diss 3d flow vis
INAUGURAL-DISSERTATION
zur
Erlangung der Doktorwürde
der
Naturwissenschaftlich-Mathematischen
Gesamtfakultät
der
Ruprecht-Karls-Universität
Heidelberg
vorgelegt von
Dipl.-Phys. Dirk Engelmann
aus Tübingen
Tag der mündlichen Prüfung: 26. Juli 2000
3D-Strömungsmessungen
mittels Stereo Bildverarbeitung
Gutachter:
Prof. Dr.
Bernd Jähne
Prof. Dr.
Kurt Roth
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Rupertus Carola University of
Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
3D-Flow Measurement
by Stereo Imaging
presented by
Diplom-Physicist:
born in:
Dirk Engelmann
Tübingen, Germany
Heidelberg, July 26, 2000
Referees:
Prof. Dr. Bernd Jähne
Prof. Dr. Kurt Roth
Summary
A new method to record three-dimensional liquid flow fields by using ‘Particle
Tracking Velocimetry’ is presented. It is based on a two-dimensional Particle
Tracking Velocimetry method. It was extended to the third space dimension in
order to include the complete physical space. This procedure allows to determine
the Lagrange-flow field and to calculate from it the Euler-velocity flow field obtained from many other flow measuring techniques.
A calibration method was developed for the wind-wave-flume which allows a high
resolution in space. The stereo camera setup and the experimental setup were
optimized for the liquid flow measurements. For the first time a liquid prism and a
Scheimpflug-camera geometry was used.
Numerical calculations using the finite element method demonstrate the complexity of the problem of dealing with free surfaces with wind-induced shear forces as a
boundary condition. They show clearly that experimental studies are indispensable
for describing phenomena such as ‘bursts’ (descending of liquid elements from
close to the surface into deeper layers).
Flow measurements were performed in a newly constructed wind-wave-flume
(AEOLOTRON) and in a smaller predecessor by using the newly developed imaging methods. In this way the flow fields of wind driven water waves could be
characterized by the velocity field and the ‘turbulence’ conditions.
Zusammenfassung
Ein neues Verfahren zur Messung des dreidimensionalen Strömungsfeldes mittels
‘Particle Tracking Velocimetry’ wird vorgestellt. Ein zweidimensionales Particle Tracking Velocimetry liegt dem Verfahren zugrunde. Es wurde auf die dritte
Raumdimension erweitert, um den gesamten physikalischen Raum zu erfassen.
Dieses Verfahren erlaubt, das Lagrange´sche Strömungsfeld zu bestimmen, woraus
auch das von vielen anderen Verfahren erhaltene Euler´sche Geschwindigkeitsfeld
berechnet werden kann.
Das für den Einsatz am Wind-Wellen-Kanal entwickelte Kalibrierverfahren ermöglicht eine hohe räumliche Auflösung. Der Stereokamera-Aufbau und der Versuchsaufbau wurde für die Strömungsmessungen optimiert. Dabei wurde erstmals
ein Flüssigkeitsprisma und eine Scheimpflug-Kamera-Anordnung eingesetzt.
Numerische Rechnungen unter Verwendung der finite Elemente-Methode zeigen
die Komplexität des Problems, freie Wasseroberflächen mit windinduzierten Scherkräften als Randbedingung zu behandeln. Sie machen deutlich, daß experimentelle
Untersuchungen unerlässlich sind, um Phänomene wie ‘bursts’ (Abtauchen von
oberflächennahen Flüssigkeitselementen in tiefere Schichten) zu beschreiben.
Strömungsmessungen wurden an einem neu konstruierten Wind-Wellen-Kanal
(AEOLOTRON) und am kleineren Vorgängermodell mit dem neuentwickelten Bildverarbeitungsverfahren durchgeführt. Die Strömungsfelder von windinduzierten
Wasserwellen konnten auf diese Weise Geschwindigkeitsfeld und “Turbulenzzustand” charakterisiert werden.
Contents
1
Introduction
1
2
Theoretical aspects of liquid flow and gas exchange
3
2.1
Description of a liquid flow field . . . . . . . . . . . . . . . . . .
3
2.2
Gas transfer processes . . . . . . . . . . . . . . . . . . . . . . . .
4
2.3
Numerical solutions and models . . . . . . . . . . . . . . . . . .
11
3
Flow visualization
19
3.1
Flow-field measurement techniques . . . . . . . . . . . . . . . .
19
3.1.1
Hot wire anemometry . . . . . . . . . . . . . . . . . . .
20
3.1.2
Laser Doppler anemometry (LDA), acoustic Doppler velocimetry (ADV) . . . . . . . . . . . . . . . . . . . . . .
20
3.1.3
Particle imaging velocimetry (PIV) . . . . . . . . . . . .
21
3.1.4
Particle tracking velocimetry (PTV) . . . . . . . . . . . .
21
Techniques for flow field visualization . . . . . . . . . . . . . . .
22
3.2.1
Seeding particles . . . . . . . . . . . . . . . . . . . . . .
22
3.2.1.1
Scattering properties . . . . . . . . . . . . . . .
23
3.2.1.2
Buoyancy . . . . . . . . . . . . . . . . . . . .
28
Hydrogen/oxygen bubbles . . . . . . . . . . . . . . . . .
30
Experimental setup for particle tracking velocimetry . . . . . . .
31
3.3.1
Traditional setup . . . . . . . . . . . . . . . . . . . . . .
31
3.3.1.1
Observation close to the water surface . . . . .
31
Stereo setup . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.3.2.1
33
3.2
3.2.2
3.3
3.3.2
Heidelberg circular wind-wave facility . . . . .
3.3.2.2
3.3.3
4
35
3.3.2.3
Volume of observation for stereo camera setup .
37
3.3.2.4
Technical data of used imaging devices . . . . .
38
Improvements for stereoscopic flow visualization . . . . .
40
3.3.3.1
41
Scheimpflug stereo camera setup . . . . . . . .
Geometry of the stereoscopic system
45
4.1
Model of the stereoscopic system
. . . . . . . . . . . . . . . . .
45
4.2
A simple camera model . . . . . . . . . . . . . . . . . . . . . . .
47
4.2.1
Homogeneous coordinate system . . . . . . . . . . . . .
47
4.2.2
Pinhole camera model . . . . . . . . . . . . . . . . . . .
48
4.2.3
Pinhole camera model including intrinsic camera parameters 49
4.2.4
The linear camera model . . . . . . . . . . . . . . . . . .
50
4.2.5
Camera model including lens distortion . . . . . . . . . .
52
4.2.6
Camera model and the Scheimpflug condition . . . . . . .
54
Multiple media geometry . . . . . . . . . . . . . . . . . . . . . .
56
4.3
5
The large Heidelberg wind-wave facility (AEOLOTRON) . . . . . . . . . . . . . . . . . . .
Image sequence analysis for stereo PTV
59
5.1
Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.1.1
The choice of the calibration target . . . . . . . . . . . .
60
5.1.2
The calibration procedure . . . . . . . . . . . . . . . . .
61
The particle tracking velocimetry (PTV) algorithm . . . . . . . .
64
5.2.1
Segmentation . . . . . . . . . . . . . . . . . . . . . . . .
65
5.2.1.1
Region oriented segmentation . . . . . . . . . .
66
5.2.1.2
Model-based method . . . . . . . . . . . . . .
69
5.2.2
Labeling and position determination of a particle . . . . .
70
5.2.3
Correspondence solving . . . . . . . . . . . . . . . . . .
71
5.2.3.1
Particle characteristics . . . . . . . . . . . . . .
72
5.2.3.2
Velocity estimation . . . . . . . . . . . . . . .
75
5.2.3.3
PTV post processing . . . . . . . . . . . . . . .
75
Stereo correspondence solving . . . . . . . . . . . . . . . . . . .
76
5.2
5.3
5.3.1
Geometric constraints . . . . . . . . . . . . . . . . . . .
77
5.3.2
Object properties constraints . . . . . . . . . . . . . . . .
79
5.3.3
Applied constraints for 3D PTV . . . . . . . . . . . . . .
83
5.3.4
Stereo correlation algorithm . . . . . . . . . . . . . . . .
84
5.3.4.1
Application of the geometric and ordering constraint . . . . . . . . . . . . . . . . . . . . . .
84
Object property constraint . . . . . . . . . . . .
86
Stereo coordinate reconstruction . . . . . . . . . . . . . .
87
5.3.4.2
5.3.5
6
7
Analysis of data and discussion of results
93
6.1
Calibration and resolution . . . . . . . . . . . . . . . . . . . . . .
93
6.2
Stereo correspondence . . . . . . . . . . . . . . . . . . . . . . .
95
6.3
Stereo particle tracking velocimetry . . . . . . . . . . . . . . . .
98
Outlook
115
A Data tables
117
B Linearized wave equation
121
C Depth of field, depth of focus
123
D Basics of the finite element method
125
Chapter 1
Introduction
The major part of the earth is covered by oceans. Gas from the atmosphere is
transported into the oceans due to exchange processes at the surface. For gases
such as CH4 , O2 or CO2 the oceans are a major sink. About 90% of the global CO2
is dissolved in the oceans. But the amount of CO2 - one of the most important gases
concerning the prediction of climatic changes - transfered from the atmosphere to
the oceans is still uncertain and under debate. Estimations deviate by ±2 Gt from
4 Gt. The concentrations of these gases in the atmosphere influence the climate
significantly. It is therefore important to understand air-sea related gas exchange
processes.
The atmospheric gas has to pass the air-water phase boundary and the amount of
gas transported per time is characterized by the transfer velocity. The transfer
through the air-water boundary is controlled by a microscopic layer (20-300 µm,
Münsterer [1996]) at the water surface in which molecular diffusion is the dominating transport process. Beneath this layer turbulent transport is the far more
efficient transport mechanism. Water waves affect the gas transport and they show
a large variety of different types. On a scale of some centimeters up to several
kilometers gravitational waves appear where gravitation is the restoring force. On
a smaller scale of millimeters up to centimeters capillary waves show up. Here the
surface tension is the restoring force. If air flows over the free surface of the liquid
and capillary waves are generated, the transfer of gas in the liquid is considerably
increased. The appearance of wind driven water waves is clearly associated with
an increase in the gas transfer rate. This effect is well known, but the underlying
physical mechanisms are not fully understood. The parameterization of the surface shape is an attempt to characterize the gas transport. The mean square slope
was found by Jähne [1985] to be an important parameter. Furthermore the liquid
flow field is of importance in order to understand the physical mechanisms in more
detail. Air-water gas transport is also important in the field of chemical engineering. For instance gas-liquid reactors are frequently used where the transport of gas
drives chemical reactions.
2
Introduction
The aim of this work was to study the liquid flow field close to the free air-water
surface. A stereo particle tracking velocimetry algorithm was applied and improved. It allows to study the liquid flow field in three-dimensional space. This
technique is a major improvement to the well established two-dimensional particle tracking velocimetry by which a two-dimensional flow field is obtained. Since
the liquid flow is a three-dimensional phenomenon it is important to study the
complete flow field by a stereoscopic method. The particle tracking velocimetry
method allows to study the Lagrange-velocity flow field and is therefore especially
well suited to study the transport of near surface water elements to the deeper water.
New techniques for the visualization of the liquid flow where introduced.
The theoretical background of liquid flow field, gas exchange processes and numerical methods are given in chapter 2. In chapter 3 frequently applied flow visualization methods are introduced and the associated techniques for the flow field
evaluation are explained. The stereoscopic geometry and the modeling of the used
system are shown in chapter 4. The image sequence processing techniques are
explained in chapter 5 and the results are presented and discussed in chapter 6. Finally future steps and tasks for image sequence processing and for improving the
experimental setup are proposed in the ‘outlook’ chapter 7.
Chapter 2
Theoretical aspects of liquid flow
and gas exchange
2.1 Description of a liquid flow field
The motion of fluids can be described in one of two ways. The way of description
implies directly the experimental techniques to be used (section 3.2).
The first way, the Euler-description of motion, regards the physical quantities such
−
−
as the velocity →
u , pressure p and density ρ as functions of position →
x and time t.
Thus
→
−
−
−
u =→
u (→
x ,t)
(2.1)
(same for p and ρ) represents the velocity at prescribed points in space-time. The
−
derivatives with respect to →
x and t represent the gradient field at a given time and
a given position.
Alternatively, the fluid motion can be described in the Lagrange way. Fluid ele−
−
−
−
ments are identified at some initial time to with position →
xo . Thus →
x =→
x (→
x o ,t −
−
−
−
to ) where →
x (→
x o , 0) = →
x o . The velocity of a fluid element is therefore the time
derivative of its position:
∂− →
→
−
−
u (→
x o ,t − to ) = →
x (−
x o ,t − to )
∂t
(2.2)
The total time derivative is written in Euler-terms as
d
∂
−
= + (→
u · ∇)
dt
∂t
(2.3)
which is the sum of the rate of change at a fixed point and a convective rate of
change.
4
Theoretical aspects of liquid flow and gas exchange
Many instruments measure fluid properties at a fixed point and provide Euler-flow
field information directly. Questions concerning diffusion and mass transport deal
with the motion of fluid elements and thus the Lagrange specification of the problem is better suited to treat them. The Lagrange description implies a marking of
fluid elements by a dye or other tracers.
The equations of motion are governed by the conservation laws of mass and momentum:
The momentum transport is expressed by the Navier-Stokes equation:
−
∂→
u
→
−
−
−
−
+ (→
u · ∇)→
u − ν∆→
u + ∇p = f
∂t
(2.4)
→
−
where f is the resultant of all forces acting on the fluid (such as the gravitational
−
force →
g ) and ν the viscosity of the fluid.
The conservation of mass (continuity equation) is expressed as
∂ρ
−
+ ∇ · (ρ→
u)=0
∂t
(2.5)
and if the density of a fluid element does not change it takes the form
−
∇·→
u = 0
(2.6)
For an unsteady, diffusive mass transport Fick’s 2nd law applies:
∂c →
+−
u · ∇c = D∆c
∂t
where c is the gas concentration and D the diffusion coefficient.
(2.7)
→
−
The flux density is related to the concentration gradient and defined as j = −D∇c.
The water (in oceanic and laboratory circumstances considered in this work) can
be regarded as an isotropic, incompressible Newtonian fluid1 .
2.2
Gas transfer processes
As the transport of atmospheric gases (inert and sparingly soluble) is the physical
background of this work, a short overview of models describing basic processes is
given.
1p
ij
∂ui
+
= −pδi j + 2νei j , with strain tensor ei j = 12 ( ∂x
j
∂u j
)
∂xi
2.2 Gas transfer processes
5
The transport of inert or sparingly soluble gases between air and water is controlled
by molecular transport (diffusion) and by turbulent transport processes. Directly
at the water surface molecular diffusion is the process controlling the gas transfer.
Turbulent transport vanishes at the water surface2 .
Molecular diffusion is caused by differences in the gas concentration. This is described by Fick´s 1st and 2nd law.
Fick´s 1st law is written as
→
−
j = −D∇c
(2.8)
→
−
and describes the proportionality of the flux density j to the concentration gradient of the gas ∇c where D is the proportionality constant.
Fick´s 2nd law is written as
dc ∂c →
=
+−
u · ∇c = −∇ j = D∆c
dt
∂t
(2.9)
and describes the diffusive transport of mass (generally of scalar tracers). The
−
transport is included in the term →
u · ∇c.
The amount of gas transported through the water surface per time is quantified by
the gas transfer velocity
→
−
j
→
−
k =
(2.10)
∆c
and ∆c = csur f ace − cbulk is the gas concentration difference between the water surface and the water bulk3 .
−
The velocity →
u is determined by the Navier-Stokes equation
−
∂→
u
1→
− 1
−
−
−
+ (→
u · ∇)→
u = ν∆→
u + f − ∇p
∂t
ρ
ρ
(2.11)
→
−
where f is the sum of outer forces, ∇p the pressure gradient and ρ the fluid density.
→
−
u · ∇u is the convective term and ν∆u the viscous term where ν is the kinematic
viscosity.
In order to understand the physical properties of the flow better, frequently a perturbation approach (see Monin and Yaglom [1975]) is used to deal with the equations
−
−
−
−
(2.9) and (2.11). The perturbation is in →
u and c, expressed by →
u =→
u +→
u 0 and
0
c = c + c (barred characters mean average over time, primed characters the fluctuating part).
|
derivative of a ‘mass element’ at the water surface z = 0 in time is ∂η
= 0.
∂t z=0
bulk’: below a reference depth the mixing of concentration is large and no concentration
gradient is present.
2 The
3 ‘water
6
Theoretical aspects of liquid flow and gas exchange
Substituting this into equation (2.9) and taking the temporal average < · > yields
∂c →
−
+−
u ∇c = −∇ (< c0 →
u 0 > −D∇c)
|
{z
}
∂t
→
−
j
(2.12)
→
−
−
−
−
(where the identity →
u ·∇c = ∇·(→
u c)−c·∇· →
u was applied). j is the flux density
which is the sum of the average molecular diffusion and the turbulent fluxes.
−
Applying the same substitution to equation (2.11) and considering the case of →
u
in x-direction and depending only on z yields
∂u
= −∇ (< u0 w0 > −ν∇u)
|
{z
}
∂t
(2.13)
=: jm /ρ

