-ficiently high Reynolds number is called 2D turbulence. Between these real three-dimensional (3D) turbulent flows and unreal pure 2D turbulence there are real flows where vertical velocities are much smaller than the horizontal ones. These flows are referred to as quasi-two-dimensional (Q2D). A typi-cal example of such a flow is the motion of air masses in the Earth’s atmo-sphere, where horizontal length scales exceed thousands of kilometers while the thickness of the troposphere measures roughly 10 km. The large ratio between these length scales together with the rotation of the Earth inhibits vertical velocities relative to the horizontal flow.
1.1.2 The theory of homogeneous 2D turbulence
“Something roughly like turbulent motion can exist in two dimensions. Weather systems on a large scale represent nearly two-dimensional flows. However, the characteristics of such flows are in many ways so different that it is perhaps unwise to include them in turbulence.” This quote from a film is-sued in the 60s by the National Committee for Fluid Mechanics Films, shows the general reluctance of some scientists towards the high Reynolds number two-dimensional flows. Various objections were formulated, see e.g. the clas-sical textbook on turbulence [79] or a recent review [77], and the discussion did not extinguish until today, see Chapter 8 in [80]. Whatever name one assigns to chaotic, unpredictable, two-dimensional motion with a very small viscosity it turned out worthwhile to study such flows.
Already in 1953 Fjørtoft [28] pointed out that the flow in two dimensions is dramatically different from its 3D counterpart. Surprisingly it took one and a half decade for significant progress in the development of 2D turbu-lence theory, presumably due to the above-mentioned reluctance. The tur-ning point seems to be the publication of now classical papers at the end of the sixties [42, 43, 7] that laid the foundation of what is known now as 2D turbulence theory.
The theory of homogeneous 2D turbulence is concerned with the stati-stical prediction of the distribution of the kinetic energy ε and enstrophy Ω among vortices of various sizes. The kinetic energy and enstrophy are defi-ned as follows ε = 1 2 Z A k~uk2dA (1.1) and Ω = 1 2 Z A ω2 zdA, (1.2)
where ~u = (u, v) is the horizontal velocity vector and ωz is the scalar 2D
vorticity. Here the concept of cascade plays a dominant role, which can be defined as a transport process of some quantity through eddies of different sizes without viscous effects. The range of vortices where the cascade process takes place is called the inertial range. The incompressible fluid motion is described by the 2D Navier-Stokes equation:
∂~u ∂t + ~u · ∇~u = − 1 ρ∇p + ν∇ 2 ~u (1.3)
1.1. Two-dimensional and quasi-two-dimensional turbulence 5
supplied with the incompressibility condition
∇ · ~u = 0, (1.4)
where p denotes pressure.
Taking the curl of (1.3) one arrives at the scalar vorticity equation: ∂ωz ∂t + u ∂ωz ∂x + v ∂ωz ∂y = ν∇ 2 ωz, (1.5)
where x and y are two horizontal coordinates of the position vector ~x = (x, y). Multiplying (1.3) by ~u and (1.5) by ωzand integrating over the full domain
one arrives at the equations for the time evolution of the kinetic energy
ǫ = −dε
dt = 2νΩ (1.6)
and the enstrophy, respectively
χ = −dΩ
dt = 2νP, (1.7)
where ǫ and χ are kinetic energy and enstrophy dissipation, respectively, and P denotes palinstrophy defined as
P = 1 2
Z
A
k∇ωzk2dA. (1.8)
Equation 1.7 can be derived only for unbounded domains.
In case of freely decaying 2D turbulence it was hypothesized by Batchelor [7] that even in the case of vanishing viscosity the enstrophy dissi-pation χ reaches a nonzero limit owing to the process of stretching of vorticity gradients, see equation (1.7). Such a hypothesis implies that the dissipation of enstrophy does not depend on viscosity for high Reynolds number flows. This observation led both Kraichnan [42] and Batchelor [7] to the concept of the enstrophy cascade: vorticity is transferred through a range of scales without viscous effects down to the smallest scales where it is eventually dis-sipated. Furthermore, as the enstrophy is bound by the initial value and can only decrease (due to viscosity), the kinetic energy dissipation goes to zero with vanishing viscosity, ǫ → 0, see equation (1.6). This excludes the possibi-lity of the cascade of the kinetic energy in freely decaying 2D turbulence. Al-though Batchelor [7] predicts the transfer of the kinetic energy towards larger scales, this is not equivalent to the cascade of kinetic energy, as the cascade
is essentially a process involving a constant nonzero dissipation rate. Howe-ver, recently Davidson [26], against a generally accepted tradition, proposed to use the term of inverse energy cascade also in case of freely decaying 2D turbulence.
In the case of forced 2D turbulence Kraichnan [42] showed that in the presence of enstrophy cascading down from the forcing scale to the dissipa-tive scales the only possibility for the transfer of kinetic energy is the upscale transport, to the larger scales of motion. This dual cascade of kinetic energy (the inverse energy cascade) and enstrophy (the enstrophy cascade) is a land-mark of the 2D turbulence theory. The concept of dual cascade implies the constancy of the kinetic energy dissipation ǫ and the enstrophy dissipation χ in the inverse and enstrophy cascade, respectively. Since a statistically statio-nary state is expected to occur, the dissipation rate is equal to the production (forcing) rate of a given quantity. In this way the dissipation of a given quan-tity (ε or Ω) is equal to the flux of this quanquan-tity in the relevant cascade range. The way to measure the amount of the kinetic energy present in a given scale of motion l is the n-th order velocity structure function, defined as
Sn(l) =< k~u(~x) − ~u(~x + ~l)kn>, (1.9)
where < · > refers to an ensemble average.
Assuming that the distribution of kinetic energy in the enstrophy cascade regime depends only on the scale l and the enstrophy flux χ, one derives by dimensional arguments:
Sn(l) ∝ χn/3ln. (1.10)
On the other hand, for vortices in the inverse cascade range the process of distribution of kinetic energy is controlled only by the scale l and the kinetic energy flux ǫ, leading to
Sn(l) ∝ ǫn/3ln/3. (1.11)
An analogous analysis can be carried out for the distribution of enstrophy among various vortex sizes by means of the vorticity structure function.
An alternative way to investigate the distribution of kinetic energy among various scales of motion is the kinetic energy spectrum E(k), defined in the k-wavenumber Fourier space, where the relation with kinetic energy is
ε = Z ∞
0
1.1. Two-dimensional and quasi-two-dimensional turbulence 7
Repeating the dimensional argument, i.e. S(l)2
∝ E(k)k, one obtains from (1.10) and (1.11) the shape of the kinetic energy spectrum in two diffe-rent regimes of 2D turbulence:
E(k) ∝ χ2/3k−3
(1.13) for the enstrophy cascade and
E(k) ∝ ǫ2/3k−5/3 (1.14)
for the inverse energy cascade.
