Dissertation NilsKrah eingereicht

Dissertation NilsKrah eingereicht
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
put forward by
Dipl.-Phys. Nils Krah
born in Bremen
Date of oral examination: 08.05.2014
Visualization of air and water-sided concentration
profiles in laboratory gas exchange experiments
Referees:
Prof. Dr. Bernd Jähne
Prof. Dr. Kurt Roth
Abstract
In this work, the transport of a tracer gas from air into water is studied experimentally
under turbulent, wavy conditions imposed by a unidirectional wind field in a laboratory
flume. To this end, a technique based on laser induced fluorescence recorded with cameras is developed to visualize the spatial tracer distribution in both phases simultaneously.
Dedicated image processing procedures are developed to determine statistically averaged
vertical profiles of absolute concentration in coordinates relative to the wavy water surface.
A thorough characterization of the technique in terms of systematic errors is presented
and a simple method to estimate the vertical probability distribution of turbulent eddies
in water is proposed. Invasion experiments are performed in which supersaturation of acetone in the flume’s air compartment is imposed giving rise to a vertical flux into the water.
The equivalence of momentum and mass transport in air, presumed by many theoretical
descriptions, is confirmed experimentally by comparing characteristic scaling properties of
wind and concentration profiles. Results also suggest that surfactants suppress gas exchange
by modifying the water-sided boundary conditions and altering the turbulent water flow
close to the surface regardless of the wind regime. Furthermore, observed discontinuity
of complete air-water concentration profiles leads to the hypothesis that kinetic processes
on molecular scale at the interface must be considered additionally to fluid mechanical
descriptions.
Zusammenfassung
In dieser Arbeit wird der Transport eines Spurenstoffgases von Luft in Wasser experimentell
untersucht. Dazu werden in einem Windkanal Turbulenz und Wellen durch ein gerichtetes Windfeld erzeugt. Es wird insbesondere eine Technik zur gleichzeitigen bildhaften
Darstellung der räumlichen Spurengasverteilung in beiden Phasen entwickelt, die auf dem
Prinzip der laserinduzierten Fluoreszenz beruht, die von Kameras aufgenommen wird. Spezielle Bildverarbeitungsprozeduren werden entwickelt, um statistisch gemittelte vertikale
Profile absoluter Konzentrationen in Koordinaten relativ zur welligen Wasseroberfläche
zu bestimmen. Eine gründliche Charakterisierung der Technik in Bezug auf systematische
Fehler wird vorgestellt und eine einfache Methode wird vorgeschlagen, um die vertikale
Wahrscheinlichkeitsverteilung turbulenter Eddies im Wasser zu schätzen. In Invasionsexperimenten wird eine erhöhte Acetonkonzentration im Luftraum des Windkanals und dadurch
ein Massenfluss ins Wasser erzeugt. Die Gleichwertigkeit von Impuls- und Massentransport
in der Luft, die vielen theoretischen Beschreibungen als Annahme zugrunde liegt, wird
bestätigt durch den experimentellen Vergleich der Skalierungseigenschaften von Wind- und
Konzentrationsprofilen. Ergebnisse legen zudem nahe, dass oberflächenaktive Substanzen
Gasaustausch unterdrücken, indem sie die wasserseitigen Randbedingungen und dadurch
den turbulenten Fluss nahe der Oberfläche verändern, unabhängig vom Windstärkenbereich.
Weiterhin führt die beobachtete Diskontinuität der vollständigen Luft-Wasser Konzentrationsprofile zur Hypothese, dass kinetische Prozesse auf molekularer Skala zusätzlich zu
fluidmechanischen Beschreibungen beachtet werden müssen.
Contents
1
2
3
Introduction
1
1.1
Scientific context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Overview of previous studies . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Theory
7
2.1
Basic fluid mechanical equations . .
2.1.1
Reynolds decomposition . .
2.1.2
Averaged equations . . . . .
2.1.3
Vertical transport equations
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. 8
. 9
. 9
. 10
2.2
Turbulent boundary layer . . . . . . . . . . . . .
2.2.1
Sublayers and scales . . . . . . . . . .
2.2.2
Turbulent viscosity . . . . . . . . . . .
2.2.3
Boundary conditions and surfactants
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. 10
. 11
. 12
. 13
2.3
Transfer resistance and transfer velocity . . . . . . . . . . . . . . . . . . . . 14
2.4
Short review of gas transfer models . . . . . . .
2.4.1
Turbulent diffusivity . . . . . . . . . . .
2.4.2
Two layer stagnant film model . . . .
2.4.3
Surface renewal . . . . . . . . . . . . .
2.4.4
Eddy renewal and surface divergence
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16
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20
2.5
Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
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Visualization technique
3.1
Laser induced fluorescence . . . . .
3.1.1
Principles of fluorescence
3.1.2
Measurement principle . .
3.1.3
Choice of tracer . . . . . .
3.1.4
Properties of acetone . . .
23
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23
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25
25
27
III
3.1.5
3.1.6
4
5
6
7
IV
Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Relative brightness images . . . . . . . . . . . . . . . . . . . . . . . 32
Experimental set-up
35
4.1
Main components . . . . . . . . . . . . .
4.1.1
Wind-wave tank . . . . . . . . .
4.1.2
Pulsed Q-switched laser . . . .
4.1.3
Cameras, lenses, and triggering
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4.2
Theoretical analysis of optical properties at the water surface . . . . . . . 38
4.3
Deployment . . . . . . . . . . . . . . . . . . . . . .
4.3.1
Optical ghosting and air-sided camera .
4.3.2
Laser set-up and spatial filtering . . . .
4.3.3
Water-sided camera . . . . . . . . . . . .
4.4
Beam profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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Auxiliary experiments
36
36
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37
41
41
42
43
45
5.1
Wind profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2
Wave-age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3
Wave slope measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4
Characterization of the pulsed UV laser
5.4.1
Beam profile . . . . . . . . . . .
5.4.2
Power output . . . . . . . . . . .
5.4.3
Intensity fluctuations . . . . . .
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. 51
. 51
. 52
. 53
Experimental procedure
55
6.1
General procedure . . . . .
6.1.1
Tracer injection . .
6.1.2
Data recording . .
6.1.3
Reference images
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6.2
Modulated wind conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3
Surfactant: Triton-X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Data processing
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59
7.1
Air-sided profiles . . . . . . . . . . . . . . . . . . . . . .
7.1.1
Quotient images and relative concentration
7.1.2
Relative coordinates and image averaging .
7.1.3
Surface detection . . . . . . . . . . . . . . . .
7.1.4
Absolute concentrations . . . . . . . . . . . .
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7.2
Water-sided profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2.1
Surface detection and image averaging . . . . . . . . . . . . . . . 65
59
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7.2.2
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70
7.3
Eddy statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
7.4
Mass balance method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2.3
7.2.4
7.2.5
7.2.6
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Results
8.1
9
Implicit dependence:
concentration and fluorescent brightness . . . . .
Implementation and stability of fit approach . . .
Absolute concentrations . . . . . . . . . . . . . . .
Systematic error due to limited optical resolution
Correction for optical sharpness . . . . . . . . . .
73
Air-sided profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1
Comparison with wind profiles . . . . . . . . . . . . . . . . . . .
8.1.2
Profile scaling - diffusive sublayer . . . . . . . . . . . . . . . . . .
8.1.3
Profile scaling - logarithmic sublayer . . . . . . . . . . . . . . . .
8.1.4
Matching Deacon model profiles . . . . . . . . . . . . . . . . . .
8.1.5
Scaling of the outer layer (wake) . . . . . . . . . . . . . . . . . .
8.1.6
Distribution of resistance within the air-sided turbulent boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.7
Bias by water-sided brightness . . . . . . . . . . . . . . . . . . .
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73
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79
80
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. 83
8.2
Water-sided profile parameters . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.2.1
Comparison with oxygen profiles . . . . . . . . . . . . . . . . . . . 85
8.2.2
Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.3
Absolute concentrations and full profiles . . . . . . .
8.3.1
Proof-of-principle . . . . . . . . . . . . . . . .
8.3.2
Matched water and air-sided profiles . . . . .
8.3.3
Matched air and water-sided concentrations
8.4
Partitioning of resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.5
Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.6
Summary: Measured transfer resistances . . . . . . . . . . . . . . . . . . . 99
8.7
Turbulent Schmidt number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Discussion
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89
89
91
93
105
9.1
Diffusive sublayer and surface roughness . . . . . . . . . . . . . . . . . . . 105
9.2
Surfactants and boundary conditions . . . . . . . . . . . . . . . . . . . . . 107
9.3
Additional interface resistance - a hypothesis . . . . . . . . . . . . . . . . . 109
10 Conclusion
115
List of symbols
118
Bibliography
124
V
VI
1
Introduction
1.1 Scientific context
System earth can be thought of as a set of interconnected compartments (e.g. atmosphere, ocean), containing various species of chemical substances. Some are
produced naturally in the biosphere, some photochemically, others are of anthropogenic origin. Chemical reactions might cause depletion, creation, or conversion
from one substances into another. Specific for each compartment, transport processes lead to a redistribution of matter, typically following the overall concentration
gradient. Of importance for environmental mass budges is the flux across the interfaces which divide the compartments. The ocean assumes particular relevance as a
huge reservoir acting as a sink or source (Takahashi et al., 2009). For example, it takes
up e.g. 30 to 40 % of the CO2 originating from fossil fuel combustion (Donelan and
Wanninkhof , 2002). The interface dividing atmosphere and ocean is obviously the
water surface. Since it is virtually impossible to directly monitor fluxes globally and
continuously, it is of interest to establish relationships with other easier to measure
parameters. This has been topic of research for the past decades and continues to be.
The water surface acts as a phase boundary both in thermodynamic and fluid
mechanical sense. For the transport of a substance from the ocean to the atmosphere
or vice versa, thermodynamics describe the processes at the interface. Gas exchange
eventually implies evaporation or condensation, respectively, leading to dependences
e.g. on temperature, pressure, humidity. Within the fluid, mass is transported by
molecular diffusion and by advection. Fluid mechanically, the surface constrains air
and water flow, leading to a suppression of turbulence in proximity to the interface.
For this reason, the otherwise comparatively inefficient diffusion becomes relevant
very close to the phase boundary. To parametrize gas exchange across the ocean’s
1
Chapter 1
INTRODUCTION
surface, it is desirable to identify transport mechanisms and suitable measurable
quantities which describe them.
Leaving aside heat-induced buoyancy, wind is the major source of mechanical forcing on the open ocean. In coastal regions, additional phenomena such as upwelling
may play a role. Naturally, wind speed emerges as possible parameter describing
the driving force. At low wind speeds, momentum transfer from the air flow into
the water occurs by means of shear stress at the surface, motivating the so-called
friction velocity as another characteristic parameter. For higher wind speed, some
studies suggest an enhancement of gas transfer due to the presence of waves, where
presumably the high frequency portion of the spectrum is most relevant (Bock et al.,
1999; Jähne et al., 1984). The formation waves might indeed lead to flow structures
too complex to be described as a unidirectional shear flow and consequently, neither
wind speed nor friction velocity are adequate universal parameters. Under very
energetic conditions, breaking waves and the associated white capping and bubbles
open yet another transport channel, and similarly, rain is believed to have an impact
on gas exchange (Wanninkhof et al., 2009).
Experimental studies have shown that naturally occurring surface active material,
e.g. originating from phytoplankton, interferes with the fluid mechanical interaction
between wind and water at the surface, possibly by altering the fluid mechanical
boundary conditions, leading to a suppression of small scale waves (ripples) and
mass transfer rates (e.g. Broecker et al., 1978; Frew et al., 1990; Hühnerfuss et al.,
1987; Lucassen, 1982; McKenna and Bock, 2006). This has also lead to speculations
whether wave attenuation is the cause for reduced gas exchange or whether both
are independent symptoms of the modified surface dynamics (Frew et al., 2004). In
any case, the effect of surfactants is not captured by either of the above mentioned
parameters.
1.2 Overview of previous studies
Research activity in the last decades has spanned quite a vast gamma of methodical
approaches. Numerous field campaigns, most of them ship-based and some from
platforms, have produced a great amount of empirically measured transfer rates for
different types of gases, (e.g. Frew et al., 2004; Ho et al., 2006, 2011; Nightingale et al.,
2000a,b; Wanninkhof and McGillis, 1999), to name only a few. Partly due to low
fluxes and cumbersome instrumentation, statistical scatter is rather prominent, as
seen from the exemplary plot in Fig. 1.1. For this reason it is intricate to formulate
quantitative relationships among parameters in a deductive way. Most empirical
parametrizations are based on a power law dependence of transfer velocity and wind
speed (Ho et al., 2006; Nightingale et al., 2000b; Wanninkhof , 1992; Wanninkhof et al.,
2009), others use sub-regimes based on wind conditions (Liss and Merlivat, 1986).
Some techniques estimate the gas flux from measurements in the air by exploiting
scaling properties of the turbulent airflow and the conceptual analogy between
2
Overview of previous studies
1.2
Figure 1.1
Exemplary plot of transfer
velocities
measured
in
field
campaigns,
depicted as a function
of wind speed. Empirical
parametrizations
are
shown as lines. The figure
is taken from Ho et al.
(2011).
momentum and mass transport (Edson et al., 2004; McGillis et al., 2001; Roether,
1983). The integrity of this approach has been verified by comparing results with
mass fluxes obtained through independent complementary methods. To my best
knowledge, however, the analogy of concentration and wind profiles, especially very
close to the water surface, has not yet been directly investigated experimentally.
Another broad methodical field comprises studies of gas exchange in wind-water
tunnels (Alaee et al., 1995; Broecker et al., 1978; de Leeuw et al., 2002; Frew et al.,
1995; Jähne, 1985; Jähne et al., 1987; McGillis et al., 2000; Ocampo-Torres et al., 1994).
While certainly idealizing the complex variable conditions encountered on the
ocean, such laboratory experiments allow to actively monitor and control parameters
like wind speed, additional mechanical wave forcing, fetch (i.e. interaction length
between wind and water), and not the least the presence or not of surface films.
Instrumentation is considerably easier to deploy than on a ship.
Advances in digital camera technology have brought about an increased use of
visualization techniques in laboratory and field experiments. A series of studies
have focused on the small scale distribution of deliberately injected tracer gases in
the uppermost centimeters below the water surface (e.g. Duke and Hanratty, 1995;
Friedl, 2013; Herlina and Jirka, 2008; Janzen et al., 2010; Jirka et al., 2010; Münsterer
and Jähne, 1998; Schulz and Janzen, 2009; Woodrow and Duke, 2001). In some cases,
turbulence was created by wind forcing, in others through stirring of the water body.
The effect of surfactants was explicitly investigated by Frew (1997); Lee et al. (1980);
McKenna and McGillis (2004). Visualization techniques have also allowed to study
geometrical and dynamical features of wind-induced wave fields (Balschbach, 2000;
Jähne and Riemer, 1990; Rocholz, 2008), turbulent structures underneath waves (e.g.
Siddiqui et al., 2001, 2002) and on the surface (McKenna and McGillis, 2004).
Experimental studies have been complemented by direct numerical simulation
(DNS) (Shen et al., 2000; Sullivan et al., 2000; Tsai and Hung, 2007; Tsai, 2001;
3
Chapter 1
INTRODUCTION
Tsai et al., 2013), and finally, a number of theoretical models have been proposed
attempting to describe air-sea gas exchange from first principles. I will explain some
of them in sec. 2.4.
1.3 Scope of this thesis
Within the spectrum of existing investigations on the matter of gas exchange, this
thesis is placed among visualization based laboratory experiments. Different from
previous studies, processes underlying the vertical mass transport in the air are of
central interest. The scope of this thesis is therefore the development and optimization of an experimental technique to investigate the spatial distribution of a tracer
gas in the turbulent airflow above the wind-driven wavy water surface. Initial efforts
have already been undertaken by Winter (2011).
In this thesis, the technique is applied in wind tunnel experiments which allow
to validate the developed methods. The purpose of these experiments is to record
statistically averaged concentration profiles in the air and to analyze them in terms
of analogy between mass and momentum transport.
The technique is extended to simultaneously visualize the distribution of a tracer
gas in the water. The goal is to study the partitioning of transfer resistance across
the interface (i.e. the macroscopic parameter describing the effectiveness of vertical
mass transport from one compartment into the other) for different wind forcing
conditions. In this context, also the effect of surfactants on the gas transfer is of
interest.
The experimental results will also allow to deduce transfer coefficients for the used
tracer gas, acetone, as a function of wind speed and friction velocity. It is stressed
here, however, that the scope is not to provide empirical data to be transfered easily
to field conditions, although this might be possible.
Other works have focused on the validation of specific transport models based
on mean vertical concentration profiles in the water (Friedl, 2013; Münsterer and
Jähne, 1998). This is not the scope of this thesis. Instead, characteristic physical
parameters are used as phenomenological indication on the relevance of physical
processes and their dependence on experimental conditions such as wind forcing
and surface active material.
4
Structure of the thesis
1.4
1.4 Structure of the thesis
This thesis is organized as follows:
• In chap. 2, I explain the basic physical notions and concepts necessary to
describe the studied phenomena. In particular, I will outline the analogy
between momentum and mass transport in boundary layer theory and I will
elaborate the different scales relevant in water and air.
• Chapter 3 is dedicated to the employed visualization technique. Basic principles of laser induced fluorescence (LIF), the choice of tracer, and how concentration profiles are obtained from fluorescent brightness images are explained.
• Chapter 4 focuses on the technical aspects of the experimental set-up, i.e.
the wind-wave tank, the main components, and their deployment. I also
present a theoretical analysis of the optical effects at the water surface and
their implications on the recorded concentration profiles.
• In chap. 5, I summarize a series of complementary experiments (and their
results) conducted to characterize the wind and wave field and the pulsed laser
system used in this work.
• The experimental procedures are explained in chap. 6.
• Chapter 7 summarizes the data processing approaches used to retrieve concentration profiles from recorded images and to convert them to absolute
concentrations. I also explain how transfer velocities are calculated based on
mass balance considerations and how streaks in the water are used to obtain
an estimate on the vertical distribution of turbulent eddies.
• Chapter 8 presents the results of visualization experiments at a wind-wave
tank on the invasion process of acetone from air into water.
• In chap. 9, I discuss three major aspects that emerged from the experimental
results.
5
2
Theory
In this chapter, I will collect and describe the basic notions necessary to theoretically
treat the transport of a passive admixture such as a gas in a turbulently mixed
medium. Since this thesis deals with the processes occurring in close proximity
to the air-water interface, the concept of a boundary layer, initially proposed by
Ludwig Prandtl, is useful. The intention is not to give a comprehensive review of
this certainly vast field and for more detailed explanations and derivations, I refer to
common textbooks such as Kundu (2008); Monin and Yaglom (2007); Schlichting
(2000).
I will further provide an overview over models used to describe gas exchange from
a small-scale point of view and the different boundary conditions at the interface
applicable to the air and water flow, respectively. Finally, I will quickly describe the
conceptual influence of surface active material on gas exchange and the partitioning
of transport resistances among the gaseous and aqueous phase according to the
solubility of a gas.
Different terms are found in literature dealing with the transport of a gaseous
substance in a fluid and through the phase boundary. These are gas exchange, gas
transfer, mass transfer, mass transport. I will use them interchangeably throughout
this thesis.
7
Chapter 2
THEORY
2.1 Basic fluid mechanical equations
In a Eulerian frame, a fluid flow is characterized by the velocity field u⃗ and the density
ρ. For the frequency of pressure fluctuations occurring under realistic environmental
conditions, e.g. on the ocean, both air and water can be treated as incompressible.
In this case, the continuity equation reads
∇ ⋅ u⃗ = 0,
(2.1)
It should be noted that close to a boundary, two-dimensional divergence can still
occur and is indeed considered by some gas exchange models to be the principle
transport mechanism.
The intermolecular forces on scales smaller than those considered in the continuum picture of fluid mechanics are described by the stress tensor,
∂u i ∂u j
) −pδ i j .
+
∂x j ∂x i
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
τi j = µ (
(2.2)
≡σ i j
Here, p is the normal pressure, µ the material specific viscosity, and σi j the viscous
stress. Note that this is valid for air and water given their Newtonian nature. The
stress is typically used as boundary condition at the surface, as will be explained in
sec. 2.2.3.
Conservation of momentum in an incompressible fluid is described by the NavierStokes equation
∇p
∂u⃗
+ (u⃗ ⋅ ∇) u⃗ = −
+ g⃗ + ν∆u⃗,
(2.3)
∂t
ρ
where ν ≡ µ/ρ is the kinematic viscosity and g i summarize the external forces. A
characteristic number associated with this equation is the Reynolds number, Re ≡
U L/ν , where U and L are typical velocity and length scales of the considered flow. The
Reynolds number quantifies the ratio of inertial to viscous forces.
Transport of a passive admixture, in this work a tracer gas, is described by the
diffusion-advection equation,
∂c
+ u⃗∇c = D∆c,
∂t
(2.4)
where D is the diffusion coefficient in the corresponding medium. As unit for the
concentration, I use mol/L. Knowledge of the flow field u⃗ is necessary to describe
the transport of a tracer in the fluid.
Viscous dissipation of momentum and diffusive dilution both refer to small scale
processes. The ratio of the related coefficients in eq. (2.3) and eq. (2.4) is defined as
the Schmidt number Sc ≡ ν/D. For most gases in the air, it is of order 1 while in the
8
Basic fluid mechanical equations
2.1
water, it is typically of order 1000.
2.1.1 Reynolds decomposition
In a turbulent flow, the quantities u⃗(x⃗, t) and c(x⃗, t) will fluctuate in time when
observed at a fixed point in space. Under stationary conditions and after sufficiently
long observation, it should be possible to establish mean values ⟨u⃗⟩(x⃗) and ⟨c⟩(x⃗),
characteristic e.g. for the distance to the fluid boundary such as the air-water interface.
This motivates to separate the variables in a mean and a fluctuating contribution,
u⃗ = ⟨u⃗⟩ + u⃗′
c = ⟨c⟩ + c ′ ,
(2.5)
(2.6)
In a strict sense, ⟨.⟩ denotes the average over all realizations in the statistical ensemble
describing the stationary flow. By the ergodic principle, it is practically calculated
by averaging over sufficiently long time periods. It should be noted that continuity
is fulfilled also by the fluctuating term, i.e. ∇ ⋅ u⃗′ = 0. If the fluctuating component
cannot be resolved in time technically, the mean term can still provide insight
into relevant mechanisms. Indeed, this work deals with the visualization of mean
concentration profiles.
2.1.2 Averaged equations
The Reynolds decomposition can be inserted in eq. (2.3) and eq. (2.4) and the average
over all terms can be taken. This results in equations describing the mean velocity
and concentration fields. The straight forward calculation can be found in all of the
textbooks cited above. The resulting expressions in indexed notation are
∇⟨p⟩
∂⟨u i ⟩
∂ ′ ′
+ (⟨u i ⟩ ⋅ ∇)⟨u i ⟩ = −
+ g i + ν∆⟨u i ⟩ −
⟨u u ⟩
∂t
ρ
∂x j i j
∂⟨c⟩
+ ⟨u⃗⟩∇⟨c⟩ = D∆⟨c⟩ − ∇⟨u⃗′ c ′ ⟩ = ∇ (D∇⟨c⟩ − ⟨u⃗′ c ′ ⟩) .
∂t
(2.7)
(2.8)
Einstein’s convention for summing equal indices is intended. The additional term
⟨u ′i u′j ⟩ is often called turbulent Reynolds stress. Henceforth, I will simply denote the
average quantities by u⃗ and c for the sake of notational simplicity.
9
Chapter 2
THEORY
2.1.3 Vertical transport equations
In certain situations, statistical homogeneity in the two lateral dimensions (x,y) can
be assumed, i.e. ∂/∂x = 0 and ∂/∂y = 0, and the above equations simplify. In a unidirectional wind field (in x-direction), like in a linear wind-wave tank, the x-component of
the mean velocity field u x is of interest. The average vertical component, uz , must be
zero on the other hand. Similarly, if a macroscopic concentration gradient between
the air and water compartment is imposed, e.g. by deliberately injecting a tracer gas,
this will result in a vertical flux into the water. In the absence of pressure gradients
and external forces, the following two expressions are obtained,
∂u x
∂
∂u x
(ν
=
− ⟨u ′x uz′ ⟩)
∂t
∂z
∂z
∂c
∂
∂c
(D − ⟨uz′ c ′ ⟩) .
=
∂t ∂z
∂z
(2.9)
(2.10)
Under stationary conditions, the time derivative is zero giving rise to expressions
for the momentum and mass flux, respectively. I will, henceforth, adopt the notational convention in literature on gas exchange by which the horizontal and vertical
velocity components are denoted by u and w, respectively.
∂u
+ ⟨u ′ w ′ ⟩
∂z
∂c
j m = −D + ⟨w ′ c ′ ⟩.
∂z
τ/ρ = −ν
(2.11)
(2.12)
The second terms on the RHS are the turbulent contribution to mass and momentum flux. Since they are associated to turbulent eddies, they are often called
eddy covariance. Some experimental techniques seek to directly determine their
value by measuring air flow and concentration at sufficiently high time resolution. A
theoretical approach is to relate the covariance terms to mean quantities, thereby
modeling the turbulent transport. I will explain this in more detail in sec. 2.2.2 and
sec. 2.4.1.
2.2 Turbulent boundary layer
In high Reynolds number flows viscosity is negligible compared to inertial forces.
On the other hand, Ludwig Prandtl showed back in 1905 that experimental findings
on drag coefficients of solid objects could only be reconciled if a no-slip boundary
condition was insisted on. Forcing the tangential velocity on a confining surface to
zero leads to strong gradients and thus to non-negligible viscous forces in proximity
of a boundary. This gives rise to the concept of a boundary layer, which is thin
compared to other scales of the flow and within which forces are to be considered
10
Turbulent boundary layer
2.2
which are negligible otherwise.
A case relevant for this work in which boundary layer flow occurs is the unidirectional airflow in a wind-wave tank. The water surface is in good approximation
rigid as timescales of deformation are much larger than those of the airflow. I will
not go through all derivations which can be found e.g. in Monin and Yaglom (2007),
but only state phenomenologically relevant facts as well as some formulæ.
2.2.1 Sublayers and scales
The turbulent boundary layer of a flow close to a smooth wall is subdivided into
three distinct regimes: the viscous sublayer, the logarithmic sublayer, and the outer
layer. Its overall thickness is given by δ.
Very close to the surface, viscous forces prevail, momentum flux is constant and
according to eq. (2.11) the wind profile is linear. A characteristic velocity scale is
given by the surface stress, i.e. momentum flux,
τs ≡ ρu∗2 ,
(2.13)
and together with the kinematic viscosity, this provides a characteristic length scale
ν/u∗ , based on which the non-dimensional wall-coordinate is defined,
z+ ≡
zu∗
.
ν
(2.14)
Note that this is conceptually a Reynolds number, but it does not describe the free
flow. The viscous sublayer is found experimentally to extend to z + = 11. The term u∗ is
called friction velocity and can be determined from wind profiles (see sec. 5.1). It may
be related to the water-sided equivalent value assuming continuity of momentum
flux across the phase boundary, i.e.
√
ρair
2
2
ρwater u∗water
= ρair u∗air
⇒ u∗air = u∗water
.
(2.15)
ρwater
I used this scaling whenever water-sided friction velocities were needed. Of course
it could also be measured directly from water-sided flow profiles, e.g. by particle
imaging velocimetry (PIV), or by momentum budget considerations (Bopp, 2011),
but that was not scope of this work.
Outside the viscous sublayer, dimensional analysis shows that the gradient of the
mean velocity is proportional to u∗ /z, leading to a logarithmic profile¹,
u(z) =
u∗
log z + C,
κ
(2.16)
where κ and C are constants which were determined experimentally and are available
¹I use log for the natural logarithm throughout this thesis, as common in mathematical notation.
11
Chapter 2
THEORY
as literature values.