u + u0
−
where →
u =  v0 .
w0

The first term is called Reynolds stress (in three dimensions this term ti j = u0i u0j is
a symmetric second-order tensor for u,i, j = {u0 , v0 , w0 }). It describes the turbulent
transport and the second term describes the viscous transport. For the stationary
case ∂u
∂t = 0, jm is a flux density of momentum - similar to equation (2.12). At
the water surface the momentum flux density is equivalent to the shear force τ,
jm |z=0 =: τ. This leads to the definition of a measure of the surface friction that
has the dimension of a velocity, the friction velocity u2∗ := ρτ . If the viscous term is
zero, the friction velocity becomes u2∗ =< u0 w0 > which expresses the correlation
of turbulent transport with the fluctuating velocity components (here in x- and zdirection). If laminar flow is considered and the turbulent term is zero, the shear
stress tensor becomes τ = νρ∇u. This can for example be caused by wind stress
on the water surface.
The terms viscous boundary layer and aqueous mass boundary layer are introduced to quantify the momentum transport and the molecular transport, respectively. For the definition of the aqueous mass boundary layer thickness z∗ the equa4
tions (2.8) and (2.10) are taken and with j = −D ∂c
∂z |z=0 at the water surface z = 0
(assuming the concentration is a function of the depth z, c = c(z)) it follows:
z∗ =
∆c
− ∂c
∂z |z=0
=
D∆c D
=
j
k
where ∆c is the concentration difference between the surface and the water bulk.
For the viscous boundary layer it follow from τ = νρ ∂u
∂z |z=0
4 At
the water surface the turbulent transport is absent.
2.2 Gas transfer processes
7
air
c
z=0
z
molecular
as
c
*
c
c, u
ws
water
u
viscous
z
*
reference concentration/ momentum (bulk)
z
Figure 2.1: Schematic graph of the molecular and the viscous boundary layers on
both sides of a gas-liquid interface. Due to the larger solubility α of the tracer gas
in water as compared to air, the tracer gas concentration is discontinuous at the
interface (cws = α · cas ).
viscous z∗ =
∆u
ρ ∂u
∂z |z=0
=
∆u
ν
=
τ/ν k
where ∆u is the velocity difference between the water surface and the velocity in a
reference depth.
The boundary layers are shown schematically in figure 2.1. The boundary layer
of molecular diffusion (for dissolved inert gases in water) is in the order of 100
to 300 µm. The range of the viscous boundary layer is larger by a factor of 100
to 2000. The Schmidt number Sc represents this relation of the boundary layer
thicknesses:
Sc =
ν
D
The turbulent transport terms of equations (2.12) and (2.13) can be replaced by
functions which depend on the depth z and the turbulent diffusion coefficients
∂u
Kc (z) ∂c
∂z and Km (z) ∂z . The equations can be integrated to obtain the concentration
and velocity profiles:
c(z) − c(0) =
Zz
0
j
dz0 ,
D + Kc (z0 )
u(z) − u(0) =
Zz
0
jm /ρ
dz0 .
ν + Km (z0 )
8
Theoretical aspects of liquid flow and gas exchange
Depending on the choice of the coefficients Kc (z) and Km (z) different models are
obtained such as the small eddy model and the surface renewal model. They are
discussed in the next sections. The models have to deal with the turbulent transport
mechanism and the gas diffusion. The simplest model (film model) assumes a
stagnant film as a surface layer of thickness z∗ where molecular diffusion appears
and turbulent transport sets in below this layer. The diffusion of gas through this
layer implies a proportionality of the diffusivity D and the gas transfer velocity
k = D/z∗ . The assumption of a linear concentration profile in the film layer is not
very realistic and experimental observations contradicted k ∼ D (a k ∼ Dn with
n < 1 was observed).
Small eddy model
This model describes the turbulent flux through the boundary layer as a cascade of
growing eddies. The turbulent diffusion coefficient Kc (z) = D − ∂cj is determined
∂z
by a Taylor series expansion applied to the concentration c(z) where the concentration and velocity fluctuations c0 ,w0 are assumed to be small compared to its mean
components. According to Coantic [1986] boundary conditions for a rigid and a
free interface5 are applied and the concentration profile c(z) is derived. It shows
a lower concentration and a larger decrease in the case of a mobile interface as
compared to the rigid interface.
For the gas transfer velocity k the Schmidt number dependency is found as
k ∼ Scn u∗
(for Sc > 100) with exponent n = 2/3 for the rigid wall and n = 1/2 for the free
interface.
Surface renewal model
The turbulent transport is based on a statistical renewal rate of the surface layer
λ = γ z p depending on the depth z. For the classical surface renewal model
(Danckwerts [1951]) the exponent p is zero, and for p > 0 the renewal rate becomes zero at the water surface and is more realistic (Jähne [1985]). The gas
is transported through the aqueous boundary layer by diffusion and parts of this
surface are exchanged with the water bulk in a statistical rate λ. Equation (2.12)
becomes
∂c
∂2 c
= D 2 − γ zp c
|{z}
∂t
∂z
∂
= ∂z
<c0 w0 >
−
interface: u0 (z = 0) = 0 and same for v0 , w0 ; ∇ · →
u0=0→
0
0
free interface: w (z) |z=0 = 0, c (z) |z=0 = 0
5 rigid
∂w0
∂z
= 0, c0 (z) |z=0 = 0
2.2 Gas transfer processes
9
0.0
normalized depth z
1.0
2.0
3.0
4.0
5.0
Legende
small eddy, n=1/2
small eddy, n=2/3
surface renewal, n=1/2
surface renewal, n=2/3
6.0
7.0
8.0
0.0
0.2
0.4
0.6
0.8
normalized concentration c
Figure 2.2: The normalized gas concentration profiles for small eddy and surface
renewal model with Schmidt number exponent 1/2 (free boundary, ‘wavy surface’)
and 3/2 (rigid boundary, ‘flat surface’).
Applying the boundary conditions6 for j yields the mean concentration profile c(z),
which is an exponential function for p = 0 and an Airy function for p = 1. The
mean concentration profile shows a larger decrease with depth than the small eddy
model which expresses the larger scale turbulence postulated by the surface renewal rate. The concentration profiles for the surface renewal and the small eddy
model are shown in figure 2.2.
The gas transfer velocity k can be predicted (Csanady [1990]) and is found as
k ∼ Scn u∗ , where n = 1/2 for p = 0 and n = 2/3 for p = 1 interpreted as free and
rigid interface respectively.
Influence of wind driven water waves on the gas transfer
If the wind speed over the water surface is increased, surface waves (capillary
waves) appear. The gas transfer velocity k changes significantly with the appearance of surface waves. This is illustrated in figure 2.3 where k (for CO2 ) is increased with the appearance of waves as compared to a smooth water surface. The
transfer velocity k for CO2 - which is controlled by the water side - is plotted versus the transfer velocity for water vapor kH2 O - which is controlled by the air side.
6 All
the fluxes through the surface at z=0 and no flux in the bulk z → ∞ :
|
= jo
j(z = 0) = D ∂c
∂z z=0
limz→∞ j(z) = 0
10
Theoretical aspects of liquid flow and gas exchange
Figure 2.3: Transfer velocity of CO2 plotted against the transfer velocity of water
vapor (measured in a circular wind-wave facility, Bösinger [1986], Huber [1984],
Jähne [1980]). Smooth water surface conditions are marked by circles, wavy conditions by stars. The solid line is the prediction of a smooth rigid wall.
The effect of waves on the turbulent transport at the water side is stronger than on
the turbulent transport at the air side and results therefore in an asymmetry seen in
figure 2.3.
With the occurrence of waves the Schmidt number exponent n decreases
(Bösinger [1986]). This indicates the enhanced gas transfer k by a factor of 3,
but can not explain the increase of k by up to a factor of 5 from the predictions.
This means, the turbulence level in the mass boundary layer is increased by a factor of 2. Pure surface increase can not explain this effect (Tschiersch and Jähne
[1980]).
The wave generation begins at low wind speed (1-5m/s) with capillary waves (wave
length in the range of cm). These capillary waves induce turbulent transport close
to the water surface, which leads to the increased gas transfer velocity. Therefore
this investigation is concerned with effects close to the free water surface in the
range of cm. Capillary waves do also appear in connection with gravitational waves
which dissipate energy from the gravitational waves (figure 2.4).
The energy flow of wind generated waves is displayed schematically in figure 2.5.
The air flow over the water surface transfers energy and momentum to the water by the shear stress tensor (which includes pressure, viscous- and Reynoldsstress). The energy is either dissipated in near surface turbulence or to the same
amount transfered to water (capillary and gravitational) waves. Due to non linear
2.3 Numerical solutions and models
11
Figure 2.4: Gravitational wave and parasitic capillary waves (small ‘ripples’ on the
leeward side of the gravitational wave). In the lower image is a view on the water
surface where the steepness of the wave is represented by the grey-value, obtained
by the imaging slope gauge (ISG, Balschbach [2000]) technique.
wave-wave interaction the capillary and gravitational waves are coupled. Energy
of gravitational waves is transfered into capillary waves (parasitic capillary waves)
and further transfered to turbulent flow, or directly to turbulent flow by wave breaking. The breaking of waves is another mechanism for turbulent flow generation.
2.3 Numerical solutions and models
The physical model describes the flow of an incompressible Newtonian fluid7 in
an area Ω and time T for a given boundary and initial conditions. The variables
−
−
−
sought-after are the velocity →
u (→
x , t) and the pressure p(→
x , t) which are func→
−
tions of time t and the location in physical space x = (x1 , x2 , x3 )T . The system
is described by the Navier-Stokes (momentum transport) equation and the incompressibility condition:
−
∂→
u
∂t
→
−
−
−
−
+ (→
u · ∇)→
u − ν∆→
u + ∇p = f
(2.14)
−
∇·→
u =0
in Ω × T
with outer forces f , the kinematic viscosity ν (ρ is included in p and f compared
to (2.11)).
7 Newtonian
fluid: The shear stress tensor is defined as Ti j = −pδi j + µ
dimensional case the shear stress is τ =
ν = µ/ρ).
µ du
dy
∂ui
∂x j
+
∂u j
∂xi
; in the one
(µ is the coefficient of viscosity, the dynamic viscosity
12
Theoretical aspects of liquid flow and gas exchange
air flow
surface stress tensor
water surface
wave generation
gravitational waves
parasitary capillary
waves
capillary waves
breaking waves
micro scale wave
breaking
turbulence
heat
Figure 2.5: Scheme of interaction processes close to the water surface: The air flow
over the water surface transfers momentum and energy via the surface stress into
the water. The surface stress generates surface waves which induce gravitational
and capillary waves (due to nonlinear interaction). Finally the energy is dissipated
to turbulent flow (small scale velocity fluctuations) and into heat.
These equations describe many occurrences in nature such as flow of water or
‘slow’8 air flow, but also serves as a basic model for more complex systems such
as combustion processes, chemical reactions, turbulence or multi-phase flow. The
mathematical problem to be solved is a time dependent three dimensional nonlinear partial differential equation (PDE). Because of the nonlinearity of the problem
direct solution methods can not be applied and iterative solution methods are required. The type of the problem can vary in time t and space Ω and this has to
−
−
−
be taken into account for the method of solution. If the term (→
u · ∇)→
u −ν∆→
u
→
−
(equivalent to Re 1), the type of PDE has a hyperbolic character, for −ν∆ u −
−
(→
u · ∇)→
u (equivalent to Re ' 1) the PDE has an elliptic character9 . The local
character of most of these problems demands also a local refinement in time and
space or other techniques which take local properties into account such as multigrid methods. Without local refinement the capacity of computer memory would
by far exceed the availability (a box of a resolution of 100x100x100 points would
require 1GByte RAM, if a typical size of 1 KByte storage per point is assumed).
8 otherwise
the incompressibility condition is not given.
order
PDE:
Auxx + 2Buxy +Cuyy + Dux + Euy + F = 0
A B
Let Z =
B C
Elliptical PDE: Z positive definite (det(Z) > 0) (i.e. Poisson equation, Laplace equation)
Parabolic PDE: det(Z) = 0 (i.e. heat conductivity equation, diffusion equation)
Hyperbolic PDE: det(Z) < 0 (i.e. wave equation).
9 Second
2.3 Numerical solutions and models
13
For the Navier-Stokes equation the elliptic type is of interest and considered here.
This leads to the linear differential operator of second order L in the region Ω
limited by the border Γ = ∂Ω, where Lu = f in Ω for a given f . The elliptic
boundary value problem is characterized by the following boundary conditions10 :
• Dirichlet boundary condition (for ‘solid walls’):
u = const on Γ
(2.15)
• Neumann boundary condition (for ‘free surface’):
∂u
= n · ∇u = const on Γ
∂n
(2.16)
If gas transport is considered, the first Fick´s law (equation (2.7)) applies
∂c →
+−
u · ∇c = D∆c
∂t
with concentration c and diffusion constant D. The equation can be solved easily,
if the velocity was determined by the Navier-Stokes equation.
Sample model
The model for a channel flow is shown in figure 2.6. At the bottom and side Γb
there are solid walls and the Dirichlet boundary condition (equation 2.15) for the
velocity is u = 0. For the surface Γb the free surface boundary conditions are valid
(equation 2.16)11 . The in- and outflow conditions are cyclic ul (z,t) |Γl = ur (z,t) |Γr
on Γl , Γr and the Neumann boundary conditions are applied.
∂u
∂n
+ σu = const on
Γ(σ 6= 0).
11 If the pressure and surface tension γ and the associated surface curvature R−1 are considered,
the dynamical boundary condition is also valid:
10 There
is also a ‘mixed’ boundary condition, the Robin boundary condition
t (i) T (u, p) n = 0
i = 1, 2
n T (u, p) n = −pa + gh + 2γR−1
T (u, p): surface stress tensor
t (i) , n: tangent, normal to Γb
pa : outer pressure
γ: surface tension
R−1 : average curvature of the free surface
g, h: gravitation, height of the water surface
14
Theoretical aspects of liquid flow and gas exchange
free boundary
Γf
periodic boundary condition
Γl
Γr
u i(z)
uo(z)
z
solid boundary
y
Γb
x
Figure 2.6: Model for channel flow showing the boundary conditions.
Methods for solving partial differential equations
There are quite a number of methods to solve partial differential equations analytically (for example: characteristic methods, variable separation, Bäcklund transform, Green´s method, Lax Pair). In many cases, however the problem can not be
solved analytically. Frequently there is not even a prove for the existence of a
solution. Numerical solution methods are therefore required such as finite difference methods, finite volume methods, finite element methods or spectral element
methods.
As far as flow dynamics is concerned, in many cases a direct simulation of the flow
−
variables velocity →
u and pressure p is not workable. In viscous flow the movement
in the fluid is conform, which means the viscous shear forces (section 2.1) are large
enough to maintain a uniform movement. This leads to so called turbulence models
where a large difference in the length scale exists. The relation between the largest
to the smallest length scale is at least in the order of 103 . This model class implies
a strong simplification of the models by empirical parameters in a limited class of
geometries and boundary conditions. Such an approach was already discussed in
−
section 2.2 where the perturbation in velocity →
u 0 and pressure p0 from their mean
−
values →
u and p was applied;
→
−
−
−
u = →
u +→
u0
−
p = p +p0
This leads to the Navier-Stokes and continuity equation averaged in time:
→
−
∂→
u →
−
−
+−
u · ∇→
u = −∇p + ∇ · (ν∇→
u − τ) + f
∂t
−
∇·→
u = 0
2.3 Numerical solutions and models
−
with the Reynolds stress tensor τ. The modeling of the tensor τ = τ(→
u , p) leads to
a number of different attempts for solutions. Some are listed in the following:
• Boussinesq hypothesis (1887):
τ is modeled like the stress tensor of viscous forces acting on Newtonian
liquids. The parameters are the velocity u and the length parameter l, which
is assumed to be constant (the numerical solutions does therefore not differ
from laminar flow).
• Mixing length theory:
The assumption is that the fluid flow dominates in one direction u = l ∂u
x .
The turbulent viscosity coefficient and the turbulent energy and the mixing
length l (free parameter) are connected.
• Energy-equation model (k-model):
The scale of fluctuating and mean velocity gradients is determined from the
Navier-Stokes equation (one equation model), which includes two empirical
constants (k: kinetic energy of turbulent motion and the length parameter l,
the same as in the mixing length theory).
• Energy-dissipation model (k-ε model):
l and k are obtained from the transport equations (two equation model) and
it follows for the energy-dissipation ε = ε(k, l) = k3/2 /l. This is one of the
most frequently used models. It implies four free parameters, and boundary
and initial conditions for k and ε are required.
Finite element method and Navier-Stokes equation
The finite element method (FEM) allows to solve the Navier-Stokes equation directly (without modeling assumptions as the Reynolds stress tensor τ mentioned
previously) and offers efficient numerical solution methods. The efficiency concerns numerical stability, convergence rate and flexibility for various posed limiting
conditions. Therefore the FEM is the most widespread technique applied nowadays
in computational fluid dynamics.
The equations (2.14) are split into different parts for solving the problem: The Poisson equation (−ε ∆u = f ), the convection-diffusion equation12 (−ε ∆u + ue∇u = f ,
ue is constant and an approximation of u) and the Ossen equation (−ε ∆u + ue∇u +
∇p = f , which is the Stokes equation if ue∇u is neglected). The equations have to
be discretized and an appropriate numerical solver has to be applied.
First the time derivative operator in the Navier-Stokes equation is applied
Müller et al. [1994]:
u − u(tn )
+ Θ(−ν∆u + (u · ∇)u) + ∇p = g(tn , tn+1 ) , ∇ · u = 0 in Ω
k
12 linearized
Burgers equation
15
16
Theoretical aspects of liquid flow and gas exchange
where k = tn+1 − tn is the time step, g(tn ,tn+1 ) is the right hand side13 and Θ characterizes the time stepping scheme. In each time step the following problem has to
be solved for u = u(tn+1 ) and p = p(tn+1 )
[I + Θ k (−ν∆ + (u · ∇)] u + k∇p = G , ∇ · u = 0
|
{z
}
S
⇔
Su + k B p = G , BT u = 0
(2.17)
where B is the gradient matrix and S the velocity matrix. u(tn ), the parameters k =
k(tn+1 ), Θ = Θ(tn+1 ) and the right hand side14 are given. This is the incompressible
Navier-Stokes equation reduced to a discrete nonlinear saddle point problem.
The spatial discretization is the next step to be applied on equation (2.17) using the
finite element method (see chapter D) and Galerkin discretization. Frequently used
FEM Ansatz functions for the Stokes elements (equation (2.17)) are approximated
by quadrilateral elements on Ω: a) linear in velocity and pressure approximations
at the vertices (Q1/Q1), b) velocity at the vertices and constant pressure in the cell
e Q0) quasi-linear velocity at the
(Q1/Q0) and c) non-conforming elements15 (Q1,
e Q0) have advantages
vertices and constant pressure in the cell. The elements (Q1,
as compared to the (Q1,Q1) and (Q1,Q0) because no additional stabilization (of
pressure) is needed (see Rannacher and Turek [1992]). The problem to be solved
for each element is the matrix equation (2.17)