1.1.3 2D versus 3D turbulence
The theory of 2D turbulence was built on concepts borrowed from the 3D turbulence theory. Richardson [67] originally envisioned the idea of the ca-scade while A. N. Kolmogorov laid the quantitative foundations for this 3D theory [40, 41].
It is known experimentally that in 3D turbulent flows the kinetic energy dissipation reaches a nonzero limit as viscosity vanishes [29]. This again im-plies that ǫ does not depend on viscosity, as was the case for χ in 2D flows.
The central difference between 2D and 3D flows is visible in the vorticity equation. For 3D it reads
∂~ω
∂t + ~u · ∇~ω = ~ω · ∇~u + ν∇
2
~
ω, (1.15)
where ~u = (u, v, w) and ~ω = (ωx, ωy, ωz) = ∇ ×~u are 3D velocity and vorticity
vectors.
In (1.15) there is an additional term ~ω · ∇~u with respect to (1.5), represen-ting vortex stretching. It causes the amplification of vorticity merely due to velocity fluctuations during the evolution of the 3D flow field. Enlarging en-strophy, this term is responsible for finite positive limit for ǫ as viscosity goes to zero, see equation (1.6), which applies also to the 3D flow field.
The constant enstrophy flux to smaller scales is no longer possible and vorticity increases while passed down to the smallest scales. The previously given dimensional argument applies, which results in the same shape of the kinetic energy spectrum, see (1.14). However, in the 3D case the cascade of ki-netic energy is not constrained by the existence of the enstrophy cascade and the energy is transferred from the forcing scale to the smallest scales where it is eventually dissipated. This shape of the kinetic energy spectrum is a well-documented feature of 3D turbulence, see e.g. [29].
The direction of the kinetic energy flux is most convincingly measured by the third-order structure function
S3(l) =< k~u(~x) − ~u(~x + ~l)k3 >, (1.16)
where ~u and ~x are the 3D velocity and position vector, respectively. In 3D turbulence the four-fifth law dictates
S3(l) = −
4
5ǫl. (1.17)
In 2D the same line of arguing leads to
S3(l) =
3
2ǫl. (1.18)
The sign of the pre-factor indicates the direction of the energy flow, i.e. a negative value indicates the downscale, while a positive value denotes the upscale transfer of the kinetic energy.
1.1.4 Geophysical flows
The primary objective of 2D turbulence studies and Q2D experimental inve-stigations remains the understanding of various types/aspects of geophysi-cal flows. Although it is clear that pure 2D turbulence is not a good model for geophysical applications, it is still a reference point in the research of large-scale planetary flows.
Large-scale flows in the Earth’s atmosphere and the ocean, generally re-ferred to as geophysical flows, are quasi-two-dimensional (Q2D), see e.g. [61]. There are three main reasons for this Q2D behaviour: the rotation of the Earth, the density stratification and the “geometrical confinement”. Density stratification occurs due to the vertical gradients of either salt concentration in the ocean or temperature in the atmosphere and oceans. The “geometrical confinement” expresses a significant difference between vertical and horizon-tal length scales.
In the context of geophysical flows one encounters various models that describe underlying dynamics. These models are quasi-two-dimensional, i.e. two-dimensional with additional modifications that attempt to take into ac-count the most important three-dimensional influences. The simplest exam-ple of such a model, disregarding the effects of rotation and stratification, is the 2D Navier-Stokes equation (1.3) extended with an additional linear dam-ping term −λ~u, where λ is the Rayleigh friction coefficient. This model is readily obtained from the 3D Navier-Stokes equation using the simplification
∂2
~u/∂z2
∝ −~u/H2
1.2. Experiments on quasi-two-dimensional flows 9
Other geophysical models that take into account the effects of rotation and/or stratification an interested reader may find elsewhere, see e.g. [61, 44].
1.2 Experiments on quasi-two-dimensional flows
1.2.1 General overview
In the past few decades many experimentalists have tried to validate predic-tions based on 2D turbulence theory. They incorporated several laboratory situations where the flow is considered, based on some assumptions, to oc-cur mainly in a plane. These laboratory experiments include: rotating fluids, e.g. [34, 86, 54], stratified fluids, e.g. [65, 47, 84], soap films e.g. [23, 31, 69, 39] and shallow flows. Studying rotation, stratification and “geometrical confine-ment” separately enables one to disentangle their influences on geophysical flows where the interplay between these three factors is usually at work.
In the presence of a strong rotation the Coriolis force strongly influences turbulent motion. Initially 3D turbulent flow quickly organizes itself into co-lumnar vortices (cigars) under the action of a strong rotation [34]. One can describe these flows as Q2D since the vertical gradients of velocity are sup-pressed in comparison with horizontal ones. The Q2D properties of rotating fluids have been investigated later, see e.g. [54, 35]. Jacquin et al. [35] inve-stigated suppression of the dissipation of the turbulent kinetic energy due to rotation, while Morize et al. [54] showed that at large scales there is a ne-gative (up-scale) kinetic energy flux towards larger scales for large rotation rates. As expected, the steepening of the kinetic energy spectrum from the 3D value (k−5/3) was observed as the rotation rate increased, but the
enstro-phy cascade scaling (k−3
) was not obtained owing to limited experimental possibilities to increase rotation even further.
In stably stratified flows a fluid parcel displaced vertically from its equ-ilibrium position experiences a restoring force due to buoyancy. This pro-perty gives stratified flows their special character. Flows tend to organize into pancake-like structures. A beautiful example of this process is shown in the experiment where a fluid of a constant density is injected into a stratified medium [89, 84, 4]. Quite rapidly the initially disordered 3D turbulent flow evolves into an organized structure, a dipolar vortex. These pancake-like structures are also observed in later times in a turbulent flow in a stratified medium [65, 30]. Although in these flows horizontal components of velocity are much larger than vertical ones, the vertical gradients of horizontal flows are significant.
films, where measurements showed many common characteristics with ty-pical features of pure 2D turbulent flows. Coherent vortex couples spontane-ously emerge in turbulent wakes created behind solid obstacles towed in ho-rizontal soap films [23]. These first investigations were only qualitative and later fostered new quantitative studies on Q2D flows. The early investigation by Gharib and Derango [31] was somewhat inconsistent with the classical prediction [7] for decaying Q2D turbulence. They reported on the expected k−3
scaling in case of a jet experiment, but claimed a simultaneous inverse and enstrophy cascade in decaying turbulence generated by a grid. Later, decaying turbulence experiments confirmed the classical predictions for the enstrophy cascade and were able to reproduce the k−3
scaling for the kinetic energy spectrum [49, 39]. The work of Rutgers [69] proved that it is possi-ble to obtain simultaneously inverse energy and direct enstrophy cascades in continuously forced experiments. He also confirmed the classical predic-tion for the decaying experiment and showed a smooth transipredic-tion of spectra between continuously forced and decaying flows. Even more convincingly, Bruneau and Kellay [11] recently showed again a simultaneous inverse and direct cascade in a forced soap film. Their approach involved computation of the energy flux using experimentally available third-order structure func-tions.