The logarithmic sublayer is found experimental to extend to 0.15 ⋅ δ. Yet above,
the so-called outer layer is identified. No closed theoretical formulation for its shape
has been found so far, but it has been confirmed experimentally that velocity profiles
are self-similar. An empirical formula is proposed for example by Hama (1954),
2
U0 − u(z) = B (z − δ) .
(2.17)
The parameter δ quantifies the overall thickness of the turbulent boundary layer. I
used this expression to fit wind profiles (see sec. 5.1).
2.2.2 Turbulent viscosity
The term ρ ⋅ ⟨u ′ w ′ ⟩ in eq. (2.11) can be interpreted as turbulent contribution to the
vertical momentum flux. Indeed, only if positive horizontal velocities u ′ appear
prevalently in correlation with negative vertical velocities w ′ , this results in a net
downward flux. Often, the approach is to express the turbulent momentum transport
in terms of mean quantities of the flow. In the turbulent viscosity approach, ⟨u ′ w ′ ⟩
is expressed as the product of the gradient of the mean horizontal velocity profile
and a height-dependent turbulent viscosity,
ρ⟨u ′ w ′ ⟩ = ρKturb (z) ⋅
∂ū
.
∂z
(2.18)
The function Kturb (z) was modeled for a turbulent pipe flow a long time ago by
Reichardt (1951). Because it is relevant for my thesis, I give a short summary of
the concepts here. In his analysis, he uses dimensionless coordinates, scaled with
the viscous length, i.e. z + = zu∗/ν. Very close to the smooth surface, turbulence
should be negligible and viscosity is the only mechanism of momentum transport,
i.e. Kturb (z + ) + ν → ν for z + → 0. Further away from the surface, Kturb (z) should
assume a form such to reproduce the logarithmic velocity profile of eq. (2.16), i.e.
Kturb (z + )/ν = κz + .
(2.19)
Reichardt proposes a parametrization that interpolates between both forms, given a
certain scale z +ℓ ,
Kturb (z + ) + ν
z+
= 1 + κ (z + − z +ℓ tanh ( + )) .
ν
zℓ
(2.20)
For small z + , the hyperbolic tangent decays linearly so that the second term becomes
zero, just as required. For large z + , the linear term dominates, reproducing the logarithmic shape upon integration. The parameter z +ℓ was determined experimentally to
be ≈ 11 and describes the scale of transition between the viscous and the logarithmic
12
Turbulent boundary layer
2.2
sublayer.
Under stationary conditions, momentum flux τ is independent of height and
may in particular be taken as the flux at the surface τs . The velocity profile can be
recovered from eq. (2.20) upon integration, following eq. (2.18) and eq. (2.11),
τs
ū(z ) =
ρu∗
+
z+
∫
ν
ν + Kturb (z + )
0
dz + .
(2.21)
Using the definition of z + = zu∗/ν, this can also be written in dimensional form as
ū(z) =
τs
ρν
∫
0
z
ν
dz.
ν + Kturb (z)
(2.22)
2.2.3 Boundary conditions and surfactants
When describing the fluid flow theoretically or by means of DNS, certain assumptions
about the boundary conditions must be made. I will not discuss the ample question
in all its completeness, but collect some conceptually important arguments and refer
to the free and the rigid boundary as two limiting cases. Here, the phase boundary
is always the air-water interface. General explanations are found in textbooks on
fluid mechanics, cited above, and also in publications related to gas exchange which
I cite where appropriate.
For flat water, the surface is usually set to z = 0. In the presence of waves, the
water height is a function of space and time, η(x, y, t), and the boundary conditions
have to hold on z = η. Some authors use curvilinear, locally orthogonal coordinates
relative to the water surface (see e.g. Sullivan et al., 2000; Troitskaya et al., 2010; Tsai
and Hung, 2007; Tsai et al., 2013), others linearize and apply boundary conditions
anyhow at z = 0 (see e.g. Coantic, 1986; Shen et al., 2000; Tsai, 1996).
Kinematic boundary conditions concern the value of the three velocity components at the water surface. For a non disrupted water surface, it is commonly
assumed that the vertical component w of the flow be zero at z = η. The no-slip
condition, already mentioned in sec. 2.2, also requires u and v to be zero everywhere.
Consequently, also the derivatives ∂u/∂x = 0 and ∂v/∂y = 0 have to vanish, and by
the continuity eq. (2.1), also the gradient of the vertical component, ∂w/∂z = 0. As a
result, divergence and convergence are suppressed close to the surface. This is a valid
approach for air above the water since the latter has much higher density and thus
inertia. From the point of view of the water, the air is much less dense and does not
create inertial resistance upon tangential replacement of water parcels at the surface.
In this case, u and v are generally not zero at z = η and the same geometric interface
does not pose rigid wall conditions on the water.
Secondly, dynamic boundary conditions require equilibrium of stress (forces) at
the surface. According to eq. (2.2), this poses conditions on the derivatives of the
velocity field at the surface. For a body of free water, no external forces act so that the
13
Chapter 2
THEORY
components of the internal viscous stress have to vanish at z = η (see e.g. Shen et al.,
2000). Over a wind-driven surface, they have to equilibrate those imposed by the
wind field, which in turn depend on the surface shape and dynamics. In principle,
the full coupled system of air and water flow would have to be solved. In DNS like
the one of Tsai and Hung (2007), an empirically motivated modulation of the wind
stress along the phase of the most prominent wave component is used.
Surface active material modifies the surface tension of clean water, σ0 , by an
additional term σ, i.e. Π = σ0 − σ. The exact chemico-mechanical understanding for
many substances is still field of active research. Generally, the surfactant-induced
tension σ depends on the local structure of the surface film. In a simple visco-elastic
approach, used e.g. by McKenna and McGillis (2004) and Tsai (1996), a so-called
Marangoni stress is deduced from horizontal gradients in the surface tension, much
like a dilated or compressed spring. This has to locally match the shear stress in the
fluid,
∂u ∂σ
=
.
(2.23)
µ
∂z ∂x
McKenna and McGillis (2004) provide good explanations of the involved effects.
Conceptually, surface tension gradients counteract tangential displacement in the
fluid and thereby immobilize the water, leading to a reduction of divergence and
vorticity close to the surface. In this sense, surfactant covered water resembles a
rigid wall. Examples of experimental studies regarding the effect of surface films on
the turbulent flow structure are (Hirsa et al., 1995; Lapham et al., 2001; McKenna
and McGillis, 2004). DNS have been performed by Tsai (1996). The modified and
reduced surface tension also leads to a suppression of small-scale waves, or ripples.
This has been studied by many investigators, e.g. by Asaki et al. (1995); Frew et al.
(1995); Wu (1989)
2.3 Transfer resistance and transfer velocity
The mass flux j across the air-water interface depends on the macroscopic concentration difference ∆c. It is therefore natural to introduce a specific flux,
k≡
j
.
∆c
(2.24)
Since it has the units of length over time, it is commonly called transfer velocity. Its
inverse is referred to as transfer resistance,
R≡
1
.
k
(2.25)
The fluids on either side of the phase boundary pose a certain resistance to a
scalar quantity transported from the air-sided to the water-sided well-mixed bulk, or
14
Transfer resistance and transfer velocity
2.3
vice-versa, much in the same way as a solid body creates resistance to heat transfer
or a wire to electric current. Indeed, the latter example provides a good analogy that
I will lean upon in the following to summarize the concept of gas transfer resistance.
According to Ohm’s law, a potential difference is related to the current by the
electrical resistance, ∆U = R⋅I and for a series of N electrical elements, the individual
resistances add up as
N
Utot = I ⋅ ∑ R i .
(2.26)
i=0
Also, the contribution of an element to the total resistance is equal to the proportional
potential difference at this element, i.e.
Ri
∆U i
=
,
Rtot Utot
(2.27)
with Rtot = ∑ R i . In gas exchange, the concentration plays the role of the potential
and the current is replaced by the mass flux. If transport is statistically homogeneous
in x and y, then one can assign a differential resistance dc(z) to each height element
dz, such that, under stationary conditions,
dc(z) =
∂c
dz = j ⋅ dR(z).
∂z
(2.28)
The concentration profile normalized by the mass flux j is therefore the integrated
resistance up to height z,
c(z) − c(z = 0) 1
=
j
j
∫
z
0
c(z ′ ) dz ′ =
∫
0
z
R(z ′ ) dz = R(z).
(2.29)
In this sense, a concentration profile is equivalent to a resistance profile times the
flux. In particular, it describes the vertical distribution of resistance.
In general, when stating the value of a transfer resistance at given flow conditions,
the reference height to which this applies has to be specified.
c(z2 ) − c(z1 ) 1
=
j
j
∫
z2
z1
c(z) dz =
∫
z2
z1
R(z) dz = R(z1 , z2 ).
(2.30)
If in a field experiment, concentration samples are taking in discrete heights, say, 2
and 5 m above the water surface, then one is really probing the transfer resistance
associated with this layer. It has to be scaled up to the entire layer of interest assuming a certain resistance distribution. In a laboratory gas exchange experiment,
in which the transfer resistance is deduced from mass balance considerations (see
sec. 7.4), the entire turbulent boundary layer contributes to the result. The lower
boundary in eq. (2.29) in this case is z1 = 0 and the upper boundary is z2 = δ (see
sec. 2.2). In experiments that visualize the scalar concentration profile, effectively the
15
Chapter 2
THEORY
contribution of each layer in the fluid to the total resistance is obtained, potentially
providing a more detailed insight in the transport processes involved.
2.4 Short review of gas transfer models
Several models have been developed starting almost from the beginning of the last
century in an attempt to describe gas transfer through the air-water interface based
on measurable parameters. A comprehensive review would certainly go beyond
the scope of this chapter and I will only outline some of the basic concepts of the
different approaches. I will pay particular attention to the turbulent diffusivity model
as it provides a conceptually valid description of the air-sided transport process.
Most models share the peculiarity of a characteristic scaling with the diffusion
constant, or the Schmidt number Sc depending on the boundary condition.
2.4.1 Turbulent diffusivity
In analogy to the momentum transport, the ⟨c ′ w ′ ⟩ term in eq. (2.11) can be replaced by
a mean expression, relating the turbulent scalar transport to the mean concentration
gradient,
∂c
∂ c̄
j = −D + ⟨c ′ w ′ ⟩ = − (D + Kdiff (z)) ⋅ .
(2.31)
∂z
∂z
A model for Kdiff (z) has been proposed by Deacon (1977) by extending Reichardt’s
description from momentum to scalar transport. He assumes the turbulent mixing
of a tracer gas to be identical to the turbulent mixing of momentum density, i.e. he
assumes the so-called turbulent Schmidt number Scturb = Kturb (z)/Kdiff (z) to be one.
By replacing the kinematic viscosity ν with its equivalent for scalar transport, the
diffusion constant D, the turbulent diffusivity becomes
Kdiff (z + ) + D
z+
= 1 + Sc ⋅ κ (z + − z +ℓ tanh ( + )) .
D
zℓ
(2.32)
Compared to eq. (2.20), the turbulent contribution is additionally scaled by the
Schmidt number, Sc = ν/D. It is this scaling that causes the transport resistance to be
distributed differently on the water and on the air-side, respectively. For acetone, the
tracer used in this work, the air-sided Schmidt number is Scair = 0.15 cm2 /s/0.105 cm2 /s =
1.43, while for the water side it is Scwater ≈ 1 × 10−2 cm2 /s/1 × 10−5 cm2 /s = 1000.
It should be noted that Deacon’s parametrization is valid for a rigid wall boundary
(see sec. 2.2.3). For the airflow over the water surface, this is certainly satisfied. For
the waterflow, this is presumably a good description in the presence of surfactants,
but only approximative for clean water. Here, the purpose of the comparison is to
illustrate the different scales involved in air and water.
Figure 2.1 (a) shows the ratio D/D+Kdiff (z) as a function of dimensionless distance
+
+
z from the air-water interface. The distance z1/2
at which D = Kdiff (z) provides a
16
Short review of gas transfer models
a
2.4
b
Figure 2.1: (a) Ratio D/D+K diff (z) as a function of dimensionless distance z + from the air-water
interface. The two curves show the molecular contribution to the overall diffusivity
for the air side (red) and water-side (blue). The distance z +1/2 at which D = K diff (z) is
indicated by a dashed line. (b) Comparison of z 1/2 for typical friction velocities of the
wind-wave tank used in this work.
scale closer to which molecular diffusion prevails over turbulent mixing. Due to
+
the larger Schmidt number in water, z1/2
is about one order of magnitude smaller in
dimensionless coordinates. It can be re-scaled to physical coordinates according to
eq. (2.2.1) and by exploiting eq. (2.15). A comparison of the resulting values of z1/2 is
shown in Fig. 2.1 (b) for typical friction velocities of the wind-wave tank used in this
work. This sets a requirement on the scales necessary to resolve in the visualization
experiments on the air and water side, respectively.
Theoretical concentration profiles can be calculated by integrating eq. (2.32) and
eq. (2.31).
js Sc
c(z ) =
⋅
u∗
+
∫
0
z+
D
dz +
+
D + Kdiff (z )
(2.33)
The result is shown in Fig. 2.2 for the water and air side separately for typical values
of u∗ . The profiles are normalized to vary between 0 and 1 and depicted in a way
to reflect an invasion experiment. The air and water-sided Schmidt numbers of a
typical gas differ by about 3 orders of magnitude and this has been seen in Fig. 2.1 to
have a significant impact on the scales up to which molecular diffusion is a relevant
transport mechanism. Consequently, the resistance in air and water is expected to
be distributed in a much different way. Figure 2.3 shows resistance profiles, R(z) =
2
−5
2
c(z)/ j s , according to eq. (2.33) for ν
air = 0.15 cm /s, ν water = 1 × 10 cm /s, Dacetone,air =
0.105 cm2 /s, Dacetone,water = 1.29 × 10−5 cm2 /s, and typical values of the friction velocity
u∗ . Both sets of profiles show a logarithmic dependence on height further away
from the surface, although for the water profiles, this is less pronounced compared
to the increase of resistance within the diffusive sublayer. Indeed, in the water, the
contribution of the logarithmic sublayer to the overall resistance is negligible. This is
17
Chapter 2
THEORY
a
b
Figure 2.2: Theoretical water-sided (a) and air-sided (b) profiles, normalized to vary between 0
and 1 at surface and bulk, respectively. The profiles are depicted in a way to reflect an
invasion experiment.
due to the much larger Schmidt number and reason for the common assertion that
the water-sided diffusive sublayer dominates the transfer resistance. It should be
mentioned though that on larger scales, i.e. further away from the phase boundary,
other physical phenomena might play a role not considered here, such as stratification,
thermal layering etc. which create a notable resistance. In the here presented analysis,
only near-surface turbulence and diffusion are taken into account.
Contrarily, on the air-side, contributions of the diffusive and logarithmic sublayers
are of similar orders of magnitude (not including the outer layer). Indeed, from
Fig. 2.3 it can be seen, that up to a distance of 1 m from the water surface, the total
resistance reaches already twice its value right above the diffusive sublayer. In a
laboratory experiment, the width of the logarithmic layer will be limited and flow
above a certain height will always be dominated by boundary conditions posed by
the tank’s geometry. On the other hand, in a well developed turbulent flow like above
the open ocean, the turbulent boundary may easily grow to larger width. Under
such conditions, the air-sided diffusive sublayer is expected to contribute negligibly
to the overall resistance. To study and verify the scaling properties of the air-sided
mass boundary layer was in part scope of this thesis.
It should be noted that although analogy of momentum and mass transport in
the air is generally accepted in literature, experimental data on the latter is scarce. In
particular, to my best knowledge no concentration profiles have so far been observed
down to the diffusive sublayer.
18
Short review of gas transfer models
2.4
Figure 2.3
Comparison of theoretical
air and water-sided acetone concentration profiles according to the turbulent diffusivity model
for typical friction velocities u∗ used in this work.
2.4.2 Two layer stagnant film model
An idealized model for gas exchange is the film model, initially proposed by Lewis
and Whitman (1924) and applied as a two layer model to gas transfer over open
waters by Liss and Slater (1974). It assumes a purely diffusive region close to the
interface of a certain thickness d and an instantly mixed bulk underneath. In this
way, the diffusion equation ċ = D∆c has a stationary solution with the boundary
condition c(z = 0) = cs and c(z = d) = cb , which is simply a linear decay. The transfer
velocity is then entirely given by this diffusive layer as k = D/d . The model is certainly
idealized as it requires an infinite flux at z = d. Also, the model does not provide any
means to deduce the layer thickness d from fluid mechanical considerations.
Also, as pointed out by Jähne et al. (1987), the model overestimates fluxes and
depends linearly on the diffusion constant contrary to other models.
2.4.3 Surface renewal
Another conceptual model was proposed by Higbie (1935) and elaborated by Danckwerts (1951) based on the intermittent nature of turbulence. He assumes the mass
transport to be diffusion controlled close to the surface. According to the diffusion
equation, a non-stationary concentration profile will then form characterized by
an increasing thickness and a flux which decreases in time. Turbulence is thought
to randomly cause patches on the surface to be entirely cleared and washed down
into the well-mixed bulk. This process is statistically associated with a renewal rate s.
Danckwerts finds the following relationship for the transfer velocity
k = (Ds)1/2
or
R = (Ds)−1/2 .
(2.34)
19
Chapter 2
THEORY
Different from the film model, it depends on the square root of the diffusion constant.
By replacing the covariance term in eq. (2.9) with a renewal term, s ⋅ c, Jähne (1985)
obtains an expression for the mean concentration profile (assuming gas flux into the
water)
z
c(z) = cb + (cs − cb ) exp (− ∗ ) .
(2.35)
z
√
The characteristic scale is defined as z ∗ ≡ D/s. Under the assumption that transport
is governed by diffusion at the surface and that this profile also provides a valid
description on those scales, it has to obey Fick’s law, j = −D ∂c/∂z and the transfer
velocity or resistance can be retrieved as
k=
D
z∗
or
R=
z∗
.
D
(2.36)
This surface renewal model does not incorporate any fluid mechanical boundary
conditions. Nonetheless, if renewal events are thought to be related with turbulent
eddies reaching the surface, the model appears conceptually compatible with a free
surface.
2.4.4 Eddy renewal and surface divergence
A series of models, extensions, and refinements are based on a concrete description
of surface renewal by turbulent eddies distributed underneath the surface. Fortescue
and Pearson (1967) were the first to propose a so-called large-eddy model. Lamont
and Scott (1970) depart from similar concepts, but allowed also smaller eddies in their
eddy-cell model. They found a dependence of the resistance on Schmidt number
as R ∝ Sc 1/2 for a free surface and R ∝ Sc 2/3 for a rigid surface. In both models,
divergences are attributed to eddies which are sourced by the shear flow.
Csanady (1990) points out that according to experimental results, shear induced
divergence is too small at higher wind speeds and that instead divergence generated
by breaking wavelets could be responsible. He proposes a model based on divergent
two-dimensional stagnation point flow at a free plane surface. The model has a free
parameter characterizing the fractional divergence coverage of the surface.
Interest in surface divergence as possible parameter to describe gas exchange
has grown recently. This also owes to an enormous increase in computational
power allowing to study small scale flow patterns in the water in detail. Examples
include the works by Tsai and Hung (2007); Tsai et al. (2013) and Banerjee et al.
(2004). Likewise, thermographic imaging is a promising tool to visualize surface
flow structure, possibly in parallel with other techniques (Asher et al., 2012; Haußecker
et al., 1995).
20
Solubility
2.5
2.5 Solubility
A quantity of importance for gas exchange is the so-called solubility, also known as
partition coefficient. Given a fixed temperature and pressure and assuming thermodynamic equilibrium, to each mole fraction of substance A mixed into substance B
there will be a corresponding mixing ratio in the gas phase. This is usually shown
in a so-called binary phase diagram. Generally, this is a non-linear function depending on the properties of the substances involved. Henry’s law is the tangential
approximation to the binary phase diagram for the limit of very dilute mixture.
The partition coefficient, or solubility, is then the ratio of concentration in the gas
and liquid phase,
cwater
.
(2.37)
cair
In literature, also the inverse is found. In the above definition, αH is dimensionless
and the concentration is usually given in mol/L. Alternatively, the gas phase fraction
can be given in terms of a partial pressure p. By the ideal gas law, this is related to a
concentration in mole per volume as c = p/RT , where R is the universal gas constant
and T the temperature. I use the dimensionless version of Henry’s law throughout
the thesis.
In air-sea gas exchange, the solubility is used to relate air and water-sided concentrations in the transport process. For a concentration cwater measured in the
water, its air-equivalent value is cwater/αH . Consequently, the air-equivalent difference
of two concentrations of which one is measured in the water and one in the air, is
∆c = cair − cwater/αH . This leads to a partitioning of transport resistances among the
gaseous and aqueous phase which depends on solubility, as explained by Liss (1971).
The resistances attributed to the transport in the air and water, respectively, are
αH ≡
Ra =
cbair − csair
j
and
Rw =
cswater − cbwater
,
j
(2.38)
where cb and cs stand for bulk (well-mixed) and surface concentration, respectively,
and j is the stationary flux. The air-equivalent values are obtained by dividing the
second of the above equations by αH . The air-equivalent resistance attributed to the
aqueous phase is then Rw/αH and total (air-equivalent) sum is given by
Rtot = Ra +
1
Rw
αH
(2.39)
When matching air-sided and water-sided concentrations, I will always express them
as air-equivalent values.
Equation (2.39) shows that for a gas with high solubility, the water-sided resistance
becomes relatively less important. In Fig. 2.3, I illustrated based on the turbulent
diffusivity model that the water-sided resistance for acetone is roughly larger than
21
Chapter 2
THEORY
the air-sided one by a factor of 1000. On the other hand, its solubility is about 800
(see sec. 3.1.4) and accordingly both phases are expected to contribute to similar
proportion to the overall resistance, i.e. Ra/Rtot ≈ Rw/Rtot . Following the line-of-thought
in sec. 2.3, this also implies that the (air-equivalent) concentration differences between bulk and surface should be similar in the water and in the air. In particular,
the air-sided concentration is expected to decay to about half its bulk value at the
water surface. For gases with low solubility, the air side is effectively irrelevant in
terms of resistance and their transport is said to be controlled by the water side.
22
3
Visualization technique
The visualization technique used in this work is based on the principle of LIF. A tracer
gas, mixed into the air-compartment of a wind-wave tank, is illuminated with laser
light and from the emitted fluorescent light intensity observed with a camera, the
concentration of the tracer gas in deduced. In this chapter, I explain the visualization
technique from the physics point of view. The experimental implementation of the
entire set-up can be found in chap. 4.
In view of the vast literature available on the topic of LIF, I present only those
aspects relevant for this study. In particular, I introduce some important characteristic parameters, review criteria for the suitability of a tracer gas, and summarize the
properties of the selected substance, acetone.
In the last section, I explain the method used in this work to deduce tracer concentration from quotient images and explain how saturation effects can thereby
effectively be taken into account.
For further reading on the principles of fluorescence, I refer e.g. to Lakowicz
(2006).
3.1 Laser induced fluorescence
3.1.1 Principles of fluorescence
A simple conceptual model for the absorption and emission processes leading to
fluorescence is a two-level (or possibly three-level) atomic system, comprised of a
singlet ground state, S0 , and excited singlet state, S1 , and possibly an excited triplet
state T1 . Such an exemplary Jablonski diagram is depicted in Fig. 3.1. While situations
might require more complicated models, this scheme is used here to explain the
23
Chapter 3
VISUALIZATION TECHNIQUE
rotational-vibrational band
vibrational
relaxation
S1
intersystem
crossing (ISC)
T1
kISC
Eσ
absorption
kf
kph knr
phosphorescent
emission
S0
0
cQkQ
quenching
Figure 3.1
Jablonski diagram for a three-level molecular energy structure with some possible transitions. S 0
is the ground state, S 1 and T1 are the excited
singlet and triplet states, respectively. Transition rates are indicated by k X . For polyatomic
molecules, rotational-vibrational modes create a
quasi continuous band.
basic principles relevant for this thesis.
Each possible transition is symbolized by an arrow and characterized by a rate
constant k i which has the unit s−1 . Its inverse can be interpreted as the lifetime of
the related state, τ i = k −1
i . Rate constants for parallel transitions add up to a total rate
constant and accordingly the total (and observable) lifetime is given by the sum of
the inverse individual lifetimes, τ −1 = ∑i τ −1
i . Consequently, the fastest transitions
dominates the lifetime of a state. Similarly, the relative occurrence of a transition i is
given by the ratio k i/∑ k i .
Absorption of photons of a given energy hc/λ cause atomic electrons to move
from S0 to S1 . If de-excitation occurs back to the ground state accompanied by the
emission of a photon, this radiation is observed as fluorescence light. For excited
states broadened, e.g., due to a band of vibrational energies, an electron can dissipate
energy internally (transiting trough vibrational states) prior to the re-emission of
a photon. This effect is commonly known as Stokes shift and causes the emission
spectrum to be red-shifted. In radiometric applications of LIF, a negligible overlap
of absorption and emission spectrum, i.e. a large Stokes shift, is desirable to exclude
the effect of re-absorption. By the so-called intersystem crossing (ISC), electrons
can move from S1 to T1 . The transition from T1 back to the ground state S0 is
quantum mechanically suppressed leading to a comparatively small rate constant
kphos , or equivalently long lifetime τphos . The light emission related to this suppressed
transition is commonly referred to as phosphorescence. In the presence of other
modes of energetic relaxation, e.g. non-radiative decay knr or collisional quenching
kQ , the phosphorescence intensity is further suppressed, kph/knr +kQ +kphos . Several other
decay modes exist, but a complete overview would be beyond the scope of this
chapter. The mechanisms specifically relevant for acetone are discussed in sec. 3.1.4.
24
Laser induced fluorescence
3.1
3.1.2 Measurement principle
The measurement principle by which the tracer concentration is determined through
LIF in this thesis is based on the dependence of the emitted fluorescent light intensity on the tracer concentration. This is easily analyzed in the regime of weak
excitation, i.e. when the majority of electrons are in ground state and only a small
fraction is excited. For higher laser irradiance, the relation becomes non-linear, but
within certain limits, this can still be taken into account through an appropriate
normalization procedure. I explain this in detail in sec. 3.1.6 and discuss the effect
of saturation and its relevance for this visualization technique in sec. 3.1.5. A general weak regime expression for the relationship between fluorescent brightness F
recorded by a camera, laser irradiance I, i.e. incident energy per time and unit area,
and tracer concentration c can e.g. be found in Thurber (1999),
F=
I
ηopt dVctracer σcs (λ, T)Φf (λ, T, P),
hc/λ
(3.1)
where ηopt is the overall efficiency of the recording optics and camera (including
vignetting), σcs the absorption cross section, and Φf the quantum efficiency. While
the cross section quantifies the probability of an incident photon to be absorbed by
the tracer molecule, Φf gives the ratio to which this absorption eventually leads to
the re-emission of a photon. Both functions depend on wavelength of the incident
light, and generally on temperature and pressure. Ideally, the latter two dependencies
are small or negligible for the purpose of retrieving the tracer concentration. Direct
calculation of the absolute tracer density from the above equation would require
exact knowledge of all the contributing terms. Instead, the approach used in this
thesis is to relate the signal to a reference image with known concentration cref , i.e.
to regard the relative signal
F
c
=
.
(3.2)
Fref cref
A more rigorous derivation is given in sec. 3.1.6 and implementation of the procedure in the data processing routine is explained in sec. 7.1.1. Equation (3.1) is
nonetheless useful as it contains all the terms to take into consideration when selecting a suitable tracer substance. This is done in the next section.
3.1.3 Choice of tracer
A tracer gas must meet certain criteria to be usable in a LIF-based visualization
experiment of scalar transport across the air-water boundary layer in a wind-wave
tank. These are explained here and a choice of tracer is deduced thereof.