S
 0

 0
BT1
0
S
0
BT2
0
0
S
BT3

B1
u1
 u2
B2 

B3   u3
0
p



G1
  G2 
=

  G3  .
G4
The treatment of this problem is shortly sketched:
The Navier-Stokes solver can be split into its tasks (Turek [1999]): In the outer
control part which is responsible for the global convergence rate and accuracy of
the overall problem and in the inner part of the solver which provides approximate
solutions within a given discrete framework.
There are two ways to split the problem into parts.
n ,tn+1 ) := Θ f (tn+1 ) + (1 − Θ) f (tn ) − (1 − Θ) (−ν ∆u(tn ) + (u(tn ) · ∇) u(tn ))
: unit matrix, right hand side G = [I − Θ1 k (−ν ∆ + (u(tn ) · ∇)] u(tn ) + Θ2 k f (tn+1 ) + Θ3 k f (tn )
with parameters Θi = Θi (tn+1 ) , i = 1, 2, 3
15 ‘non conforming’ means: 1) boundary conditions are only approximated or 2) V ⊂ V or 3) the
h
bi-linear and linear forms are only approximated numerically.
13 g(t
14 I
2.3 Numerical solutions and models
1. Linearize first the outer iteration problem (adaptive fixed point defect correction - or quasi Newton method and extrapolation of time), then solve the linear, indefinite problem in a inner iteration procedure (applying for instance
the coupled multigrid techniques)
2. Solve first in the outer iteration the definite problem in u (Burgers equation) and p (linear pressure-Poisson equation), then solve the nonlinear subproblems in u (applying nonlinear iteration or linearization techniques).
Most Navier-Stokes solver are constructed from this scheme where details of implementation are of course various. However, the effective implementation is difficult in general situations. Modern solvers are based on multigrid and adaptive grid
techniques and provide good convergence rates and high stability even for complex
geometries of the posed problem.
Even if computing power improves quickly, computational fluid dynamics is still
one of the big challenges and far from being applied in most ‘real world’ situations. Currently the available software has no proven control mechanism to determine deviations of the calculated result from the (unknown) exact solution. Beside
this are many problems concerning numerical simulations. The large amount of
required memory is one problem if three dimensional problems are considered.
Local variation of the problems character result in difficulties in respect to stabilization (of convective terms) and require tricky methods of adaptivity in time and
space. Boundary layers and complex geometries demand local mesh refinement
methods.
Even ‘simple’ posed benchmark problems such as benchmark simulations of laminar (three-dimensional) flow around a cylinder (Schäfer and Turek [1996]) in a
channel with solid walls pose the difficulties in obtaining quantitative results (even
for low Reynolds numbers in the order of 102 to 103 ). A free boundary problem
such as the model considered in this work is far more complicated. If a two-phase
flow problem is considered with an air flow over a (wavy) water surface, the shear
forces and pressure on the water surface can be determined rather easily. But calculations of the flow field close to the free water surface have not yet been carried out
successfully. Even for low shear forces a high spatial resolution of the discretization mesh is required, particularly for high Reynolds numbers as considered here
(in the order of 104 to 105 ). Due to the free surface condition (Neumann boundary)
the surface location varies with each time step and requires a complete renewing
of the mesh after a few time steps already. Computational efforts and memory
requirements are therefore enormous and not even a qualitative solution is guaranteed. The main improvements needed are of algorithmic nature. It is of crucial
importance to treat the local character of the problem. First steps in this direction
are for instance the local refinement of meshes by using adaptive grid techniques
and combined multigrid techniques (Becker and Braack [2000]). The improvements in computational power and computer memory will even in the far future
17
18
Theoretical aspects of liquid flow and gas exchange
not solve these problems without essential improvements of numerical algorithms
and methods.
Chapter 3
Flow visualization
3.1 Flow-field measurement techniques
In this section a number of frequently applied methods for liquid flow field measurements are mentioned. They differ in the way they can be applied and in the
type of physical data they supply.
Classical point measuring methods provide information of the flow field at only a
single point in the physical space, such as heat wire anemometry, Laser Doppler
velocimetry and acoustic Doppler velocimetry (next sections 3.1.1, 3.1.2).
Most other methods use tracer material. The tracers can be either continuous or
discrete providing information on the two- or three-dimensional flow fields.
A continuous tracer material (molecular tracer, fluoresceine) is applied in the Laser
induced fluorescence method (Münsterer [1996]). This technique is also used in
biological sciences (Uttenweiler and Fink [1999]) and in observing mixing
(Koochesfahani and Dimotakis [1986]), (Koochesfahani and Dimotakis [1985])
and exchange processes (Eichkorn et al. [1999]). A frequently used continuous
tracer for air flow field measurements is a dye aerosol injected into the air flow.
The Schlieren method uses inherent properties of the liquid (or gas) such as the refraction number dependency on the density: the deflection of the optical ray results
in an intensity modulation of the image. Flow field properties which are related to
these inherent properties (i.e. heat, density and refraction number) can therefore be
deduced (Oertel and Oertel [1989]).
The most frequently methods used for liquid flow field measurements use (macroscopic) seeding particles. These methods are divided into two branches (see also
section 2.1) and determine the type of method for the flow visualization as discussed in section 3.2. One type of methods are region oriented such as Laser
speckle velocimetry (LSV) or particle imaging velocimetry (PIV) and provide a velocity vector field (section 3.1.3). The other type of methods detects single (seed-
20
Flow visualization
ing) particles and persecutes them in time (particle tracking velocimetry PTV, section 3.1.4).
3.1.1
Hot wire anemometry
A thin wire, surrounded by a liquid medium, is connected to a constant electrical
voltage source and heated. The flow of the liquid around the wire determines the
rate of the heat transport from the wire to the liquid. The temperature change of
the wire causes a change in the electrical current through the wire. This current is
a measure of the flow speed of the liquid (Klages [1977]).
The thinner the wire (some microns), the lower the heat capacity and the better the
time constant of the temperature adaption. Due to this time constant, the sampling
frequency is typically up to 50 kHz. If several wires are positioned crosswise, even
two- and three-dimensional velocities can be determined (Tsinober et al. [1991]).
This is an invasive method and frequently used in commercially available devices.
The precision is better than 1%.
3.1.2
Laser Doppler anemometry (LDA), acoustic Doppler velocimetry (ADV)
The LDA method is based on the Doppler effect of small particles (seeding particles or naturally found particles) in the liquid for two crossed Laser beams.
A Laser beam is divided into two single beams which are crossing each other at
the volume of interest. In this volume (typically 0.15 · 0.15 · 2 mm3 ) an interference
pattern of light and dark stripes occurs (Wiedmann [1984]). The particles crossing
this volume change their scattering intensity in a certain frequency according to
the distance of the stripes and to the velocity of the particles. A photo-detector
records this changing intensity and the velocity is then readily calculated (by a signal processor). If more Laser-beam crossings are used (of different wave length1 ),
the three-dimensional velocity can be determined with a precision of about 1% and
the sampling rate can exceed 100 kHz.
The ADV method is very similar to LDA but uses an acoustic signal. For three
dimensional measurements three acoustic generators are necessary. They are positioned in a plane with equal distance from each other generating an interference
pattern. An acoustic sensor measures the intensity changes of moving particles due
to the interference pattern.
Both methods, LDA and ADV, are invasive and are restricted to one single point.
1 Such
as Argon-Ion Laser lines with 514, 488 and 476 nm.
3.1 Flow-field measurement techniques
3.1.3
Particle imaging velocimetry (PIV)
This method yields dense velocity vector fields in Euler-coordinate representation
(2.1). Either continuous or dense particles are required. The PIV method (for
review see Grant [1997]) takes a spatial limited window of the image and crosscorrelates this in time to the subsequent image. The result is a displacement vector,
averaged over the correlation window. Moving the window over the whole image
yields the velocity vector field. The correlation procedure with a window of a certain size is a low pass filtering procedure - depending on the size of the correlation
window - and therefore information on small scale fluctuations are lost. Usually
this method requires a high seeding particle density (several thousands per image)
to gain a good spatial resolution - typically a correlation window should contain
some tenths of particles. At the Heidelberg wind/wave facility PIV measurements
were performed close to the aqueous boundary layer by Dieter et al. [1994].
3.1.4
Particle tracking velocimetry (PTV)
This method follows single (seeding) particles in time over a sequence of images.
The result is therefore the Lagrange velocity field of the flow (2.2) which consist of
particle trajectories (Hering [1996]). Most PTV techniques use streak photography
as a tool for determination of the flow field (Hesselink [1988]). The velocity can be
obtained by measuring the length, orientation and location of each streak (Gharib
and Willert [1988]). Interpolating the velocity of the particles onto a regular grid
allows to calculate the Euler-velocity field (equation (2.3)).
This method is quite critical concerning the density of particles (number of particles per volume/image). The higher the density and the higher the speed2 of
particles the more likely is the overlap of the streaks. Overlap of streaks means a
loss of the particle tracking in the image sequence. Either the trajectory is interrupted or has to be reconstructed by some interpolation algorithms. The maximum
number of particles per image is limited to about 1000-2000.
An important aspect is the type of velocity information to be extracted. There are
basically two different approaches: The particle-imaging velocimetry (PIV) and
the particle-tracking velocimetry (PTV) techniques. Continuous or discrete tracers
can be applied for both of these techniques. The best spatial resolution is achieved
by using discrete tracers.
The PIV and PTV techniques are distinct in respect to the observation of the velocity flow field (see section 2.1). The PIV technique yields a ‘snapshot’ of the flow
−
−
−
field, which is →
v i=→
v (→
x o , t − to ) |t=ti , the Euler-representation of the velocity vi
−
at a certain time ti for each spatial location →
x o.
In contrast to this, the PTV technique follows a ‘mass point’ - actually a particle - in time. It yields the Lagrange-representation of the velocity, which is
2 The
higher the speed, the longer the streak image of the particles.
21
22
Flow visualization
→
−
−
−
−
−
x =→
x (→
x o , t − to ) where the particle was at position →
x =→
x o at the time t = to
→
−
and is moving in a trajectory to position x (ti ) at time ti .
The Lagrange-representation
allows also to calculate the Euler-flow field by simply
.
differentiating ~x in time. The purpose of this work was to determine the Lagrange
velocity field of seeding particles. Therefore the choice of the method was the PTV
method. More details about the PTV techniques are found in section 5.2.
3.2
Techniques for flow field visualization
The flow field can be visualized by using either discrete tracers, such as seeding
particles or bubbles, or by continuous tracers, such as fluorescent dyes using Laser
induced fluorescence (Eichkorn et al. [1999]). The visualization techniques discussed here use tracer particles which are either mixed into the liquid before or
locally added to the flow. Properties and characteristics of tracer particles such as
seeding particles and hydrogen bubbles are discussed in section 3.2.1 and 3.2.2.
Depending on the physical properties of the liquid to be examined the type of
tracer has to be chosen carefully. To investigate the velocity flow field of the liquid, discrete tracers are suitable. Either seeding particles (perspex, polycrystalline
material) or hydrogen bubbles generated by electrolysis are chosen.
3.2.1
Seeding particles
There are several criteria which the seeding particles have to fulfill. The main
criteria are: they should disturb the measurement respectively the liquid flow in a
minimal way and they should be sufficiently visible for the observing device (CCD
camera3 ).
The following points are important:
• The size/diameter d of the particle:
The smaller, the less disturbing; but the visibility (refractivity/ reflectivity)
is reduced (proportional to the area). Another important effect of the size is
the ability of the particles to follow the flow field of the fluid. This can be
deduced from the equation for the motion of small spherical particles, first
introduced by Basset, Boussinesq and Ossen (Hinze [1959]). The analysis of
this equation shows the ability of particles to follow a fluctuating flow field
with frequency ω. The deviation is given by
ε(ω) = const. ω2 d 4
The choice of smaller particles reduces this deviation, but the visibility of the
particles is proportional to d 2 and therefore larger particles are better visible.
3 CCD
stands for ‘Charge Coupled Device’.
3.2 Techniques for flow field visualization
23
• The material properties:
– the intensity of the scattered light, depending on the observation and
illumination angle, where the intensity should be as large as possible
(next section 3.2.1.1)
– the density which should be rather close to the liquid density to allow
for a minimum buoyancy
• The particle density (number of particles per volume) which is usually not
critical because the average free length is much smaller.
3.2.1.1
Scattering properties
In this section the scattering cross section σ(Ω) and its dependency on the steradian
Ω is briefly discussed.
The scattering cross section4 is defined as
σ(Ω) =
dI(Ω)
dΩ
which is the scattered light intensity dI in the differential steradian dΩ.
The seeding particles are large compared to the wavelength of the light source.
Therefore the Mie scattering theory has to be applied.
The scattering is described by the Maxwell equations written in a symmetric form
→
−
→
−
(Born and Wolf [1984]) using the vector potential5 Π e and Π m :
→
−
→
−
→
−
E = ∇ × ∇ × Π e − ikc Π m
n2 →
→
−
−
→
−
H = ∇ × ∇ × Π m + ikc 2 Π e
c
where
n: refraction index
c : light speed
k : wave number
4 more
precisely: the differential scattering cross section
→
−
→
−
magnetic field potential expressed as A = ikc cn2 Π e + ∇ × Π m
→
−
and the electric field potential Φ = −∇ · Π e
5 The
(3.1)
24
Flow visualization
The particle shape (can be assumed as spherical) leads to spherical symmetry of
the electromagnetic waves. Therefore the vector potentials can be written as the
Debey potential
→
−
−
Π e/m = Πe/m · →
x
(3.2)
−
where →
x is the radial direction of the wave extension.
The equations (3.1) can be solved through (3.2) written in spherical coordinates
(r, ϕ, ϑ) with
Πe/m (r, ϕ, ϑ) = Ylm (ϕ, ϑ) · zl (r)
(3.3)
where the equation contains an angular part with Ylm (ϕ, ϑ), the spherical surface
function6 , and radial part with the spherical Bessel function zl (r). This spherical
Bessel function zl (r) can be written with the Bessel function Zl+ 1 (r) for uneven
2
pπ
Zl+ 1 (r). Zl+ 1 (r) has to be chosen according to the
number l + 12 as zl (r) = 2r
2
2
initial conditions as the Bessel function of the first type Jl (r), the Bessel function of
the second type (Weber function) Wl (r) or a linear combination of both Hl (r)(1,2) =
Jl (r) ± iWl (r) (Hankel function).
The Debey potentials can then be written as
Πe/m (r, ϕ, ϑ) = e
iωt
∞
2l + 1
∑ (−i) l(l + 1) Pl1 (cos ϑ) jl (kr)
l=1
l
cos ϕ, for e
.
sin ϕ, for m
The solution of the incoming wave scattered on a sphere (particle) has to be divided