An exhaustive review of these kind of experiments can be found in a re-cent review [38]. Experiments in shallow fluid layers and in stratified turbu-lence are reviewed by Clercx and van Heijst [21].
1.2.2 Experiments in shallow fluid layers
There are two types of shallow-flow experiments reported in the literature: first in homogeneous layers and second in two-layer systems, where discon-tinuous density stratification is expected to additionally promote Q2D dyna-mics, see e.g. [58]. While some authors argued that the “geometrical confi-nement” is not enough to ensure Q2D dynamics [4], others claimed the op-posite [75, 81]. Although this thesis deals only with homogeneous shallow fluid layers, the phenomena in two-layer systems are very closely related with those encountered in homogeneous layers.
Flow initialization in shallow flows
Various different methods have been utilized to perform experiments in a shallow fluid layer. A pair of monopoles can be created by the abrupt
mo-1.2. Experiments on quasi-two-dimensional flows 11
vement of a piston [45]. Flow can be produced applying a pressure gradient while turbulence is obtained by placing cylinder(s) into the liquid [15]. A sud-den injection of a small volume of a fluid can produce a dipolar vortex [76]. Lastly, the motion can be set up in electrolyte solution by means of electro-magnetic forcing where the electro-magnetic field of a permanent magnet combined with an electric current gives rise to a Lorentz force. A single magnet creates a dipolar vortex [2] while an array of magnets may produce a turbulent flow with many interacting vortical structures [90, 78, 27, 57, 24]. A specific arran-gement of magnets controls the range of scales where the kinetic energy is injected, compare [59] and [57]. Also, one permanent magnet, translating just above the free surface, can produce vortex streets or wakes [3].
Cascades
At the end of the nineties a series of papers [59, 60, 57] was published that showed that the flow dynamics at the free surface of a continuously forced shallow fluid layer is consistent with the picture of the double cascade by Kraichanan [42]. Firstly, the inverse energy cascade was claimed in ho-mogeneous and isotropic flow with the classical k−5/3 scaling and with the
Kolmogorov constant consistent with previous numerical computations [59]. As an extension of this result the authors reported on the lack of intermit-tency in the inverse cascade range [60]. This non-intermittent statistics was indicated by structure function scaling exponents, which were found to be consistent with those resulting from dimensional analysis. The authors re-ported on the Gaussian statistics for both longitudinal and transverse velo-city increments for separations k~lk well in the inverse cascade range. The computed spectral energy flux was negative in the inertial range, indicating an upscale transfer of kinetic energy. However, it failed to be constant, which should be the actual proof of the cascade, as it was shown in soap film expe-riments [11]. The process of aggregation of vortices was proposed as a me-chanism driving the inverse cascade. Secondly, the enstrophy cascade was claimed later based on the classical k−3
scaling in the inertial range between forcing and dissipation scales [57]. Also in this experiment non-intermittent statistics was reported. Probability density functions of vorticity increments were nearly Gaussian, while corresponding vorticity structure functions were nearly scale-independent. The spectral enstrophy flux was shown to be po-sitive in the inertial range, indicating downscale transport of enstrophy, but still it failed to be constant.
di-scontinuous density stratification is supposed to promote Q2D dynamics [58]. However, the experimental observation of the enstrophy cascade was also claimed in a homogeneous shallow layer [81] based only on the scaling expo-nent for the energy spectra. The spectral energy flux was not presented.
Later it was shown in two-layer experiments that the scaling exponent in the enstrophy cascade has to be modified according to the vertical dam-ping [9], in remarkable agreement with previous theoretical considerations and numerical simulations [55, 8]. The energy spectrum has to be modified as E(k) ∝ χ2/3k−3−ξwhere ξ is proportional to the Rayleigh damping
coeffi-cient λ. Similar conclusions were reached in case of experiments with rotating fluids where the exponent in the direct enstrophy range has to be modified depending of the strength of the Ekman damping [91]. Only in case of vani-shing dissipation coefficient ξ the classical k−3
scaling is expected. This re-sult directly implies that as the depth of the layer decreases the discrepancy between experiments and 2D theory increases. These results are not neces-sarily contradictory with the clean k−3
scaling observed in a homogeneous layer [81] where much larger fluid depths were used. On the other hand they clearly contradict the observations by Paret and Tabeling [60] where the clas-sical exponent was reported, although their total fluid thickness was much smaller, H = 5 mm.
The mechanism of the inverse energy cascade through clustering of like-sign vortices [59] has been challenged recently by another explanation [17]. The authors provided an indication that the inverse energy cascade is driven by a process of elongation and thinning of small-scale vortices due to large-scale strain.
The experimental observation of the simultaneous double cascade, as in the case of the afore-mentioned experiments in soap films, was never repor-ted in a shallow fluid layer. A possible explanation for this fact is the difficulty to achieve a relatively high Reynolds number flow in the laboratory.
Vortex dynamics
In 1991 a theory concerning vortex statistics in freely evolving 2D turbulence has been proposed [13], which was consistent with 2D numerical simulations performed with hyperviscosity [51]. This theory assumed two dynamical invariants during the freely evolving phase, namely the kinetic energy and the supremum vorticity, and derived from this assumption the scaling laws for geometrical and dynamical quantities, like the number density of vorti-ces and enstrophy. These scaling laws were formulated based on only one
1.2. Experiments on quasi-two-dimensional flows 13
unknown coefficient ξ. If the number density of vortices decreases with time as t−ξ
then it follows that the enstrophy decays as t−ξ/2
, the mean vortex size increases as tξ/4, and both the mean separation between vortices and the vorticity kurtosis evolve as tξ/2.
The applicability of the theory by Carnevale and co-workers [13] to shal-low fshal-lows that are driven electromagnetically has been tested during the nineties. Starting from their paper in 1991, Tabeling and co-workers found consistently a good agreement between their experimental results and this theory [78, 48]. In a summarizing short communication [33] additional indi-cations were provided suggesting that the scaling theory is valid for freely evolving shallow flows.