For the visualization of scalar distributions in air, and possibly water, close to the
air-water interface, the absorption spectrum of a suitable tracer should be accessible
with a commercially available laser system and offer at least a reasonable quantum
yield. The Stokes shift (see sec. 3.1.1) should be large enough to avoid overlap between
25
Chapter 3
VISUALIZATION TECHNIQUE
absorption and emission spectrum, i.e. to avoid self-excitation. Fluorescence in
the visible range of the spectrum is preferred as this allows to easily monitor an
experiment by eye and to employ standard cameras and optics as opposed to UVtransmissive elements. Tracers with self-quenching properties have the disadvantage
of introducing non-linear, implicit relations between tracer concentration and observed brightness. Fluorescence lifetime should be short (on the order of ns) to
minimize the effect of collisional quenching typical for long-lived, phosphorescent
tracers and to avoid complications due to saturated stimulation, as explained in
sec. 3.1.5.
In terms of thermodynamic properties, a suitable tracer should be sufficiently
volatile to allow for good seeding in the air-flow during an invasion experiment.
On the other hand, the solubility needs to be high (αH ≥ 500) for the air-side to
contribute significantly to the overall resistance (see sec. 2.5). Not the least, toxicity
has to be carefully considered.
An experimental study of possible tracers in this kind of experimental setting
has already been performed by Winter (2011). A comprehensive literature review of
tracers for planar LIF can be found in Lozano (1992). While Lozano’s application
is in the field of flow visualization in flames and combustion, thus at much higher
temperatures than relevant for an environment-like experiment, his very systematic
analysis provides useful information also for the application in this work. I will
follow his structure, stating conclusions in a rather compact way, and in part refer to
Winter’s results.
Diatomic molecules can be ruled out, mainly due to technically demanding excitation, quenching, and sometimes toxicity. As exception, NO has been used in planar
laser induced fluorescence (PLIF) experiments, but is potentially lethal when inhaled.
Polyatomic molecules, in general, have wide rotational-vibrational (henceforth rovibrational) bands so that absorption and emission spectra are typically continuous.
Since thermal relaxation of an excited state within the rovibrational manifold is fast
compared to the radiative decay, the Stokes shift is often large and overlap of absorption and emission spectra is small or negligible. Inorganic polyatomic molecules
used for LIF are rare, with the exception of NO2 and SO2 , where the former is lethal
already in small amounts and the latter shows strong self-quenching.
Another group of polyatomic molecules are aromatic compounds. They often have
exceptional quantum efficiency of Φf ≥ 0.5, but volatility (and solubility) is unacceptably low due to the size and structure of the molecules. Benzene and derivatives
represent the smallest (and thus most volatile) candidates in this category, but usually they are carcinogenic. Additionally, solubility is too low. Among small organic,
non-aromatic compounds, aldehydes, (di-)ketones, and amines show luminescence.
Lozano provides a list with some physical parameters. In general, these substances
are interesting as their boiling point is low and their volatility is fairly high. Some
of these molecules are ruled out because of (self-)quenching, others due to their
toxicity, such as formaldehyde and acetaldehyde.
From this analysis, two potential LIF tracers emerge, acetone and diacetyl. The
26
Laser induced fluorescence
a
3.1
b
Figure 3.2: (a) Absorption spectrum of acetone, taken from Lozano et al. (1992). (b) Emission
spectrum of acetone corrected for detector response. Units are normalized to the
maximum value. Figure taken from Bryant et al. (2000).
former has already been tested by Winter (2011) and found to be rather applicable.
Notably, diacetyl has a very high phosphorescence yield of Φ = 0.15, about 60
times greater than its fluorescence yield. On the other hand, the long lifetime
of 1.5 ms suggests strong susceptibility to quenching. Indeed, in air and under
standard atmospheric pressure, the phosphorescence signal is suppressed by a factor
of ≈ 6000 requiring, e.g, a nitrogen atmosphere. Anyhow, the long lifetime might
create difficulty when attempting to record frozen images of the turbulent airflow.
Diacetyl has therefore been rejected as tracer substance.
In this thesis, acetone was selected as tracer gas, since it meets nearly all of the
above listed requirements. Its properties are described in the following section.
3.1.4 Properties of acetone
A summary of physical properties of acetone is given in Tab. 3.1 and the electronic
levels schematically correspond to the Jablonski diagram in Fig. 3.1.
Acetone can be excited in the UV regime, e.g. at λ = 266 nm, and in this work,
a frequency quadrupled neodymium-doped yttrium aluminum garnet (Nd:YAG)
laser with λemission = 266 nm was used. Therefore, all wavelength dependent parameters are provided for this value. An absorption spectrum, taken from Bryant
et al. (2000), is shown in Fig. 3.2 (a). The extinction coefficient at λ = 266 nm is
є266 nm = 11.7 cm L/mol. The broadband character is evidence for the dense rovibrational manifolds in the excited and ground states.
The ISC of acetone is close to 100 %, so that most excited electrons are in the T1
state from where the phosphorescent radiative decay is quenched. Of the remaining
ones in the S1 state, many decay radiation-less and only a small portion actually
causes luminescent light emission. This is the reason for the relatively low quantum
yield of Φf = 0.2 %. An emission spectrum is given in Fig. 3.2 (b) and shows a brought
peak in the blue range of the visible spectrum. This allows to employ standard optics
27
Chapter 3
VISUALIZATION TECHNIQUE
Table 3.1: Summary of relevant physical properties of acetone at T = 25 ○C and p = 1 atm
Parameter
Value
Source
abs. cross section σcs
extinction coefficient єair
extinction coefficient єwater
diffusion constant Dwater
4.4 × 10−20 cm2
11.7 L/(mol cm)
15.2 L/(mol cm)
1.29 × 10−9 m2 /s
diffusion constant Dair
fluorescence life time τ
vapor pressure
quantum yield of Φf
Solubility αH
0.105 cm2 /s
2.7(3) ns
0.27 atm
0.2 %
732, 800
Lozano et al. (1992)
Lozano et al. (1992)
Feigenbrugel et al. (2005)
Grossmann and Winkelmann
(2005)
Lugg (1968)
Breuer and Lee (1971)
Ambrose et al. (1974)
Lozano et al. (1992)
Benkelberg et al. (1995)
Betterton (1991)
and cameras.
Thurber (1999) and Bryant et al. (2000) show that there is no appreciable dependence of cross section and fluorescence yield on temperature in a range of
T = (23 ± 10)○C, so that this effect can be neglected for the visualization experiments presented in this work.
Ambrose et al. (1974) report a vapor pressure of acetone at room temperature of
0.27 atm. This high value indicates that injection into the wind-wave tank can easily
be achieved, e.g. by flushing air through a bubbling flask. Indeed, I adopted this
simple but efficient technique in the visualization experiments (see sec. 6.1.1). On
the other hand, the high vapor pressure guarantees condensation of acetone in the
tank not to be a relevant phenomenon even if some elements of the tank structure
were cooler than the air by some centigrade.
Acetone is known to dissociate to some degree under the influence of UV radiation
(Gandini and Hackett, 1977; Hansen and Lee, 1975; Heicklen, 1959; Ho et al., 1975). In
particular, the molecule dissociates to produce an acetyle radical,
CH3 COCH3 + hν Ð→ CH3 CO + CH3 .
Two radicals can then recombine to form diacetyl
CH3 CO + CH3 CO Ð→ CH3 COCOCH3 .
In a small, static cell, Lozano (1992) observed some build-up of diacetyl when
exposed to the light of a 308 nm XeCl excimer laser, although the scope of his experiment was merely to illustrate the effect qualitatively. An estimate on the relevance of
photolytic diacetyl formation during the visualization experiments presented in this
thesis can be made based on the diacetyl production yield of 0.2 given in Gandini
28
Laser induced fluorescence
3.1
and Hackett (1977). For typical concentrations of acetone of 10−3 mol/L, a lifetime of
2.7 ns, and a laser irradiance of 50 mJ/pulse, about 1016 photons are absorbed along
a distance of 90 mm in the tank. For a production yield of 0.2, this corresponds
2 × 10−9 mol/pulse of produced diacetyl, or 4 × 10−8 mol/s with a laser frequency of
flaser = 2 Hz. To produce a total amount of 10−3 mol diacetyl, equivalent to 1/20 of the
assumed acetone concentration for an air volume of 220 L, would require about on
the order 100 hours of constant irradiation under these conditions. Diacetyl formation can therefore be neglected for visualization experiments especially since acetone
is constantly transported into the water and re-flushed into the air compartment of
the tank.
3.1.5 Saturation
In eq. (3.1), the weak excitation limit was assumed, i.e. most electrons are in ground
state and only a few are excited. For high enough laser irradiance, this is not true
anymore and the emitted brightness is less than proportional to the incident photon
flux. In the extreme case of infinite photon flux, state occupation is in equilibrium
and no additional laser photon can be absorbed. Fluorescence then becomes fully
independent of laser irradiance. The effect of saturation is well-described e.g. in
Crimaldi (2008).
More generally, the fluorescence intensity is related to the laser irradiance by
Isat
.
Isat + I
F(c, I) ∝ c ⋅ I
(3.3)
Here, Isat is a constant defined by F(Isat )/F(I→inf) = 1/2, which depends on the tracer.
From this definition and analyzing the rate equations which describe the radiative
decay, the saturation irradiance in J/m2 /s can be related to the cross section σcs and
the lifetime of the emitting state τ as
Isat =
hc
,
σcs τ λ
1
⋅
(3.4)
with hc/λ the energy of a single photon.
Note that for I ≪ Isat , eq. (3.3) reduces to F ∝ c ⋅ I, i.e. the weak regime expression.
The absorption cross section of acetone at λ = 266 nm can be read from Fig. 3.2 (a)
as σcs = 4.4 × 10−20 cm2 and the fluorescence lifetime is given in Tab. 3.1 as τ = 2.7 ns,
so that Isat ≈ 2 × 108 J/mm2 /s. For a pulse duration of 8 ns, this corresponds to
Isat ≈ 0.5 J/mm2 /pulse.
(3.5)
To verify the assumption I ≪ Isat , I numerically calculated the expected fluorescence signal based on this value for a realistic beam geometry. The result helped to
estimate and exclude systematic influences of the laser irradiance on the extracted
29
Chapter 3
VISUALIZATION TECHNIQUE
x
en
l. l
cy
laser sheet
d=150 mm
d=5 mm
y
s
z
projection
camera
d=0.1 mm
1
L=
Figure 3.3
Sketch of the model set-up used to simulate the
attended effect of laser saturation.
m
0m
concentration profiles and to appropriately choose a data processing approach (see
sec. 7.1.1). The sketch of the idealized set-up used for this analysis is shown in Fig. 3.3.
I assume a laser sheet of length 10 mm and initial width d of 5 mm which narrows linearly to 100 µm at 150 mm distance and then diverges again linearly. This is
certainly idealized because real (Gaussian) laser beams have a characteristic waist
around their focal plane reciprocal to the beam convergence (Träger, 2007). For
the purpose of modeling the effect of saturation here, this geometric approximation seems adequately precise, however. The water height in an experiment would
coincide with the focal plane. The emitted fluorescence is observed with a camera
at 90° and image recording is modeled by projecting, i.e. integrating, the emission
density along the line of sight (assumed as the y-axis). The problem is treated as
homogeneous in x-direction. This geometry closely resembles the experimental
set-up presented in sec. 4.3.1.
In the following, small curly letters indicate density and capital letters its projection
along the y-axis, e.g. I(z) = ∫ ι(y, z) dy. The origin of the vertical z axis is set to
the initial position of the unfocused sheet with z increasing towards the focus point.
The maximum power of the used laser system is 50 mJ / pulse so that the irradiance
at z = 0 is ι0 = 1 × 10−3 J/mm2 /pulse. A homogeneous concentration of realistic
value cєair ≈ 2 mm−1 is assumed and Lambert-Beer absorption is treated as a monoexponential decay, i.e.
I(z)/I0 = exp (−єair cz) ,
(3.6)
and a Gaussian sheet profile in y-direction is assumed,
ι(y, z) = N(z) ⋅ exp (−
2d
with N chosen such that ∫−2d ι(y, z) dy = I(z).
30
y2
),
2d 2
(3.7)
Laser induced fluorescence
a
b
101
10
-2
i(y =0,z)
isat
d
10
10-3
10-4 0
50
100
Distance z in mm
0
Normalized projected brightness
10-1
1.00
Sheet width d in mm
100
Energy density i in J/mm2 /pulse
3.1
-1
20010
150
0.95
0.90
0.85
0.80
0.75
0.700
Flinear(z)
Fnon−lin.(z)
50
100
Distance z in mm
150
Figure 3.4: (a) Modeled irradiance for a laser sheet of length L = 10 mm and Gaussian shape in
y-direction. The width d as a function of z is plotted relative to the right ordinate. The
horizontal line gives the saturation irradiance for comparison. The red line depicts
the local irradiance in the center of the laser sheet, i.e. at y = 0. (b) Modeled projected
fluorescence signal, according to eq. (3.8) (red) and in linear approximation F ∝ c ⋅ I
(blue). The maximum systematic deviation is about 2 %.
The modeled fluorescence signal F(z) is then calculated according to eq. (3.3) as
F(z) = c
∫
2d
−2d
ι(y, z) ⋅
ιsat
dy.
ιsat + ι(y, z)
(3.8)
Note that for notational consistency, I used small letter ιsat instead of Isat , as in
eq. (3.3).
In the left panel of Fig. 3.4, the irradiance ι(y = 0, z) in the sheet center is shown as
a function of vertical coordinate z. For reference, the sheet width d is plotted relative
to the right ordinate. As expected, the decreasing width causes a strong increase
in irradiance which however does not exceed the value ιsat . Also, the change in
irradiance due to Lambert-Beer is small compared to the impact of beam geometry.
The right panel shows the projected fluorescence signal F(z)/F(z=0), according to
eq. (3.8) (red) and in linear approximation F ∝ c ⋅ I (blue). The maximum systematic
deviation in the focal plane is about 2 %. This shows that saturation effects are still
weak for the such a beam geometry and laser power, but should be taken into account
in the procedure by which tracer concentration is deduced from images.
31
Chapter 3
VISUALIZATION TECHNIQUE
3.1.6 Relative brightness images
In this section, I outline the principle of retrieving tracer concentrations from relative
fluorescent brightness. I use the same notation as in the previous section, but the
considerations are obviously valid also for a real, non-idealized laser beam. I will
also note some not so obvious subtleties and how expressions can be simplified. The
implementation of the normalization principle in the data processing routines is
explained in sec. 7.1.1.
An effective way to deduce the concentration distribution c(z) from the fluorescence signal is to record reference images without water in the wind-wave tank and
consequently homogeneous tracer distribution, i.e. cref (z) = cref , and to match them
with data images (i.e. with water, c(z) ≠ const.) according to the brightness in the
bulk region away from the water surface. The brightness ratio can then be calculated
from eq. (3.8) as
ι sat
dy
∫−2d c(x, y, z) ⋅ ι(x, y, z) ιsat +ι(x,y,z)
F(x, z)
,
=
2d
Fref (x, z)
cref ∫−2d ιref (x, y, z) ιsat +ιrefιsat(x,y,z) dy
2d
(3.9)
where I have added explicitly the horizontal coordinate x to account for the 2D character of the image. As on average, concentration depends on the vertical coordinate
z only, c(y, z) can be pulled out of the integral¹.
As a first subtlety one should note that in general saturation effects related to the
reference image need not be the same as for the data image because local fluctuations
of c(z) with respect to cref lead to different absorption, thus to different irradiance
and thus to different degrees of saturation. However, Fig. 3.4 (a) shows that for
the here assumed realistic acetone concentrations the influence of Lambert-Beer
absorption on the irradiance modulation is small compared to the sheet geometry.
The saturation effect arising from ιsat/ι+ιsat < 1 will therefore be a function of height z,
well-mixed bulk concentration cb , and laser power P only, i.e. Sat(z; cb , P). It is not
important to specify the functional relationship, but sufficient to observe that only
macroscopic parameters are involved. In a quotient image, this function will then
cancel out.
As a second subtlety one should observe that in the presence of water, the acetone
concentration might decay to half of its bulk value within the boundary layer, (see
sec. 2.5) causing absorption to be weaker and consequently ι(z ≈ zsurf ) < ιref (z ≈
zsurf ). The relative systematic error induced by this approximation is expected to be
on the order ∆ι/ι ≈ єcb δ, with δ an estimate for the turbulent boundary layer thickness.
For typical values of these parameters, є ≈ 1 L mm/mol, cb ≈ 1 × 10−3 mol/L, and
δ ≤ 10 mm, this is only about 1 %. Summarizing, Lambert-Beer absorption can be
well described based on the bulk concentration cb and treated as independent of
local fluctuations.
¹Note that spatially averaging is equivalent to ensemble averaging in an ergodic sense.
32
Laser induced fluorescence
3.1
A last subtlety regards the projection integral. While irradiance in theory is a local
quantity expressing photon or energy flux per unit area, all data in the camera-based
visualization experiment in this work are projected values. Therefore, and in view of
the above mentioned approximations, the fluorescence brightness F(z) is written as
a multiplicative function of projected factors. Equation (3.9) then reads
F(x, z)
c(z) I(x, z; cb ) Sat(z; cb , P)
=
⋅
⋅
,
Fref (x, z)
cref I(x, z; cref ) Sat(z; cref , P)
(3.10)
and by the matching condition cref = cb ,
c(z)
F(x, z)
=
.
Fref (x, z)
cb
(3.11)
To summarize: By calculating the quotient of a data image and a reference ground
truth image, concentration profiles normalized to their bulk value can be retrieved.
33
4
Experimental set-up
The experimental set-up consists of the following main components:
• A wind-wave tunnel with a turbine to generate wind
• A pulsed laser pointing downwards from the top through the air and water
• Two cameras looking cross-wind to observe the tracer distribution in the air
and water, respectively
• An optical set-up to form a focused laser sheet
• Readout and trigger electronics
In the following, I will first describe the main components in sec. 4.1 and then,
before explaining their deployment in sec. 4.3, I will present some theoretical considerations regarding the optimal arrangement and systematic limitations in sec. 4.2.
Finally, I will explain two methods to verify and monitor the beam profile in-situ in
sec. 4.4.
35
Chapter 4
EXPERIMENTAL SET-UP
Figure 4.1
Overview sketch of the wind-wave
tank with the turbine to the left, the
experimental section, and the tubing for air back flow. From Warken
(2010).
4.1 Main components
4.1.1 Wind-wave tank
The wind-wave facility is described in detail in Warken (2010) and Winter (2011). Here,
I will only summarize the most important aspects relevant for this work. Figure 4.1
shows an overview sketch of the wind-wave tank with the turbine to the left, the
experimental section in the center, and the tubing for air back flow. The turbine is
fitted with a honeycomb homogenizer to achieve a well defined homogeneous wind
profile.
A sketch of the experimental section of the tank is shown in Fig. 4.2 including
the geometric dimensions. The total air volume is about Vair = 220 L and the water
volume at standard filling height of 3.5 cm is Vwater = 23 L. Walls, top, and bottom
are made from Schott Borofloat 33©glass allowing for easy optical access. The top
lid in the experimental section has two windows made from fused silica glass with
high UV transmittance. The laser sheet is introduced through one of them.
Results of experiments to characterize the wind field and water waves under
operating conditions are presented in chap. 5.
4.1.2 Pulsed Q-switched laser
A pulsed frequency quadrupled Nd:YAG laser with a wavelength of 266 nm was
used as illumination source for LIF. The model CFR200 by Quantel BigSky lasers
offers a maximum repetition rate of 20 Hz, a maximum power of 50 mJ / pulse, and
a pulse duration of about 8 ns. The laser is actively Q-switched and output power
can be adjusted by changing the Q-switch (QS) delay. The relation between the delay
value and the output power is non-linear and not specified by the manufacturer, but
was characterized in the framework of this thesis (see sec. 5.4). The laser can be
externally triggered with a standard TTL signal. An experimental characterization
of the laser was performed and is presented in sec. 5.4.
36
Main components
4.2
laser optics
pulsed laser
fetch zero
175 cm
25 cm
air flow
9 cm
camera
water side
water level
3.5 cm
cam
wa era
t er
sid
e
Figure 4.2: Sketch of the experimental section of the tank including the geometric dimensions.
Adapted from Warken (2010). The cameras’ axis is perpendicular to the direction of
the wind.
4.1.3 Cameras, lenses, and triggering
Two monochrome scientific CMOS cameras (pco.edge by PCO, Kelheim, Germany)
were used in rolling shutter mode to minimize readout noise. This 16 bit camera
technology offers very high dynamic range (27000:1) and very low noise (1.1 electrons) at the same time. Both are valuable properties for the purpose of recording
the dim air-sided LIF signal in proximity of the bright water surface. Pixel size is
6.5 × 6.5µm2 .
Both cameras were controlled with a frame grabber card in exposure control mode.
In this modality, the frame grabber issues a high signal during the entire exposure
time (35 ms) and a low signal in between frames, thus controlling both, exposure
time and recording frequency (2 Hz). The trigger signal was also looped trough a
function generator into the pulsed laser. In this way, synchronization was guaranteed
and illumination of the entire image was achieved by appropriately delaying the
laser trigger on the output channel of the function generator with respect to camera
exposure.
On both cameras, a Nikon Micro Nikkor, 105 mm, 1:2.8, was used. For the air-side,
aperture was set to 4.0 as a compromise between maximum signal intensity and low
optical aberration. For the water-side, an additional Canon 500D close-up lens was
used to allow for a sufficiently short working distance of about 20 cm. Since image
brightness was less critical than optical aberration, aperture was set to 8.0.
37
EXPERIMENTAL SET-UP
laser
Chapter 4
ζ
beamprofile
camera
air
water
α
(x0, z0)
z
y
β
θ
x
Figure 4.3
Sketch to illustrate the optical refraction at the water surface. The laser
sheet is described by a Gaussian profile across and treated as homogeneous in the long direction y.
4.2 Theoretical analysis of optical properties at the water surface
The scope of the air-sided visualization set-up was to obtain acetone concentration
distributions as a function height as close as 100 µm to the water surface (see sec. 2.3).
To this end, I present a few careful definitions and preliminary considerations which
are necessary to define an optimal set-up. A sketch which graphically complements
the explanations is shown in Fig. 4.3 with flat water for simplicity.
Acetone vapor in the air is excited by laser light shaped to a sheet of finite length
and width. As reasonable approximation, I assume the sheet to be homogeneous in
the long direction and Gaussian shaped in the short cross-direction. The width in
Fig. 4.3 is exaggerated for clarity. The camera observes the scene as a projection of
fluorescent light across the laser sheet onto the camera sensor along the light-of-sight.
Profiles should be expressed in terms of relative height above the wavy water
surface. As origin for this coordinate system, the intersection of the water surface
and the central laser sheet plane, (x0 , z0 ), is an appropriate choice. The coordinates
x and z in the laboratory frame are related to the vertical coordinate on the camera
sensor by ζ = cos z and ζ = sin x. To avoid optical shading by wave crests adjacent to
this point, the camera needs to point downwards by some angle α. In such a setting,
even for a point with z > z0 , light originating from the water will be projected onto
the camera sensor, as seen in Fig. 4.3, overlaying in this way the air-sided signal.
Given the solubility of acetone of αH ≈ 750 (see sec. 3.1.4), the water-sided tracer
concentration, and thus the fluorescent brightness density, is expected to be 750
times higher than in the air leading to a very high dynamic contrast in the image.
The recorded air-sided profiles are therefore biased below a certain minimal distance
38
Theoretical analysis of optical properties at the water surface
4.2
from the water surface which depends on laser sheet width σ and viewing angle α.
Also, the refraction at the water surface will pose a limit on the precision by which
the point (x0 , z0 ) can be determined from the images.
To analyze and quantify this systematic effect, I calculated the attended brightness
observed by the camera based on a modeled fluorescent light distribution as follows:
The shape of the beam profile is described by a Gaussian function of width σ and
the brightness decay in the water is modeled by a mono-exponential function with
characteristic scale z ∗ . The problem is treated as homogeneous in the direction
parallel to the long axis of the laser sheet (y-axis in Fig. 4.3).
The fluorescent brightness density in the water is then given by
F(x, z) ∝ exp (−
z
x2
) ⋅ exp ( ∗ ) ,
2
2σ
z
(4.1)
with the origin at (x0 , z0 ). This brightness density is projected onto the water surface
under an angle θ, defined through Snell’s law of refraction sin(90°−α)/sin(θ) = nwater , with
nwater = 1.33 the refractive index of water. This can indeed be solved analytically by
appropriate straight forward substitution of variables and completion of the square,
but since the calculation is lengthy and irrelevant for the conclusion, I only state the
result.
The projected fluorescent brightness P w at a point xs on the water surface is
1
P (xs ) ∝
b
w
√
π
x2
a2
a
exp (− s 2 + 2 ) ⋅ erfc ( √ ) ,
2
2σ
2b
b 2
(4.2)
with a = xs sin θ/σ 2 + cos θ/2z∗ and b = sin θ/σ .
For the air side, I used Deacon’s model, described in sec. 2.3, for a friction velocity
of u∗ = 20 cm/s and modulated the light intensity in x-direction according to the
Gaussian laser sheet profile. After sec. 2.5, the surface concentration is set to half the
bulk value, cs = 0.5cb as an estimate. I calculated the projected fluorescent brightness
P a numerically by interpolating and summing in the direction given by the angle α.
The vertical brightness profile observed by the camera is then given by a superposition of P w and P a . Following the considerations further above, the water-sided
surface brightness density is set to αH = 750 times the air-sided bulk brightness
density.
At this point, it still has to be considered that only a portion of the light originating
from the water is transmitted through the surface while another portion is reflected.
Therefore, the transmittance T, following from Fresnel’s equation for dielectric media,
has to be applied as an additional factor. With the here defined angles, this is given
by (Jackson, 1998)
tan θ
T=
tan β
⎡
2
2⎤
⎢
⎥
2nwater cos θ
2nwater cos θ
⎢
) +(
) ⎥⎥ .
⋅ ⎢(
cos θ + nwater cos β ⎥
⎢ nwater cos θ + cos β
⎣
⎦
(4.3)
39
EXPERIMENTAL SET-UP
a
Brightness normalized to air-sided bulk value
b
3.5
FWHM = 50 µm
FWHM = 100 µm
FWHM = 150 µm
FWHM = 200 µm
FWHM = 300 µm
FWHM = 500 µm
3.0
2.5
Threshold for surface detection
2.0
1.5
Air-sided bulk brightness
1.0
0.5
101
102
Height above water in µm
103
Brightness normalized to air-sided bulk value
Chapter 4
3.5
3.0
2.5
Threshold for
surface detection
2.0
α=1 ◦
α=2 ◦
α=3 ◦
α=4 ◦
α=5 ◦
α=6 ◦
α=7 ◦
α=8 ◦
α=9 ◦
α = 10 ◦
1.5
Air-sided bulk brightness
1.0
0.5
100
Air-sided surface brightness
101
102
Height above water in µm
Figure 4.4: Selection of brightness profiles observed by the camera, (a) for a viewing angle of
α = 5°, a decay scale of z ∗ = 200 µm, and varying laser sheet thickness, and (b) for a
full width at half maximum (FWHM) of 80 µm and varying viewing angle.
The brightness profile observed by the camera is then
P(ζ) = P a (ζ) + αH ⋅ T ⋅ P w (ζ).
(4.4)
In Fig. 4.4 (a), I show a selection of such profiles for a viewing angle of α = 5°, a
realistic decay scale of z ∗ = 200 µm, and varying laser sheet thickness. The profiles
are normalized to the air-sided bulk brightness and for clarity, only a section close to
the water surface is shown. The plot underlines the importance to adequately focus
the laser in order to record an unbiased brightness as close to the surface as possible.
For the set-up used in this work, beam profile measurements yielded a FWHM of
the laser sheet of about 80 µm, i.e. σ = 35 µm (see sec. 4.4), and the observing angle
was α = 5°.