up into the inside of the sphere (refraction index n) and the outside of the sphere.
Outside the sphere the Ansatz function is:
Πe/m (r, ϕ, ϑ) = e
iωt
∞
2l + 1
(2)
∑ (−i) l(l + 1) Pl1 (cos ϑ)hl (kr)
l=1
l
al · cos ϕ, for e
bl · sin ϕ, for m
(3.4)
(2)
where the Hankel function related Ansatz function hl (kr) has to be chosen (bel+1
(2)
cause the Ansatz function has to be a spherical wave limr→∞ hl ∼ i kr e−ikr ). The
coefficients al , bl , cl , dl are determined by the boundary conditions on the sphere
(transition from vacuum to matter).
Inside the sphere the Ansatz function is:
6Y m (ϕ, ϑ) =
l
q
2l+1 (l−m)! m
imϕ , where Pm (x) is the Legendre function to the associated
l
2π (l+m)! Pl (cos ϑ)e
Legendre polynomial Pl (x); l ≥ |m| ≥ 0.
3.2 Techniques for flow field visualization
Πe/m (r, ϕ, ϑ) = e
iωt
∞
25
2l + 1
∑ (−i) l(l + 1) Pl1 (cos ϑ) jl (nkr)
l=1
l
n · cl cos ϕ, for e
n · dl sin ϕ, for m
(3.5)
where the Bessel function related to Ansatz function jl (nkr) has to be chosen (because limr→0 Wl (r) → ∞ and limr→0 Hl (r) → ∞).
The main interest is the intensity characteristics of the scattered light in a large
(2)
distance (large compared to the sphere/particle size). Substituting limr→∞ hl ∼
il+1 −ikr
in equation (3.4) yields
kr e
ieiωt
Πe/m (r, ϕ, ϑ) =
kr
∞
2l + 1
∑ l(l + 1) Pl1 (cos ϑ)
l=1
al cos ϕ, for e
.
bl sin ϕ, for m
(3.6)
The electrical and magnetic fields can be expressed as
i
→
−
→
−
E ϑ = c H ϕ = e−i(kr−ωt) cos ϕ S1 (ϑ)
kr
(3.7)
i
→
−
→
−
− E ϕ = c H ϑ = e−i(kr−ωt) sin ϕ S2 (ϑ)
kr
(3.8)
and
which is a product of an outgoing spherical wave and the form factors S1,2 (ϑ):
∞
S1 (ϑ) =
Pl1 (cos ϑ)
∂Pl1 (cos ϑ)
2l + 1
a
+
b
l
l
∑
sin ϑ
∂ϑ
l=1 l(l + 1)
(3.9)
∞
∂Pl1 (cos ϑ)
Pl1 (cos ϑ)
2l + 1
S2 (ϑ) = ∑
al
+ bl
∂ϑ
sin ϑ
l=1 l(l + 1)
(3.10)
2
−
→
Finally the intensity I(ϑ) = E /2 is
I(ϑ) =
|S1 |2 + |S1 |2
Io
2k2 r2
(3.11)
26
Flow visualization
CCD camera
water
Probing volume
(seeding particles in water)
scattering
angle ϕ
Light source
Lens
75mm
300mm
Figure 3.1: Principle of the setup for measuring the scattering angle.
Comparison of Mie-scattering theory with experimental results
The scattering properties of some frequently used seeding particles such as polycrystalline seeding particles (Optimage company), hollow glass spheres and perspex seeding particles of 30µm, 35µm and 180µm in diameter are shown in figure
3.2. The size (diameter) distribution of the particles is, due to production conditions, a normal distribution (20%-50% full width half length). The dependency of
the scattered light intensity (expressed in mean grey-value) on the scattering angle
is shown in figure 3.2 and was measured by Garbe [1998] and Hering et al. [1998].
A white light source (glow-discharge lamp, Cermax) was used and an angle of
about 320◦ (excluding 40◦ of direct view into the light source) was covered. The
intensity measurement was done by a CCD camera (50 images were averaged).
The principle of the experimental setup is shown in figure 3.1. An inner cylinder
contains water and the seeding particles. Close to the water-filled outer cylinder
a CCD camera and a light source were mounted. The outer cylinder reduces the
refraction effect due to a smaller curvature.
The results of the light intensity on the scattering angle show a good agreement to
the calculated curve (solid lines in fig. 3.2) from the minimum angle (about ±20◦ )
up to ±80◦ . Calculated values deviate by about ±10% for angles of |±ϕ| ≥ 80o .
The differences in scattering intensity of hollow glass spheres and the other seeding
particles in between the range of ±(50◦ -80◦ ) scattering angle are caused by the
different refraction indices. The hollow glass spheres behave like air bubbles (glass
volume is about 1/10 of the air volume).
The consequences for the measurements are:
• The scattering angle around 90◦ shows a plateau-like minimum in intensity.
3.2 Techniques for flow field visualization
27
600
500
mean grey-value
400
300
200
100
0
-150
-100
-50
0
50
100
150
ϕ
mean grey-value
scattering angle ϕ
scattering angle j
Figure 3.2: Scattered light intensity (expressed in mean grey-value) dependency on
the scattering angle. Solid lines: calculated values according to the Mie-scattering
theory averaged for wavelengths λ = 400 . . . 800nm. Top left: Hollow glass spheres
(diameter d = 35µm, refraction index n = 1.0); top right: polycrystalline seeding
particles (d = 30µm, n = 1.6), (diploma thesis by Garbe [1998]); bottom: perspex
seeding particles (d = 180µm, n = 1.6), (by Hering et al. [1998]).
28
Flow visualization
Therefore the traditional illumination technique (section 3.3.1) where usually an angle of slightly less than 90◦ was used is not well suited.
• The intensity raises rapidly for scattering angles from 80◦ to the minimum
angle. A better choice for particle tracking velocimetry (PTV) is therefore
to take the angle close to forward scattering. A drawback is the large change
of intensity with a small change in scattering angle which limits the usage
of the continuity of optical flow as a correspondence correlation criterion for
PTV7 .
3.2.1.2
Buoyancy
It is required that the particle velocity does not deviate significantly from the fluid
velocity in order to avoid distortion of the fluid motion. This is the case for the
particles used in this work.
Nevertheless the entrainment of the particles by the liquid, especially in the case
of hydrogen bubbles, has to be taken into account. The movement of solid spheres
can be calculated by the Basset-Boussinesq-Ossen equation which is in the one
dimensional case described by:
mp
du p
dt
= |{z}
G −
g(m p −mF )
|W
{z }
+ mF · dudtF + χ mF
3πd µF (u p −uF )
R
√
πρ f µF · tto
d(u p −uF )
−
dt
(3.12)
d(u p −uF )/dt 0 0
√
dt
t−t 0
where the meaning of the symbols is:
up:
3
πd /6:
mp = ρp
mF = ρF πd 3 /6:
g:
uF :
d:
µF :
χ:
velocity of the particle
mass of the particle with density ρ p
superseded mass of the fluid with density ρF by the particle
gravitational constant
flow velocity
diameter of the particle
the kinematic viscosity νF = µF /ρF
parameter which describes the dissipation of kinetic
energy to the fluid
The meaning of the terms of equation (3.12) (balance equation for the forces acting
on the particle) is the following:
7 Especially
if the light source is mounted above the water surface the refraction angle - and
therefore the scattering angle - changes with the steepness of the waves. Small capillary waves show
a large steepness and therefore the scattered light intensity fluctuates quite heavily.
3.2 Techniques for flow field visualization
29
The first term G is the sum of the external forces acting on the particle which is the
difference of the weight of the masses m p and mF . The second term W is in this
case the Stokes air resistance (for uF constant). If uF is not constant, then three
further terms have to be added. The third term is the inertia of superseded mass
of the fluid from particle. The fourth term takes the dissipation of kinetic energy
toward the liquid into account with the factor χ for the partition of the superseded
mass of the liquid by the particle. The last term is caused by the shear force of the
particle from the beginning of the particle movement (where u p 6= uF ) at time to to
the present time t.
Equation (3.12) can be simplified. Assuming that the last term (the integral) is
small compared to the other terms and setting χ = 1/2 the dissipation of kinetic
energy towards the liquid is taken into account with half of the mass from the
superseded liquid. Defining u0 = u p − uF for equation (3.12) it follows
ρ p − ρF
du0 u0
duF
+ =
(g −
)
dt
τ
ρ p + ρF /2
dt
(3.13)
with the characteristic time
τ=
d 2 (ρ p + ρF /2)
.
18µF
(3.14)
If the liquid medium is at rest (uF = 0, duF /dt = 0), equation (3.13) is written as
du p u p
ρ p − ρF
+
=
g
dt
τ
ρ p + ρF /2
(3.15)
du
Because limt→∞ ( dtp ) −→ 0, the velocity u p is constant in the limit of t → ∞. The
buoyancy velocity is therefore
u p∞ = u p (t → ∞) =
⇔
u p∞ =
ρ p − ρF
gτ
ρ p + ρF /2
d 2 (ρ p − ρF )
g
18µF
(3.16)
where u p∞ is in the direction of the gravitational force g if ρ p > ρF and against g if
ρ p < ρF such as for H2 bubbles.
Taking u p∞ from equation (3.16) the equation (3.15) results in
d(u p∞ − u p )
dt
=− .
u p∞ − u p
τ
(3.17)
30
Flow visualization
A
Figure 3.3: Hydrogen/oxygen generator: A wire frame with jagged platinum
coated tungsten wire connected to a current source. The bubbles are generated
at the inflection points.
If the initial condition is assumed as u p (t = 0) = 0 the integration of equation (3.17)
is
u p = u p∞ (1 − e−t/τ ).
(3.18)
In the case of ρ p ρF the buoyancy velocity is u p∞ = −d 2 ρF g/18µF and the time
constant τ = d 2 ρF /36µF and therefore u p∞ = 2gτ (if ρ p ρF it follows u p∞ = gτ).
3.2.2
Hydrogen/oxygen bubbles
Compared to solid particles hydrogen and oxygen bubbles have the major advantage that they are easy to produce by electrolysis and that they do not pollute or
disturb the environment. They can also be placed rather precisely into the measuring volume. A disadvantage of this method is the disturbance induced by the
hydrogen/oxygen generating device.
The device to generate the bubbles (fig. 3.3) is mounted in some distance from the
measuring volume. The distance has to be chosen appropriately depending on the
velocity range of the along-wind (horizontal) velocity of the water in the measuring
volume. De-ionized water is used and KCl is added (concentration 0.24·10−6 M)
in order to get a sufficiently high conductivity. The size of the hydrogen and oxygen bubbles is in the range of the wire8 diameter. To produce bubbles more regularly in time the oxygen cathode has to be separated and current pulses have to
be applied (up to 100 Hz). Oxygen is more irregularly produced as compared to
hydrogen. The conditions prevailing in this work did not require to separate the
oxygen from the hydrogen bubbles and the cathodes and anodes were placed just
opposite to each other in the wire frame. If the medium of the fluid is water, and
the electrolysis of water produces H2 and O2 (where ρH2 ρH2 O and ρO2 ρH2 O
with ρH2 O = 998.2 kg/m3 , ρH2 = 8.379 · 10−2 kg/m3 and ρO2 = 1.341 kg/m3 ), the
8 platinum
coated tungsten
3.3 Experimental setup for particle tracking velocimetry
Figure 3.4: Scheme of the experimental setup used for classical PTV. A light sheet
is generated by a cylindrical lens (from Hering [1996]).
kinematic viscosity is µH2 O = 1.004 · 10−3 kg/ms, and from equation (3.16) the
buoyancy (ascending) velocity is calculated as
u p∞ (d = 30µm) = 0.49 mm/s
for H2 and O2 bubbles of 30µm in diameter. The buoyancy velocity is almost
immediately achieved with a time constant τ = 25µs.
3.3 Experimental setup for particle tracking velocimetry
3.3.1
Traditional setup
The two dimensional - traditional - flow visualization uses a light sheet to cut out
a small slice of a fluid volume. The light sheet can be generated for example by a
Laser scanner or by an optical system (figure 3.4).
3.3.1.1
Observation close to the water surface
The main interest of this study is the influence of small scale waves, such as capillary waves, on the flow field. The typical wave length is in the order of cm. There-
31
32
Flow visualization
fore the aim is to get as close as possible to the water surface and to obtain a high
spatial resolution9 . The experimental setup has to
• minimize disturbances due to light reflections at the water surface and
• optimize the scattering angle of the seeding particles (see section 3.2.1.1)
with the angle of light source and the camera.
There are basically two ways for the camera and illumination to obtain the volume
of observations close to the water surface. In the first method a camera is looking
from the side (figure 3.4), inclined in a small angle away from the water surface,
beneath the water surface. In the second method the camera looks from the bottom
to the water surface (optical axis of the camera and lens system perpendicular to
the water surface). Each method has its advantages and disadvantages:
An important advantage of the first method is the possibility to follow the water
surface by a wave follower which shifts the observation area/volume according to
a certain reference point at the water surface.
Wave follower
This method was established and applied for Laser induced fluorescence (LIF)
measurements (Münsterer [1996]) but could also be applied for PTV measurements. The setup is shown in figure 3.5. The goal of the experiment was to visualize the two-dimensional concentration profile of gas in the aqueous boundary
layer. In this experiment the water contained fluoresceine where the fluorescence
intensity is related to the dissociated HCl concentration in the water bulk.
A Laser beam10 and a scanner mirror generate a light sheet perpendicular to the
water surface and excites the fluorescence of fluoresceine. This intensity profile is
seen by a CCD camera where the imaging area was shifted by the wave follower
scanning mirror. To avoid optical distortions caused by the scanning mirror a so
called f-theta lens system was used instead of an achromatic lens. The scanning
mirror was controlled by the line array camera attached to a micro controller (Sorcus Multilab 2) which is in turn connected to a host PC computer. The line array
camera takes a profile image perpendicular to the water surface which images the
height of the water surface. This system allows to follow waves of 5 cm amplitude.
A major drawback of this technique for application in PTV is the measurement of
the scanning mirror position; precise position measurement is required for PTV.
Deviations in the determination of the mirror position increases the error of the
particle and trajectory position in the image. Another drawback is the inertia of the
9 A high spatial resolution means a small volume of observation for a given lens system and a
given number of CCD-elements (CCD-sensor).
10 1W Argon ion laser at 488nm wavelength.
3.3 Experimental setup for particle tracking velocimetry
33
wind paddels
line array CCD
fixed
mirror
Laser beam
f-theta lens
scanner mirror
for Laser scanner
collimator
lens
Laser beam fiber
achromatic
lens
wave follower
scanning mirror
CCD camera
moving belt
Figure 3.5: Wave follower installed at the Heidelberg wind-wave facility. The
moving belt moves in the opposite direction of the flow and reduces the average
flow speed.
scanning mirror which does not allow to follow the water surface precisely - this
reduces the size of the clipping area of the image.
3.3.2
Stereo setup
Experiments were carried out at the circular Heidelberg wind-wave facility and the
newly constructed AEOLOTRON circular wind-wave facility, which is about three
times larger in the extend to the former. This section explains the stereoscopic
setups constructed for the stereo PTV at the wind-wave facilities.
3.3.2.1
Heidelberg circular wind-wave facility
The (small) wind/wave facility at the Institute of environmental Physics at the University of Heidelberg has a cylindrical shape and the measures are as given in
figure 3.6. The wind is generated by a rotating paddle ring. The main reason for
the choice of a cylindrical shape is the infinite fetch which simulates optimally the
conditions on the ocean. Because the interest of the studies lies in small scale processes such as capillary waves and related effects, which are in a range of 1 − 10 cm
in wave length, the resonant waves as a boundary effect are only a minor distur-
34
Flow visualization
Figure 3.6: Circular wind-wave facility (previous) at Heidelberg University and
the stereo PTV setup. Diameter (outer) = 4m, width = 0.3m, height = 0.7m, typical
water height = 0.25m.
bance. This is demonstrated by a good agreement between wave spectra analysis
of field and laboratory measurements (Klinke [1996]).
A major disadvantage of a circular flume is the wind induced momentum into the
water bulk which causes a water current in the wind direction. The water current
amounts up to 30 cm/s at the water surface. The effects on the flow induced by
the Corriolis-force are of minor importance. But the large (average) speed of the
water bulk reduces the time of observation: A typical area of observation is of a
size of 5 cm, which means the time of observation is about 1/6 of a second only.
To increase the time of observation a moving bottom has been installed. It rotates
with a speed of up to 0.6 m/s against the wind direction, and the mean water bulk
velocity can therefore be compensated. The dependency of the moving belt speed
to the induced surface velocity is almost linear (Münsterer [1996]).
The two CCD-cameras look from beneath the water surface through the transparent bottom and enclose symmetrically to the bottom an angle of about 90◦ . This