Additionally, Cardoso et al. [12] found, what they called, liquid turbu-lence. It was shown that after the merging phase the system was densely filled with clusters of vortices. In contrast, the theory of scaling laws assumes, based on numerical simulations [51], that the system should be di-lute. Vortices ought to occupy a relatively small area. Three-dimensional effects were held responsible for these discrepancies.
Later, Hansen et al. [32] pointed out differences between the theory and 2D simulations on one side and experimental data on the other. They showed a slight decay of extremum vorticity and the evolution of the kurtosis of vorticity was different than predicted by the theory. They proposed the rescaling of the time in order to reconcile the experimental re-sults obtained from shallow fluid layers with 2D theory predictions. Clercx et
al. [22] criticized their approach as ineffective. They reported that the
resca-ling of the time does not change power-law exponents for the first stage of the flow evolution and any clear power law was found for later times. On the other hand it was shown [22] that the method proposed by Hansen et
al. [32] can be useful for vortex statistics of decaying 2D turbulence. Power
laws describing the evolution of vortex statistics become bottom-friction in-dependent provided the compensated velocity and vorticity are introduced.
Summarizing, the attempts to validate the 2D inviscid theory [13] with the experiments in shallow fluid layers seem rather inconclusive. This is most probably due to the limited Reynolds number attainable in the labo-ratory as increasing the Reynolds number further introduces unavoidable three-dimensional secondary flows.
Instead of focusing on statistical quantities as the number density of vorti-ces, energy spectra and structure function scaling or particle-pair separation there are numerous investigations that are concerned with the dynamics of relatively simple vortical structures (monopoles, dipoles) that are either em-bedded in a turbulent sea or in laminar flows. Sous et al. [75, 76] investigated
the transition from a deep to a shallow layer. They showed that in case of a shallow layer coherent dipolar vortices emerge from initially 3D flow field, while in a deep layer this emergence does not occur. They also reported on the 3D frontal recirculation cell in front of the propagating dipole. This characteristic vertical motion was also observed in case of two monopolar vortices created by an abrupt movement of a piston [45]. Only recently this frontal recirculation roll has been directly measured in the case of a freely evolving dipole [5, 6]. Moreover the authors showed a very complex verti-cal profile of the horizontal velocity, which casts doubts on the feasibility to use the Rayleigh damping friction in 2D Navier-Stokes equation in order to simulate shallow flows.
1.3 Concluding remarks
The large variety of topics in numerical and experimental investigations on shallow fluid layers is impressive. Naturally, this PhD thesis covers only a small fraction of possible research paths. In all experiments presented in this thesis electromagnetic forcing has been utilized in order to set the fluid in motion. The material presented here can be divided in two parts. One is devoted to a study of a dipolar vortex, both for the case of a collision with a lateral no-slip wall and for the case of the dipole propagating over a sloping bottom. The second part focuses on the classical problem of freely decaying turbulence in a shallow fluid layer. The approach to this problem is different than the usual one. Here the emphasis is placed more on the phenomenology of vertical flows and their relation with the horizontal motion and not on the statistical quantities like the kinetic energy spectra and structure functions.
2. EXPERIMENTAL AND NUMERICAL TECHNIQUES
2.1 Chapter outline
In this chapter the experimental and numerical tools are presented, together with a detailed characterization of the experimental setup. Sec. 2.2 descri-bes the experimental setup. Sec. 2.3 gives details of measurement techniques used. Finally, Sec. 2.4 describes the numerical method used in this study.
2.2 Experimental setup
The experimental setup is similar to the ones used by previous researchers, see e.g. [78, 68] and consists of a square perspex tank containing either a thin layer of fluid of uniform thickness H (Chapter 3 and Chapter 5) or a layer of fluid with varying depth due to an inclined bottom (Chapter 4). In the latter experiments the tank is inclined with an angle α to the horizontal plane. The thickness of the bottom plate is 1 mm. The dimensions of the container are 52.0 × 52.0 × 2.0 cm3
(length × width × height). The PVC bottom is painted black to avoid light reflections from any possible source.
Permanent neodymium flux (NdFeB) magnets (type GSN-35, fabricated by Goudsmit Magnetic Supplies BV) are placed below the bottom. Each magnet used in the experiments has a circular-disc shape, a diameter of 25 mm and a thickness of 5 mm, and it produces an axially symmetric magnetic field strength of the order of 1 tesla. The magnet is assumed to be uniformly ma-gnetized along its axial direction. A single magnet is used to produce a dipo-lar vortex (Chapter 3 and 4), while the array of 10 × 10 magnets creates the array of 10 × 10 dipolar vortices (Chapter 5). The details of the magnetic field produced by a single permanent magnet are presented in the section that is concerned with the 3D numerical technique (Sec. 2.4).
In the experiments presented in Chapter 3 and in Chapter 5 three velo-city components were measured in a single horizontal plane at the distance h above the bottom by means of Stereoscopic Particle Image Velocimetry (SPIV). In the experiments presented in Chapter 4 two horizontal velocity components were measured at the free surface by means of Particle Image
Velocimetry (PIV). For these velocimetry techniques the fluid is seeded with passive particles, which are illuminated by a horizontal laser light sheet. Fur-ther details are given later in this chapter. The specific experimental arran-gements are shown from the top views in Fig. 2.1 and side views in Fig. 2.2. Throughout this thesis the fluid motion is described with respect to a Carte-sian coordinate system, with the origin on the bottom of the tank and either above the centre of the magnet (Chapter 3 and Chapter 4) or above the centre of the tank (Chapter 5), see Fig. 2.1. The z-axis is always perpendicular to the horizontal measurement planes, see Fig. 2.2, which does not necessarily coincide with the axis of the magnet (Chapter 4).
Figure 2.1: Top views of the setup arrangements for experiments discussed in
Chap-ter 3 (a), ChapChap-ter 4 (b) and ChapChap-ter 5 (c). The area where the flow evolu-tion was measured is shown by a dashed rectangle. The horizontal laser light sheet enters the measurement area through side glass wall.
The flow is set in motion by means of non-intrusive electromagnetic for-cing. A salt solution (NaCl, 27.8%Brix) serves as the fluid. Two electrodes are placed on opposite sides of the tank and are connected to the power supply. The Lorentz force results from the interaction between the magnetic field of the magnet(s) and an electric DC current that is driven through the fluid by an external power source. The DC current ensures that the forcing does not change in time.