Figure 4.4 (b) show the brightness profiles observed by the camera for a FWHM of
80 µm and varying viewing angle. The following conclusions can be drawn from this
plot. First, the surface can be defined to very good precision by a simple threshold
criterion, indicated as a dashed line. The implementation of the automatic surface
detection is explained in sec. 7.1.3. Second, given the dependence on the viewing angle
with respect to the water surface, the precision with which the position (x0 , z0 ) can
be estimated is intrinsically limited in the presence of waves regardless of the surface
detection algorithm. For the here used realistic laser sheet width and admitting the
non-ideal conditions in a real experiment, the zero point in a profile should anyhow
be precise up to a few ten µm. Third, the observed minimal air-sided brightness
close to the surface does not significantly depend on the viewing angle so that the
presence of waves should not have any relevant influence.
It should be noted that in the eventually employed set-up, the laser was tilted
with respect to the mean water surface in order to be perpendicular to the camera
40
Deployment
R2=0.64%
Δh
R=8%
R=8%
α
d=3 mm
4.3
era
cam
Figure 4.5
Sketch to illustrate the effect of
optical ghosting at the side glass
pane of the wind-wave tank. The
non-zero reflectivity leads to double internal reflections that appear
shifted in height in the camera image.
axis (see sec. 4.3.1). This slightly changes the geometrical consideration presented
here. As the length of the projection integral within the slanted water-sided laser
sheet in this case decreases, the projected light intensity originating from the water
is systematically lower. The estimates on the minimal unbiased distance observable
in the air given here can thus be regarded as upper bounds.
The recording optics introduce some optical aberration, especially at the open
aperture end. This blurring will cause the harsh brightness gradient at the surface
to be slightly washed out. Regardless, of the refractive properties at the surface,
additional interference with the air-sided signal is therefore expected.
The pulsed CFR200 UV laser used in this work does not have a Gaussian beam
profile (see sec. 5.4.1), but shows higher spatial modes resulting in non-zero irradiance
even at greater lateral distance from the focus (> 5σ). To suppress this effect, I used
a spatial filter set-up explained in sec. 4.3.2. Beam profile measurements showed
that the resulting laser sheet was indeed very clean in the focal plane (see sec. 4.4) so
that the principle conclusions drawn from the theoretical analysis presented in this
section should safely apply also for the real laser set-up.
4.3 Deployment
In the following, I explain the deployment of the various components at the windwave facility along with some optical considerations. Related overview sketches of
the set-up are shown in Fig. 4.2 and Fig. 4.6.
4.3.1 Optical ghosting and air-sided camera
Optical ghosting occurs at the sidewalls of the wind wave tank due to double reflection within the glass panes. This is illustrated in Fig. 4.5. For the reflectivity at
450 nm wavelength (blue fluorescent light) of Borofloat 33 glass of 8 %, still 0.64 %
of the initial brightness reaches the camera after being reflected twice within the
3 mm thick panes. As a result, the camera captures a second “ghost” scene attenuated
by about a factor 100, shifted upwards by ∆h = 2d tan(α), which for α = 5° would
41
EXPERIMENTAL SET-UP
first surface mirror
wind
laser sh
ee t
Chapter 4
fused silica
window
air
2.5°
camera
water
5°
25 cm
cam
era
Figure 4.6
Sketch of the camera set-up in a
cross-sectional, i.e. long-wind view.
Drawing is not to scale.
amount to ∆h = 0.5 mm. On the other hand, the brightness contrast at the water
surface for acetone is typically also on the order 100. This seemingly negligible optical
effect can therefore significantly modify the observed air-sided brightness profile.
As a remedy, the camera was positioned with its optical axis perpendicular to the
tank wall because in this case, tan(α = 0) = 0, and image and ghost image precisely
overlap. To anyhow observe the water surface under a slanted viewing angle of 5°,
a first-surface mirror was positioned on the opposite sidewall of the tank with an
angle of 2.5°. This is shown in Fig. 4.6.
Magnification of the air-sided camera was 21.2 µm/px with a field of view of
21 × 54mm2 . The extension in the vertical direction was necessary to capture the
entire turbulent boundary layer whose thickness under the used conditions was
about 40 mm (see sec. 5.1).
4.3.2 Laser set-up and spatial filtering
The laser was formed to a slightly divergent sheet of average length 10 mm and
focused to the water surface. The optical set-up is illustrated in Fig. 4.7. A sheet
(2D) was preferred over a beam (1D) geometry for several reasons. First, focusing
the laser in two directions rather than only one would have led to a much higher
irradiance and consequently difficult to treat saturation effects (cf. sec. 3.1.5). Second,
images obtained with a laser sheet contain valuable two-dimensional information,
e.g. about the surface shape. Third, in particular on the water-side, a sheet geometry
facilitates surface detection because the total reflection is visible at every point (cf.
sec. 7.2.1). For a thin laser beam, on the other hand, this is not always the case at
higher winds impeding easy detection (Friedl, 2013).
Focusing and sheet formation was achieved with a plano-convex (f=200 mm)
and a cylinder lens (f=63 mm), both made from fused silica for UV transmittance.
A short-pass filter was placed in front of the laser to remove residual green light
(532 nm) remaining after wavelength doubling. A slit filter, placed in the focal plane
of the plano-convex lens, helped to filter out higher spatial modes by narrowing
42
Deployment
laser
high
pass
filter
Long-wind view
mirror
Cross-wind view
4.4
laser
high pass filter
plano-convex lens, f = 200 mm
adjustable slit
cylinder lens, f = 63 mm
fused silica window
water
wind
wind
water
Figure 4.7
Sketch of the optical set-up used
to form the laser sheet and to focus it onto the water surface, seen
from cross-wind (left) and longwind (right) direction.
down the slit width (note that the focal plane is equivalent to the optical Fourier
plane). This led to a cleaner beam profile and effectively minimized brightness halos
in the focal plane on the water surface.
For technical reasons, the laser was positioned horizontally and deviated downwards by a mirror. The two-axis mirror mount also allowed to precisely align the
laser beam with the central axis of the optical set-up. While the distance between
plano-convex lens and slit had to be kept fixed for the spatial filter to work properly,
the cylinder lens could be displaced to focus the sheet onto the water surface.
The laser sheet was slanted by 5° to be oriented perpendicular to the air-sided
camera axis. This allowed for a sharp image at any height despite the shallow depth-offield given the large aperture of 4.0. For the purpose of recording mean concentration
profiles, a slanted laser sheet is absolutely admissible since the scalar transport is
assumed to be statistically homogeneous in the cross-wind direction. This also
allowed to avoid optical components such as a Scheimpflug set-up which in this case
would not have brought any advantage yet significant technical complications.
4.3.3 Water-sided camera
The camera was mounted beneath the water level pointing upwards by 15°, corresponding to 11.5° in the water due to refraction at the sidewall of the tank. According
to the wave slope measurements presented in sec. 5.3, this value guaranteed optical shading by waves not to occur. As an additional advantage of air-sided mirror
set-up (cf. sec. 4.3.1), the laser sheet was slanted towards the water-sided camera
rather than away from its axis, thus improving the depth-of-field. This is clear from
Fig. 4.6. Magnification of the water-sided camera was 12.0 µm/px with a field of view
of 12 × 31mm2 .
43
EXPERIMENTAL SET-UP
y-dimension in µm
Chapter 4
150
100
50
0
50
100
1000 2000 3000 4000 5000 6000 7000 8000
x-dimension in µm
Figure 4.8
Profile of the laser sheet recorded insitu with a PCO Sensicam UV. From
a Gaussian fit, the across-width was
determined as σ = 35 µm or a
FWHM of 82 µm.
4.4 Beam profiling
In sec. 4.2 I showed that a well-focused laser sheet is important in order to avoid
the water-sided fluorescence to systematically bias the air-sided measurements. To
assess the performance of the laser optics, beam profiles were measured in-situ in
two ways.
In a first approach, a UV sensitive camera (PCO Sensicam UV) was mounted
inside the empty wind-wave tank facing upwards at the position of the laser sheet.
The vertical position of the camera was adjusted so that the sensor plane matched
the mean water level. Neutral density filters were placed in front of the bare sensor
without any other optics. In this way, the Sensicam effectively served as a beam
profiling unit. A recorded profile image in shown in Fig. 4.8. Although the pixel
resolution is limited to 8 µm, this allowed to estimate the quality of the beam focus.
From a Gaussian fit, the across-width was determined as σ = 35 µm or a FWHM of
82 µm.
To obtain a qualitative impression of the beam profile prior to an experiment
with the entire set-up mounted, an Allied Vision Guppy F503B Pro camera with a
50 mm lens was placed on the cover lid of the experimental tank section and pointed
downwards to observe the water surface through the second window. Some acetone
was given into the water together with ascorbic acid as strong absorber in the UV
regime. In this way, laser light was absorbed quickly and fluorescence light was
emitted only from the uppermost surface layer. The laser imprint on the water could
then easily be observed with the camera from above providing an in-situ measured
laser sheet profile. I used this auxiliary set-up as guidance for fine-tuning the laser
optics.
44
5
Auxiliary experiments
To be able to compare the experimental results of this thesis with data of other studies
and with numerical simulations, it was desirable to retrieve certain fluid mechanical
parameters characteristic for the used wind-wave tank. In particular, knowledge
of the wind profiles provided the air-sided friction velocity which in turn served
as characteristic scale. For the wave field, the slope distribution and the typical
wave amplitudes were determined from which mean square slope (mss) could be
calculated. A wave age parameter was determined with the help of the estimated
wave phase speed.
Additionally, the pulsed laser system used in this work was characterized in terms
of output power and intensity fluctuations.
5.1 Wind profiles
A simple and well-established instrument to measure flow velocities in the air is the
so-called Pitot tube. Essentially, it measures the stagnation pressure at the tip and
the static pressure along the side walls. From their difference, the air speed can be
calculated following Bernoulli’s law. A step-motor driven vertical translation stage
allowed to automatically record wind speed profiles in steps of 1 mm for the entire
ranges of wind turbine settings used in this work. A collection of such profiles is
shown in Fig. 5.1 (a).
To determine the friction velocity u∗ , a fit to the data in the range of the logarithmic sublayer would be necessary according to eq. (2.21). This would require
wind speed data to be available below 5 to 7mm which in the presence of waves is
not possible. Instead, the self-similar properties of the outer wake portion of the
turbulent boundary layer were exploited to match wind profiles at different wind
45
Chapter 5
AUXILIARY EXPERIMENTS
a
b
50
1.4
5.0Hz
6.0Hz
7.0Hz
8.0Hz
9.0Hz
10.0Hz
11.0Hz
12.0Hz
13.0Hz
14.0Hz
15.0Hz
16.0Hz
17.0Hz
18.0Hz
19.0Hz
20.0Hz
21.0Hz
22.0Hz
23.0Hz
24.0Hz
25.0Hz
26.0Hz
27.0Hz
28.0Hz
29.0Hz
30.0Hz
1.2
1.0
40
0.8
30
U0 −u(z)
βu
Height from water surface in mm
60
0.4
20
0.2
10
0
100
0.6
0.0
200
300 400 500
Windspeed in cm / s
600
700
0.2 -2
10
10-1
Dimensionless height z/δ
100
Figure 5.1: (a) Wind speed profiles as a function of height above the mean water surface measured with a Pitot tube. (b) Scaled wind speed profiles (U 0 −u(z))/βu∗ as a function of
dimensionless height η = z/δ above the mean water surface measured with a Pitot
tube. The plot shows data points from a total of 25 profiles for bulk wind speeds
ranging from 150 to 650 cm/s.
speeds and to recover u∗ from a simultaneous fit. This approach is used e.g. in
Troitskaya et al. (2010).
In a first step, according to eq. (2.17), the following function was fitted to the data
u(z):
2
u(z) = U0 − βu∗ (z − δ) ,
(5.1)
with the free parameters U0 , (βu∗ ), and δ, where the latter determines the overall
thickness of the turbulent boundary layer. While the wake portion of the profile
readily provides the bulk velocity U0 , u∗ alone may not be extracted.
By scaling the wind speed profiles with the determined wake parameters, i.e.
U 0 −u/βu∗ as a function of z/δ , they collapse onto each other, underlining their selfsimilar properties. This is shown in Fig. 5.1 (b). The following logarithmic expression
was fitted to the scaled profiles simultaneously:
η
U0 − u(η) 1 U0 1
1
= ( − log(η)) = − log ( ) ,
βu∗
β u∗ κ
βκ
η0
(5.2)
with κ the von Karman constant, and β and η0 free fit parameters. For each wind
speed, u∗ was calculated from the wake portion as u∗ = (βu∗ )/β. The roughness
parameter z0 = δ ⋅ η0 was calculated as
κU0
).
(5.3)
u∗
It should be noted that by this method, the friction velocity u∗ is not directly
determined from the logarithmic sublayer for higher wind speeds, but effectively
z0 = δη0 = δ exp (
46
Wave-age
b
800
35
700
30
Friction velocity u in cm/s
Free stream velocity U0 in cm/s
a
5.3
600
500
400
300
200
1000
5
10 15 20 25
Turbine frequency in Hz
30
35
25
20
15
10
5
00
5
10 15 20 25
Turbine frequency in Hz
30
Figure 5.2: (a) Free stream velocity U 0 as a function of turbine frequency. (b) Friction velocity u∗
as a function of turbine frequency f wind .
extrapolated based on the self-similar scaling properties of the turbulent boundary
layer. An explicit experimental determination would require, e.g., PIV measurements
which were beyond the scope of this work. For the purpose of establishing u∗ as
scaling parameter, the here outlined approach was sufficient.
The left panel of Fig. 5.2 shows U0 as a function of turbine frequency fwind with
a nearly linear relation. The right panel depicts u∗ as a function of the turbine
frequency fwind . The value of u∗ starts to grow stronger with turbine frequency at
around fwind = 15 Hz which can be attributed to the onset of waves and the increased
momentum dissipation. This is indicated in the figure by the two linear functions
chosen to match at the transition point.
Figure 5.3 shows the roughness parameter z0 (a) and the thickness of the turbulent
boundary layer δ (b) as a function of turbine frequency. Again, the increase starting
at around fwind = 15 Hz coincides with he onset of waves and demonstrates how
the increased roughness acts as a source of turbulence. This tendency played an
important role in the interpretation concentration profiles at higher wind speeds.
5.2 Wave-age
As another characteristic parameter, the wave-age c/u∗ was determined for turbine
frequencies above fwind = 21 Hz by relating the friction velocities to the wave’s phase
velocity. The latter was measured by observing the time needed for a wave to travel
a predefined, marked distance in the wind-wave tank. For all turbine settings, the
wave-age was found to be on the order of c/u∗ ≈ 1 − 2. This parameter was particularly
useful for the comparison of experimental concentration profiles with the results of
numerical simulations of the turbulent flow of a wavy surface.
47
Chapter 5
AUXILIARY EXPERIMENTS
a
b
2.5 1e 2
Boundary layer thickness δ in mm
52
Roughness z0 in mm
2.0
1.5
1.0
0.5
0.0
0
5
10 15 20 25
Turbine frequency in Hz
30
35
50
48
46
44
42
40
380
5
10 15 20 25
Turbine frequency in Hz
30
Figure 5.3: (a) Roughness parameter z 0 as a function of f wind . (b) Thickness δ of turbulent boundary layer as a function of turbine frequency f wind .
5.3 Wave slope measurements
For the air-sided turbulent flow, the wavy water surface acts as a rough moving
boundary. To estimate its impact on the flow at least qualitatively, the characteristic steepness of the waves for a given wind speed is needed. For the water-sided
near-surface flow wave slopes provide a way to qualitatively compare to results of
numerical simulations and to estimate the occurrence of micro-scale wave breaking.
Various techniques are available to visualize the momentary wave slopes, even over
an entire 2-dimensional surface patch (Balschbach, 2000; Rocholz, 2008). For the
mere purpose of obtaining statistical slope distributions, the well-established technique called laser-slope-gauge (LSG) was used (Lange et al., 1982; Tober et al., 1973).
The experimental work was performed within a bachelor thesis that I supervised
during my PhD (Platt, 2011) and I refer to that document for a detailed description.
Here, I will only outline the peculiarities of the experimental implementation at the
wind-wave tank and report the results relevant for this thesis.
The principle of a LSG is to have a laser beam penetrate the wavy water surface
from the air-side and to visualize its impinging point underneath the tank (through
the glass bottom) on a frosted glass pane. The lateral deviation (∆x, ∆y) from the
center point depends on the water surface slope and the water height at the point of
penetration. To account for the latter, in general, a telecentric optical set-up is used
between the bottom of the tank and the frosted glass pane. As the small amplitudes
of up to 3 mm occurring in the wind wave tank at maximum wind speed induce a
negligible systematic error, this optical element was avoided.
Typically, an LSG is implemented in a radiometric fashion in so far as the camera
integrates accumulated brightness at each point on the frosted pane. This will be
proportional to the probability of the laser to imping the pane in this particular
coordinate, i.e. to the related water surface slope. This approach, however, requires
48
Wave slope measurements
a
5.3
b
11 Hz
16 Hz
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
1e 3
Along-wind slope
c
3
2
1 0
1
2
Along-wind slope
3
4
1e 2
d
18 Hz
0.15
0.10
21 Hz
0.05 0.00 0.05
Along-wind slope
0.10
0.15
e
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4
Along-wind slope
f
25 Hz
0.6
0.4
30 Hz
0.2 0.0 0.2
Along-wind slope
0.4
0.6
0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8
Along-wind slope
Figure 5.4: (a)-(f ) Exemplary wave slope distributions, along-wind, for different settings of the
wind turbine. Slopes ak begin to be appreciable at around 15 Hz. In (e) and (f ), the
skewness is quite noticeable, manifesting the appearance of small scale gravity and
capillary waves.
49
AUXILIARY EXPERIMENTS
Mean square slope (mss)
Chapter 5
0.05
mssx
mssy
mss
0.04
0.03
0.02
0.01
0.00
0
5
10 15 20 25
Turbine frequency in Hz
30
Figure 5.5
Mean square slope as a function of
turbine velocity f wind . Slopes grow
rapidly above f wind = 15 Hz, consistent to the development of the surface roughness and thickness of the
turbulent boundary layer δ shown
in Fig. 5.3.
a precise radiometric calibration of the camera set-up and of the emission characteristics of the frosted glass. To avoid this complication, we measured in a binary,
position-detection mode. The camera was operated at a sufficiently short integration
time of T = 10 ms to minimize motion blur. The laser was set to maximum power
to deliberately saturate the image around the center of the laser spot on the frosted
glass to allow for a simple threshold-based detection and segmentation. For each
image, the center of mass (center of mass)center of mass\relax of the spot was then
easily retrieved and a series of images yielded a series of lateral deviation (∆x, ∆y).
Local curvature of the water surface generally leads to geometrical distortion of the
laser beam profile at the point of penetration. This will cause the laser spot on the
frosted pane not to appear circular, as assumed by the coordinate determination
based on the center of masscenter of mass\relax . To avoid this effect, the laser was
focused on the water surface to a FWHM
√ of2 about 100 µm, which was smaller than
the typical scales of surface curvature, (∂ η/∂x i ∂x j )−1 .
Figure 5.4 shows a series of exemplary along-wind slope distributions for different
settings of the wind turbine frequency fwind . Up to about fwind = 16 Hz, the water
surface is merely undulated and slopes are below ak = 0.01. For higher wind speeds,
slopes rapidly increase and reach values of ak ≈ 0.2. This is in coherence with the
observed wind speed profiles and the flow parameters deduced thereof in sec. 2.2.
From about fwind = 21 Hz onwards, skewness of the slope distributions becomes
apparent. This is to be attributed to the appearance of small scale gravity and capillary
waves and their characteristic asymmetric shape. Together with observations of the
wave field by eye, this suggests that micro-scale wave breaking might set-in in this
regime.
The parameter mss is essentially defined as the variance of the slope distribution.
Its dependence on wind speed is shown in Fig. 5.5, where the along-wind, crosswind, and total mss is depicted. This is in excellent agreement with the measured
50
Characterization of the pulsed UV laser
5.4
Figure 5.6
Lateral unfocused beam profile of the CFR200
pulsed UV laser measured with a Dataray WinCamD beam profiler. The overall diameter is
about 5 mm.
roughness parameter z0 and the air-sided boundary layer thickness δ in Fig. 5.3.
Cross-wind mss reaches a maximum of 0.01 at fwind = 30 Hz, corresponding to a
slope of ak = 0.1. This serves as a lower limit for the optimal air-sided camera
viewing angle (see sec. 4.3.3).
5.4 Characterization of the pulsed UV laser
As light source for the LIF experiments, a pulsed Q-switched laser¹ was used (see
sec. 4.1.2). To provide quantitative information for an efficient design of the experiments, the laser was characterized in terms of beam shape, power output, and
intensity fluctuations.
5.4.1 Beam profile
To measure the lateral beam profile, a UV sensitive beam profiler (Dataray WinCamD) was used². A combination of several absorbing and reflecting filters was
needed to attenuate the intensity of the laser in order to avoid damage to the beam
profiler sensor. Figure 5.6 shows the profile of the unfocused beam with an overall
diameter of 5 mm at about 50 cm distance from the laser head.
The non-uniform, non-Gaussian shape of the profile is evident and indicates the
existence of multiple higher spatial modes. In terms of Fourier optics, these show up
in the focal plane as higher order maxima around the center of the focused beam. To
anyhow obtain a clean laser sheet profile, I used a spatial filter set-up, as explained in
sec. 4.3.2. I also verified the quality of the beam focus in-situ before every experiment
(see sec. 4.4).
¹Model CFR200 by Quantel Lasers
²In kind collaboration with Dr. Andreas Rudolf of TU Darmstadt/GRK-1114
51
Chapter 5
AUXILIARY EXPERIMENTS
Figure 5.7
The dependence of the induced
fluorescent brightness of a white
paper target illuminated with the
pulsed UV laser was measured. By
relating the maximum brightness
to the maximum energy output of
50 mJ/pulse, this provided a onepoint calibration of the laser power
output in terms of Q-switch delay.
5.4.2 Power output
The power of a Q-switched pulsed laser can be modulated by adjusting the QS
delay with respect to the flash pump (see sec. 4.1.2). The correspondence is in
general non-linear and not specified by the manufacturer of the laser system. To
estimate, e.g, the effect of saturation, it was desirable to obtain a calibration curve
to relate QS delay to laser output power. In principle, the output power of a laser
system can be determined with the help of a so-called power meter which in essence
measures the increase in temperature on an internal surface due to the impinging
laser beam. In the case of the pulsed UV laser used in this thesis, the maximum spatial
power density would have exceeded the destruction threshold typically indicated
by the manufacturer. Indeed, the maximum power output can be estimated as
50 mJ/8 ns ≈ 6 × 105 W for a pulse length of 8 ns. Assuming a circular profile of radius
r = 2.5 mm, this amounts to an average spatial power density of 3 × 105 W/mm2 .
Instead, the output power calibration was performed in the following alternative way.
The laser beam was expanded by a concave fused silica lens to a diameter of about
1 m, reducing the maximum averaging power density to only 8 W/mm2 , and shone
onto a standard white paper sheet. Due to the presence of optical brightening agents,
commercial paper emits blue fluorescent light induced by UV radiation, which for
the expanded beam could safely be assumed to be proportional to the incident laser
power. The emitted fluorescent light was recorded with a camera and the average
image brightness was determined as a function of Q-switch setting of the laser. By
relating the maximum brightness value to the highest nominal energy output of
50 mJ/pulse, this provided an effective one-point calibration, shown in Fig. 5.7.
52
Characterization of the pulsed UV laser
5.4
6.5 1e 3
Relative variance Var(SN)
6.0
Figure 5.8
The variance of the image
brightness over an otherwise homogeneous fluorescent scene was determined by first calculating
the average over N pixels
in each individual image.
While the camera noise
contribution decays with
1/N , the contribution due
to laser intensity fluctuations is constant in each
image.
Var(SN)/SN2 =Var(I)/I2 + N1 Var(Ψ)
S2
5.5
N
5.0
4.5
4.0
3.5
0
20
40
60
Number of pixels N
80
100
5.4.3 Intensity fluctuations
The calibration routine for the air-sided images required a brightness-based matching
to reference images. To minimize inaccuracies in this procedure due to variable
illumination, it was useful to quantify the intensity fluctuations of the laser system.
To this end, the set-up explained in sec. 4.3 was used. In general, the brightness
variation over a series of images originates from different independent sources of
fluctuations. Based on the following rationale, the contribution of the laser could
nonetheless be extracted.
The image brightness S can be thought of as a sum of fluorescent signal I ⋅ c and
some camera noise Ψ, i.e.,
S = I ⋅ c + Ψ,
(5.4)
with ⟨S⟩ = ⟨I ⋅ c⟩ and ⟨Ψ⟩ = 0. Here, ⟨.⟩ indicates the average over a long series of
images. Consequently, the variance is given as
Var(S) = c ⋅ Var(I) + Var(Ψ),
(5.5)
where the acetone gas is assumed to be well-mixed and constant in time, i.e. Var(c) =
0. The second term, indicating the camera noise, is thought to be independent for
each individual pixel. Statistically, in terms of camera noise, it is therefore equivalent
to average over N different pixels in one single image, or over one single pixel in N
images. The laser intensities, on the other hand, are identical for different pixels in
one single image. Therefore, the relative variance of the image brightness averaged
over N pixels in each image of a series, is given as
Var(⟨S⟩N−pixel )/⟨S⟩2 = Var(I)/⟨I⟩2 +
1 Var(Ψ)
.
N c ⋅ ⟨I⟩2
(5.6)
53
Chapter 5
AUXILIARY EXPERIMENTS
Following this approach, I recorded a series of 500 images without water in the
tank and acetone vapor well-mixed at high turbine frequency. Since the tank was
kept closed, the concentration remained constant in time. By fitting eq. (5.6) to the
variance data obtained from the 500 images, the relative laser fluctuation could be
determined to be about 15 %. This is shown in Fig. 5.8.
54
6
Experimental procedure
Experiments were conducted as a gas invasion, i.e. the air was initially supersaturated
with acetone causing a flux into the water. This chapter briefly describes the procedure
of the visualization experiments.
6.1 General procedure
Before each run, the wind-wave tank was thoroughly rinsed to remove any unwanted
surfactants which might have accumulated over time and to guarantee for the initial
water-sided tracer concentration to be zero. The tank was then filled with fresh,
demineralized clean water. The wind turbine was turned on to the desired frequency
a few minutes prior to the experiment. Considering the small size and fetch of the
tank, this was certainly long enough to assume the wave field to be in equilibrium, as
confirmed by observation. Continuous tracer injection was started, as will explained
in sec. 6.1.1, and kept on for typically 20 min. In this way, a presumably stationary
condition was reached with a roughly constant air-sided concentration governed by
the equilibrium of injection rate, flux into the water, and losses to the release tube.
Subsequently, injection was stopped, effectively turning the tank into a closed system,
and data acquisition continued for typically another 15 min. Concentrations recorded
during this phase were used to calculate transfer resistances from mass balance
considerations, explained in sec. 7.4. Leakage under neutral pressure conditions was
determined by Niegel (2013) and could be neglected to about 2-3 % precision.
55
Chapter 6
EXPERIMENTAL PROCEDURE
6.1.1 Tracer injection
Acetone was injected into the back flow tubing of the tank in order that any plume
first passed once through the turbine before getting in contact with the water surface
in the experimental section. Liquid acetone was filled in a 250 ml bubbling flask
through which dry pressurized laboratory air was flushed. To compensate for the
strong cooling due to acetone’s negative evaporation enthalpy, the flask was kept
in an actively temperature-controlled water bath at about 23 ° ○C. To keep pressure
constant during injection, a release tube was connected to the exhaust system of the
wind-wave facility.
Injections rates were determined by measuring the weight difference of the liquid
acetone in the flask after 10 min of flushing. Typical values were 11 ml/min or, equivalent at room temperature and atmospheric pressure, 0.13 mol/min. Exact values
varied dependent on the filling height of the flask and consequently the rising time
of the bubbles. The here given average value is estimated to be precise to about 10 %.