implies a significant refraction and therefore a dispersion of the (white) light rays
hitting the camera CCD-plane. The dispersion was reduced by the usage of a color
filter. For this reason the filter had to be chosen in such a way, that the spectral
sensitivity of the CCD is high and the light source is of sufficient spectral intensity.
The sensitivity of the CCD and the spectrum of the light source are both around
500 nm. A red-filter was used, where the band width was large enough to obtain a
sufficient intensity of the light passing through. The refraction has to be taken into
account for the model of the optical system and is discussed in section 4.3.
3.3 Experimental setup for particle tracking velocimetry
35
inner diameter 8.68m
channel width 613mm
heigth 2.4m
light fiber
termic isolated
walls
dispersion plate
water heigth
inner CCD
1100mm
outer CCD camera
hydrogen generator
Figure 3.7: Scheme for the stereo PTV and the measures at the AEOLOTRON
circular wind-wave facility at the Institute of environmental Physics, Heidelberg.
3.3.2.2
The large Heidelberg wind-wave facility (AEOLOTRON)
In the scope of this work the AEOLOTRON wind-wave facility was constructed.
The measures are shown in figure 3.7. The inner diameter is 8.68 m and the inner width of the channel is 0.614 m, and the water fill height was 1.1 m from the
channel bottom.
The walls of the channel are isolated by a 90 mm thick thermal insulator11 , which
allows to set a water temperature range of 10◦ C to 50◦ C. An inner layer of a foil
provides gas tightness of the channel and a second layer of an aluminum coated
bitumen foil on the innermost side guarantees water tightness. Similar to the small
wind-wave facility a belt driven paddle ring serves as the wind generator (0 to 25
m/s).
The two CCD-cameras look from the inner and the outer side of the channel, enclosing an angle θ = 26◦ to the flat water surface, through a pipe into the channel.
Inside the pipe a plane window of optical glass, perpendicular to the optical axis
of the lenses, works as a ‘liquid prism’ to minimize the refraction and dispersion
effects of the light. The angles and position of the cameras can be moved as shown
in figure 3.8.
11 Styrodur,
BASF AG.
36
Flow visualization
x,y
z
z
CC
q
D
x
y
CCD-camera
liquid
prism
q
wall
water
water
Figure 3.8: CCD-Camera mount at the lateral walls of the channel. The camera is
mounted on a x,y,z-shift device, the viewing angle θ of the camera can be adjusted.
The camera view is directed against the inner of the channel through a pipe. A
plane window of optical glass perpendicular to the optical axis of the lenses constitute a liquid prism. Left: Scheme of a single camera mount setup. Right: One of
the two camera mounts.
3.3 Experimental setup for particle tracking velocimetry
37
light fiber
camera holes
wire frame for hydrogen
bubble generator
Figure 3.9: Stereo PTV setup. The optical fiber guides the light into the channel,
the dispersive plate homogenizes the light and a wire frame generates the hydrogen
bubbles.
A liquid optical fiber connected with a glow-discharge lamp guides the light into
the channel (see figure 3.9). A dispersive plate homogenizes the light intensity
in the volume of observation. The hydrogen/oxygen bubble generator is mounted
beneath the volume of observation. It is slightly shifted in the opposite direction to
the mean flow, depending on the speed of the flow and the velocity of the uprising
hydrogen bubbles.
The setup for the calibration procedure is displayed in figure 3.10. The calibration
target consist of an aluminum frame mounted between the walls of the channel and
contains two grid planes separated by a given distance in z-direction. The two grid
planes consist of intersecting lines at each grid point. The two planes are separated
by a different orientation of the grid lines for each plane.
3.3.2.3
Volume of observation for stereo camera setup
The volume of observation is determined by the intersection of the two camera perspectives (figure 3.11). The optical axes of the cameras form an angle ϕ of about
26◦ in respect to the surface normal of the bottom window. It can be calculated
from the parameters of the camera (which are determined by the calibration procedure) and the projected coordinates from the edge points of the two camera sensor
planes to the world coordinates. The projected coordinates result in a ray (see section 5.3.5) and the world coordinates are determined by the top and bottom range
38
Flow visualization
Calibration target
(two grid planes)
scattering disk
hydrogen generator
(wire frame)
Figure 3.10: Calibration setup for stereo PTV measurements. The view is from the
bottom of the channel.
of the focal depth (see section C.1). Due to lens distortions and other disturbances
the rays are not straight, but slightly bent. The rays starting from the sensor and
cutting the intersection volume determine the edges of a pyramid.
If the cameras are assumed to be aligned exactly opposite to each other with the
optical axes enclosing an angle ϕ to the surface normal, and assuming furthermore the extend of the volume of observation being small in comparison to the
distance between the sensor and the volume of observation, the pyramid lines
(edge lines in figure 3.11) can be assumed to be parallel. This case can be reduced to two dimensions as shown in figure 3.12. The area A is then determined by
A = ∆x · ∆y = 4D2 · cos(ϕ) sin(ϕ) which is maximized at ϕ = 45o . The same considerations can be applied to a single CCD-sensor element. The volumetric resolution of the stereoscopic camera setup is then determined by the intersecting
volume given by the CCD-sensor element. The choice of the angle ϕ depends on
the requirements of resolution and on the restrictions given by the conditions of the
experimental setup.
3.3.2.4
Technical data of used imaging devices
The technical data of the used lens system, cameras and frame grabber are listed in
table 3.1. The frame grabbers have to have the capability to connect two cameras.
The cameras have to be synchronized in time in order to get image pairs for each
time interval of an image frame. The camera Pulnix TM 6701 AN is able to operate
3.3 Experimental setup for particle tracking velocimetry
Intersecting volume of
stereo camera set-up
View of camera 1
View of camera 2
Figure 3.11: Volume of observation given by the intersection of the two camera
views. The dashed and solid lines mark the views of camera 1 and camera 2,
respectively. Bottom and top (Z-axis) are determined by the focal depth range of
the two cameras.
∆y
ϕ
∆x
ϕ
D
Figure 3.12: Volume of intersection as a function of the angle ϕ in two dimensions,
simplified.
39
40
Flow visualization
Camera
CCD-type
Pixels
CCD-cell size
CCD size
Scanning frame rate
Pixel clock
Lens system
Aperture
Frame-grabber
Input channels
Pulnix TM 6701 AN
1/2”
progressive scanning interline
transfer CCD
648 (H)*484 (V)
9.0 µm*9.0 µm (square pixels)
5.83 mm*4.36 mm
60 Hz full frame (non interlaced)
25.49 MHz
Tamron Zoom 35-70 mm
(modified with shift and tilt device
for Scheimpflug setup)
3.5-4.5
Mikrotron INSPECTA 2
3*8 Bit RGB
Sony XC-73CE
1/3”
752 (H)*582 (V)
6.5 µm*6.25 µm
6.0 mm*4.96 mm
30 Hz (non interlaced)
14.1875 MHz
fixed focal length 25mm
2.8
Eltec PCEYE 4
3*8 Bit RGB
Table 3.1: Technical data of lens system, camera and frame-grabber.
in full resolution (648 horizontal and 484 vertical pixels) mode at 60Hz (standard
video is 30 Hz). The higher the frame rate, the higher are the particle velocities
which can be extracted. Therefore the frame rate is an important parameter for the
dynamic resolution of the flow field.
The Sony XC-73CE was used for the determination of the hydrogen bubble velocity distribution where the frame rate (30 Hz) was of no importance.
3.3.3
Improvements for stereoscopic flow visualization
Compared to the classical PTV setup, the following improvements for the experimental setup were made:
• illumination from the top of the water surface
• liquid prism
• Scheimpflug camera setup
The classical PTV used a light source from the bottom of the channel (see section
3.3.1). This resulted in light reflections at the water surface which disturb the
images taken by the CCD camera. An illumination from the top avoids this effect
but results in light intensity variations depending on the steepness of the water
surface.
3.3 Experimental setup for particle tracking velocimetry
A main disadvantage of the stereo PTV setup in 3.3.2.1 was the refraction of light at
the air-water interface of the channel (air/ perspex/ water interfaces) due to the large
angle of refraction (about 45◦ ). This caused a bending of the epipolar lines which
could not be compensated well enough by a multiple media module in the calibration and stereo coordinate evaluation module. Therefore a liquid prism (see figure
3.8) was introduced in the AEOLOTRON wind-wave facility (section 3.3.2.2) and
the angle of refraction was minimized.
A further improvement is the Scheimpflug setup (discussed in the next section)
which increases the depth of field range. This depth of field is a crucial parameter
for the spatial resolution of the stereo PTV method. The depth of field for a usual
lens system is only about 10-30 mm whereas the volume of observation should
exceed a depth range of about 30 to 150mm.
3.3.3.1
Scheimpflug stereo camera setup
Three different typical stereoscopic setups are shown in figure 3.13 (a) to (c). For
the translational method 3.13 (a) all its planes (object, lens and image) are parallel.
For the angular method 3.13 (b) the image and lens planes are parallel, the object
plane is turned by an angle of θ towards these planes.
The accuracy of the out of object plane component z depends on the distance S,
where the translational system imposes an upper bound of the value S and therefore
a limit on the visible area in the object plane. The angular method implies a larger
angle θ which has no upper bound. Therefore the accuracy of the out of plane
component of the angular system is, in principle, higher. Furthermore the visible
area in the object plane is larger in the angular system due to the CCD sensor in
the image plane, which is in standard cameras symmetric to the center of the lens.
Obviously the objects in the object plane are no longer in the focal plane in the
case of the angular method. This leads to an increase of the depth of field of the
optics. The depth of field is given by
∆z = 4(1 + M −1 )2 O2 λ
where M is the magnification of the object, O is the f-number (section C) and λ is
the wave length of the light source.
A sketch of the Scheimpflug setup is shown in figure 3.13 (c). The Scheimpflug
condition is described in the following: The center of the coordinate system is
placed to O, the z-distance of the planes from the lens plane through O is
for the lens plane
z=0
for the object plane
41
42
Flow visualization
z = x · tan(θ) − do
and for the image plane
z = −x · tan(α) + di .
3.3 Experimental setup for particle tracking velocimetry
Obejct Plane
θ
S
Image Plane
∆X
Camera 1
Camera 2
(a)
Obejct Plane
θ
θ
Le
ns
Pl
an
e
Im
ag
e
Pl
an
e
∆X
Camera 2
Camera 1
(b)
Obejct Plane
θ
θ
α
Le
do
ns
P
lan
e
Im
age
ne
Pla
di
∆X
Camera 1
Camera 2
(c)
Figure 3.13: Different stereoscopic methods. (a) translational method, (b) angular
method, (c) Scheimpflug method.
43
44
Flow visualization
Chapter 4
Geometry of the stereoscopic
system
To handle the system numerically, a model has to be specified. This is done in this
chapter by splitting the system into its single components. Basically the system
looks as shown in figure 4.1. Each component requires a description. The stereo
camera setup used here requires two of these basic systems. The exactness of the
model description is of crucial importance in order to obtain a high accuracy in the
calibration procedure.
4.1 Model of the stereoscopic system
The stereoscopic system requires at least two different views of one scene. This
can be obtained either by two or more cameras for a stationary or non stationary
object, or by a moving camera where the object is stationary. A two camera system
including the used coordinate systems is shown in figure 4.2.
→
−
The object is located at U in the world coordinate system (U1 ,U2 ,U3 ). The object
→
−
point U is projected by the perspective transform through the center of projection
−
−
(‘pinhole’) to the image planes of the left and right camera →
u l and →
u r respectively.
The origin of the sensor coordinate system (xl,r , yl,r ) is given by the intersection
→
−
→
−
point - the principal point P l on the left and P r on the right image plane - of
the optical axis and the image plane. The image plane is in a distance of the focal
length Zl,r = f from the center of perspective. The sensor coordinates are transformed to the the image coordinate system (ul,r , vl,r ), which is usually chosen with
the origin at the top left corner.
Principally the camera model consists of intrinsic and external parameters. The
external parameters describe the geometry and position of the camera system. This
is shown for the simplest model, the pinhole model, in section 4.2.2. The intrinsic
46
Geometry of the stereoscopic system
Camera
Lens System
Mutliple Media
Object
CCD Sensor
Figure 4.1: Basic system and its components for each camera setup.
+, ,+
Pr
u
principal
Point
ur
v
r
)* r
x l xr
P
l
y
ul r
coordinate system
camera
center
of
perspective
for right camera
y O
r v
l
l
X
Y
"
r O
center of perspective
r
l"
!
!
!
!
!
!
!
!
"
"
"
"
"
"
X
Z
l r
Y
l
camera coordinate
system
Z left
ray
l right
ray
for left camera U
%
%
&&
&&%&% 3
%&
%
%
%
&
&
%&
&
&%&%&% %
%
&
%&
&
&%&%&% %
%
&
%
%
&
&
%
%
&
&
world coordinate
system
'(
'(
(
%$# ('('(' (
&
&%$
# # # # # # $
$
$
$
$
$# '(
'
'
'
(
''' (
''' ('('' (
U=(U
object
,U
,Ucoordinate
)
( (
( ( (
U2
image plane for left camera
image plane for right camera
image coordinate system
U
1
ul
1 2 3
~ is
Figure 4.2: Basic system for the pinhole camera model. The world coordinate U
projected to the image coordinate ~xl on the left, and ~xr on the right image plane.
4.2 A simple camera model
47
image plane
image
pinhole
object
Figure 4.3: Pinhole camera model.
parameters are not related to position and geometry but to camera and lens system specifics. It transforms the sensor coordinate system to the image coordinate
system (the ‘user’ coordinates). This is discussed for the pinhole model in section
4.2.3 (see also Faugers [1993]).
4.2
A simple camera model
Looking from the geometric point of view, the simplest model is the pinhole camera
model. Figure 4.3 shows the setup; the first screen has been punched with a small
hole where the rays of the emitted or reflected light passes.
The pinhole camera model represents an ideal camera, free from any distortions
and other artifacts. The model is a rigid body transformation and a projective
transformation applied afterwards. The pinhole camera is explained in section
4.2.2. The model can be described in a matrix style applying the homogeneous
coordinate system which is shown in the next section.
4.2.1
Homogeneous coordinate system
→
−
The perspective projection P maps the camera coordinate system X to the image
→
−
−
−
coordinate system →
x (P : { X ∈ IR3 } 7−→ {→
x ∈ IR2 }). Therefore, the relation be→
−
T
−
tween the object coordinate X = (X,Y, Z) and the image coordinate →
x = (x, y)T
is:
48
Geometry of the stereoscopic system
x
y
⇔
f
=−
Z
− Zf =
x
X
− Zf
y
Y
X
Y
(4.1)
(4.2)
=
The homogeneous space is introduced to be able to write equation (4.1) in a matrix
form in the same way as the rotation and translation can be written in a matrix from.
This allows to express the transformations in terms of the linear algebra and offers
a way to deal with the methods and rules of linear algebra. The homogeneous space
is an embedding of the n-dimensional physical space into the (n + 1)-dimensional
−
−
e
space by introducing a scaling factor t: →
x : {(x1 , x2 , . . . , xn )T ∈ IRn } 7−→ →
x : {t ·
T
n+1
→
−
e
(x1 , x2 , . . . , xn , 1) ∈ IR } where x are called homogeneous coordinates. For
example considering a plane ((x, y)T ∈ IR2 ), this plane is in the homogeneous space
((x, y,t)T ∈ IR3 determined up to a translation on a straight line (in the direction
perpendicular to the plane) by the scaling factor t. The physical space is obtained
from the homogeneous space for t = 1.
→
−
The physical space X : {(X,Y, Z)T ∈ IR3 } is embedded into the four dimensional
→
−
e : {t · (X,Y, Z, 1)T ∈ IR4 , t ∈ IR}.
homogeneous space by X
The projective transformation P (equation (4.1)) expressed in homogeneous coor→
−
e ∈ IR4 } 7−→ {→
−
e
dinates (P : { X
x ∈ IR3 }) is written as:



−f
xe
 ye  =  0
0
s
|

X
0 0 0
 Y 

−f 0 0  t ·
 Z 
0 1 0
1
{z
}


(4.3)
P
where xe = x · s, ye = y · s. The physical coordinate system refers to the homogeneous
coordinate system for t = 1.
The matrix form (4.3) is written as
→
−
e.
→
−
e
x =PX
4.2.2
(4.4)
Pinhole camera model
The transformation equation describing the pinhole camera model is
→
−
e
→
−
e
x = P·R·T · X
(4.5)
4.2 A simple camera model
49
where T represents the translational part, R the rotational part and P the perspective
projection transformation coordinates.
The rotation in the physical space is


c(γ) · c(β) −s(γ)c(α) + s(γ)s(β)s(α) s(γ)s(α) + c(γ)s(β)c(α)
c(γ)c(α) + s(γ)s(β)s(α) −c(γ)s(α) + s(γ)c(α)s(β) 
R3 =  s(γ)c(β)
−s(β)
c(β)s(α)
c(β)c(α)
where α, β, γ are the Euler angles, c(. . .) stands for cos(. . .) and s(. . .) for sin(. . .).
In the homogeneous space the rotation is
R3 0
R=
0T 1
with 0 = (0, 0, 0).
The translation in the physical space is given by the vector T3 = −(xo , yo , zo )T and
is written in homogeneous coordinates as
1 T3T
T=
0 1
with 1 the unit matrix in IR3 .
4.2.3
Pinhole camera model including intrinsic camera parameters
−
−
e
e
The transformation from the sensor coordinates →
x to the image coordinates →
u
(usually the origin is the upper left corner of the image array) is described by the
transformation matrix B. The intrinsic camera parameters include the (effective)
focal length f , the image center (‘principal point’) (Cx , Cy ) - which is defined as
the intersection point of the optical axes of the lens system and the CCD sensor
plane, and parameters related to the CCD sensor including
• Nx , Ny , the effective number of sensor elements in an image line (x-direction)
and the effective number of lines (y-direction),
• Sx , Sy , the effective size of the sensor in x and y direction and
• α, a scaling factor which considers horizontal pixel sensor deviation due to
synchronization of frame-grabber and CCD camera.
The transformation matrix for the internal parameters is given by
→
−
−
e
e
u = B·→
x
50
Geometry of the stereoscopic system
where the 3 × 3 matrix B is
α NSxx

B= 0
0

0
Ny
Sy
0

Cx

Cy  .
1
(4.6)
The model is then finally written as
→
−
e
→
−
e
u =F·X
where F is the complete transform matrix (‘fundamental matrix’) with
F = B · P · R · T.
4.2.4
(4.7)
The linear camera model
A linear camera model is obtained quite naturally when applying homogeneous
coordinates. This model is described by the Direct Linear Transformation (DLT):
→
−
e
→
−
e
u i = AU
i
(4.8)
where A is a 3 × 4 projective transformation matrix, projecting the vector
→
−
−
→
−
e = (→
U
U i , 1)T (with world point U i = (U1,i , U2,i , U3,i )T , to the image point
i
−
→
−
−
e
u i = (ui , vi )T in homogeneous coordinates →
u i = t · (→
u i , 1)T (see equation (4.3)).
Equation (4.8) is in detail:

 
 U
ui ti
a11 a12 a13 a14  1,i
 vi ti  =  a21 a22 a23 a24   U2,i
 U3,i
ti
a13 a23 a33 a34
1



.

The parameters a11 , . . . , a34 can be solved by eliminating ti and solving the matrix
equation
La = 0
where
4.2 A simple camera model

U1,1
0
..
.
y1
0
..
.
z1
0
..
.





 U1,i U2,i U3,i
L=
 0
0
0

 ..
..
..
 .
.
.

 U1,N U2,N U1,N
0
0
0
51
1
0
0 U1,1
..
..
.
.
1
0
0 U1,i
..
..
.
.
0
U2,1
..
.
0
U3,1
..
.
0
1
..
.
−U1,1 u1
−U1,1 v1
..
.
−U2,1 u1
−U2,1 v1
..
.
−U3,1 u1
−U3,1 v1
..
.
0
U2,i
..
.
0
U3,i
..
.
0
1
..
.
−U1,i ui
−U1,i vi
..
.
−U2,i ui
−U2,i vi
..
.
−U3,i ui
−U3,i vi
..
.
0 −U1,N uN
1 −U1,N vN
−U2,N uN
−U2,N vN
−U3,N uN
−U3,N vN
1
0
0
0
0 U1,N U2,N U3,N
and
a=
a11 a12 a13 a21 a22 a23 a13 a23 a33
.
The parameters a11 , . . . , a34 do not have any physical meaning. Physical parameters
can be extracted by a method based on RQ decomposition proposed by Melen
[1993]:
The transformation describes an extension of the pinhole model, including additionally the
1. correction of the measured image coordinates in the principal point (image
−
center) →
u = (uo , vo )T
2. and difference in scale and lack of orthogonality between the image axes by
(small) compensation coefficients b1 and b2 :
0 ui
1 + b1
b2
ui
uo
=
−
v0i
b2
1 − b1
vi
vo
|
{z
}
=: B
or
→
−
−
−
u 0 = B · (→
u −→
u o)
In homogeneous coordinates this can be equivalently expressed by
 
 
 

u0i
ui
1 + b1
b2
0
1 0 −uo
 v0i  =  b2
1 − b1 0  ·  0 1 −vo  ·  vi 
1
0
0
1
0 0
1
0
|
{z
}

=:V
or
0
→
−
−
e
e
u = B ·V · →
u
−u1
−v1
..
.






−ui 

−vi 

.. 
. 

−uN 
−vN
52
Geometry of the stereoscopic system
0
−
−
e
e
The inverse transformation, from →
u →→
u , is needed, where B−1 and V −1 has to
exist.