In all the experiments the flow was forced for 1 s. This choice follows from a need to have a well-defined dipolar vortex or an array of dipoles at the beginning of the flow evolution. Longer forcing times than 1 s produce vortices that are too elongated. On the other hand, forcing shorter than 1 s
re-2.2. Experimental setup 17
Figure 2.2: Side views of the setup arrangements for experiments discussed in
Chap-ter 3 (a), ChapChap-ter 4 (c,d) and ChapChap-ter 5 (b). In case of the inclined tank the cross-section through the y = 0 axis (c) and the cross-section through the x = 0 axis (d) are shown.
sults in motions that are not sufficiently strong. After the forcing is switched off the time is set to zero.
One of the goals of this thesis is to study the evaluation of the possible influence the fluid depth H has on the vertical structure of the flow. For that purpose two different fluid depths are examined, both experimentally and numerically: H = 6.2 mm for a shallower fluid layer and H = 11.4 mm for a deeper fluid layer. As a lower bound, one would be interested to set up a layer as thin as possible. However, this is practically restricted by the damping due to the bottom friction. Additional difficulties arise with very shallow layers. Firstly, the free surface deformations are becoming relatively more important as the depth H decreases. Secondly, the shallower the fluid layer the noisier the experimental results; this is presumably due to passive particles at the free surface. On the other hand, the deeper layer thickness should be substantially larger than the shallower layer thickness in order to have a clear indication of the fluid depth influence. Secondly, the fluid depth should not be too large in order to prevent full three-dimensional dynamics. Additional 3D numerical simulations have been performed for the dipole-wall collision problem in a H = 3.0 mm fluid layer (see Chapter 3).
2.2.1 Electrolysis
After applying a current to the electrodes the process of electrolysis starts. Po-sitively charged sodium ions will drift towards the negative electrode and ne-gatively charged chloride ions will drift towards the positive electrode. From the chemical point of view at the positive electrode negatively charged chlo-ride ions release electrons and form chlorine gas:
2Cl−
→ Cl2 ↑ +2e −
(2.1) Chlorine partially dissolves in water and forms an acid environment; be-sides oxygen is produced:
2Cl2 ↑ +6H2O → 4H3O++ 4Cl −
+ O2 ↑ (2.2)
Due to the presence of the hydronium ions H3O+a slight acidity of about
pH = 5 was confirmed near the positive electrode by measurements with a simple pH sensitive paper. This reaction depends on the salt concentration. In a highly concentrated salt solution chlorine will dissolve only a little. As a consequence the more concentrated the salt solution the less oxygen will be released.
On the other side of the tank at the negative electrode water is reduced to hydrogen gas and hydroxide ions. The sodium hydroxide is produced. The whole process is described by the reaction:
2H20 + 2N a +
+ 2e−
→ 2N a++ 2OH−
+ H2 ↑ (2.3)
Naturally, due to the production of the hydroxide ions the pH paper measurements indicated a very strong alkaline environment (pH ≈ 12) in the close neighbourhood of the negative electrode. The above-described pro-cesses are presented in Fig. 2.3.
Electrodes arrangement
Due to the process of electrolysis gases are released on both sides of the tank close to the electrodes. The production of these gases creates an additional forcing mechanism that is essentially three-dimensional. In order to have a well-controlled experiment one would like to avoid this side effect. Additio-nally, one would like to keep a constant ion concentration inside the working fluid during the experiment. This would be problematic if the ions produced at the electrodes were swept away immediately by the surrounding turbulent flow and transported further into the bulk of the flow.
2.2. Experimental setup 19
The arrangement that avoids this additional effect is shown in Fig. 2.3. Between the electrodes and the bulk of the flow there is a region where the turbulent transport is inhibited. Turbulence in this region is avoided by con-necting it to the other flow domains by narrow gaps (b = 2 mm). These gaps are large enough to ensure a uniform electric field. Such a solution was tested and even after half an hour of strong continuous forcing no change of pH was measured inside the tank. This indicates that the region between the electrodes and the experimental flow domain is dominated by diffusion.
Figure 2.3: A cross-section of the experimental setup. A special arrangement of
elec-trodes is seen on both sides of the tank.
2.2.2 Electromagnetic forcing
In all the experiments presented in this thesis the flow was forced by means of electromagnetic forcing. This body force ~f reads
~ f = κ
ρJ × ~~ B, (2.4)
see Appendix A for the derivation of this equation.
Throughout this thesis it is assumed that the electric current is uniform, i.e.
~
J = J0e~x. (2.5)
The argument for this approach comes from the specific geometry of the experimental setup: the current inhomogeneities are negligible due to
shal-lowness of the fluid layer [78]. From the assumption (2.5) follows that the forcing ~f has only two components: a horizontal component, perpendicular to the current direction
fy = −
κ
ρJ0Bz (2.6)
and a vertical component
fz = κ
ρJ0By. (2.7)
2.3 Experimental techniques
2.3.1 Velocimetry: the principle and settings
In order to measure the velocity in a shallow fluid layer the working fluid is seeded with PMMA (polymethyl methacrylate, density: 1.19 kg/m3
) passive particles of diameters between 250 − 300 microns. When the fluid is electro-magnetically forced these particles should follow the flow passively as the Stokes number is approximately 2 · 10−4
. Moreover, the sedimentation of these particles is avoided by matching their density with the density of the fluid. The latter was adjusted by changing the salt concentration.
During the flow evolution these passive particles are illuminated by a Neodymium:YAG laser. Every 1/15 s a short (10−8
s) laser pulse is created. The laser consists of two cavities. This allows to set a delay time (∆t1)
be-tween two illuminating laser pulses according to a specific need. The delay time is controlled by a delay generator.
The circular laser beam traverses horizontally from the laser output. Two lenses were used in order to change this beam into a thin (≈ 2 mm) horizontal laser sheet: one lens converging the laser beam in a vertical plane and one lens diverging the laser beam in a horizontal plane. In order to obtain a laser sheet as thin as possible the two lenses were mounted approximately 1.5 m from the measurement area according to the focal length of the positive lens (f = 1.5 m). In the PIV measurements this horizontal light sheet illuminated the particles at the free surface, while in the SPIV measurements the laser sheet was traversing the bulk of the fluid at hLabove the bottom of the tank.
The laser sheet is entering the fluid through a lateral glass wall, see Fig. 2.1. The internal side of this glass wall was covered with an anti-reflective coating in order to prevent possible illumination inhomogeneities in the laser sheet. The thickness of the laser sheet produced by two lenses is too large for the total fluid depths used. In order to further reduce the thickness of the laser
2.3. Experimental techniques 21
sheet, directly in front of this glass wall there is a black painted obstacle with a 1 mm narrow horizontal window, through which the laser sheet traverses. Mounting such a window in front of the glass wall produces a thinner and well-defined laser sheet. The vertical position of the window is controlled accurately using micrometer screws.