6.1.2 Data recording
The components and their deployment was described in detail in chap. 4. Images of
the fluorescent brightness in the water and air were recorded simultaneously. The
two cameras were electronically coupled to the pulsed laser for proper synchronization. A recording frequency of f = 2 Hz as compromise between data volume and
sufficient time resolution. Note that typical decay times after closing the tank could
be estimated to be on the order of 1-5 min. For higher recording frequency, the
resolved structures in the water would have overlapped from on image to the other.
Treating each of them as independent statistical realization would anyhow have
been questionable. Temperature was monitored manually and remained constant at
25(1) ○C.
6.1.3 Reference images
Reference images (see sec. 3.1.6) were recorded with the same recording procedure.
The tank was dried and no water was filled in. Acetone was injected over a course of
about 10 min as explained in sec. 6.1.1, while images were recorded. Concentrations,
and consequently image brightness, increased at a decreasing rate owing to the losses
to the pressure release tube. Afterwards, injection was turned off and the tank was
connected to the actively pumped air-exhaustion system causing concentrations
to gradually fall back to zero. This is shown in Fig. 6.1. In this way, reference
images covered the entire range of concentrations encountered during an invasion
experiment.
56
Modulated wind conditions
6.3
Reference mean bulk brightness
160
140
120
100
80
60
40
20
00
100
200
300
Reference image index
400
Figure 6.1
Mean bulk brightness of reference
images given in gray value (GV).
6.2 Modulated wind conditions
A few experiments were conducted under modulated wind conditions. To this end,
the turbine was switch to e pre-chosen low and high setting in periods of 50 s-50 s.
Modulation was synchronized to image recording.
6.3 Surfactant: Triton-X
Experiments were conducted to investigate the effect of surfactant on gas exchange
and the typical vertical distribution of tracer. As surface active material, Triton-X¹,
was chosen. It is a soluble detergent commonly used in laboratories and has been
employed in the past in gas-exchange experiments, e.g. by Krall (2013); Kräuter
(2011). An amount of 250 mg was first dissolved in demineralized water and then
added to the water in the tank. This quantity was chosen to guarantee effective
suppression of waves even at the higher wind speed setting.
¹IUPAC name: polyethylene glycol p-(1,1,3,3-tetramethylbutyl)-phenyl ether
57
7
Data processing
7.1 Air-sided profiles
The data processing procedure for the air-sided images consists of the following
subsequent steps, which I explain in detail in the following sections:
1. To each data image, a matching reference images with homogeneous tracer
distribution is selected.
2. From these two, a quotient image is formed which directly provides the relative
tracer distribution c(x,z)/cb .
3. The water surface is automatically detected in each images with a dedicated
low-level image processing routine.
4. From each image, a vertical concentration profile is obtained by horizontally
averaging in coordinates relative to the surface.
5. An intrinsic calibration allows to convert relative to absolute concentration
values.
59
Chapter 7
DATA PROCESSING
7.1.1 Quotient images and relative concentration
After sec. 3.1.2, the local fluorescence brightness intensity depends on the tracer
concentration and the laser irradiance. I showed in sec. 3.1.2 that for the pulsed
Nd:YAG laser used in this work, small systematic effects of saturation occur close
to the focal plane of the laser sheet and in sec. 3.1.6 I explained how this can be
effectively accounted for by regarding the relative brightness of two images. For
typical tracer concentrations of 1 × 10−3 mol/L, the saturation effect can be described
as multiplicative damping term. Here, I explain the implementation of this approach
in the data processing routine.
Reference images were recorded without water in the tank as described in sec. 6.1.3.
They were averaged over subsequent 20 frames, equivalent to 10 s and the mean bulk
brightness was calculated from image pixels corresponding to a fixed reference
height about 40 mm above the water. This set of bulk brightness values covered a
range from virtually 0 gray value (GV) to about 160 GV in steps of about 1.5 GV (see
Fig. 6.1 in sec. 6.1.3).
From each data image, i.e. with water in the tank, bulk brightness was computed
for the same image pixels and the reference image with the closest matching value
was selected, i.e. cref ≈ cb . Following from sec. 3.1.6, the ratio of such two images is
given as
c(x, z) ⋅ I(x, z; cb ) ⋅ Sat(z; cb , P) ⋅ Vig(x, z)
F(x, z) − Ψ
=
Fref (x, z) − Ψ
cref ⋅ I(x, z; cref ) ⋅ Sat(z; cref , P) ⋅ Vig(x, z)
(7.1)
Obviously, the non-zero noise signal Ψ of the camera (dark image) was subtracted
before calculating the quotient. From here on, I will simply drop the noise term in
any equation for simplicity. It should be noted here, however, that any other diffuse
background light contributes an additive term to the observed signal and causes
the multiplicative factors not to cancel precisely. In other words, background light
leads to unwanted residual artifacts in the quotient image so that the visualization
technique only works in dark conditions. The Vig(x, z) factor describes shading by
the camera optics, i.e. vignetting. This term automatically cancels in the quotient
image and radiometric camera calibrations are unnecessary with this approach.
Likewise, the Sat(z; cb , P) and ⋅I(x, z; cb ) terms cancel if cref = cb .
Summarizing, relative concentration images were obtained as
F(x, z) − Ψ
c(x, z)
=
.
cb
Fref (x, z) − Ψ
(7.2)
An example for air raw image and the corresponding quotient image is shown in
Fig. 7.1. Since outside of the laser sheet brightness quickly drops to zero, the quotient
image becomes very noisy in that region. Only the inner portion, indicated by the
two red lines, is used for further processing.
60
Air-sided profiles
a
7.1
b
Figure 7.1: (a) Example of a raw image for turbine frequency f wind =11 Hz. Only the image portion
contained between the two red lines was used. (b) Quotient image F/Fref = c/c b obtained
by matching the average bulk brightness. Note the large noise on either side of the
laser sheet where virtually zero signals are divided. Images are stretched in horizontal
direction for clarity.
7.1.2 Relative coordinates and image averaging
Over the wavy water surface, the choice of coordinate system is not unique. A
Cartesian laboratory reference frame would suit to represent tracer distribution
sufficiently far away from the water. On the other hand, close to the surface on
scales similar to the wave amplitudes, it would certainly have a limited physical
meaning. A mathematically rigorous choice of reference frame would be curvilinear
coordinates, which are locally orthogonal with the vertical axis locally perpendicular
to the water surface. This choice would require rather cumbersome geometric image
transformations to be determined for each individual image. As wave slopes do
not significantly exceed ak = 0.2 (see sec. 5.3), the merit of this procedure for the
physical interpretation of concentration profiles is anyhow questionable. For this
work, I chose an intermediate way in which the vertical coordinate was linearly
shifted according to the height of the water surface and the lateral coordinate was
kept, i.e.
ζ = z − zsurf and ξ = x.
(7.3)
From each image, a profile was calculated by horizontally averaging in this relative
coordinate system,
c(ζ) = ∑ c(ξ, ζ).
(7.4)
ξ
Series of images could further be averaged depending on the scope of the specific
analysis. For the sake of notational simplicity, I will retain x, z as coordinates instead
61
Chapter 7
DATA PROCESSING
a
c
b
Figure 7.2: (a) Example for an air-sided binary image obtained from a quotient image F/Fref applying a threshold of 2. (b) The three largest connected objects in the binary image,
identified as potential surface candidates. The blue one is selected for its highest
center of masscenter of mass\relax . (c) The detected water surface defined as the
upper edge of the selected binary object.
of ξ, ζ, implicitly intending relative height with respect to the water surface.
7.1.3 Surface detection
To apply the averaging procedure described above, the water surface needed to be
detected in each image. In sec. 4.2, I have shown that the position of the surface
is sufficiently well defined based in a threshold criterion. Since the detection was
performed on quotient images F/Fref = c(x,z)/cb , values fluctuated around c(z)/cb = 1 and
a threshold of 2 proved to be robust and effective.
The detection algorithm performed the following successive steps. First, a binary
image was created in which pixels with values above the threshold were set to 1
and all others to 0. Connected pixels were joined to objects applying the structure
element
1 1 1
1 1 1
1 1 1
The so-identified objects were regarded as potential surface candidates. An example
of such a binary image is shown in Fig. 7.2 (a).
Under certain conditions, reflections were visible in the image which could potentially have led to spurious surface detection. Such reflections are seen in Fig. 7.1 (a)
and (b) and consequently in the binary image Fig. 7.2 (a). Note that due to the limited
resolution of the figures, small structures appear connected to larger ones. The here
62
Air-sided profiles
7.1
Figure 7.3
′
Calibration plot relating Fb/є air Fb = c
to the bulk brightness Fb .
For the extinction coefficient,
є air =11.7 L/mol/cm was used
(Lozano et al., 1992). The dashed
line is a linear fit to the calibration
data points.
shown binary image actually contains almost 60 separate objects corresponding
to pixel values above the threshold. Two selection criteria were applied. First, the
surface was expected to be continuous, extending almost across the entire image.
The size of the corresponding object in the binary image should therefore be large
compared to most reflections. Correspondingly, only the largest five binary objects
were admitted as surface candidates. An example is shown in Fig. 7.2 (b). Second,
by geometric considerations, reflections could only originate from the water patch
between the laser sheet and the camera and thus result in bright spots beneath the
height of the water surface in the image. Among candidate objects, the one with the
highest center of masscenter of mass\relax was preferred. From this, the physical
water surface was determined as the position of their upper edge, as illustrated in
Fig. 7.2 (c). The here described algorithm proved to be very robust and detection
precision was as good as 1 px.
7.1.4 Absolute concentrations
The reference images provided a simple way to obtain a calibration curve to convert
image brightness in physical tracer concentration. This was possible because the
camera observed the brightness decay due to Lambert-Beer absorption over a certain
height. Since the extinction coefficient is known, the concentration could be derived
from this decay.
In the reference images, the tracer concentration is homogeneous and the fluorescent brightness as a function of vertical coordinate is given as
F(z) = cb I(z) = cb I0 exp (−єair cb z) .
(7.5)
In particular, one can define the bulk fluorescent brightness
Fb ≡ F(zb ) = cb I0 exp (−єair cb zb ) ,
(7.6)
63
Chapter 7
DATA PROCESSING
where zb denotes the height of the bulk. The derivative at z = zb is then
Fb′ ≡
∂F
(z = zb ) = −єair I0 cb2 exp (−єair cb zb ) = −єair cb ⋅ Fb .
∂z
(7.7)
In other words, from the relation
cb =
−Fb′
Fbєair
←→
Fb ,
(7.8)
a calibration curve can be obtained relating image brightness to concentration.
Numerically, the derivative Fb′ was calculated as tangent fit to the brightness profile
F(z) in the bulk portion of the image. The set of reference images covered the
entire brightness, i.e. concentration range encountered in any data image. The
obtained calibration curve is shown in Fig. 7.3 together with a linear fit c = a ⋅ Fb ,
with a = 2.21 × 10−5 mol/L/GV.
Absolute concentration profiles could then be calculated from relative brightness
profiles and bulk brightness values as
c(z) =
F(z)
⋅ F(z = zb ) ⋅ a,
Fref (z)
(7.9)
with F(z) and Fref (z) data and reference profiles, respectively, as in eq. (7.2). One
should note that absolute concentrations could also directly be obtained from each
single data profile by performing the above described tangent fit in the bulk portion.
While both methods in principle yield the same result, this latter approach is less
robust due to local fluctuations in single images.
7.2 Water-sided profiles
The data processing procedure for the water-sided images was similar as for the air
side and consisted of the following subsequent steps, which I explain in detail in the
following sections:
1. The water surface was automatically detected in each image with a dedicated
low-level image processing routine.
2. From each image, a vertical fluorescent brightness profile was obtained by
horizontally averaging in coordinates relative to the surface.
3. Concentration profiles were calculated by fitting an exponential model combined with Lambert-Beer absorption to the vertical brightness distribution
averaged over 16 000 pixel columns corresponding to 10 s time steps.
4. This procedure yielded a time series of surface and bulk concentration as well
as characteristic boundary layer scale.
64
Water-sided profiles
7.2
7.2.1 Surface detection and image averaging
To obtain mean brightness profiles, images had to be averaged horizontally across
the laser sheet in coordinates relative to the water surface. To this end, the water
surface had to be detected automatically in each individual image. This was achieved
by low-level image processing departing from the observation that fluorescence is
brightest right at the surface and that for the occurring wave slopes, the condition of
total reflection was always fulfilled. The principle idea was to identify the surface as
the longest (horizontally) and at the same time brightest structure in the image. The
implementation of the algorithm is shortly explained in the following.
First, the dark image is subtracted and vignetting is corrected for with a flat field
image recorded off an Ulbricht sphere. The image is smoothened to suppress small
scale noise and local maxima are determined column-wise and used to form a binary
image.
To remove isolated pixels resulting from noise, a Hit-or-Miss filter with the structure element
0 0 0
0 1 0
0 0 0
is applied to the binary image. Since local maxima are determined independently
for each column, binary pixels along the water surface might be shifted vertically
relative to each other and would therefore not combine to a single labeled structure. Therefore, three consecutive dilatations were applied using the following two
structure elements which were adapted to the curvy elongated shape of the wavy
surface:
0 0 0
1 1 1
0 0 0
1 0 0 0 1
1 1 1 1 1
1 0 0 0 1
In the dilated binary image, connected pixels are combined to labeled objects and
for each of them, the area and mean brightness is determined and normalized to the
greatest occurring value, respectively. An example of a structured image is shown in
Fig. 7.4 (a) where the fluorescence brightness is given in false colors. The object with
the largest sum of normalized area and mean brightness is selected as water surface.
Due to the dilatations, the surface objects were typically 3 to 5 pixels wide and
their geometrical centerline was defined as physical water surface to achieve pixel
precision. This is shown in Fig. 7.4 (b) for the example image in (a). Taking the
centerline as estimate of the surface is reasonable as the brightness is expected to be
symmetric around the water surface due to total reflection.
All models for concentration profiles referenced in sec. 2.4 are applicable to mean
tracer distributions. On the other hand, invasion of acetone in water takes place
65
Chapter 7
DATA PROCESSING
a
0
2100
200
2090
Pixel position of surface object
Vertical pixel coordinate
b
400
600
800
1000
1200
1400
0
200 400 600 800
Horizontal pixel coordinate
upper edge
center line
lower edge
2080
2070
2060
2050
2040
2030 0
200
400
600
800
Horizontal image coordinate
1000
Figure 7.4: (a) Example for a water-sided image with the dilated local maxima overlaid as color
image. (b) Upper and lower edge of the surface object and the geometric centerline
used as an estimate of the water surface position.
on fairly short time scales, i.e. on the order of minutes. As a compromise, intensity
profiles were calculated by averaging horizontally in coordinates relative to the water
surface and over 20 consecutive images, i.e. over a total of 20 × 800 = 16000 image
columns. Only for lower wind speeds of up to u∗ = 10 cm/s and consequently lower
water drift velocities, subsequent images overlapped by about 15 %, but turbulent
structures had considerably changed. For higher wind speeds, there was no overlapping between pictures. According to Taylor’s principle of frozen turbulence, each
mean brightness profile was thus based on 1.6 × 104 realizations of the statistical
ensemble describing the turbulent mixing. It was therefore reasonable to apply mean
profile models to this averaged, but still temporary data.
7.2.2 Implicit dependence:
concentration and fluorescent brightness
In the LIF experiments conducted in this work, the tracer gas acetone is also the absorbing substance at the same time. This is different in studies where a water soluble
fluorescent dye is illuminated by a laser and the concentration of some tracer gas,
typically oxygen, is measured through the quenching effect (Friedl, 2013; Münsterer
and Jähne, 1998; Walker and Peirson, 2008, e.g.). As the acetone concentration decreases further away from the water surface, the Lambert-Beer absorption decreases
accordingly yielding a non-monoexponential profile. For an arbitrary concentration
profile c(z), the differential irradiance decrease is
dI = −єI(z)c(z) dz.
(7.10)
dz̃ ≡ c(z) dz
(7.11)
Defining
66
Water-sided profiles
7.2
eq. (7.10) can be re-written as
dI(z̃) = −єI(z̃) dz̃.
(7.12)
Upon integration, this yields the irradiance profile for general tracer distributions,
I(z) = I0 exp (−є
∫
0
z
c(z ′ ) dz ′ ) .
(7.13)
Leaving aside optical vignetting by the objective, which is anyhow corrected for, the
camera essentially observes the spatial distribution of the fluorescent brightness
F(z). This is proportional to the product of irradiance and tracer concentration, so
that in general
F(z) = I0 c(z) ⋅ exp (−є
∫
0
z
c(z ′ ) dz ′ ) .
(7.14)
For the sake of simplicity, all constant factors are summarized in I0 . The task therefore
consists in extracting the tracer concentration profile according to the above equation.
One possible attempt could be to solve eq. (7.14) iteratively by adjusting an estimated
profile cest (z) which minimizes the difference Fobserved (z) − F(z; cest ). At z = 0, the
fluorescent brightness is simply given as F(z = 0) = I0 ⋅cs , suggesting an optimization
routine that follows the integration. The pre-factor unfortunately is not known to
good precision, as it depends on the exact local irradiance, absorption cross section
of acetone, quantum efficiency, and the optical characteristics of the camera system.
A global, parallel optimization approach, on the other hand, would require a suitable
scheme by which to adjust the estimated profile cest (z) at each z during iteration
while retaining realistic smoothness and good converging properties. To come along
these numerical, algorithmic difficulties, I decided to adopt a simpler, yet suitable
approach, which is explained in the following section.
7.2.3 Implementation and stability of fit approach
The concentration was a-priori parametrized by an exponential function as
c(z) = cb + (cs − cb ) exp (−
z
),
z∗
(7.15)
Such an exponential profile corresponds to the prediction by the surface renewal
model (see sec. 2.4) for free surface boundary conditions, but here it is used as a
generic functional description for the vertical tracer distribution close to the water
surface without explicitly intending the conceptual validity of the surface renewal
model. It contains all relevant parameters, i.e. two concentration values (cb and cs )
and a scale parameter (z ∗ ), to characterize the tracer profile which can independently
of any model assumed to gradually decay from its surface value towards the wellmixed water bulk (in an invasion experiment).
67
Chapter 7
DATA PROCESSING
b
cb = 0.0 mm−1
cb = 0.1 mm−1
cb = 0.2 mm−1
cb = 0.3 mm−1
cb = 0.4 mm−1
cb = 0.5 mm−1
cb = 0.6 mm−1
cb = 0.7 mm−1
cb = 0.8 mm−1
cb = 0.9 mm−1
1.4
1.2
1.0
0.8
0.6
Fluorescent brightness in arbritrary units
Fluorescent brightness in arbritrary units
a
0.4
0.2
0.0
0
1
2
3
Depth in mm
4
5
cb = 0.0 mm−1
cb = 0.1 mm−1
cb = 0.2 mm−1
cb = 0.3 mm−1
cb = 0.4 mm−1
cb = 0.5 mm−1
cb = 0.6 mm−1
cb = 0.7 mm−1
cb = 0.8 mm−1
cb = 0.9 mm−1
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
1
2
3
Depth in mm
4
5
Figure 7.5: Fluorescence model profile for a range of realistic values of c˜b for a trivial homogeneous tracer distribution (a) and a profile with a boundary layer width of z ∗ = 200 µm
(b).
With eq. (7.15) and eq. (7.14), an explicit form of the fluorescence profile can be
given in terms of the model parameters,
z
z
)] ⋅ exp [−єcb z − є(cs − cb )z ∗ (1 − exp (− ∗ ))] .
∗
z
z
(7.16)
Observing that I0 is anyhow a free parameter that may contain an additional factor
of є, the concentration terms can be redefined as c˜b ≡ єcb and c˜s ≡ єcs . This provides
an explicit expression with four free parameters which can be used with any standard
least-square algorithm.
The fluorescence model profile is shown in Fig. 7.5 for different realistic values
of c˜b for a homogeneous tracer distribution (a) and a profile with a boundary layer
width of z ∗ = 200 µm (b).
It should be observed, that eq. (7.16) contains two scale parameters, i.e. z ∗ and
(єcb )−1 , related to the extent of the diffusive sublayer and the degree of light absorption, respectively. If both scales coincide, i.e. (єcb )−1 ≈ z ∗ , the two parameters are
essentially degenerated and the least-square fit is expected to become instable. For
an extinction coefficient of acetone of є ≈ 1.52 L/(mol mm) (see sec. 3.1.4) and a
rather thick boundary layer of z ∗ = 1 mm occurring at the lowest wind speeds, the
above condition would require a concentration of cb ≈ 0.5 mol/L. This is equivalent
to a total amount of 600 g of acetone in the water which was never reached in any
experiment.
When an invasion experiment approaches an equilibrium state in the absence of
influx and leakage, the flux across the air-water interface ceases and (cs − cb ) → 0.
In this case, eq. (7.16) reduces to
F(z) = I0 [cb + (cs − cb ) exp (−
F(z) = I0 cb exp (−єcb z) ,
68
(7.17)
Water-sided profiles
7.2
and becomes insensitive to the fit parameter cs . To guarantee anyhow sensible results,
the fit is pre-conditioned by the value cb obtained from fitting eq. (7.17) to the bulk
portion (z ≪ z ∗ ) of the fluorescence profile. Indeed, for z/z∗ ≫ 1, again eq. (7.16)
reduces to eq. (7.17), as can be easily checked.
The complete fit routine therefore consists of two consecutive steps. First, the bulk
tracer concentration cb is obtained by fitting eq. (7.17) to the brightness profile in
the range of 4 mm < z < 8 mm corresponding to the bulk portion of the image. In
a second step, cb is used as a fixed value and the full function eq. (7.16) with only
three free parameters is fitted to the entire brightness profile up to the surface. This
proved to give stable results even in near-equilibrium conditions.
7.2.4 Absolute concentrations
For the water-sided profiles, conversion into physical values with unit mol/L was
straight forward. Indeed, the model-based fit approach in the previous section
already includes a description of the Lambert-Beer absorption. Physical concentrations were obtained by dividing the fit parameters (єcb ) and (єcs ) by the extinction
coefficient for acetone, є = 1.52 L/(mol mm) (see sec. 3.1.4).
7.2.5 Systematic error due to limited optical resolution
As explained in sec. 2.4.1, the water-sided diffusive sublayer is expected to become
as thin as 80 µm-100 µm for the highest wind speeds in the wind-wave tank used in
this work. On the other hand, the effective resolution of the camera system is limited
by optical aberration. To quantify the impact of this limitation on the experimental
results and to possibly correct for them, I estimated the systematic error induced
in the profile parameters due to the optical imperfections of the system. In this
context it is useful to observe that the total reflection angle for a light ray at the phase
boundary water-air is about 49°. For a camera observing the water surface from
underneath at a angle of −10° from the horizontal plane, water surface slopes of at
least tan(39°) = 0.8 would be required for the surface to become transparent. Such
values are never reached in any wind condition as is evident from the wave slope
measurements presented in sec. 5.3. In other words, the water surface always acts as
a perfect mirror when observed by the camera.
Secondly, it should be noted that optical aberration amounts to a smoothing
of the image, i.e. a spatial convolution. This has the mathematical property of
conserving image brightness. For a narrow, pronounced brightness profile such as in
Fig. 7.5 (b), optical aberration would therefore lead to a mere spatial redistribution of
the brightness. Without correction for this effect, the maximum value at the surface
would however be systematically underestimated. I quantified this by creating a
synthetic brightness profile according to eq. (7.16), reflecting it at z = 0, and applying
a spatial convolution with a Gaussian function of varying FWHM. The result is
shown for exemplary concentration parameters in Fig. 7.6 (a). For unrealistically
69
DATA PROCESSING
Observed fluorescence brightness in a.u.
a
b
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.00.0
FWHM = 115µm
FWHM = 230µm
FWHM = 345µm
FWHM = 460µm
FWHM = 575µm
True profile
Relative deviation (Ftrue−Fobs)/Ftrue
Chapter 7
0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.2
0.4
0.6
0.8
Depth in mm
1.0
1.2
z∗
z∗
z∗
z∗
z∗
z∗
z∗
z∗
z∗
0.0
0.1
0.2
0.3 0.4 0.5
Depth in mm
0.6
= 100µm
= 200µm
= 300µm
= 400µm
= 500µm
= 600µm
= 700µm
= 800µm
= 900µm
0.7
0.8
Figure 7.6: (a) Simulated fluorescence profiles for єc b = 0.2 mm−1 , єc s = 0.8 mm−1 , and z ∗ =
0.1 mm for different degrees of optical smoothing. (b) Relative systematic deviation
of the observed fluorescence profile for a Gaussian smoothing function with a FWHM
of 120 µm for different boundary thickness z ∗ . No correction for optical smoothing is
applied.
strong optical smoothing, the observed fluorescence profile is significantly altered
with respect to the ground truth. For reasonable value for the FWHM of the Gaussian
smoothing kernel, the surface appears systematically darker but the overall brightness
profile is still preserved.
Optical aberration affects the profile stronger the thinner the boundary layer
is. To obtain a quantitative impression of this dependence, I created synthetic
observed fluorescence profiles for a fixed FWHM of 120 µm and different values of
the boundary layer scale z ∗ . The relative deviation from the ground truth profile is
depicted in Fig. 7.6 (b). For the thinnest expected boundary layers (see sec. 2.4.1),
the water surface might appear up to 25 % darker due to optical smoothing if no
correction is applied.
7.2.6 Correction for optical sharpness
The brightness conserving properties and the total reflection at the water surface
suggest a simple method to compensate for the systematic underestimation of the
surface brightness due to optical aberration. During an iteration step of the leastsquare fit algorithm, after calculating the model profile for a set of test parameters,
(cs , z ∗ , I0 ), but before passing the result to the optimization routine, the model profile
is reflected around z = 0 and convolved with a Gaussian kernel of fixed FWHM. In
this way, optical aberration is explicitly taken into account and the systematic error
in the retrieved concentration profile parameters is minimized. The smoothing scale
was estimated from target images and manually set rather than adding an additional
free parameter to the algorithm. Depending on how correct the smoothing scale
is chosen, the fit parameters retain a certain degree of systematic deviation. This is
illustrated in Fig. 7.7, where concentration profiles with different values of z ∗ and
70
Relative surface parameter (cs,fit−cs,true)/cs,true
Eddy statistics
0.4
0.3
0.2
z∗
z∗
z∗
z∗
= 100µm
= 200µm
= 300µm
= 400µm
0.1
0.0
0.1
0.2
0.3
90
100
110
120
130
140
FWHM of Gaussian convolution function in µm
150
7.3
Figure 7.7
Relative deviation of the
surface concentration єc s
retrieved from a synthetic
profile for different values of z ∗ . The dashed
lines correspond to the
systematic error when no
correction for the optical
aberration is done. The
solid lines show the systematic error if the fit function is smoothened with
a Gaussian kernel of a certain size prior to the leastsquare optimization.
smoothened with a FWHM of 120 µm were used as ground truth. The dashed lines
correspond to the relative systematic error in the retrieved surface concentration
cs when no correction for the optical aberration is done. The solid lines shows
the relative systematic error with optical correction as a function of applied kernel
size. Evidently, for an adequately chosen FWHM value, the systematic error in cs
can efficiently be reduced. A rather conservative choice of the smoothing scale is
preferable, however, as for too large values, cs may easily be overestimated greatly.
Based on tests with recorded images, I have chosen a FWHM of 100 µm. According
to this analysis, measured water-sided surface concentrations are estimated to be
precise to about 10 % or better.
7.3 Eddy statistics
The surface detection algorithm explained in sec. 7.2.1 can be modified to yield useful
information about the eddy distribution underneath the water surface. The physical
reasoning is that streaks in the fluorescent image correspond to connected regions
of locally enhanced tracer concentrations. These are originally detached from the
surface and carried downwards into the bulk. Statistically, the distribution of streaks
should therefore provide a good estimate in the occurrence of turbulent eddies.