b1 − 1 −b2 0
1
 −b2 1 + b1 0 
B−1 =
1 − b21 − b22
0
0
1


1 0 uo
V −1 =  0 1 vo 
0 0 1
The matrix A (equation (4.8)) can therefore be written as
A = λ ·V −1 · B−1 · F · M · T
(4.9)
(scaling factor λ 6= 0, affects only t from homogeneous coordinates, equation (4.3)).
Altogether the physical model1 has eleven degrees of freedom - eighteen minus
six parameters from the orthogonality constraint of M and minus the scaling factor λ transforming to the physical space). The DLT matrix A can be computed
by equation (4.9) provided the eleven physical camera parameters are given. The
opposite problem, the extraction of the eleven physical camera parameters and the
scaling factor λ from a given DLT matrix was solved by Shih and Faig [1987] and
Faugeras and Toscani [1987] who developed complete and exact decomposition
methods. Melen [1993] proposed an improved method based on the RQ decomposition method. If all object points are located on a flat plane, zi = 0 can be chosen
and the 4 × 4 DLT reduces to a 3 × 3 matrix.
4.2.5
Camera model including lens distortion
A significant distortion is the radial distortion of the lens and the decentering distortion. The effect on a regular grid is illustrated in figure 4.4 and figure 4.5.
−
The pinhole camera model is extended with a shift of the origin →
u 0i and a term
→
− →
δ (−
q 0i ) which models the linear and the lens distortion (Fryer [1989]):
→
− −0
→
−
−
−
ui=→
v +→
u 0i + δ (→
u i)
1 The physical model is obtained by the projection of homogeneous coordinates to the physical
space.
4.2 A simple camera model
53
⇓
⇓
Figure 4.4: Lens decentering distortion (x-direction).
−
where →
u 0i are the undistorted coordinates from equation (4.4). The most widespread
→
−
term for δ in close range photogrammetry is
δu
b1 u0i + b2 v0i
b3 u0i ri2 + b4 u0i ri4 + b5 u0i ri6
2b6 u0i v0i + b7 (ri2 + 2u02
)
i
=
+
+
0 0
δv (u0i v0i )
−b1 v0i + b2 u0i
b3 v0i ri2 + b4 v0i ri4 + b5 v0i ri6
b6 (ri2 + 2v02
i ) + 2b7 ui vi
|
{z
} |
{z
} |
{z
}
linear
radial
tangential
02
where ri2 = u02
i + vi . The linear distortion is described by b1 and b2 , the radial
distortion by b3 to b5 and the tangential distortion by b6 and b7 . Depending on the
amount of distortion and the required accuracy the number of coefficients for the
radial distortion may be reduced or increased. In this work the first two (b3 and b4 )
were used. Thus four parameters modeling lens distortion are included into this
extended camera model.
The parameters of this non linear camera model can not be determined analytically in contrast to the DLT discussed previously. Instead, a numerical method to
minimize the camera transform F has to applied
Res = ∑ (F(Xi ) − xi )2 → minimum
i
54
Geometry of the stereoscopic system
(a)
(b)
Figure 4.5: Radial lens distortion barrel (a) and pincushion type (b).
where Xi are the world coordinates given by the calibration points of the calibration
target (such as seen in figure 3.10), the xi are determined in the 2D calibration procedure (section 5.1) and F(Xi ) is the projection to the image coordinates. The parameters are determined by a Least-Square optimization such as the gradient based
Marquardt-Levenberg method. Start values are obtained from the DLT method.
They guarantee vicinity to the global minimum and thus a high probability not to
reach a local minimum.
4.2.6
Camera model and the Scheimpflug condition
For the Scheimpflug method the three planes - image plane, lens plane and object
plane - are not parallel and intersect in a point (section 3.3.3.1). The geometry of
the Scheimpflug method is shown in figure 4.6. The object plane with coordinate
system (x, y) encloses an angle α with the optical axis and the image plane an angel
θ.
For the Scheimpflug method the following relations can be derived from the geometry:
On the object side the following equations are valid:
o = do − ∆o
∆o = x · cos(α)
O = x · sin(α)
(4.10)
4.2 A simple camera model
optical axes
Θ
O
ο
d
ob
jec
ge
e
ima
lan
pla
tp
y´
∆ι
i
lens plane
P(x,y)
P´(x´,y´)
I
x´
ne
α
x
ι
do
∆ο
y
55
Figure 4.6: Geometry of the Scheimpflug projection. The y-axis is perpendicular to
the view, di is the image distance and do the object distance. Further explanations
are found in the text.
and for the image side:
i = di + ∆i
∆i = x · cos(θ)
I = x · sin(θ).
(4.11)
Taking the known projection equation
O o
=
I
i
and substituting equations (4.10) and (4.11), the Scheimpflug transformation relation of the image coordinates (x0 , y0 ) and the object coordinates (x, y) are found:
x0 =
di · sin(α)
x
d · sin(θ) − x · sin(α + θ)
|o
{z
}
trans f ormation f actor
(4.12)
56
Geometry of the stereoscopic system
Figure 4.7: Effect of the Scheimpflug transformation (equations (4.12) and (4.13)
on a regular grid pattern.
y0 =
di · sin(α)
y
do · sin(θ) − x · sin(α + θ)
(4.13)
The transformation factor (equation (4.12)) is a function which depends on x only.
The same transformation factor is also applied for the transformation2 in y, equation (4.13).
This transformation includes four parameters di , do , α and θ. The parameters have
to be included into the calibration procedure. The object distance di and the angle
to the object θ were already included in the camera model. Thus the two parameters
of the image distance do and the angle α to the image plane have to be determined
additionally. Altogether the camera model has 17 parameters (15 from the camera
model with lens distortion and from the Scheimpflug condition another two).
The result of this transformation (4.12) and (4.13) on a regular grid is shown in
figure 4.7.
4.3
Multiple media geometry
If different media are involved in the optical setup then the optical geometry has
to be taken into account. Three different media were used here: water, glass or
perspex (window) and air. The larger the refraction angles, the larger the deviation
→
−0
−
→
∆R , which is the distance from the projected image point xe from the image point
2 The independence of the transformation factor is due to the parallel invariance in the y-direction.
4.3 Multiple media geometry
57
Z
X
world
Z=0
X
n
Z
3
Z
3
b
world
Z
3
β
Z2
β
1
Y
b
n
X 2
n
Z2
2
Z
1
1
n
1
n
2
β
3
n
3
2
1
X,Y
Z
1
x´
camera
x´
camera
~
∆R
~
x´
∆R
x´
β
~
x´
Figure 4.8: Multiple media geometry. A cross section through the volume in X, Zplane is shown on the right side.
−
without different media →
x 0 (shown in figure 4.8). From the geometry of figure 4.8
the following equation is valid:
R = Z1 · tan β1 + Z2 · tan β2 + Z3 · tan β3 , R = (Z1 + Z2 + Z3 ) · tan β1
and for the refraction (Snell´s law)
n1 · sin β1 = n2 · sin β2 = n3 · sin β3 .
This equation system has to be solved numerically3 in order to obtain the coordi→
−0
nate xe .
Dispersion is another significant effect which has to be considered if a non-monochromatic light source is used (such as the glow-discharge lamp used in this work).
The dispersion for visible light varies significantly stronger in water (1.4%) as
compared to air (0.008%), see Höhle [1971]. Light of shorter wavelength (blue)
is refracted more strongly than light of longer wavelength (red). This effect leads
to a color hem at the object projected on the image plane. If the sensitivity of the
sensor is not constant, as is true for the CCD sensor where the sensitivity for red is
larger than for blue, the center of mass of the projected object is moved. This effect
can be reduced either by avoiding large refraction angles (such as liquid prisms do,
3 For
− 0
→
− →
example in Maas [1992]; start value is Ro = X − xe , then β1,2,3 and ∆R are calculated.
The iteration is done with Rn+1 = Rn + ∆R.
58
Geometry of the stereoscopic system
water
blue
n
red
1
n
air
2
image plane
red
blue
Figure 4.9: Dispersion of light in water for multiple media geometry and the effect
of refraction at the water-air-water interface. The refraction index of water is n1 =
1.33 and of air n2 = 1.
applied in the stereo setup in the AEOLOTRON) or by reducing the bandwidth of
the light source by a color filter (applied in the Heidelberg wind-wave facility).
Further deviations can be caused by inhomogeneities of the media in respect to the
refraction index (due to temperature or pressure) or due to glass windows being
not exactly plane. All of them are of minor importance for the setup as used in this
work.
In fact, if the multiple media deviations were included into the calibration procedure, the convergence rate and residue for the parameter estimation (section 4.2.5)
of the camera model (applied for the Heidelberg wind/wave facility, see Engelmann
et al. [1998]) became worse. Because a real camera position in world coordinates
is not required, the camera location is virtual.
Chapter 5
Image sequence analysis for
stereo PTV
The main parts of the stereo-particle tracking velocimetry (stereo-PTV) method is
shown in figure 5.1. For an overview of the different PTV methods see Engelmann
et al. [1999b]. After the optical visualization of the flow field the two camera setups supply two image sequences for each camera perspective. These sequences
are then processed separately. First the segmentation of particles - which are imaged as streak lines - is performed, as discussed in section 5.2.1. Next the particle
tracking velocimetry method is applied (section 5.2). Finally the stereo correlation algorithm (section 5.3) is applied for the two processed image sequences and
provides the trajectories in three-dimensional space (section 5.3.5).
To obtain quantitative results in image processing the calibration procedure is an
essential part. The calibration procedure is discussed in detail in the next section.
5.1 Calibration
The aim of the stereo calibration procedure is to obtain the parameter set of the
camera model 4.2. This is required to apply the stereo correlation algorithm and to
reconstruct the three-dimensional coordinates of the particle trajectories (sections
5.3 and 5.3.5).
The camera calibration procedure consists of two basic steps which are discussed
in the following sections:
• The 2D calibration procedure and
• the parameter estimation of the 3D camera model.
60
Image sequence analysis for stereo PTV
Flow visualization
acquisition
Image
Image sequence 2
Image acquisition
Image sequence 1
Segmentation
of particle streaks
Segmentation
of particle streaks
Particle tracking velocimetry
Particle tracking velocimetry
Stereo correlation
of multi-image streaks
3D-trajectorie
reconstruction
Figure 5.1: Overview of the stereo particle tracking velocimetry.
5.1.1
The choice of the calibration target
Investigations where performed at the Heidelberg wind-wave facility and the large
Heidelberg wind-wave facility. According to the different properties such as geometry and accessibility, the mounting and calibration target was appropriately
chosen. In both cases a grid pattern was used as a calibration target. The calibration procedure requires calibration marks not only in one plane, but at least
in two different planes to solve the equations of the camera model. The quality
of the calibration target is essential for the resolution obtained by the calibration
procedure.
The experimental setup of the Heidelberg wind-wave facility was described in section 3.3.2.1. The calibration target is a quartz glass plate with grid lines imprinted.
The grid lines where imprinted onto the glass by a Lift-Off technique: The glass
surface is coated with a photographic layer and this layer is exposed to light through
a photographic mask (grid pattern). The light exposed part (the grid lines) is then
removed and a metallic layer (60-80 nm thick aluminum or chromium) is put onto
the surface. Removing the photographic layer results in a grid pattern of the metal
layer. The precision of the grid lines depends mainly on the used mask. The
mask was produced by a photographic printer (Linotype) which avoids thermic
deformation of the substrate which usually appears if Laser printers are used. The
resolution of the mask and the grid is 100 nm.
5.1 Calibration
The calibration grid (a size of 5 × 5 cm2 ) was mounted on a linear shifting table and
moved in z-direction, perpendicular to the flat water surface. A light source (glowdischarge lamp, Cermax) was mounted on the top. The light was guided through a
collimator lens and a red filter was used to reduce the chromatic dispersion effects
due to the refraction at the water surface. Images of the grid were taken in several
distances from the water surface at z = 0 to a depth of z = 5 cm for both cameras.
In the case of the AEOLOTRON wind-wave facility (section 3.3.2.2) the shifting
device could not be used. Similar to the calibration target used for the Heidelberg
wind-wave facility a grid was used as a calibration target (see figure 3.10). The
device consists of two grid planes separated in a given distance in z-direction (30 −
60 mm). A foil with a grid pattern of a size of 20 × 20 cm2 was used. The Lift-Off
technique could not be used because of its size limitations. As a light source the
same glow-discharge lamp was used, but a color filter was not necessary due to
the liquid prism construction (which minimizes chromatic dispersion effects at the
air-window-liquid interface).
The two layer calibration grid requires an extended calibration procedure, as compared to the single grid plane moved by the shifting device. The next section explains this 2D-calibration procedure. In principal the procedure is the same in the
case of the shifting device calibration, but the grid plane separation is skipped.
5.1.2
The calibration procedure
The aim of this 2D calibration was to identify the image coordinates with the world
coordinates of the object, where the object is the 2D grid pattern. For the 3D
calibration procedure two grid planes are required. There are two possibilities:
1. One plane which is moved in z-direction - at least two positions - where at
each position a (temporally averaged) image is taken.
2. Two planes in fixed position and (z-) distance (figure 3.10), where the image
includes both image planes.
The first method requires a precise moving facility such as a stepper of micrometer
precision. There is no separation of the two planes necessary as in case 2. For
the calibration target used in the large wind-wave facility method 2 was applied.
The advantage of this method is the simpler setup. Furthermore it requires less
physical space for mounting the calibration target. Here the cross line patterns of
two grid planes are projected into the single camera image. Therefore a procedure
is required which separates the two planes.
The following procedure was used to find the center of the intersecting lines (figure
5.2):
61
62
Image sequence analysis for stereo PTV
Segmentation
Mask images
Grid 1
Grid 2
Cross correlation
Segmentation
...
same as grid 1
Cross detection
proceed wirth
subpixel detection
Figure 5.2: Separation of two grid planes and localization of the intersecting lines.
5.1 Calibration
63
Figure 5.3: The cross function equation (5.1) with parameters w1 = w2 = 0.2,
o1 = o2 = 0, ho = 100, h1 = 80, θ1 = −60o , θ2 = 40o .
The first step is the background segmentation. It takes the inhomogeneous illumination into account.
The basic idea is to use the two cross types (for instance rotated by 45◦ ) as template
masks and to perform a cross correlation between the image and the two masks. As
the second step the cross type is matched with the image and yields the two cross
planes separately (second step in figure 5.2). The two cross types are now clearly
visible and the cross correlation shows a maximum at the line intersection of the
crosses. After background separation, the third step is a local maximum search1
which yields the line intersections (bottom left in figure 5.2).
Taking the maximum from step three, the center of mass is determined in step four
and serves as a starting point for the center point of the intersecting lines from
which the direction (angle) of the cross arms is found by a line matching method.
In step five the matching is done in a certain circular region where the middle
point is the previously determined center point (bottom right in figure 5.2, region
marked with a circle). The line is rotated stepwise around the line intersections and
matched to the image data. The maximum of this matching characterizes the line
and therefore its angle.
The previous procedure supplies the start parameters for step six, where the center
of the intersecting lines is determined with sub-pixel precision. This is done by a
minimization routine (modified Marquart-Levenberg method) where the function
to be minimized is:
f (x, y) = ho +h1 ·(1−e−w1 ((y−o2 )·cos θ1 −(x−o1 ) sin θ1 ) )(1−e−w2 ((y−o2 )·cos θ2 −(x−o1 ) sin θ2 ) )
(5.1)
2
and the cross is shown in figure 5.3. The parameters are ho + h1 for the background
(plateau), ho being the height (or ‘depth’) of the intersecting lines, the intersec1 The
maximum in a certain region, where the region size is in the order of the cross distance.
2
64
Image sequence analysis for stereo PTV
Figure 5.4: Streak image of particles. Due to the time of exposure ∆t, spherical
particles are not imaged as circular objects but ‘smeared’ over the time interval ∆t
and therefore imaged as streaks.
tion point (o1 , o2 ) and the two angles θ1 and θ2 of the two intersecting lines with
thickness w1 and w2 respectively.
With the knowledge of the cross distance and the z-distance of the two planes the
world coordinates and the corresponding coordinates for the image plane are obtained. This list of image plane coordinates and world coordinates are the input for
the 3D calibration procedure discussed in section 4.1 and the parameter estimation
for the stereo camera model is done as shown in section 4.2.5.
5.2
The particle tracking velocimetry (PTV) algorithm
This section describes the two dimensional (classical) particle tracking velocimetry
(PTV) algorithm which is a part of the image sequence analysis shown if figure
5.1. The purpose of the PTV is to trace each individual particle in an image over
a sequence of images. The particles themselves are imaged as streaks2 (figure
5.4). These streaks have to be identified in the images. First the streaks have to be
found in the images which involves a segmentation procedure (discussed in section