Illuminated passive particles were recorded by digital camera(s) at the time instant t. At time instant t + ∆t1 these particles in different positions
were recorded in a separate exposure. This arrangement is provided via the delay generator, which synchronizes the laser pulses with releasing the shutter of the camera(s). The resolution of the CCD of each camera was 1600 × 1200 pixels. Nikon 28 mm f /2.8 lenses were used.
Finally, the recorded particle positions at the two time instants t and t + ∆t1 are further post-processed with either the PIV or the SPIV technique in
order to obtain either two or three velocity components in a single horizontal plane.
2.3.2 Particle Image Velocimetry
The Particle Image Velocimetry (PIV) method is used to measure the projec-tion of the local velocity vector into the plane of the light sheet. It yields two velocity components in the plane of illumination. Here only a basic de-scription of this method is presented together with a short justification for the specific settings used. For a complete description the reader may consult [66]. One camera mounted above the tank is used to capture particle positions. Then the digital PIV recordings are divided in small subareas called interro-gation windows. The image Itof particles recorded at time t has to be
cross-correlated with the image It′ of those particles captured at time t ′
= t + ∆t1.
This cross-correlation is performed more efficiently in Fourier space via the correlation theorem that relates the cross-correlation of two functions with a complex conjugate multiplication of their Fourier transforms. Firstly, images It and It′ are transformed through the Fast Fourier Transform (FFT)
algo-rithm. Secondly, the complex conjugate multiplication ˆIt· ˆIt∗′ is performed.
Finally, the resulting function is transformed back through the inverse FFT, which yields the cross-correlation function.
In the cross-correlation plane a distance which has the highest value of the cross-correlation function is the most probable PIV displacement in pixels in the interrogation window applied. This distance in pixels (x,y) has to be mapped into the distance in physical units (X,Y ). The mapping function is obtained through a calibration procedure. In this procedure a calibration target with equally spaced dots (white painted on the black background) is
photographed in the plane of the PIV measurement. A large set (more than one hundred) of dots positions in physical and pixels units allows one to fit a polynomial mapping function. Finally, the most probable displacement in the cross-correlation plane (X,Y ) divided by the time gap applied (∆t1)
yields the two average velocity components for the given interrogation area. As the size of the interrogation window defines the spatial resolution one would like to have as small a window as possible. The possible options for the choice of the interrogation window are power-of-two sizes owing to the FFT operations performed. However, the smallest size of the window is re-stricted by a number of particles that need to be recorded in such a window. Typically 5 − 8 particles should be captured inside the interrogation window. This condition is very difficult to fulfil when the interrogation window was 16 pixels. Therefore, in all the experiments performed the 32-pixel interro-gation window was used. Then 50% overlap of interrointerro-gation windows was applied, which yields a velocity vector every 16 pixels. These choices, toge-ther with the resolution of the camera and the size of the measurement area, resulted in a spatial resolution of about 1.5 mm. These settings apply to all experiments presented in this thesis.
As indicated before the PIV technique results in two velocity components, forming the projection of the local velocity vector into the horizontal plane of illumination. These two velocity components are usually not identical with the horizontal velocity components; only in case of the out-of-plane velocity component identically equal to zero this difference vanishes. In general, the higher the out-of-plane velocity the larger this difference. For the experi-ments presented in Chapter 4 this difference is negligible as the vertical di-splacements (free surface deformations) are much smaller than the horizontal ones.
2.3.3 Stereoscopic Particle Image Velocimetry
The Stereoscopic Particle Image Velocimetry (SPIV) technique is a natural extension of the PIV method and yields all three velocity components in a single plane of illumination. This extension is realized by adding a second camera. Here only a general description of SPIV is given. The reader may consult [82] for a thorough explanation of the code together with an accuracy discussion.
Two cameras are mounted above the tank under an angle ≈ 30o to the vertical axis. Due to this angle, without any other arrangement it would be impossible to focus on all the illuminated particles in the horizontal field of view. In this case the Scheimpflug criterion must be satisfied in order to be
2.3. Experimental techniques 23
able to focus on all the particles. This condition demands that the CCD, lens and measurement plane must be collinear. This is accomplished by mounting each camera and the accompanying lens on the Scheimpflug adapter, which allows tilting the CCD plane with respect to the lens.
The calibration procedure has to be extended with respect to the one used for PIV since an additional vertical displacement has to be determined. The same grid plate as in the PIV technique has to be photographed for a few different heights, around the measurement height.
The raw measurement images are distorted due to the specific camera positions. The area of overlap for images from two different angles is only a fraction of the recorded field of view (usually about 80%). In order to get rid of perspective distortions this common area for images from two different cameras is projected onto a rectangular area for an arbitrarily chosen verti-cal position Z0. This procedure is referred to as dewarping and results in
measurement images without perspective distortions.
An additional complication arises with respect to the PIV technique from the fact that the laser sheet is not perfectly aligned with the vertical positions of the calibration plates. This misalignment results in a shift between the de-warped images from different cameras. Cross-correlating the de-warped particle image from one camera with the dede-warped particle image from the other camera for the same time instant results in this shift, the so-called disparity map.
If the laser sheet was perfectly aligned with the vertical positions of the ca-libration plates and the position of the middle of this laser sheet was exactly known and given as Z0in the dewarping procedure then the disparity
map would be identically zero. As this is not the case one has to apply an ad-ditional procedure in order to correct for that. Using the disparity maps the triangulation method projects the dewarped images into the real (inclined) plane of the laser sheet.
The resulting measurement images can now be cross-correlated with ima-ges delayed by a certain time gap for each camera separately. In this way one obtains two PIV displacements fields, one for each camera. Finally, for each PIV displacement a perspective ray is computed based on the mapping functions obtained from the calibration procedure. The intersection of two corresponding perspective rays defines the 3D displacement vector. Divi-ding this displacement vector by the time gap applied results in a 3D velocity vector.
Figure 2.4: Snapshots of the vertical velocity w in mm/s from the experiment
pre-sented in Chapter 5. The resulting velocity field (a) was obtained by com-bining velocity fields displayed in Figs. 2.4b,c,d. The other velocity fields were obtained by using time gaps ∆t1(b), ∆t2(c) and ∆t3= 1/15 s (d) as the time gap between two successive images. (For specification of these time gaps, see text.)