Practically, the same steps outlined in sec. 7.2.1 to find local maxima and to reduce
erroneous detections were performed resulting in binary objects which represent the
streaks. This was shown in Fig. 7.4 (a). For all objects, their upper edge was detected
with a gradient mask and for all edge pixels, their vertical position relative to the
water surface above was calculated. This could be performed on a series of images
yielding a distribution of vertical distance of streaks from the interface. Following
the above argumentation, this procedure allowed to estimate the minimal approach
71
Chapter 7
DATA PROCESSING
distance of turbulent eddies to the water surface under different forcing conditions.
The results were particularly useful for interpreting the mechanisms leading to a
reduction of gas exchange in the presence of surfactants (see sec. 8.5).
7.4 Mass balance method
A commonly used approach to determine transfer resistances, or equivalently transfer
velocities, is to monitor the well mixed bulk tracer concentration in a wind-wave tank
(e.g. Krall, 2013; Nielsen, 2004). As explained in sec. 2.3, the so-determined resistances
are of integral character and provide no direct insight in the local, near-surface
turbulence. Nonetheless, this approach provides a straight forward complementary
method that has also been used in this work. Here, I shortly outline the formulae
needed for the data analysis in this work.
The wind-wave tank is thought of as an air and a water compartment of respective
volumes Vwater and Vair , separated by an interface of area A. In closed conditions,
i.e. neglecting leakage and with acetone injection turned off, water and air-sided
concentrations are coupled through two differential equations. In matrix form they
read
(
k
− k
c
ċair
α H Vair
) ( air )
) = A ⋅ ( Vk air
k
cwater
ċwater
Vwater − α H Vwater
(7.18)
This can be solved by computing the eigenvalues of the matrix and posing the
initial conditions cair (t = 0) = cair,0 and cwater (t = 0) = cwater,0 , yielding
cwater,0 + Vair /Vwater cair,0
αH cair,0 − cwater,0
1
1
) t]
+
exp [−kA (
+
αH + Vair /Vwater
αH + Vair /Vwater
Vair αH Vwater
(7.19)
cwater,0 + Vair /Vwater cair,0
cwater (t) =
1 + Vair /(Vwater αH )
Vair αH cair,0 − cwater,0
1
1
) t] .
(7.20)
−
⋅
exp [−kA (
+
Vwater αH + Vair /Vwater
Vair αH Vwater
cair (t) =
From an exponential fit to the decay, the integral transfer velocity k = R −1 can thus
be easily determined. The parameters Vwater , Vair , A, and αH are given in sec. 4.1.1
and sec. 3.1.4.
72
8
Results
8.1 Air-sided profiles
Air-sided profiles were obtained from quotients of data images and reference images,
consequently yielding normalized concentrations c(z)/cb denoted as c̄ + . The visualization concept was described in sec. 3.1.6 and the implementation in the data processing
routines in sec. 7.1.1. In this section, I summarize aspects that may be deduced from
the concentration profiles alone, i.e. without regarding the simultaneously observed
water side. For nomenclature, I refer to chap. 2.
In sec. 2.4.1, I explained how concentration profiles can be calculated based on
a parametrization of the turbulent diffusivity Kdiff (z) which interpolates between
pure diffusion close to the boundary and the linear dependence on z leading to
the logarithmic shape (see sec. 2.2.1). A basic assumption of this approach is the
conceptual equivalence of momentum and mass transport through the turbulent
boundary. To my best knowledge, profiles of passive scalars have so far not been
observed experimentally down to scales of the diffusive sublayer.
8.1.1 Comparison with wind profiles
If momentum and mass transfer are assumed conceptually equivalent, a diffusive
sublayer should exist below z + ≈ 11(≡ z11 u∗/ν) and a logarithmic sublayer for about
(z11 < z < 0.15δ), where δ is the overall thickness of the air-sided boundary layer.
To make the analogy clear, I restate the relevant equations from sec. 2.2. According
to eq. (8.1), the wind profile derived from Reichardt’s turbulent viscosity model is
73
Chapter 8
RESULTS
ū(z + ) − uplate =
τs
ρu∗
∫
ν
z+
dz + ,
ν + Kturb (z + )
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
0
(8.1)
R(z + )
where I have included explicitly a term to allow for a bottom plate moving relative
to the airflow. In terms of momentum transport, its value determines the strength
of the momentum sink and for uplate = 0, the wind profile decays to zero in the
laboratory frame. If on the other hand, the plate moves with exactly the bulk wind
speed U0 , no momentum is dissipated and ū(z) = U0 , i.e. the profile is flat.
In analogy, for a non-vanishing surface concentration one has according to
eq. (2.33),
z+
js ν
D
⋅
dz + .
(8.2)
c(z) − cs =
Du∗ 0 D + Kdiff (z + )
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
∫
D(z + )
The term D(z + ) contains all the scaling properties of the profile and can be calculated
numerically by integration.
To compare profiles for different wind speeds and to retrieve the transfer resistance,
knowledge of the surface concentration is necessary. Different from the velocity
of the moving bottom plate in the case of wind profiles, it is not easily measurable
as separate experimental parameter because it depends on the unknown relative
flux in the underlying water. Therefore, cs has to be determined from the profiles
themselves.
To investigate the scaling properties and self-similarity of the concentration profiles, I exploited their presumed characteristic shape within the diffusive and logarithmic sublayer, respectively. The exact scaling procedures are explained in the
following two sections. Equation (8.2) is used in sec. 8.1.4 to fit full Deacon profiles.
I present the deduced transfer resistances in sec. 8.6.
8.1.2 Profile scaling - diffusive sublayer
Very close to the surface, i.e. for z → 0, the integrand in eq. (8.2) is 1 and thus
D(z + ) ≈ z + . In this regime, eq. (8.7) reduces to a linear expression of the form
c̄ + (z) = A ⋅ z + B.
(8.3)
The two parameters A and B can be used to scale and non-dimensionalize the
profiles as
c̄ + (z/δ) − B
clin (z/δ) ≡
.
(8.4)
A
By this definition, clin (0) = 0.
Figure 8.1 shows measured non-dimensional concentration profiles clin (z/δ) for
74
Air-sided profiles
a
b
c
d
c
d
8.1
Figure 8.1: Air-sided concentration profiles c + (z) = (c(z)−c s )/c b scaled by a tangent fit to the
+
diffusive sublayer, i.e. c lin (z/δ) = c̄ (z/δ)−B
. The surface gradient is shown as a red
A
dashed line. The green dashed line indicates the coordinate of minimal measured
concentration and the blue one z + = 11. Turbine frequencies are (a)-(f ): 5, 8, 11, 15, 18,
21 Hz.
75
Chapter 8
RESULTS
different wind speeds. The blue shades indicate profiles averaged over 10 s and the
black curve shows the average over a series of 3 min. The surface gradient was fitted
for z + < 11 and is shown as a red dashed line. The green dashed line indicates the
coordinate of minimal measured concentration and the blue one the theoretical
transition from diffusive to turbulent regime (z + = 11).
It should be noted that for the lowest wind speed (a), the measured profile levels
off more quickly than at higher wind speeds. Specifically, the mean profile value
at z + = 11 is less than extrapolated by the gradient. This is to be ascribed to the
non-developed turbulent flow under these conditions. Indeed, it is easily checked
that z11 > 0.15δ so that no logarithmic sublayer is expected to exist.
Figure 8.2 shows the same data and fitted gradients, however in logarithmically
scaled vertical coordinate z. In this representation, the diffusive (z < z11 ) and the
logarithmic sublayer (z11 < z < 0.15δ) are visible. Below a certain minimal distance, measured concentrations are biased by the overlaying recorded brightness
originating from the water side (see sec. 4.2). I will comment on this in sec. 8.1.7.
Matching profiles based on their surface gradient has two major drawbacks. First,
the linear approximation is valid only asymptotically for z → 0 and the numerical
fit result of eq. (8.3) will conceptually depend on the choice of the upper boundary
(here z11 ). Second, the diffusive sublayer becomes thinner for higher winds so that
effectively ever fewer data points are available causing the approach to be less robust.
8.1.3 Profile scaling - logarithmic sublayer
Analogously to wind profiles (see sec. 2.2.1), also concentration profiles within the
turbulent boundary layer are expected to have a distinct logarithmic shape in the
sublayer for (z11 < z < 0.15δ), i.e.
c̄ + (z) = A log(z) + B.
(8.5)
Since it is much wider than the diffusive sublayer and thus contains more data points,
this regime should be well-suited for scaling measured profiles.
The above expression can be re-ordered to yield
clog (z/δ) ≡
c̄ + (z/δ) − B
.
A
(8.6)
By determining A and B from a logarithmic fit for z11 < z < 0.15δ and scaling the
vertical coordinate by the boundary layer thickness δ determined from wind profiles
(see sec. 5.1), the profiles clog (z/δ) should become self-similar.
This is shown in Fig. 8.3 for six wind speeds. The blue shades indicate profiles
averaged over 10 s and the black curve shows the average over a series of 3 min. The
data are taken from the initial stage of an invasion experiment since at later times,
the water-sided brightness overlays the signal in the diffusive sublayer (see sec. 8.1.7).
The height z/δ = 1 corresponds to the beginning of the well-mixed bulk region.
76
Air-sided profiles
a
b
c
d
c
d
8.1
Figure 8.2: The figures depict the same data as in Fig. 8.1, with the vertical coordinate scaled
logarithmically. The surface gradient is shown as a red dashed line. The blue dashed
line indicates z + = 11 and the green one z = 0.15δ. Turbine frequencies are (a)-(f ): 5, 8,
11, 15, 18, 21 Hz.
77
Chapter 8
RESULTS
a
b
c
d
e
f
Figure 8.3: Air-sided concentration profiles scaled by a fit to the logarithmic sublayer. The logarithmic portion is shown as a straight red dashed line. Additionally the surface
gradient of the average profile (black) is shown as curved red dashed line. The blue
dashed line indicates z + = 11 and the blue one z = 0.15δ. Turbine frequencies are
(a)-(f ): 5, 8, 11, 15, 18, 21 Hz.
78
Air-sided profiles
8.1
The two dashed red lines indicate the logarithmic and the linear regime, respectively. For (a)-(e), their point of intersection matches well the expected scale of
transition z11 , indicated by the blue dashed line. This demonstrates, that profiles resulting from scalar mass transport show analogous scaling behavior as wind profiles
resulting from momentum transport. The green dashed line represents z = 0.15δ, i.e.
the upper boundary of the logarithmic sublayer. Indeed, roughly above, the averaged profiles start to become more shallow towards the upper edge of the turbulent
boundary layer at z = δ, especially for low and intermediate wind speeds. In (e) and
(f), the logarithmic sublayer appears to extend beyond z = 0.15δ. For these higher
wind speeds, the diffusive sublayer appears less pronounced or almost inexistent in
(f). This might be due to the overlaying water-sided brightness (see sec. 8.1.7), but
could also indicate an alternation of the flow profile above the wavy surface. I will
discuss this in sec. 9.1.
8.1.4 Matching Deacon model profiles
Under the assumption of stationarity, the mass flux js can be expressed in terms of
transfer resistance, i.e. js = Ra−1 (cb − cs ). For the normalized concentrations c̄ + = c/cb
extracted from the quotient images (see sec. 7.1.1), eq. (8.2) can then be stated in the
compact form
(cb − cs )ν
cs
c̄ + =
⋅ D(z + ) + .
(8.7)
cb u∗ DRa
cb
It should be noted that the surface concentration influences both offset and overall
slope of the normalized concentration profiles. The above expression can be used to
match measured profiles with Deacon’s model and to retrieve transfer resistances
therefrom.
In practice, I used the following rather technical procedure. According to eq. (8.7),
two parameters γ1 and γ2 are necessary such that
c̄ + = γ1 D(z + ) + γ2 .
(8.8)
The function D(z + ) is evaluated numerically at those heights corresponding to
the data points z = zu∗/ν and a standard least-square algorithm is used to fit this to
the data and obtain γ1 and γ2 . According to the coefficients in eq. (8.7), the air-sided
transfer resistance Ra can be calculated as
Ra =
1 − γ2
ν.
Du∗ γ1
(8.9)
I present the transfer resistances determined by this approach in sec. 8.6. It should
be noted that by using information from the logarithmic sublayer, the results are less
susceptible to bias due to water-sided fluorescence. It should also be observed that
resistances obtained by fitting Deacon’s model to the data do not entirely depend
on the assumption of diffusive transport at the surface, as the diffusive sublayer
79
Chapter 8
RESULTS
Figure 8.4
Scaled air-sided concentration profiles (c b −c)/Ω as
a function of dimensionless height z/δ . The parameters were obtained
by a fit to the outer layer,
2
c̄(z) = Ω (z − δ) + c b .
Profiles match reasonably
well for z > 0.1δ, asserting
the self-similarity in the
outer layer.
contributes only partially to the air-sided resistance. The slope, containing the mass
flux as factor, is chosen such that model and data profile match up to the upper edge
of the logarithmic sublayer. In this sense, this approach is conceptually more robust
than fitting the surface gradient only, as explained in sec. 8.1.1. Since j = R −1 (cb − cs ),
the resistance does depend, however, on the extrapolated surface concentration.
8.1.5 Scaling of the outer layer (wake)
For wind profiles, the logarithmic sublayer extends to about 15 % of the total turbulent
boundary layer thickness, i.e. up to z < 0.15δ (see sec. 2.2). Beyond, no ab initio
model description is available, but empirically, the outer layer was found to be selfsimilar. The obvious question arises, whether concentration profiles associated with
mass transport show an equivalent behavior.
For the determination of the friction velocity u∗ in sec. 5.1, I used a quadratic
function as simplest self-similar expression. In analogy to eq. (5.1), I fitted the
following function to concentration profiles for all wind speeds used in this work,
2
c̄(z) = Ω (z − δ) + cb
(8.10)
and used the retrieved parameters to rescale the profiles. Figure 8.4 depicts the
scaled concentration (cb −c)/Ω as a function of dimensionless coordinate z/δ. The plot
suggests self-similarity in the wake-region (z/δ > 0.15). Closer to the surface, some
discrepancies are visible. These are partially to attributed to the non-developed
turbulent flow for lower wind speeds and partially to the profiles at higher wind
speeds which appear flatter than expected from smooth wall flow.
80
Air-sided profiles
8.1
Figure 8.5
Sketch to illustrate the
partitioning of resistance
within the air-sided turbulent boundary layer. The
black line is a theoretical
Deacon profile for u∗ =
10 cm/s. The three sublayers are indicated by the
color filling and the horizontal lines show the resistance up the corresponding height above the water surface.
Summarizing, the analysis based on experimental results presented in the preceding sections demonstrates that indeed air-sided scalar transport may be treated in
close analogy to momentum transport.
8.1.6 Distribution of resistance within the air-sided turbulent boundary layer
In sec. 2.3, I underlined the point of view that concentration profiles essentially
describe the vertical distribution of transfer resistance. In this light, an interesting
question is to analyze the contribution of the three sublayers to the total resistance
of the air-sided boundary layer.
A concentration profile varies from cs at the surface to the value cb which is reached
at z = δ. Normalizing with these two parameters yields a function defined on the
unity interval which is equivalent to the relative resistance distribution (see eq. (2.27)
in sec. 2.3).
R(z)
c(z)
=
Rtot
cb − cs
∈ [0, 1]
(8.11)
In Fig. 8.5, this is shown as an example based on the Deacon profile for acetone, a
friction velocity u∗ of 10 cm/s and a boundary thickness δ of 40 mm. For simplicity
and the lack of any theoretical description of the outer layer, I used a logarithmic
shape also for z > 0.15δ.
To each layer, a partial resistance can be attributed by the following definitions:
81
Chapter 8
RESULTS
b
a
Figure 8.6: (a) Theoretical partitioning of resistances within the air-sided turbulent boundary
layer for a Deacon profile as depicted in Fig. 8.5 and extending the logarithmic form
in to outer layer. (b) Resistance partitioning calculated from logarithmically scaled
profile depicted in Fig. 8.3.
R(z11 )
c(z11 )
=
Rtot
cb − cs
R(z = 0.15δ) − R(z = z11 ) c(z = 0.15δ) − c(z11 )
=
Rlog. ≡
Rtot
cb − cs
c(z = δ) − c(z = 0.15δ)
R(z = δ) − R(z = 0.15δ)
=
Router ≡
Rtot
cb − cs
Rdiff . ≡
(8.12)
(8.13)
(8.14)
This is indicated in Fig. 8.5 by the three different colors.
It is well-known from boundary layer theory that for increasing wind speed the
logarithmic sublayer expands downwards causing the diffusive sublayer to become
slimmer (e.g. Schlichting, 2000). Also, the overall thickness of the turbulent boundary
layer δ generally depends on wind speed, fetch and roughness. Indeed, a slight
increase of δ has been observed from wind profile measurements with the onset of
waves (see sec. 5.1).
To determine the relative resistance contributions defined in eq. (8.12), I used
the logarithmically scaled profiles, such as shown in Fig. 8.3. The upper value cb
at z = δ could readily be taken from the data and the surface concentration cs was
extrapolated from the linear fit shown as curved dashed red lines in the plots. The
values c(z11 ) and c(0.15δ) were taken at the corresponding wind speed dependent
coordinates.
The result is shown in Fig. 8.6 (a) together with the theoretical values obtained
from Deacon’s model profile shown in Fig. 8.5. The agreement is quite satisfying and
confirms the theoretical expectation that for growing wind speed, i.e. thickening
82
Air-sided profiles
8.1
Figure 8.7
Brightness in the region
underneath the detected
surface coordinates in the
air-sided images, normalized to the bulk brightness. For lower wind
speeds, the brightness ratio approaches a value between 700 and 800, i.e. approximately the solubility
of acetone.
logarithmic sublayer, the diffusive contribution to the air-sided resistance plays an
inferior role. Because δ also grows with fetch, under field conditions, the diffusive
impact on the air-sided transport should often be expected to become negligible.
In a laboratory experiment, the size of the turbulent boundary layer is limited
by the geometry of the wind-wave facility and in particular by the height of the air
compartment.
8.1.7 Bias by water-sided brightness
Due to the high solubility of acetone, its concentration in the aqueous phase, and
consequently also the fluorescent brightness projected onto the camera sensor by
optical refraction at the surface, is up to 800 times higher. This overlays the air-sided
signal and thus induces a bias on the measured surface-near concentration values.
Additionally, optical aberration of the camera objective causes some blurring of the
high brightness contrast at the air-water interface. For a more detailed description
of the geometric properties, I refer to sec. 4.2 where I have analyzed the dependence
of this optical effect on parameters such as water surface angle and laser sheet width.
During an invasion experiment, acetone continuously accumulates in the aqueous
phase resulting in a rise in concentration which causes the water-sided fluorescence
to become brighter. A typical raw image can be seen in Fig. 7.1 of sec. 4.2. To
obtain a qualitative impression of the time evolution of the water-sided brightness
observed by the air-sided camera, I determined the maximum brightness in the
region underneath the detected surface coordinates and normalized to the air-sided
bulk brightness. The result is shown in Fig. 8.7. For lower wind speeds, the brightness
ratio approaches a value between 700 and 800, i.e. approximately the solubility of
acetone. At higher wind speeds, values are systematically lower. This might to some
extent be associated with optical distortion of the light originating from the water. It
83
Chapter 8
RESULTS
a
b
c
Figure 8.8: Air-sided concentration profiles, scaled according by their logarithmic sublayer as
explained in sec. 8.1.3, for different instances during the course of an invasion experiment. Time is color-coded, in seconds. Each profile is an average over 50 images, i.e.
25 s. Blue and green dashed lines depict z + = 11 and z = 0.15δ, respectively.
seems reasonable that this spreads out the observed spot on the surface resulting in
a less pronounced signal in the projected air-sided image. On the other hand, also
water-sided surface concentrations were generally observed to be much lower under
more energetic conditions (see sec. 8.3.3) which would explain the dimmer signal.
In Fig. 8.8, air-sided profiles are shown at different times during an invasion experiment for fwind =8 Hz (a), 15 Hz (b), and 21 Hz (c). To match the profiles, I used the
procedure explained in sec. 8.1.3. Note the logarithmic scale on the abscissa. As time
progresses, the minimum distance of unbiased signal moves upwards as expected.
Also, at higher wind speeds profiles are more biased. Recalling the dependence of
refraction on water surface angle demonstrated in sec. 4.2, this is probably due to
increased wave slopes.
Two conclusions can be drawn from these figures. First, to analyze the air-sided
profiles in terms of structure and scaling properties down into the diffusive sublayer,
data from the initial stage of an invasion experiment are best suited. Second, despite
the growing bias, the outer layer (z > 0.15δ) is still clearly captured even at high wind
speeds and large times. This allows to estimate the surface concentration based on
the value at z = 0.15δ and the scaling properties described in sec. 8.1.6.
8.2 Water-sided profile parameters
Images of the water-sided tracer distribution were recorded with the set-up described
in sec. 4.3.3. Because acetone is fluorescent tracer and absorber at the same time,
the observed brightness profiles depend implicitly on the unknown distribution
of acetone. I explained this in detail in sec. 7.2.2. To address this issue, I used an
exponential function to describe the tracer concentration profile and derived a
model for the observed brightness thereof which could then be fitted to the data.
The implementation along with considerations regarding correction for optical
aberration were presented in sec. 7.2.3. Shortly speaking, the water-sided images yield
three physical parameters, the bulk and surface concentration, cb and cs , respectively,
84
Water-sided profile parameters
a
8.2
b
Figure 8.9: Example of a water-sided brightness profile in linear (a) and logarithmic (b) scale.
Data points are shown as gray dots. Green line: fit to bulk portion of profile; red line:
convolved best fit function; blue line: optically corrected best fit brightness profile.
and a scale parameter z ∗ characteristic for the thickness of the diffusive sublayer.
Figure 8.9 shows an example of a water-sided brightness profile for a turbine
frequency of fwind = 25 Hz, in linear scale (a) and in logarithmic scale (b). Data
points are depicted as gray dots. The green curve shows the fit result to the bulk
portion of the profile, the red line the overall fit, convolved with a Gaussian function
to correct for optical aberration, and the blue curve is the optically corrected fitted
brightness profile. In general, the fit routine proved to be robust under forced
invasion conditions, i.e. with continuous acetone influx. Once tracer injection
was turned off, the surface concentration quickly approached the bulk value. As
explained in sec. 7.2.2, for cs = cb , the fit is insensitive to the parameters cs and z ∗ .
Thus for this later stage during an experiment, only the bulk concentration provides
sensible physical information.
8.2.1 Comparison with oxygen profiles
In the same wind-wave facility and under similar wind conditions, oxygen profiles
were measured in the framework of a Bachelor thesis supervised during my PhD. For
details about instrumentation and experiments, I refer to Platt (2011). The method
has been extensively used, e.g. by Friedl (2013); Herlina and Jirka (2008); Schulz and
Janzen (2009); Woodrow and Duke (2001). In short, a phosphorescent dye is mixed
into the water and illuminated with a blue laser. The emitted light is quenched in
the presence of oxygen and from the light attenuation, its local concentration can be
retrieved. Here, Platt’s averaged oxygen profiles are used for comparison with the
parameters obtained in this work. This is shown in Fig. 8.10. In order to simulate the
same data processing, an exponential function was fitted to the oxygen profiles and
85
Chapter 8
RESULTS
a
b
Figure 8.10: (a) Water-sided oxygen concentration profiles measured with the quenching technique described in Platt (2011), shown for reference. Dots are data points. Solid
lines are exponential profiles fitted including optical correction. (b) Acetone profiles
reconstructed based on z ∗ .
optical aberration was included in the same way as in this work. Note that profiles
are normalized to cb = 0 and cs = 1. Profiles appear generally steeper for acetone,
but it should be kept in mind that the exponential shape has been a-priori imposed.
The precise distribution might in reality vary slightly. The effect can also be due
to a different dynamical behavior of the molecules compared to oxygen. Overall
agreement of the retrieved scale parameters z ∗ is good, however, suggesting the
validity of the method used in this work.
8.2.2 Stationarity
Conceptually, the transfer resistance is introduced as a macroscopic quantity relating
a concentration gradient ∆c with a mass flux j given certain fluid mechanical conditions. Either based on empirical findings or theoretical models, such conditions may
be described with parameters such as wind speed, friction velocity, wave slope. This
has been explained in more detail in sec. 2.3. In any case, the assumption of stationarity tacitly lies at the base of the definition. In other words, given a concentration
gradient, the resulting mass flux is assumed to be independent of time.
In the limiting case of pure diffusion, this is necessarily not satisfied. In fact, the
diffusion equation admits only linear terms in the concentration profile under the
requirement of stationarity,
∂2 c
! ∂c
0=
= D 2.
(8.15)
∂t
∂z
On the other hand, a linear profile leads to j = −D ∂c/∂z ≠ 0 even at the bottom of the
tank which certainly violates physically sensible boundary conditions. Generally,
for pure diffusion, a concentration profile must evolve in time. Danckwerts (1951)
86
Water-sided profile parameters
a
b
8.2
c
Figure 8.11: Water-sided boundary layer scale parameter z ∗ over time for three different wind
settings, f wind = 5 Hz (a), 15 Hz (b), and 21 Hz (c). For the lowest wind speed, z ∗
increases in time suggesting non-stationary flux conditions.
gives a solution to the above equation, e.g. assuming a constantly saturated surface
√
and finds c ∝ erfc (z/2 Dt). In other words, the saturated layer grows in time and
the flux decreases.
It is only in turbulent conditions, i.e. upon adding a depth dependent term
Kdiff (z) to the diffusion equations, as explained in sec. 2.4.1, that a (quasi-) stationary
concentration profile is possible. Experimentally, there should be a transition from
purely diffusive to turbulent conditions.
This can be observed from the time evolution of the scale parameter z ∗ , shown
in Fig. 8.11. In (a), the wind speed is very low ( fwind = 5 Hz) causing minimal
shearing of the water body. The water-sided boundary layer scale z ∗ constantly
increases in time, characteristic for a diffusion dominated regime. Similarly, under
these conditions, there is no logarithmic sublayer in the air-flow yet (see sec. 8.1.3).
Panel (b) corresponds to turbulent condition in the air-sided flow ( fwind = 15 Hz) so
that water flow is expected to be turbulent to some degree due to the surface shear
exerted by the wind. For (c), fwind = 21 Hz, waves already reach slopes of ak ≈ 0.2 and
show some skewness (see sec. 5.3) suggesting that tracer is almost homogeneously
mixed into the deeper water bulk by turbulence. In good conceptual agreement, z ∗
grows only slightly in time in (b) and is constant in (c).
Similar complementary conclusions can be drawn from the surface concentration
cs , shown in Fig. 8.12, under modulated wind speeds (see sec. 6.2) alternating from
fwind = 5 Hz and 25 Hz. Acetone was injected at a constant rate of 2.2 × 10−3 mol/s.
For low wind speed, the surface concentration quickly increases while the bulk
concentration remains constant, implying a higher mass flux at the surface than in
the lower layers. Once the wind speed is increases, turbulence and possibly waveinduced coherent flow rapidly transports tracer into the bulk, thus re-setting the
surface concentration to a lower value.
The non-stationarity implies that averaged fluxes under modulated conditions not
only depend on the magnitude of the parameters used to characterize flow conditions,
but also on the modulation frequency T −1 . In the simple example above, assuming
constant bulk concentrations in either phase, one can assign two characteristic
87
Chapter 8
RESULTS
Figure 8.12
Evolution of water-sided
surface (red) and bulk
(blue) concentration under oscillating wind conditions. Gray shaded bars
correspond to high wind
( f wind = 25 Hz) and white
bars to low wind ( f wind =
5 Hz).
fluxes, j1 and j2 , to the two wind speeds. If both conditions were stationary, the
average flux would just be the arithmetic mean j = ( j1 + j2 )/2, independent of T.
If instead condition 1 is non-stationary, the flux will decrease in time described by
some function f (t), with f (0) = 1. The average flux over a modulation period T is
then
j2
1 T/2
+ j1 ⋅
f (t ′ ) dt ′ .
(8.16)
2
T 0
In other words, the average flux does depend on the modulation frequency T −1 and
decreases as T increases.