Time gap between two laser pulses
The choice for the time gap (∆t1) between two successive laser pulses is
re-stricted by maximum horizontal and vertical displacements permitted. When these displacements are exceeded the measurement error substantially increases. For the horizontal displacement one demands it to be smaller than a quarter of the interrogation window, which in all the experiments per-formed was 8 pixels. For the vertical displacement one demands it to be smaller than a quarter of the laser sheet thickness, which in all the
experi-2.3. Experimental techniques 25
ments performed was, 0.25 mm. These maximum displacements follow from the necessity to prevent a substantial in-plane or out-of-plane particle loss.
Therefore, the time gap (∆t1) was applied for each kind of experiment
differently according to the velocities being measured. At first its value was set so that the condition for the horizontal displacement was fulfilled at the moment when the forcing was switched off when horizontal motion is stron-gest. After the experimental data were obtained it was checked whether the maximum vertical displacement did not exceed at any time of the flow evo-lution the necessary condition. Otherwise the time gap (∆t1) was shortened
and the experiment repeated. The time gap (∆t1) applied in different
experi-ments was between 6−20 ms, depending on the conditions of the experiment. The shortcoming of the PIV or SPIV technique is a narrow dynamic range of the displacements which can be measured reliably. Suppose the error is 0.1 pixel, then this range is between about 1 − 8 pixels, not even one decade. To improve the quality of data and extend the range of reliable displacements a longer time gap (∆t2= 1/15 s−∆t1) was used.
Naturally, the displacement fields (VL) obtained from cross-correlating
images delayed by ∆t2 contain many spurious vectors as the time gap was
too long for these relatively large local displacements. In displacement fields (VS) obtained from cross-correlating images delayed by ∆t1 these vectors
were measured accurately. On the other hand, there are many areas where vectors were detected correctly in VLas the local displacements were
relati-vely small, while in VSthese displacements are too small to be reliable.
There-fore, the aim was to combine VL and VS in order to obtain a final
displacement field, where large displacement vectors are taken from VS and
small displacement vectors from VL.
At first, each displacement vector ~xS = (xS, yS, zS) [pixels] in VS is
exa-mined. If at least one of the two conditions is fulfilled
xS ∆t2 ∆t1 > 8 (2.8) yS∆t 2 ∆t1 > 8 (2.9)
then this vector is taken for the final displacement field. Otherwise, the vector ~xL= (xL, yL, zL) [pixels] obtained from VLis taken for the final displacement
field provided the vertical displacement zL, mapped into physical unit, does
not exceed 0.25 mm. In case this condition is not satisfied one is forced to use the ~xS for the final displacement field. The result of this procedure is shown
in Fig. 2.4a for one velocity field at t = 9 s obtained from the experiment presented in Chapter 5. At this time instant the flow is slow enough to use
mainly vectors measured in VL, see Fig. 2.4c. PIV displacements are too short
(≈ 1 pixel) in VS (Fig. 2.4b) to be considered as reliable. One can see that
relying only on measurements from VSwould substantially overestimate the
vertical velocities. Additionally, cross-correlating images delayed by even longer time gap (∆t3 = 1/15 s) does not lead to a further data improvement,
see Fig. 2.4d.
2.3.4 Estimation of differential quantities
Raw measurements of velocity fields need to be further post-processed to obtain velocity gradients. With the PIV or SPIV techniques one can compute only horizontal gradients of velocity components. It means that from the vor-ticity vector only the vertical component of the vorvor-ticity ωz = ∂v/∂x − ∂u/∂y
is available from these measurements. The vertical vorticity is obtained using the definition of the circulation and the Stokes theorem.
The circulation Γ is defined as a line integral of fluid velocity ~u around a closed curve σ according to
Γ ≡ Z
σ
~u · d~s, (2.10)
where d~s is a vector tangential to the closed curve σ. The numerical integra-tion of (2.10) for a specified point (i,j) yields
Γi,j = 1/2∆x(ui−1,j−1+ 2ui,j−1+ ui+1,j−1) +
+1/2∆y(vi+1,j−1+ 2vi+1,j+ vi+1,j+1) +
−1/2∆x(ui−1,j+1+ 2ui,j+1+ ui+1,j+1) +
−1/2∆y(vi−1,j−1+ 2vi−1,j+ vi−1,j+1) (2.11)
where u and v are horizontal velocity components measured in grid points as indicated in Fig. 2.5a, ∆x and ∆y are grid spacings in x- and y-direction, respectively.
The Stokes theorem relates the surface integral over an open surface A to the line integral (2.10) around the boundary curve σ of this surface A.
Z σ ~u · d~s = Z A ∇ × ~u · d~a (2.12)
where d~a is a vector normal to the surface A. From relation (2.12) the vertical vorticity component is obtained as the circulation divided by the area
(ωz)i,j =
Γi,j
2.3. Experimental techniques 27
Figure 2.5: (a) A scheme for calculation of the circulation Γi,j around a specified point (i,j) and (b) points where free surface deformations were measured (A, B, C, D). Black circles indicate magnet positions below the bottom and taken from the experiments presented in Chapter 5.
The implementation of such an algorithm is valid only when the inter-rogation window is small enough, so that the velocities across this window do not change drastically. This condition is fulfilled for all the experiments presented in this thesis. For the grid points at the edges of the velocity fields the numerical scheme should be adjusted suitably since one does not have the full information about surrounding velocities.
The application of the above scheme leads to a minimum error [66] in comparison with other methods like Richardson extrapolation or least squ-ares technique. Additionally, the errors resulting from this scheme due to over-sampling of velocity fields data are not as large as in the case of other methods. For a more extensive discussion on the comparison between diffe-rent methods the reader may consult [66].
Also, ∂v/∂y and ∂u/∂x have to be obtained for the calculation of strain components in Chapter 5. These gradients are approximated through
∂u ∂x ≈
1
(2.14)
and ∂v ∂y ≈
1
8∆y[(vi+1,j+1+ 2vi,j+1+ vi−1,j+1) − (vi+1,j−1+ 2vi,j−1+ vi−1,j−1)] . (2.15)
It has to be noted that using the above differentiation operators for
∂v ∂x ≈
1
8∆x[(vi+1,j−1+ 2vi+1,j + vi+1,j+1) − (vi−1,j−1+ 2vi−1,j + vi−1,j+1)] (2.16)
and ∂u ∂y ≈
1
8∆y[(ui+1,j+1+ 2ui,j+1+ ui−1,j+1) − (ui+1,j−1+ 2ui,j−1+ ui−1,j−1)] (2.17)
results in relation (2.13) with (2.11).