The tacit assumption of stationarity should be carefully considered in field studies,
however, where variable wind and turbulence conditions are inevitable. It should be
stated for completeness here that the above considerations also hold if turbulence is
not induced by wind. Also, the reasoning is very similar to the concept of the surface
renewal model by Danckwerts (1951), outlined in sec. 2.4, but conceptually different.
The surface renewal model is an attempt to describe turbulent mixing into the water
bulk based on the statistical intermittent nature of turbulence. The time scales
involved are short compared to other time scales of the transport phenomenon and
the result is a statistically averaged, observable transfer velocity and concentration
profile. Here, the modulation period T is macroscopic and describes the transition
from one statistical condition to the other.
The effect of non-stationarity is also seen in the time evolution of resistance
partitioning, presented in sec. 8.4.
jT =
88
∫
Absolute concentrations and full profiles
8.3
8.3 Absolute concentrations and full profiles
In sec. 7.1.4 and sec. 7.2.4, I explained how measured concentration profiles could
be converted into absolute concentrations. As water and air-sided images were
recorded simultaneously, this offers the unique possibility to match profiles across
the air-water interface and to obtain timely information on the local surface-near
concentrations during a gas invasion experiment.
8.3.1 Proof-of-principle
To assess the consistency of the absolute concentrations in the water, I compared the
integrated amount of tracer with the known injection rate. In well-mixed conditions,
the bulk concentration cb determined from the water-sided profiles provides an
estimate of the average concentration in the water compartment of the wind-wave
facility. With the known volume of 23 l, the total amount N in mol at a given time
t is simply N = cb ⋅ Vwater . On the other hand, for a constant injection rate Ṅinj of
2.2 × 10−3 mol/s, the injected amount of acetone is Ninj = 2.2 × 10−3 mol/s ⋅ t. The
injection rate was known to about 10 % precision (see sec. 6.1.1).
In Fig. 8.13, this is shown for six exemplary wind speeds. The shaded cone indicates
the approximate injected amount. The blue line reflects the amount of acetone in the
water and the black curve the amount in the air (with respect to the right ordinate).
For conditions (c)-(f), measured and injected amount coincide reasonably well
with the given uncertainties. For low wind speed in (a) and (b), the concentrations
deduced from the profiles overestimate the injected amount. This is to be attributed
to the non-stationary nature of the mass transport at these low wind speeds, discussed
in sec. 8.2.2. In those conditions, the tracer is not perfectly mixed throughout the full
water body and concentrations are systematically higher closer to the surface in the
region observed by the camera. Indeed, after injection is turned off, the estimated
amount starts to drop towards the attended value as tracer is slowly mixed into the
deeper bulk water. Even at higher wind speeds, a slight decrease is still visible.
A proof-of-principle of the conversion to absolute concentrations for the air-side
is given in Fig. 8.14. Here, the cbwater was divided by the solubility αH to yield airequivalent value. In equilibrium, both should match which is satisfied to about
10 %. Considering the uncertainty on the solubility of acetone of similar order (see
sec. 3.1.4), this seems acceptable.
89
Chapter 8
90
RESULTS
a
b
c
d
e
f
Figure 8.13: Time evolution of total amount of acetone in the air and water compartment. Calculated from bulk concentrations by multiplying with respective volume, N = c ⋅ V .
The green-shaded cone indicates the total amount estimated from the approximate
acetone influx as N = j ⋅ t. For f wind =5 Hz (a) and f wind =8 Hz (b), the amount of tracer
in the water increases faster than expected, indicating that transport is not entirely
homogeneous in depth so that tracer accumulates faster in the 20 mm beneath the
surface observed by the camera than it is transported into the deeper bulk. The
other four figures are for 11 Hz (c), 15 Hz (d), 18 Hz (e), and 25 Hz (f ).
Absolute concentrations and full profiles
a
8.3
b
1e 3
1e 3
cbw
cba
1.2
1.0
Concentration in mol/L
Concentration in mol/L
1.0
0.8
0.6
0.4
0.8
0.6
0.4
0.2
0.2
0.0
cbw
cba
500
1000
1500
Time in s
2000
2500
0.0
500
1000
1500
Time in s
2000
2500
Figure 8.14: Two exemplary plots of air (red) and water-sided (blue) bulk concentrations over time.
The water-sided values are divided by the solubility α H to convert to air-equivalent
concentrations. In equilibrium, both values match to about 10 %, validating the
procedure to convert to absolute concentrations.
8.3.2 Matched water and air-sided profiles
By converting concentrations into absolute values and by scaling the water-sided profiles with the solubility αH , full profiles across the air-water interface were obtained.
This is shown in Fig. 8.15 for six different wind speeds. Gray dots are air-sided data
points, the blue curve is the exponential model profile with experimentally determined parameters, and the red curves represent Deacon’s model fitted according
to the procedure described in sec. 8.1.4. For fwind = 5 Hz, the gradient is depicted
instead, as at this low wind speed, no logarithmic sublayer exists and Deacon’s profile
does not describe the data (see sec. 8.1.3). The profiles are obtained by averaging data
from the first three minutes of an invasion experiment.
Having in mind that concentration profiles are essentially resistance distributions,
it can immediately be asserted that for acetone, both phases contribute to similar
proportions to the overall resistance of the air-water boundary. This is in accordance
with the estimates presented in sec. 2.5 and thereby explicitly confirms the concept of
resistance partitioning across the interface. It should be noted that the reconstructed
surface concentrations do not exactly match as would be expected when assuming
instantaneous thermodynamic equilibrium at the phase boundary. One reason could
be a systematic underestimation of the water-sided surface concentration, possibly
due to insufficient optical resolution. Also, the air-sided profiles are intrinsically
biased close to the surface, as explained in sec. 8.1.7. The fitted Deacon profiles
compensate for this by extrapolating into the diffusive sublayer.
On the other hand, the discontinuity might have a physical reason. In particular,
it might suggest that an additional interface resistance has to be taken into account. I
will discuss this in detail in sec. 9.3. It should be noted that the here depicted profiles
91
Chapter 8
RESULTS
a
b
c
d
e
f
Figure 8.15: Full air-water concentration profiles for different wind speeds normalized to the
air-sided bulk concentration. Air-sided data points are shown as gray dots. The red
line is the fitted Deacon profile; the blue line the reconstructed water profile. Green
dashed lines indicate characteristic scales in the air-sided boundary layer. Turbine
frequencies are (a)-(f ): 5, 8, 11, 15, 18, 21 Hz.
92
Absolute concentrations and full profiles
8.4
are normalized to the air-sided bulk concentration. Therefore, they only contain
information on the relative resistance partitioning but not on the overall absolute
resistance. This is instead presented in sec. 8.6.
8.3.3 Matched air and water-sided concentrations
The full matched profiles provide a unique momentary view of the tracer distribution
across the air-water interface. The time evolution of absolute tracer concentrations
in the bulk and at the surface is shown in Fig. 8.16 for six different wind speeds. For
each data point in these time series, 20 subsequent profiles corresponding to 10 s
were averaged. Water-sided bulk and surface concentration as well as air-sided bulk
concentration were readily obtained from the corresponding parameters and profiles,
respectively. The air-sided surface value is increasingly biased during an invasion
experiment (see sec. 8.1.7). It was therefore reconstructed in the following three
different ways. For the red data points, Deacon’s model was fitted to the averaged
profiles and cs was determined as described in sec. 8.1.4. For the black and green
lines, the scaling properties of the air-sided turbulent boundary layer were exploited.
In sec. 8.1.6, I have shown that the relative resistances and concentrations at specific
points within the boundary layer are in fixed relation to each other characteristic
for the wind speed. For example according to Fig. 8.6, at u∗ = 13 cm/s, c(z =
0.15δ) − cs = 0.8(cb − cs ) so that cs = (c(0.15δ)−0.8cb )/0.2, and analogously for c(z11 ).
Since the concentration profiles remain less affected by the water-sided brightness at
z = 0.15δ, especially c(z = 0.15δ) provides a relatively robust estimate of the surface
concentration cs . This latter approach relies on the scaling properties of the turbulent
boundary layer in a similar way as the fit routine used in sec. 5.1 to extract u∗ from
wind profiles.
Figure 8.16 shows that all three methods yield consistent results compared among
each other. For higher wind speeds, the value retrieved from c(z11 ) is overestimated
due to the brightness induced bias (see sec. 8.1.7). When tracer injection is turned off,
the air-sided signal quickly starts to decay. This portion is used to calculate the total
transfer resistance from mass balance considerations (see sec. 7.4). The air-sided
bulk concentration assumes a roughly constant plateau value characterized by the
balance of acetone influx and transport into the water. Consequently, this steady
state concentration is lower for higher wind speeds, i.e. faster transport into the
water. The precise value varies in time as the acetone filling in the bubbling flask
(see sec. 6.1.1) decreases causing the injected air to be less saturated. The air-sided
surface concentration during the exponential decay lies above the bulk values. This
is because the fluorescent brightness quickly becomes dimmer while the water-sided
fluorescence remains unchanged causing the bias to be relatively higher. As already
seen in the profiles in sec. 8.3.2 and discussed there, csa and csw do not match. Generally,
csw is lower for increasing wind speed possibly due to a more efficient transport of
acetone away from the water surface.
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Chapter 8
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a
b
c
d
e
f
Figure 8.16: Absolute concentrations over time for different wind speeds. Water-sided values are
divided by solubility α H . Air-sided surface concentrations (red, green, black) were
determined by three different methods (see text). Turbine frequencies are (a)-(f ): 5,
8, 11, 15, 18, 21 Hz.
94
Partitioning of resistances
8.5
8.4 Partitioning of resistances
From the simultaneously measured absolute concentrations, the resistance partitioning across the air-water interface can be calculated. The individual contributions are
given by
Rw cswater − cbwater
= air water
Rtot
cb − cb
cbair − csair
Ra
=
Rtot cbair − cbwater
(8.17)
(8.18)
The value of csair was determined by the three methods described in the previous section. Figure 8.17 shows the relative resistances over time for different wind
speeds under continuous acetone injection. The three air-sided contributions agree
well, again underlining the validity of the approach. For higher wind speeds, Ra
determined from the concentration at z + = 11 is affected by water-sided brightness.
For the two lowest wind speeds, the effect of non-stationarity, discussed in sec. 8.2.2,
is visible. Initially, the air-sided resistance is larger, but in time the water-sided
contribution grows as tracer accumulates near the surface and the flux ceases. Again,
for higher wind speeds the resistances do not add up to one as expected suggesting
either methodological systematics, explained in sec. 8.3.2, or an additional interface
resistance, discussed in 9.3.
8.5 Surfactants
Identical invasion experiments were performed with Triton-X added to the water,
as explained in sec. 6.3, to investigate the effect of surfactants on gas exchange and
the partitioning of resistances. Figure 8.18 shows full, continuous concentration
profiles across the air-water interface generated in the same way as for clean water.
Evidently, the air-sided profiles are much less pronounced compared to their watersided counterparts. In other words, the water-sided resistance is comparatively
larger with Triton-X added to the water than in the clean case. This effect has already
been observed in other studies, e.g. by Kräuter (2011) and Mesarchaki et al. (2014).
Interestingly, the resistance is greater not only under energetic wind conditions,
associated with the formation of waves in clean water (Fig. 8.18 d, e, f), but also for
lower wind speeds. This suggests that the suppression of waves is not the principal
mechanism by which surfactants affect the transfer velocities in this experiment.
It seems instead more plausible that near surface turbulence is damped by additional surface stress associated with gradients in the surface tension ∂σ/∂x , as proposed
by Davies (1966). In this way, surfactants modify the dynamic tangential boundary
conditions, as I have explained in sec. 2.2.3. McKenna and McGillis (2004) have
95
Chapter 8
RESULTS
a
b
c
d
e
f
Figure 8.17: Relative contributions to the overall resistance of the air-water interface. Air-sided
values (red, green, black) were determined by three different methods (see text). Note
that in (a) the water-sided relative resistance grows in time due to non-stationarity
(see sec. 8.2.2). Turbine frequencies are (a)-(f ): 5, 8, 11, 15, 18, 21 Hz.
96
Surfactants
a
b
c
d
e
f
8.5
Figure 8.18: Full air-water concentration profiles for different wind speeds and Triton-X added
to the water normalized to the air-sided bulk concentration. Air-sided data points
are shown as gray dots. The red line is the fitted Deacon profile; the blue line the
reconstructed water profile. Green dashed lines indicate characteristic scales in the
air-sided boundary layer. Turbine frequencies are (a)-(f ): 8, 11, 15, 18, 21, 25 Hz.
97
Chapter 8
RESULTS
a
b
Figure 8.19: (a) Water-sided boundary layer scale parameter z ∗ in mm over time for clean water
(blue) and Triton-X added (red). Note the stronger increase in time suggesting
that non-stationarity is more pronounced with Triton-X. (b) Vertical distribution of
schlieren, serving as proxy for turbulent eddies, relative to the water surface for
clean water (solid) and with Triton-X (dashed).
shown experimentally that the presence of surfactants is correlated with a vanishing
of surface velocity divergences. In consequence, the air-water interface resembles
a rigid wall rather than a free surface, as in the clean water case. Tsai (1996) finds
from the results of direct numerical simulations that the apparent viscous sublayer
becomes thicker in the presence of surfactants and that vortices are less connected
to the interface, as it frequently occurs in free turbulent flow.
In this work, no explicit flow visualization was performed. Some insight can
anyhow be gained from the following observations. In Fig. 8.19 (a), a comparison
of the water-sided boundary layer scale z ∗ as a function of time is shown for clean
water (blue) and water with additional Triton-X (red), both at fwind = 15 Hz. For
the latter case, the overall thickness is larger by about a factor of 2 which can be
attributed to Tsai’s observation of a wider viscous sublayer.
In sec. 7.3, I have explained how information can be gained from the water-sided
images about the probability to observe streaks in a certain relative depth underneath
the water surface. As such streaks reflect locally enhanced concentration fields swept
away from the surface by the turbulent flow, statistically they should reflect the
occurrence of eddies at a given distance from the interface. In Fig. 8.19 (b), the streak
distribution is shown for different wind speeds with clean water (solid) and with
Triton-X added (dashed), generated from 400 water-sided images, corresponding to
200 s. Evidently, the minimal approach distance of turbulent eddies is much larger
with surfactants, supporting the simulation results of Tsai (1996). In both cases,
eddies come closer to the surface with increasing wind speed.
Concluding, Triton-X leads to a thicker water-sided boundary layer and the
suppression of turbulent eddies close to the water surface. I will discuss these aspects
98
Summary: Measured transfer resistances
8.6
Figure 8.20
Exponential decay over time of airsided bulk concentration for various
wind speeds and surface conditions,
normalized to c(t = 0). The straight
lines indicate fit results according
to the mass balance method explained in sec. 7.4.
in more detail in sec. 9.2.
8.6 Summary: Measured transfer resistances
The visualization experiments allowed to determine absolute and relative transfer
resistances in various ways, as described in the previous sections of this chapter. Here,
I summarize and compare the results and comment on their possible implications
in chap. 9. The reported resistances are based on data averaged over 10 min of an
invasion experiment under roughly constant acetone injection. It should be kept in
mind that transfer velocities can be easily determined as the inverse, k = R −1 .
In Fig. 8.21, I present a comparison of (partial) resistances as a function of airsided friction velocity u∗ for clean water (a) and with Triton-X added (b). Numerical
values are given for concentrations measured in the gas phase (air-equivalent, see
sec. 2.5). As explained in sec. 2.3, air and water-sided resistances can be summed to
supposedly yield the total resistance. They are therefore shown as stacked columns.
Groups of columns are aligned with the abscissa according to their left edge.
The gray bars represent values obtained from the exponential brightness decay after
acetone injection was turned off according to the mass balance method explained
in sec. 7.4. They reflect the total average resistance of the air-water boundary. The
exponential decay of concentrations is shown in Fig. 8.20. The volume of the air
compartment of the wind-wave tank enters the mass balance fit as parameter. It sets
the accuracy of the resistances to about 5 %
Red bars show the air-sided resistance derived by fitting Deacon’s model to the
measured concentration profiles, as explained in sec. 8.1.4. According to eq. (8.9), D,
ν and u∗ directly enter the formula as parameters, resulting in about 5 % uncertainty.
Another 5 % uncertainty is estimated to derive from the fit procedure itself. In
this context, it should also be noted that the Deacon profile assumes the diffusive
sublayer to exist just like above a smooth wall. At higher wind speeds, the observed
concentration profiles appeared however flatter than the model prediction sec. 8.1.3.
99
Chapter 8
RESULTS
a
b
Figure 8.21: Partial and total transfer resistances determined by different methods (see text) as
a function of friction velocity u∗ . Groups of columns are aligned with the abscissa
according to their left edge. (a) clean water and (b) with Triton-X added.
If this is not due to biased concentrations, the air-sided resistances are expected to be
overestimated by 10 to 20 %. Based on the data, an affirmative or negative conclusion
is not possible.
Yellow and blue bars depict the air and water-sided contribution, respectively,
obtained as proportion of the mass balance value according to the partitioning described in sec. 8.4. Specifically, the partial contributions, depicted in Fig. 8.17, were
multiplied with the total average resistance. For the lower wind speeds, where effects
of non-stationarity had been observed, temporal average values were used. Uncertainties are difficult to estimate as they mainly depend on the unknown systematic
optical errors.
Finally, the scale parameter z ∗ was used to provide an independent estimate of
the water-sided transfer resistance. To that end, mass flux was assumed to be entirely
diffusive at the surface, so that j = −D ∂c/∂z at z = 0 (cf. sec. 2.4). For the exponential
profile used as model distribution, this yields Rw = z∗/D and is shown as green bars in
Fig. 8.21. At low wind speeds, effects of non-stationarity were observed, causing z ∗
to increase in time. Again, the temporal average was used. The resistances derived
from z ∗ are seemingly overestimated in the case of clean water. This indicates that
the exponential model might not accurately reflect surface concentration gradient.
Also systematic errors due to limited optical resolution cannot fully be excluded
despite the adopted correction procedures. These resistances should be regarded as
references only.
As expected, resistances strongly decay with increasing wind speed. The turbulent
diffusivity model predicts an inverse proportionality to the friction velocity u∗ . To
quantify the relationship, I fitted the following reciprocal expression to Rtot exploiting
the readily available parameter (u∗ ):
100
Turbulent Schmidt number
Rtot (u∗ ) =
A
+ B.
u∗
8.7
(8.19)
The resistance for the lowest wind speed was excluded as the turbulent flow is not
fully developed. The fit result is shown as dashed line. The parameters are A = 34.77,
B = 0.066 s/cm for clean water, and A = 48.0, B = 0.414 s/cm for Triton-X. The
parameter B plays the role of an additive resistance as will be discussed in sec. 9.3.
With Triton-X, resistances in general are about 30 % higher than in clean water. Notably, the air-sided resistances determined by fitting Deacon’s model are
almost identical. This is certainly expected, at least for smooth water surface, as the
turbulence in the water should not affect the air flow. In other words, surfactants
influence the water-sided mass transport only. Additional observations regarding the
underlying mechanism have been presented in sec. 8.5. There is some controversy
about the conceptual impact of surfactants on gas exchange and about adequate
parameterizations. I discuss this aspect in more detail in sec. 9.2.
Not all sums of and water-sided resistance yield the full total value recovered
from the mass balance calculation. This is related to the non-matching surface
concentrations described in sec. 8.3.2. If experimental systematics cannot fully
account for this discrepancy, it suggests that an additional transfer resistance has to
be considered equivalent in magnitude to the observed difference. This hypothesis
will be discussed in sec. 9.3.
8.7 Turbulent Schmidt number
One objective of this thesis was to experimentally investigate the commonly assumed
analogy between momentum and mass transport through the turbulent air boundary
layer. As explained in sec. 2.4.1, this is nurtured by the near equality of kinematic
viscosity and diffusion constant for many gases.
The measured air-sided acetone profiles show indeed to good precision the same
scaling properties as wind profiles for the given friction velocities. In particular, the
model of turbulent diffusivity by Deacon (1977) proved to describe the data well up
to intermediate wind speeds. This also implies that rigid wall boundary conditions,
which are at the base of this parametrization, are a correct description of the water
surface. It should be noted, however, that for testing the analogy I assumed a-priori
the turbulent Schmidt number Scturb. to be one (see sec. 2.4.1).
For Scturb. ≠ 1, the turbulent diffusivity within the logarithmic sublayer would read
(see eq. (2.19))
Kdiff (z + )/ν = Scturb. −1 κz + ,
(8.20)
and consequently the additional factor would appear in the logarithmic scaling in
eq. (8.5) as
c̄ + (z) = Scturb. −1 A log(z) + B.
(8.21)
101
Chapter 8
RESULTS
It seems tempting to explicitly determine the value of Scturb. from the air-sided
concentration profiles. However, the prefactor A also contains the acetone flux,
or equivalently, the resistance. Without an independent measurement, Scturb. can
therefore not be retrieved.
A possible approach is to compare resistances calculated with the mass balance
m.b.
Deacon
method, Rtot
and by fitting Deacon’s profile to the air-sided data Rtot
. If fetch
dependence is not too strong, both methods should yield similar results, i.e. their
ratio should be one. At most, the profile-derived resistances should be somewhat
smaller given their local character and the general tendency that turbulent mixing
becomes stronger with increasing fetch (at least not the contrary).
I explained the procedure of fitting Deacon’s profile to the data in sec. 8.1.4. The
air-sided resistance was retrieved from the two fit parameters according to eq. (8.9),
which I restate here
1 − γ2
Ra =
ν.
(8.22)
Du∗ γ1
By the partitioning across the air-water interface (see sec. 8.4), the total resistance
can be obtained thereof as
Deacon
Rtot
cbair − cbwater 1 − γ2
1 − cbwater/cbair
ν=
ν,
= air air ⋅
cb − cs
Du∗ γ1
Du∗ γ1
(8.23)
with γ2 = csair/cbair .
Since in the initial phase of an invasion experiment the water-sided bulk concentration is still low, an approximative version of the above expression can be obtained
by neglecting cbwater . Equation (8.23) reduces to
Deacon,app.
Rtot
=
ν
.
Du∗ γ1
(8.24)
This does not depend on the water-sided concentration and thereby reduces systematic errors.
Figure 8.22 (a) shows a comparison of total resistances, non-dimensionalized by
u∗ , obtained from eq. (8.23) and eq. (8.24). In particular for higher winds, where the
turbulent contribution to diffusivity becomes important, the values derived from a
model fit are systematically higher than the mass balance results. This trend is seen
both, for the exact and the approximative formulæexcluding systematic errors in csair
to be at the root of this discrepancy. Following the above argumentation, it should
rather be attributed to a non-zero turbulent Schmidt number.
For clarity, I restate the integral eq. (8.2) explicitly including Scturb. ,
js ν
c(z) − cs =
⋅
Du∗
∫
0
z+
D
D + Scturb. −1 Kdiff (z + )
dz + .
(8.25)
Outside the diffusive sublayer, i.e. z + ≫ 11, molecular diffusion is negligible compared to turbulent mixing so that D can be omitted from the denominator and Scturb.
102
Turbulent Schmidt number
a
8.7
b
Figure 8.22: (a) Total dimensionless resistance R tot u∗ , obtain from mass balance fit (black squares),
by scaling up air-sided resistances obtained from precise Deacon fit (red upwards
triangles) and by approximating c bw ≈ 0 (blue downwards triangles). (b) Estimate of
turbulent Schmidt number Scturb. obtained as ratio of resistances from mass balance
and from air-sided profiles. The dashed lines indicate the average for both datasets.
can be pulled out of the integral. Consequently, the turbulent Schmidt number
m.b.
Deacon
appears as additional factor in eq. (8.22). Therefore, the ratio of Rtot
and Rtot
,
where for the latter Scturb. = 1 had been used, should yield an estimate of Scturb. . This
Deacon
is shown in Fig. 8.22 (b) for the exact and approximative form of Rtot
. The plot
does not imply any functional dependence on friction velocity, but u∗ should simply
be regarded as data index. Here, all values for clean water and with surfactants are
used. The dashed lines indicate the average ratio, i.e. the best estimate of Scturb. ,
which is about 0.7. This is in agreement with the value mentioned by Deacon (1977).
Scatter is considerable, but as Monin and Yaglom (2007) point out, experimental data
on concentration profiles of passive admixture (including heat) close to a boundary
are very rare and generally poor. The here obtained estimate should therefore be
considered as rather valuable.
103
9
Discussion
Some of the experimental findings have already been discussed in the previous
chapter. Here, I concentrate on three aspects which emerge from the experimental
results of this work and which deserve particular attention.
9.1 Diffusive sublayer and surface roughness
Uncertainty has arisen regarding the precise shape of the diffusive sublayer for higher
wind speeds. The experimental data suggest a flatter decay of acetone profiles than
expected from the turbulent diffusivity model (see sec. 8.1.3). I have already discussed
the influence of water-sided fluorescent brightness on the air-sided signal and this
undoubtedly limits interpretation of data close to the surface. Nonetheless, it is
worthwhile considering a possible physical reason for the observed discrepancy.
Indeed, the turbulent diffusivity model inherits the assumption of smoothness of the
boundary from the wall flow in hydrodynamics. However, wave slope measurements
have shown that for the higher wind speeds in question, considerable wave steepness
of ak ≈ 0.2 occurs so that roughness may be a relevant factor influencing the near
surface tracer distribution. In Fig. 9.1, I compare an estimate of the aerodynamic
roughness, taken as three times the standard deviation of the wave amplitudes,
with the viscous length. Intuitively, if roughness elements protrude throughout the
viscous, or diffusive, sublayer they should potentially alter it. Schlichting (2000)
gives as criterion for intermediate roughness 4ν/u∗ < h0 < 60ν/u∗ for a flow over a
fixed rough surface. This is met for higher wind speeds in the wind-wave tank. This
treatment assumes the rough surface too be fixed.
Sullivan et al. (2000) performed DNS of turbulent airflow over an idealized sinusoidal water wave with steepness ak = 0.1 and ak = 0.2. A water wave constitutes
105
Chapter 9
DISCUSSION
Figure 9.1
Comparison of roughness height
h 0 , obtained as three times the
standard deviation of the wave amplitudes, with multiples of viscous
length. According to Schlichting
(2000), 4ν/u∗ < h 0 < 4ν/u∗ corresponds to an intermediate roughness regime.
a complicated system in that the roughness elements (the waves) move with phase
velocity c while the no-slip condition matches the air-flow to the fluid surface velocity,
dictated by the orbital motion underneath the waves. It is for this reason that the
authors parametrize the relative motion of the rough boundary by the so-called
wave-age, defined here as c/u∗ . Interestingly, the viscous sublayer virtually disappears
in the simulated wind profiles for ak = 0.2 and small, but none-zero wave age. The
smallest value tested by the authors is c/u∗ = 3.9, roughly corresponding to the value
estimate that I obtained for the wind-wave tank (see sec. 5.2). Although results of
DNS generally have to be considered with care, the possibility should not be precociously rejected that roughness due to waves have caused the observed alteration of
concentration profiles for higher wind speeds on scales of the diffusive sublayer. For
a decisive conclusion on this question, dedicated flow visualization experiments, e.g.
PIV, are probably more suited than the here presented visualization technique.
A precise knowledge of the flow structure above small scale water waves would be
of interest as this impacts the momentum dissipation, thus the dynamic interaction
of the wind and wave field. This was not the scope of this thesis however. In terms
of mass transfer resistance on the other hand, the possibly altered structure of the
diffusive sublayer over waves is of minor significance for the following reason. The
length of 120 cm of the wind-wave tank used in this work is certainly short compared
to scales relevant to typical situations on the open ocean. It is also known from
boundary layer theory (e.g. Schlichting, 2000) that the overall width δ of the turbulent
boundary grows monotonously with fetch. In view of the resistance distribution
within the boundary layer, verified from the experimental results, it can be concluded
that the diffusive contribution to the air-sided resistance does not play any significant
role. In other words, in a field measurement it would be sufficient to probe the
concentration profile within the logarithmic sublayer.