2.3.5 Measurements of free surface deformations
In a shallow fluid layer experiment one might be concerned with the impor-tance/unimportance of free surface deformations. The measurements of free surface deformations have been performed in four characteristic points for the experiments presented in Chapter 5, see Fig. 2.5b. These turbulent flows were chosen for these measurements as in this case deformations were expec-ted to be highest as compared to all other experiments. Points A, B, C and D were situated inside the area indicated by the dashed region in Fig. 2.1c, where also measurements of velocity fields were carried out.
In these measurements a light beam propagates obliquely to the free sur-face and is reflected from it. The reflected ray is collected by a photosensi-tive element (photodiode), which measures the incoming light intensity. The measured electric current is proportional to the instantaneous free surface elevation.
The calibration curve is obtained through measuring the current ampli-tude from the photodiode at two unforced states: the first one with fluid height of an actual experiment and the second one with a slightly increased height (0.5 mm).
2.4. Three-dimensional numerical simulations 29
The main measurement error results from the fluctuations of the electric current around a mean value. The standard deviation of these fluctuations equals 0.1 mA, which in the experiments performed accounts for 0.015 mm uncertainty. The results of these measurements are presented in Fig. 5.17 in Chapter 5.
2.4 Three-dimensional numerical simulations
The numerical simulations of the 3D Navier-Stokes equation including a for-cing term ~f
∂~u
∂t + ~u · ∇~u = −1/ρ∇p + ν∇
2
~u + ~f (2.18)
with the incompressibility condition
∇ · ~u = 0 (2.19)
were conducted with a finite element package COMSOL MultiphysicsTM[1].
The magnetic field
In order to model a realistic electromagnetic forcing (2.6) and (2.7) one may derive an analytical solution for the magnetic field ~B of a single circular magnet. The derivation starts with the Biot-Savart law, relating the magnetic field to its source, the electrical current. Applying this law to a single circular current loop with radius 0.5 results in the magnetic field ~BLproduced by this loop: BzL= B0h K(m) + 0.25 − x2− y2− (z + z 0)2 (0.5 −px2+ y2)2+ (z + z 0)2 E(m) ! (2.20) ByL= −B0 y(z + z0)h x2+ y2 K(m) − 0.25 + x2 + y2 + (z + z0)2 (0.5 −px2+ y2)2+ (z + z 0)2 E(m) ! (2.21) with h = q 1 (0.5 +px2+ y2)2+ (z + z 0)2 , (2.22)
where z = −z0denotes the vertical position of the current loop with respect to
the origin of the (x, y, z) coordinate system. K(m) and E(m) are the complete elliptic integrals of the first and second kind, respectively,
K(m) = Z π/2 0 dψp 1 1 − m2sin2 ψ (2.23) E(m) = Z π/2 0 dψ q 1 − m2sin2ψ (2.24)
where m is defined through
m = s 2px2+ y2 (0.5 +px2+ y2)2+ (z + z 0)2 . (2.25)
The reader may consult [72] for the full derivation of these results. Fur-thermore, a permanent magnet is modeled here by a stack of current loops. Thus the total magnetic field is obtained through an integration of the above current loop solutions as follows
~ B(x, y, z) = Z h2 h1 ~ BL(x, y, z, z0)dz0, (2.26)
where −h1 and −h2 indicate the vertical positions of the lower and upper
surface of the permanent magnet normalized to the diameter of the magnet. This integration is performed numerically prior to solving the full 3D Navier-Stokes equation in which the resulting forcing field is used. The result (2.26) defines the magnetic field up to a multiplicative constant B0in (2.20) and (2.21).
Therefore the strength of ~B(x, y, z) has to be derived with the help of the experimental data. Combining (2.26) with (2.6) and (2.7) results in the for-cing ~f defined up to an unknown multiplicative constant, C = κB0. This
constant is found by matching the experimentally measured vertical vorti-city ωz with the one obtained in the 3D simulations. The resulting forcing
field above a single magnet with the current density used in the experiment presented in Chapter 3 is shown in Fig. 2.6. It is clear that the forcing used for the experiments is fully three-dimensional.
In case of the dipole-wall collision (Chapter 3) and a dipole propagating over a sloping bottom (Chapter 4) under the assumption of symmetry with respect to the vertical plane x = 0, the computations were carried out for only
2.4. Three-dimensional numerical simulations 31
Figure 2.6: The axisymmetric forcing field fy(a) and fzfor x = 0 (b) above a perma-nent magnet obtained from numerical computation of the magnetic field.
one dipole half. A symmetry condition was applied at the symmetry plane of the dipole. This simplification was validated by additional simulations performed in a full domain and without imposing the symmetry condition. The full domain simulations showed minor differences as compared to the half-domain simulation.
In all the 3D simulations the upper boundary is taken to be rigid, which implies that free surface deformations are not taken into account. In case of a dipole freely evolving in a shallow fluid layer gravity waves can be neglected as shown by Akkermans et al. [6]. The unimportance of these deformations in case of decaying turbulence in a shallow fluid layer has been confirmed by direct measurements and described in Sec. 5.9.
3. DIPOLE-WALL COLLISION IN A SHALLOW LAYER
3.1 Chapter outline
1This chapter is devoted to the study of a dipolar vortex in a homogeneous shallow layer impinging on a no-slip vertical wall. The influence of the wall on the approaching dipole is examined by directly comparing two cases: the collision case with the free evolution of the vortex. The complex vertical structure of a shallow-layer dipole becomes even more complex during the collision process. It was also observed that the vertical velocities increase during the collision irrespective of the fluid layer thickness. The relative in-crease of vertical velocities may even become stronger as the thickness of the fluid layer decreases. The phenomenon of enhancement of vertical motion is mainly accounted for by the stretching of the x-component of the vorticity. 3D numerical simulations provide the possibility to extend the analysis and to validate the main explanation of the phenomena.
The chapter is organized as follows. Sec. 3.2 gives the state-of-the-art of the problem. Sec. 3.3 consists of details of all the experiments and simulations performed. Sec. 3.4 describes in detail experimental and numerical results obtained from a deeper (H = 11.4 mm) layer. Sec. 3.5 reports on experimental and numerical results obtained from shallower (H = 6.2 and H = 3.0 mm) layers. Finally, the main findings are summarized in Sec. 3.6.
3.2 Introduction
A dipolar vortex has been recognized, together with a monopole, as a very frequently occurring coherent flow structure in the Earth’s atmosphere and oceans. At the large scales these flows are influenced by the background ro-tation of the Earth. Additionally, the density stratification, caused by either the temperature or the salt concentration gradients, plays a crucial role in the dynamics of these geophysical flows. Finally, “geometrical confinement” is
1Partially, the contents of this chapter have been published in the separate paper [18]. The