106
Surfactants and boundary conditions
9.2
9.2 Surfactants and boundary conditions
Simultaneously to concentration profiles in the air, the water-sided acetone distribution was visualized. While concentration fields were not explicitly captured,
parameters describing a model profile yielded some valuable insight into the coupled
mass transport through the air-water interface. Conversion to absolute concentrations allowed to match profiles at the phase boundary. While the water side alone
has been observed, e.g. by Friedl (2013); Herlina and Jirka (2008); Janzen et al. (2010);
Jirka et al. (2010); Schulz and Janzen (2009); Woodrow and Duke (2001), to my best
knowledge this is the first time that both phases were simultaneously investigated
bringing about some interesting results.
For clean water, air and water-sided boundary layer contribute to roughly equal
proportions to the overall resistance as expected from the solubility of acetone of
αH ≈ 800 (see sec. 2.5). These results explicitly confirm the conceptual model by
Liss and Slater (1974) that resistances are partitioned across the phase boundary
according to the solubility. As a tendency, the aqueous component decays more
rapidly with increasing wind speed than the gaseous which is in accordance to
recent mass balance based observations by Mesarchaki et al. (2014) and Kräuter
(2011). Furthermore, the experiments with surfactants added to the water (Triton-X)
showed a considerable increase of the water-sided resistance only, while the air-sided
value, as expected, remained basically constant. This itself is no surprising discovery,
obviously. Significant is the observation that surfactants affect the mass transport at
any wind speed and in particular in the flat water as well as wavy regime.
Indeed, there has been disagreement (and a clear full picture is maybe still lacking) on the mechanism by which surfactants suppress gas exchange. Experimental
investigations by Jähne (1985) Broecker et al. (1978) have demonstrated the enhancing
effect of waves on gas exchange. Motivated by this supposition, Frew et al. (2004)
have suggested that it is the dynamic damping of waves due to surfactants which
leads to an increase of the resistance. Consequently, surfactants should attenuate
gas exchange only in a regime in which waves would normally develop in clean
water. This is, however, incompatible with the findings in this work (see sec. 8.5). A
more consistent description is provided based on the so-called eddy-renewal model,
initially proposed by Fortescue and Pearson (1967) and Lamont and Scott (1970) and
elaborated by Csanady (1990). The conceptual idea is that turbulent eddies reach a
region close to the aqueous diffusive sublayer and locally transport mass into the
deeper bulk water. The sublayer is thereby momentarily thinned, leading locally to
steeper surface gradients and higher fluxes. Subsequently, the diffusive sublayer is
replenished from the air phase. The intermittent nature of this process motivates to
picture it as random surface renewal event. It should be noted, however, that the
early conceptual description of Danckwerts (1951), while sharing the name, does not
provide a concrete renewal mechanism and is in that sense less complete. Inevitably,
locally downwards pointing velocity due to an eddy is related to non-vanishing diver-
107
Chapter 9
DISCUSSION
gence and vorticity close to the surface, whose amplitude and direction is influenced
by the boundary conditions. This is explained in detail in Csanady (1990).
Two specific cases are the free surface and rigid wall (see sec. 2.2.3). In short, the
former allows eddies to reach closer to the air-water interface thereby enhancing
the scalar transport. A surface active substance adds an internal tangential stress
to the otherwise stress-free clean surface, the so-called Marangoni stress τ = ∂σ/∂x .
Here, σ is the surface tension. If surfactants are abundant, the internal stresses can
virtually immobilize the water surface, like a rigid wall. This is the mechanism which
Csanady (1990) proposes.
McKenna and McGillis (2004) have performed PIV based visualization experiments in a grid stirred tank and found marked surface divergence and vorticity,
even in the absence of waves. With surfactants added to the water, the divergences
(almost) vanished and transfer velocities, measured in parallel, were significantly
lower.
In my experiments, although not specifically designed for flow visualization, I
could show that Triton-X leads to a systematic shift of the eddy depth distribution
indicating an increase of the minimal distance to the water surface at which eddies
occur (see sec. 8.5). This is in conceptual accordance with Csanady’s model and
the results of McKenna and McGillis (2004). It also agrees with the theoretical
considerations of Davies (1966). In short, surfactants reduce surface mobility, thus
causing a change of boundary conditions which suppress surface divergence and
reduce the eddy renewal. As a consequence, transfer resistance increases.
It should be noted that Csanady’s divergence model based on stagnation point
flow yields a transfer resistance dependent on the Schmidt number as
R ∝ Scn ,
(9.1)
with n = 1/2 for a free surface (clean water) and n = 2/3 for a rigid wall (strong
surfactant) effect. The former corresponds to the prediction of surface renewal
models (Danckwerts, 1951; Fortescue and Pearson, 1967, e.g.) and the later well reflects
results of laboratory measurements of flow over solid surfaces (Deacon, 1981).
Depending on concentration and effectiveness of a surface active substance, the
resemblance to the rigid wall case might be only approximate. To this end, McKenna
and McGillis (2004) use Csanady’s model to provide a formula which interpolates
the Schmidt number exponent between the free and the rigid wall case based on the
magnitude of surface stress.
It should be pointed out that the choice of n is related to the boundary condition
and in this sense suited to account for effects of surfactants. On the other hand,
it is conceptually independent of the geometric shape the water surface. In other
words, a clean water surface is anyhow free in the above sense, be it smooth or wavy.
From this point of view and from the above considerations, no change in Schmidt
number exponent should be expected at the transition to a wavy regime in clean
water. Laboratory measurements in wind flumes may easily lead to this erroneous
108
Additional interface resistance - a hypothesis
9.3
conclusion, however, for the following reason. As Jähne et al. (1984) point out,
contamination of wind-wave tanks with surface active material is a frequent issue
in experimental routine. When measuring at low wind speeds, the anyhow smooth
water surface makes it difficult to identify such a contamination. At stronger winds,
straining forces will eventually disrupt the surfactant film and possibly wash part of
it into the deeper water. In the latter case, the effectiveness of the contamination will
also depend on the re-adsorption rate at the surface. In any way, the water surface
becomes increasingly “free”. Boundary conditions then allow for stronger surface
divergence and eddy renewal, leading to enhanced transfer rates. The Schmidt
number exponent is observed to shift from 2/3 to 1/2. It is then tempting to deduce
a conceptual relationship with the onset of waves, while these merely cause the
unobserved change of boundary conditions.
9.3 Additional interface resistance - a hypothesis
The full air-water concentration profiles showed a discontinuity at the water surface,
particularly with clean water and for higher winds (see sec. 8.3.2). This is contrary
to the assumption that at the surface concentrations obey Henry’s law (see sec. 2.5).
Comments regarding experimental systematics have already been made in the previous chapter. The water-sided surface concentration might be underestimated as
an effect of optical aberration in the images, as discussed in sec. 7.2.5. This is in
principle corrected for in the data processing routine. In the air, the brightness bias
leads to an overestimation of the observed surface concentration. Likewise, fitting
the Deacon profile corrects for this by effectively extrapolating the diffusive sublayer.
If waves cause indeed alterations of the air-sided concentration profile, as discussed
in sec. 9.1, the profile discontinuity would still be larger.
Therefore, it seems appropriate to provide a hypothesis on possible physical effects
explaining the observations. If concentration profiles reflect the vertical distribution
of transfer resistance (see sec. 2.3) then a gap at the surface suggests that an additional
resistance has to be attributed to the interface itself.
In models describing the uptake of substances and aerosol particles by droplets,
e.g. in clouds or fog, such an interface term is indeed included. A recent review
on the matter can be found in Davidovits et al. (2006). It should be noted though
that a small droplet is much different from the open water in terms of turbulence.
Woolf (2002) has already speculated on the relevance of an interface resistance for
gas exchange at the sea surface and more detailed explanations can be found therein.
I will shortly outline the possible mechanisms and their implications.
Two effects can be identified regarding the surface kinetics. First, a gas molecule
striking the water needs to be geometrically incorporated into the uppermost molecule
layers. According to Pollack (1991), the opening of a surface cavity to accommodate
a monomolecular gas requires energy, typically 30 times greater than the kinetic
109
Chapter 9
DISCUSSION
energy at room temperature. This leads to an inhibited adsorption¹. For larger
molecules such as acetone this mechanism is presumably even more pronounced.
As a consequence, the effective adsorption rate dn/ dtis smaller than expected from
ideal gas theory by a factor γ, i.e.
p
dn
= γ√
,
dt
2πmkT
(9.2)
with m the mass of one molecule, p the pressure, T the temperature, and k the
Boltzmann constant. Such a formulation is used, e.g. in Knipping et al. (2000). The
factor γ is often termed mass accommodation coefficient in literature. Pollack (1991)
conceptually relates it to the work W required for opening the surface cavity as
γ = exp (−W/kT ), where W however remains to be determined. In short, the mass
accommodation coefficient cannot easily be deduced from first principles but needs
to be determined experimentally.
The second inhibiting effect regards the mobility of a molecule very close to the
interface. Woolf argues that diffusion might be different compared to the free liquid
due to altered geometry and a change in intermolecular forces. The spectroscopic
study by Tarbuck and Richmond (2006) provides evidence for modified relative
orientation of molecules and the formation of complex structures in the presence of
SO2 and CO2 . Venables and Schmuttenmaer (2000) report changes in the hydrogen
bonding environment in mixtures of water and acetone based on spectroscopic
data and molecular dynamics simulations. This active field of research is certainly
too ample to be discussed here, but in essence dynamical effects in the liquid on
molecular scales close to the surface might be of importance.
Conceptually, both above explained mechanisms motivate the inclusion of a
resistance to account for interface processes, specifically
Rtot = Ra +
1
Rw + RIF .
αH
(9.3)
Krop (2003) performed a theoretical study in the framework of linear non-equilibrium thermodynamics on the relevance of an interface resistance for describing
transfer velocities. He applied his approach to data by Liss and Slater (1974) and
found good agreement.
According to eq. (9.3), the interface process becomes increasingly relevant with
enhanced transport in the two fluid compartments as forcing, e.g. by wind, increases
and for higher solubility. Woolf limits his analysis to sparingly soluble gases controlled by the water side and concludes that for αH = 1, γ must be of order 10−5 for
the interface resistance to be relevant in comparison.
He uses transfer velocities of sulfur hexafluoride and chlorinated volatiles mea¹Note that while describing a non-equlibrium process, mass accommodation is also related to
equilibrium concentration ratios (Pollack, 1991).
110
Additional interface resistance - a hypothesis
9.3
sured by Alaee et al. (1995) and estimates the order of magnitude of the interface
resistance by fitting the following expression,
k600 = (
−1
100
)
+
R
.
IF
u∗water
(9.4)
Note that data are scaled to Schmidt number 600, i.e. k600 = (Sc/600)1/2 k, and are
measured in the water. To convert them into air-equivalent values, the solubility αH
has to be multiplied (see sec. 2.5).
Depending on the dataset, Woolf finds values of 13.3 s/cm to 37 s/cm. In much
the same way, the offset B = 0.066 s/cm which I used in eq. (8.19) also assumes the
role of an interface resistance. Scaled to Schmidt number 600 and divided by αH ,
it is RIF = 50 s/cm. Considering that the acetone is of similar size as the volatiles
considered by Woolf, the agreement seems plausible.
In Fig. 8.21, a discrepancy between Rtot obtained from mass balance calculations
and the value determined as sum of air and water-sided resistances was observed.
If this difference is to be attributed to an interface resistance, it would suggest an
order of 0.3 s/cm, i.e. greater than the value B obtained as offset from averaged mass
balance calculations. I will comment on the comparability on local and averaged
measurements further down. In any case, an interface resistance might lie at the root
of the non-continuous concentration profiles.
It is worthwhile discussing briefly the relevance of an interface resistance for
laboratory and field experiments. At sufficiently high turbulent mixing, one expects
measured transfer velocities to level off like in the data considered by Woolf. As
conditions become more energetic, an interface resistance implies an upper limit on
transfer rates. Certainly, wave breaking will also set in, giving rise to an enhancement
of mass transport.
In the intermediate regime, a plateau should be visible when transfer velocities
are depicted relative to a forcing parameter such as U0 or u∗ . Some examples of
this kind can indeed be found, notably the studies by Asher and Wanninkhof (1995);
Donelan and Drennan (1995); Komori and Shimada (1995). For reference, I show
their results in Fig. 9.2.
The most striking results are those by Komori and Shimada (1995) in (a, solid
circles). There is a clear plateau followed by a fast incline at U0 ≈ 10 m/s associated
with wave breaking. In the data by Asher and Wanninkhof (1995) in (b), white
capping sets in at about U0 ≈ 13 m/s and the authors depict a model (solid line)
supposedly accounting for it. Nonetheless, it can not be negated that data would also
allow to suggest a leveling off, as qualitatively indicated by the red line. Lastly, also
the plot (c, open circles) from Donelan and Drennan (1995) shows a similar behavior.
At the same time, other experimental findings do not suggest such a plateau so
clearly. In this context, two observations should be made, similar to the argumentation presented in Komori and Shimada (1995). First, an interface resistance would
gain relevance under stronger forcing when resistances associated with the fluid
111
Chapter 9
DISCUSSION
a
b
c
Figure 9.2: Experimental data on transfer velocities. (a) by Komori and Shimada (1995) for CO2
(solid circles), (b) by Asher and Wanninkhof (1995) for Helium; red line indicates hypothesized qualitative dependence if interface resistance is included, and (c) by Donelan
and Drennan (1995) for CO2 (open circles). The plots are shown to illustrate how an
interface resistance could be reconciled with experimental data, or vice versa, how
the latter might even suggest such a term.
mixing decrease, as outlined above. On the other hand, waves notably grow with
fetch while mass balance methods effectively yield transfer velocities averaged over
the entire length of a wind-wave tank. In particular in long facilities, this might mask
the existence of a plateau. Secondly, Komori and Shimada (1995) argues that the
inhomogeneous distribution of gas in depth is source of systematic underestimation
of transfer rates at low wind speeds. A conclusive answer seems difficult, but the
results of this thesis are in favor of an interface resistance model.
Assuming an interface resistance entirely dominated by the adsorption process
and neglecting possible reduced mobility in the water, the mass accommodation
coefficient can be estimated according to eq. (9.2). Using B = 0.066 s/cm as value for
RIF , one obtains γ ≈ 1.8 × 10−3 and attributing the discontinuity in the full profiles
in sec. 8.3.2 to RIF ≈ 0.3 s/cm, leads to γ ≈ 4 × 10−4 . Duan et al. (1993) measured
γ ≈ 5 × 10−3 for the adsorption on small falling droplets. An exact quantitative
comparison seems difficult, but the rough order of magnitude agreement is plausible.
Surface kinetics as possible source of resistance to gas transfer to a liquid has also
come to the attention among some physical chemists. Noyes et al. (1992, 1996); Rubin
and Noyes (1992) measured the uptake of gases such as CO2 and N2 in a small closed
container about half filled with gas and half with water. Turbulence was created by
mechanically stirring the water. The experimental design owed to their usual field
of research. The authors observed a steady increase in gas transfer with increasing
stirring rates which leveled off beyond a certain threshold before rising again once
vigorous steering caused deformation and disruption of the water surface. This
tendency shows surprising resemblance in particular to the data by Komori and
Shimada (1995) described above. Assuming interface processes to contribute the
112
Additional interface resistance - a hypothesis
9.3
only relevant resistance at high stirring rates, the authors determine γ ≈ 5 × 10−8 for
CO2 and yet an order of magnitude less for N2 .
As the authors themselves admit, neglecting transport phenomena in the fluids
and exclusively considering the interface process is certainly idealized and does not
fully describe the underlying physics. Realistically, mass transport prescribed by
the boundary layer flow in liquid and gaseous phase (depending on solubility) will
contribute to resistance as well as interface dynamics. The results of my visualization
experiments would reconcile such a proposition.
113
10
Conclusion
The scope of this thesis was the development and application of a visualization technique to study the distribution of a passive admixture in the air above the water
surface. Experiments were conducted in a laboratory wind-wave tunnel. Specifically,
vertical concentration profiles of a tracer gas were recorded. To this end, supersaturation in the air compartment was imposed by deliberate injection resulting in a mass
flux into the water. A unidirectional wind field created shear-induced turbulence
in the water and waves at sufficiently high wind speeds. Interest in this topic had
arisen because of the conceptual importance of air-sided transport processes for gas
exchange across the interface between atmosphere and ocean.
The developed technique is based on the principle of LIF by which molecules
fluoresce when excited by laser light of a suitable wavelength. As tracer gas, acetone
was selected since it met a series of required criteria. In terms of gas exchange, it is
an interesting substance as the transport is controlled equally by the gaseous and
aqueous phase due to its solubility. Acetone can be excited in the ultraviolet regime
and emits in the blue range of the spectrum. As source of illumination, a pulsed
266 nm Nd:YAG UV laser was used pointing downwards into the wind-wave tank.
The laser was formed to a sheet and spatially filtered to remove higher modes. The
fluorescent brightness was observed with a camera from the cross-wind direction,
i.e. perpendicular to the main air-flow. Special attention was paid to optical effects
at the sidewalls of the tank and at the refracting water surface. This was important
because of the weak signal originating from the air-sided fluorescence compared to
the high brightness in the water.
Relative acetone concentration profiles in the air were recorded, normalized to
their value in a height above the water surface where tracer was well-mixed. This
was achieved by calculating quotients of images and reference images. By this ap-
115
Chapter 10
CONCLUSION
proach, effects which contribute to the spatial modulation of the observed fluorescent
brightness like vignetting, Lambert-Beer absorption, and marginal saturation, were
automatically considered. At the same time, concentration profiles could be converted to absolute values by a calibration procedure based on the characteristic
Lambert-Beer decay in the reference images. An automatic algorithm was developed
which allowed to detect the water surface in individual images and to express and
average profiles in coordinates relative to the water surface.
The visualization technique was extended to simultaneously obtain information
on water-sided acetone profiles. A numerical approach was developed to resolve the
implicit dependence of fluorescent brightness and tracer concentration. This allowed
to determine the characteristic boundary layer scale z ∗ and the concentrations in
the bulk and at the surface. In this context, correction for optical aberration was
included in the data processing routines and their effectiveness was analyzed. As in
the air-sided images, the position of the water surface was automatically detected
with the help of dedicated image processing routines developed for this purpose. By
a simple modification of the algorithm, the vertical distribution of tracer streaks was
retrieved and used as proxy for turbulent eddies.
Invasion experiments were conducted in which acetone was continuously injected
into the air compartment of a wind-wave tank. This caused a supersaturation in the
air and thus forced a mass flux into the water. Wind speed was varied in a range from
1.5 to 6.5 m/s. To study the influence of surface active material on the gas exchange,
the detergent Triton-X was added to the water in some experiments.
Measured air-sided concentration profiles were analyzed in terms of their scaling
properties and good similarity to wind profiles was found. This confirms experimentally the assumed conceptual analogy between mass and momentum transport in the
air. The scaling was also used to deduce the partitioning of transfer resistances within
the air-sided turbulent boundary layer. Comparison with theoretical predictions
showed good agreement. The turbulent diffusivity model by Deacon (1977) was used
as a fit function to obtain air-sided transfer resistances. The influence of fluorescence
originating from acetone in the water on the air-sided signal was discussed. For
greater wind speeds accompanied by steeper (ak ≈ 0.2) and asymmetric waves, the
measured air-sided concentration profiles in the region of the diffusive sublayer
were shallower than expected from theory for smooth wall flow. This is partially
attributed to experimental systematics. DNS by other authors suggest alterations of
wind profiles induced by water waves of similar amplitude as in this work (a few mm).
No assertive conclusion could be reached based on the experimental data. For the
dynamic interaction of wind and waves, these small scales are presumably important
suggesting the use of dedicated flow field visualization in future experimental studies.
Based on the resistance partitioning within the air-sided turbulent boundary layer it
was concluded, however, that the diffusive sublayer does not play a major role for
gas exchange, especially under field conditions.
The surfactant Triton-X was found to suppress gas exchange by about 30 % regardless of the wind speed regime. It was concluded that wave damping was not the
116
10.0
principal causal factor as suggested previously by some other studies. The statistical
distribution of acetone streaks in depth relative to the water surface showed that
the presence of Triton-X was concurrent with a layer below the surface privy of
turbulent eddies which was wider than in clean water. It was further concluded that
surfactants lead to a transition of boundary conditions from free surface to rigid
wall. This is in accordance with other theoretical and experimental studies which
have identified surface divergence as a potentially relevant parameter. This suggests
the complementary use of techniques to visualize the surface flow structure in future
laboratory studies. These include PIV or the use of thermographic imagers.
Conversion of concentration profiles to absolute concentrations and use of the solubility αH allowed to represent them as continuous distributions across the air-water
interface. When normalized to the difference of air and water-sided bulk concentrations, these full profiles allowed to investigate the resistance partitioning across the
phase boundary. Results generally agreed with previous studies of other authors confirming acetone to be controlled almost equally by the gaseous and aqueous phase.
Normalized profiles were found not to be continuous at the water surface which
was particularly evident for higher wind speeds. Systematic experimental errors
were discussed as possible cause for this discrepancy. At the same time, a physical
explanation for the observations was attempted. As hypothesis was proposed that an
additional transfer resistance might need to be considered to account for processes
at the water surface not described by fluid mechanics. Specifically, this refers to the
adsorption and solvation of gas molecules at the air-water interface. A short review of
existing literature on the matter was provided and selected exemplary experimental
results of other authors were shown to illustrate the possible signature of an interface
resistance in gas exchange data. Figures of merit based on the current results were
compared with another publication and a reasonable agreement was found. The
analysis supports the proposed hypothesis that additional physical process might
need to be considered to fully and coherently describe gas exchange, but for an
affirmative conclusion, further investigations are desirable.
117
List of symbols
Isat Saturation laser irradiance. 29, 31
Kdiff (z + ) Turbulent contribution to vertical scalar flux. 16, 17, 74, 101, 102
Kdiff (z) Turbulent contribution to vertical scalar flux. 16, 17, 73, 87, 120
Kturb (z + ) Turbulent contribution to vertical momentum flux. 12, 13, 74
Kturb (z) Turbulent contribution to vertical momentum flux. 12, 13, 16, 120
RIF Transfer resistance associated to the air water interface.. 110–112
Ra Transfer resistance associated to the air sided fluid.. 21, 22, 79, 95, 102, 110
Rtot Total transfer resistance, i.e. of air and water-sided boundary together.. 101–103
Rw Transfer resistance associated to the water sided fluid.. 21, 22, 95, 100, 110
U0 Free stream air flow velocity. 12, 47, 74, 111
Vair Volume of the air compartment. 36, 72
Vwater Volume of the water compartment. 36, 72, 89
Φf Quantum efficiency of fluorescence, i.e. ratio of emitted to absorbed photons..
25–28
αH Solubility (a.k.a. Partitioning coefficient) cwater = αH ⋅ cair . 21, 26, 28, 38–40, 72,
89, 91, 94, 107, 110, 111, 117
⟨c ′ w ′ ⟩ Turbulent contribution to vertical scalar flux. 16
⟨u ′ w ′ ⟩ Turbulent contribution to vertical momentum flux. 12
c̄(z) Mean concentration profile. 80, 120
c̄ + Normalized mean concentration profile, c̄ + (z) = c̄(z)/cb . 73–76, 79, 101
119
List of Symbols
ū(z + ) Mean velocity profile in dimensionless units. 13, 74
ū(z) Mean velocity profile. 13, 74, 120
δ Width of the turbulent boundary layer.. 11, 12, 15
єair Optical extinction coefficient of acetone vapor, intended at λ = 266 nm. 28, 30,
63, 64
єwater Optical extinction coefficient of dilute acetone in water, intended at λ =
266 nm. 28
∂ c̄
∂z
Gradient of the mean concentration profile, c̄(z). 16
∂ū
∂z
Gradient of the mean velocity profile, ū(z). 12
κ Von Karman constant, typical literature values are κ ≈ 0.370.41. 11, 12, 16, 101, 120
D Diffusion constant, usually in cm2 /s. 8–10, 16, 17, 19, 20, 74, 79, 99, 102, 120
Re Reynolds number, Re ≡ U L/ν.. 8, 120
Scturb. Turbulent Schmidt number, defined as ratio of turbulent viscosity and diffusivity, Scturb. = Kturb (z)/Kdiff (z). 101–103, 120
Sc Schmidt number, defined as ratio of kinematic viscosity and diffusion constant,
Sc = ν/D. 8, 16, 17, 108, 111, 120
c/u∗
A parameter quantifying empirically the wave age, with c the wave’s phase
velocity and u∗ the friction velocity.. 47
νair Kinematic viscosity of air. 17
νwater Kinematic viscosity of water. 17
ν Kinematic viscosity, defined as ν = µ/ρ. 8–13, 16, 73, 74, 79, 99, 101, 102, 105, 106,
120, 121
ρ Density. 8
σcs Absorption cross section. 25, 28, 29
τs Vertical momentum flux at the surface. 11, 13, 74, 121
τ Vertical momentum flux. 13
ak Wave slope, with a the wave’s amplitude and k = 2π/λ. 49–51, 61, 87, 105, 106, 116
cb Bulk concentration of the tracer.. 19–21, 32, 33, 39, 59–64, 67–70, 73, 75, 79–82,
84–86, 89, 93, 95, 102, 103, 120
120
List of Symbols
cs Surface concentration of the tracer.. 19–21, 39, 67–71, 74, 75, 79–82, 84–87, 93, 95,
102
flaser Flash frequency of the pulsed UV laser. 29
fwind Frequency of the wind turbine in Hz. 47, 48, 50, 51, 61, 84, 85, 87, 88, 90, 91, 98
js Vertical scalar flux at the surface. 17, 74, 79, 102
j Vertical scalar flux. 15, 16
nwater Refractive index of water. 39
u∗ Friction velocity defined from moment flux at the boundary as τs = ρu∗2 . 11–13,
17, 19, 39, 45–47, 66, 73, 74, 79–81, 93, 99–103, 105, 106, 111, 120, 121
z ∗ Characteristic scale of an exponential profile, c(z) ∝ exp(z/z ∗ ).. 20, 39, 40,
67–71, 85–87, 98, 100, 116
z + Dimensionless vertical coordinates z + = zu∗/ν. 11–13, 16, 17, 73–79, 84, 95, 101, 102,
119, 120
z11 Scale of transition between diffusive and logarithmic sublayer, z + = z11 u∗/ν = 11 .
73, 76, 79, 82, 93
121
List of abbreviations
DNS direct numerical simulation. 3, 13, 14, 105, 106, 116
FWHM full width at half maximum. 40, 44, 50, 69–71
GV gray value. 57
ISC intersystem crossing. 24, 27
LIF laser induced fluorescence. 5, 23–26, 36, 37, 51, 66, 115
LSG laser-slope-gauge. 48
mss mean square slope. 45, 50, 51
Nd:YAG neodymium-doped yttrium aluminum garnet. 27, 36, 60, 115
PIV particle imaging velocimetry. 11, 47, 106, 108, 117
PLIF planar laser induced fluorescence. 26
QS Q-switch. 36, 52
123
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Acknowledgments
I am very grateful to Prof. Bernd Jähne for supporting me during my PhD work,
financially and scientifically.
In this context, I also acknowledge funding of the German Research Foundation
(DFG) through the GRK1114 Graduiertenkolleg.
I thank Prof. Kurt Roth for co-reviewing my thesis and Prof.s Ulrich Uwer and
Björn Malte Schäfer for complementing the exam commission.
Many thanks also to Karin Kruljac for any support with bureaucracy.
Thanks to all my colleagues in our group for a very enjoyable and interesting time
together.
Thank you in particular to Christine, Daniel, Dasha, and Felix for proof-reading
parts of my thesis, and to Wolfgang for support with anything regarding framegrabbers, hardware drivers, and Co.
Doppelkeks is a miraculous drug to get you through a long night of writing.
Thanks Felix for providing me with it and not only!
E grazie per innumerevoli piccole cose a chi non vuol’ essere nominata.
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