5.2.1). This separates the pixels belonging to the streaks from the background. The
segmented streaks are identified by a labeling procedure which marks each pixel
belonging to a particle-streak uniquely. Finally the identified streaks have to be
traced to the following images by solving the correspondence problem.
The steps of the PTV algorithm are shown in figure 5.5 and discussed in the subsequent sections.
−
−
the particles grey-value distribution at location →
x is g(→
x ) the streak is integrated over the
R to +∆t →
0 0
0.
−
−
time of exposure: G(→
x (t)) = t=t
g(
x
(t
))
dt
o
2 If
5.2 The particle tracking velocimetry (PTV) algorithm
image number 1
Segmentation
of particle streaks
next
image number
Identifiy particle streaks by
labeling
Correspondence problem
solving
until end of sequence
trajectories
Figure 5.5: Overview of the PTV-Algorithm.
5.2.1
Segmentation
The separation of the object from its background is called segmentation. For a
pixel at location (x, y) in the original image g the segmented image gs is given by
the following operation:
gs (x, y) =
1 : g(x, y) ∈ object (streak)
0 : g(x, y) ∈ background
Segmentation is the most critical step. The more streaks found, the better the segmentation works. Not segmented streaks are lost for further processing.
The segmentation should consider the following points for PTV:
• Variable size range of particles: The size distribution of the particles can
vary up to a factor of 4 - typically the particles show, due to manufacturing
processes, a normal distribution; i.e. for polycrystalline Optimage seeding
particles (30µm average diameter) it is about ±20% FWHM. Likewise, the
size of the hydrogen bubbles can vary as shown in figure 6.8.
• Ignore light reflections: The walls and the water surface can be the cause
for light reflections which are imaged as large ‘objects’. A size criterion
65
66
Image sequence analysis for stereo PTV
(upper threshold for the number of pixels of a streak) for the object avoids
this problem.
• Inhomogeneous illumination: This is caused by a non-ideal homogeneous
light source and by light reflections. If the illumination is from above the
water surface the illumination becomes inhomogeneous inside the water bulk
due to refraction at the water surface. This occurs if the water surface turns
wavy and the refraction angle changes - the main reason for inhomogeneity.
If the time scale of the wave frequency is in the range of the frame rate of
the camera, this inhomogeneity avoids taking the continuity of flow (section
5.2.3.1) as a criterion.
• Dynamic grey-value range: The grey-value of a streak depends - beside the
illumination intensity - on the speed of the particle. Moving particles are
imaged as streaks due to the time of exposure (chosen as maximally). If the
velocity is higher, the streak length is larger and due to the continuity of the
optical flow the grey-value intensity is therefore lower (see section 5.2.3.1)
Global segmentation methods are not well suited for the PTV. Local properties of
the particles/streaks have to be taken into account.
Two methods are introduced in the following part, region oriented methods and
model based methods. In this work the region growing method was used because
of lower computational costs and therefore lower processing time.
5.2.1.1
Region oriented segmentation
Region growing method
This segmentation method looks for regions of similar characteristics in an image
and fuses them together. By Matas and Kittler [1995] this method was proposed
and applied by Wierzimok and Hering [1993]. The principle is illustrated in figure
5.6:
• The n × n, n = 1, 2 . . . N square surrounding of a central pixel is enlarged
gradually until no other pixel of the object (streak) is found. If the square
intersects with another object in its 8-neighborhood, the two objects are considered as connected.
• The extent of the object (streak) is determined by a significance level (threshold) g of the grey-value.
Local properties of the segmented streaks are extracted (see figure 5.7): The
absolute height gmax , the minimum of the height difference
5.2 The particle tracking velocimetry (PTV) algorithm
central pixel
n=3
2
Streak 1
1
Streak 2
intersection
Figure 5.6: Principle of the region growing method.
∆g = mini (gmax − gmin,i ) where gmin,i , i = 1 . . . 4, are the minima (in horizontal and vertical direction). An interpolation of the grey-value threshold
gi, j is done by an ellipse (figure 5.7 (b))
gx,y =
s
g2i
∆y2
∆y2 + ∆x2
+ g2j
∆x2
∆x2 + ∆y2
where the threshold height in horizontal and vertical direction is given by
gi = gmin,i + s∆g , i = 1 . . . 4 (where s is the relative threshold level). A pixel
is considered as belonging to a streak if its grey-value exceeds the threshold
gi and the width w does not exceed a given threshold.
Adaptive threshold segmentation
This method is quite similar to the previously described region growing method.
The background image is calculated by a smoothing operation (for example a Gaussian kernel) which levels out small objects such as the particles. The size of the
objects which are filtered by the smoothing operation depends on the smoothing
operation (size of the filter mask). This background image is subtracted from the
original image and a threshold is applied to remove objects (streaks) of very low
intensity. This procedure is a high-pass filtering of the image which enforces the
local grey-value contrast. The size of the filtering mask must be substantially larger
than the object size. The object size is typically in the range of 3 × 3 to 15 × 15
pixels which requires a filter size of about 20 × 20 to 35 × 35. The resulting image
67
68
Image sequence analysis for stereo PTV
Grey-value
g
max
g
∆g
4
w
g(x,y)
g
G
g
min,1
g
∆y
∆x
1
g
3
min,2
Location x,y
(a)
g
2
(b)
Figure 5.7: Characteristic properties used for region growing. (a): Grey-value of a
streak depending on the location x, y on the image plane. (b): Interpolation of the
grey-value threshold g(x, y) on elliptic segments.
is binarized and a morphological dilation operation3 enlarges the object. The result
is a mask image4 which is multiplied with the original image and therefore contains all segmented particles (figure 5.8). This method was applied for flow field
measurements at the AEOLOTRON wind/wave facility. This method could also be
applied for a very simple but efficient image compression. The ratio of the number
of pixels belonging to a streak Ns to the number of pixels of the image N is small
and the compression rate is a factor of Ns /N (usually about 100-200). To avoid a
loss of data which is relevant for the determination of the streak after compression,
the streak area should be enlarged in its neighborhood (i.e. by dilation).
The local orientation method (see Jähne [1997] chapter 15.5 for details) is another
segmentation method which considers local properties of the objects region. This
method yields a certain angle of the grey-value and the associated coherency measure within a given window size5 from the second order momentum tensor. For
the angle and the coherency measure a threshold can be applied - if the calculated
value exceeds this threshold, then the associated pixel belongs to the object. This
method was used for example as a pre-segmentation procedure for a model-based
segmentation described in the next section. The disadvantage of this method is the
3 Definition
of morphological dilation operation: Input set X and result set Y , and K a structural
element (mask). K is shifted over all elements y ∈ Y , and called Ky for each y. The dilation operation
T
⊕ is defined as: Y = X ⊕ K = {y|Ky X 6= 0} (unification of mask Ky and image X , X not empty).
4 If a pixel is set, it belongs to the object/streak.
5 The window size is determined by the filter size.
5.2 The particle tracking velocimetry (PTV) algorithm
grey-value
5.0
5.0
0.0
Histogram
filtered
10.0
10.0
69
0
10
20
x,y
Histogram
streak
background
30 40
0.0
0
10
20
(a)
x,y
30
40
(b)
Figure 5.8: (a) Grey-value histogram of the original image and the background,
subtraction results in (b) where two streaks are found.
large sensitivity on background signals such as defocused particles which are out
of the volume of observation or light reflections from the wall.
5.2.1.2
Model-based method
This method considers the structure of the streak. The grey-value distribution is
−
−
modeled by a particle moving in time. The particle grey-value (intensity I(→
x ,→
x o ))
6
can be approximated by a Gauss function (according to Hering et al. [1998]) of
width σ
−
1
−
−
e
I(→
x ,→
x o) = √
2πσ
→
−
−
x −→
x o )2
(
4σ2
−
where the particle starts at location →
x o at time to . If the particle moves with a (con→
−
→
−
stant) velocity v to location x (t) the streak is then approximated by the integral
−
gs (→
x)=
1
t1 − to
Zt1
−
−
−
I(→
x −→
v · t, →
x o ) dt
(5.2)
to
This distribution can be split into a time dependent and a location dependent part
due to separability and rotational symmetry of the Gaussian function I 7 :
6 On
−
−
condition that the particles are rotational symmetric: I(→
x ) = I(|→
x |)
−
−
of rotational symmetry and separability: I(|→
x −→
v · t|) = I
→
−→
−
v . Separability: I(x, y) = I(x) · I(y).
−
I |→
v | t − x→
−
v2
7 Because
r
→
−
−
−
−
x 2→
v 2 −(→
x→
v )2
→
−
v2
!
·
70
Image sequence analysis for stereo PTV
=⇒
→
−
−
x (t1 )→
n
q
1
2
→
−
→
−
→
−
→
−
2
gs ( x ) = I
x −( x n )
2d
d·t1
t1 −to −
Z
I(τ) dτ
(5.3)
→
− →
−
d·to
t1 −to − x o n
→
− −
−
where the distance between the endpoints is d = →
x (t1 ) − →
x (to ) and
→
−
d
→
−
n = →
− .
d
Applying a gradient based minimization method (such as the Marquardt-Levenberg
method) for equation (5.3) on the streak pixels yields the parameters for the function. A pre-segmentation step is necessary to obtain start values for the minimization. The method of local orientation was used by Leue [1996].
Compared to the region growing method this model-based method yields very precise results for the end points of the streak. Its disadvantage is the large computational effort necessary to find start values and to obtain the model parameters
by a minimization method. Usually the finding of the endpoints is not very critical because the correspondence solving (section 5.2.3) does not need a (sub-pixel)
precise endpoint.
5.2.2 Labeling and position determination of a particle
The unique identification and marking of each streak is necessary for the following
image processing steps. The method called labeling marks all pixels belonging to
one streak with a unique number. The background is marked with a zero. The
number of pixels with the same number is the area of a streak. This immediately
allows to apply an area size criterion to get rid of objects which can not belong to
a streak. An upper limit for the number of pixels thresholds the size of the streaks
and reflections and other effects resulting in large objects are therefore filtered.
The moving (symmetric) tracer particles are imaged as streak lines due to the time
of exposure8 ∆t = t1 − to . This grey-value distribution is expressed with equation
(5.2).
To determine the position of a streak, the center of mass is a good measure
−
(Hering et al. [1998]). The center of mass →
x s of the streaks grey-value is
R∞ →
−
−
−
x 0 gs (→
x 0) d→
x0
−∞
→
−
x s = R∞
−∞
8 Chosen
−
−
gs (→
x 0) d→
x0
.
as maximum which is the inverse of the frame rate of the camera.
(5.4)
5.2 The particle tracking velocimetry (PTV) algorithm
71
Substituting equation (5.2) into equation (5.4) and exchanging the integral over t
−
and →
x results in
Rt1 R∞ →
−
−
−
−
x 0 I(→
x 0, →
x o (t 0 )) d →
x0
t −∞
→
−
x s = o t1 ∞
R R
to −∞
(5.5)
−
−
−
g I(→
x 0, →
x o (t 0 )) d →
x0
−
The center of mass of the grey-value distribution →
x o is defined as
R∞ →
−
−
−
−
x 0 I(→
x 0, →
x o (t 0 )) d →
x0
−∞
→
−
x o = R∞
−∞
−
−
−
I(→
x 0, →
x o (t 0 )) d →
x0
1
=
Io
Z∞
→
−
−
−
−
x 0 I(→
x 0, →
x o (t 0 )) d →
x0
−∞
−
and this →
x o substituted into equation (5.5), results in
t1
→
−
xs=
R− 0
Io →
x o (t ) dt 0
to
Io
Rt1
dt 0
to
=
1
∆t
Z t1
→
−
x o (t 0 ) dt
t
| o {z
}
−
<→
x o>
(5.6)
Equation (5.6) shows that the center of mass over the pixel grey-values of a streak is
−
equivalent to the expectation value < →
x o > of a particle moving during the time of
→
−
exposure ∆t. x s can therefore be taken as the (sub-pixel precise) position for each
streak. This procedure is done for each streak of an image through the whole image
sequence. To identify the particle (streak) in time uniquely the correspondence
problem (in time) has to be solved. This is discussed in the next section.
5.2.3
Correspondence solving
−
The purpose of the PTV method is to obtain the trajectory →
x i for each particle i.
The particles are imaged as streak-lines in each image. After segmentation, the
correspondence problem of identifying the same streak/particle in the consecutive
image frames has to be solved.
The particles and therefore also the streaks are almost indistinguishable, the characteristics are only the grey-value, size (number of pixels per streak) and the orientation (length and width). Even the grey-value is a very poor criterion if the light
intensity changes during the time of exposure ∆t - which is the case in the experimental setups for the wind/wave facility AEOLOTRON and the small Heidelberg
wind/wave facility (section 3.3.2.2).
72
Image sequence analysis for stereo PTV
center of mass
streak i
center of mass
streak i+1
enlargement from
dilation operation
∆x
image frame n
image frame n+1
n+2 ...
Θ
t +∆ t
o
t
o
t + 2∆t + Θ
o
Figure 5.9: Corresponding streaks. The particles are imaged as streaks during
the time of exposure ∆t with their center of mass as the (average) position of the
particle. Θ is the vertical synchronization time between two consecutive image
frames. From a morphological dilation operation the pixel expansion of the streaks
is enlarged (outer shaded area of the streaks).
The principle of the correspondence matching is shown in figure 5.9. The dark
grey symbolizes the pixels of the streaks i imaged during the time interval to to
to + ∆t and the subsequent streak i + 1 imaged during the time interval to + ∆t + Θ
to to + 2∆t + Θ, where ∆t is the time of exposure and Θ the vertical synchronization
time of the video signal. The expansion of the streak is enlarged by a morphological
dilation operation (light grey).
The two consecutive streaks i and i + 1 are marked as corresponding9 if the dilated
streaks (light grey) do overlap (this is equivalent to a logical and operation). The
−
→
direction of the vector ∆x is given by the known temporal order of the image frames
and the (pixel grey-value) center of mass of the streak. The velocity is therefore
−
→
→
−
v = ∆x . This procedure is done for the whole image sequence. The consecutive
∆t
overlapping streaks and their corresponding grey-value center of mass constitute
−
the particles trajectories →
x i (t).
To avoid unnecessary clustering of the streaks the dilation is not calculated simultaneously for all objects in an image but for each object individually.
5.2.3.1
Particle characteristics
The dilation operation is a sufficient criterion for the correspondence match if the
particle concentration (number of particles per image) does not exceed about 100
particles (streaks) per image. If the concentration exceeds this limit, the probability for ambiguities in the correspondence match increases. If no correspondence
is found or an ambiguity of correspondences occurs, further properties of the particles/streaks have to be taken as correspondence criteria. The characteristic properties are then taken to calculate a confidence measure. A confidence measure
9 ‘caused
by the same particle’
5.2 The particle tracking velocimetry (PTV) algorithm
73
was implemented by F. Hering with usage of Fuzzy-set theory (details in Hering
[1996]). Five characteristic properties (µi , i = 1 . . . 5) are taken as fuzzy variables:
1) The distance of the current streak from the streak in the previous image, 2) the
distance from the current streak from the estimated position in the next image, 3)
the difference of the current velocity of the streak from the velocity in the previous
image, 4) the grey-value difference of the streak in the current image from the previous and 5) the difference of the streak area in the current image from the previous.
These five values are then linked by Fuzzy-logic lookup tables (in all combinations
of µi ) with the confidence measure Con f (µi ) and the resulting confidence measure
is K =
∑5i=1 Con f (µi )
.
∑5i=1 µi
The candidate of the largest K is taken as corresponding.
The characteristic properties of the particles are discussed in the following part of
this section.
Area and grey-value sum of a streak
The number of pixels constituting a streak can be taken as a characteristic of a
streak. The particles are solids and can be assumed as rotational symmetric - and
therefore they do not change their shape. But the projection of the particle on the
image plane of the camera changes the particle image according to the point spread
function. If the particle gets more defocused, the size of the image increases. This
case is typical for the stereo-PTV measurements where the experimental setup is
done in such a way that the range of the focal depth of field is large (to increase the
intersecting volume of observation of both cameras, see section 3.3.2.3) compared
to the classical PTV technique where a thin light sheet is used and the focal depth
is in the range of the light sheet thickness.
Continuity of the optical flow
−
For moving particles with a velocity →
u and a given time of exposure of the CCD
sensor ∆t the grey-value g of the imaged particle is not a constant but a function
−
−
g(→
u ,t). If the grey-value g is only a function of location →
x and time t then the
total differential equation for the optical flow is written as
∂g
∂g
−
dxi + dt = 0
dg(→
x ,t) = ∑
∂t
i ∂xi
⇔
−
where →
u =
∂g →
+−
u · ∇g = 0
∂t
−
d→
x
dt .
(5.7)
74
Image sequence analysis for stereo PTV
100
grey-value/area
grey-value/area
12000
8000
4000
0
0
10
20
30
40
time [frame number]
75
50
25
0
50
0
10
20
30
40
time [frame number]
50
Figure 5.10: Grey-value sum of a streak over time (left) and the grey-value sum
per area of the streak (right) for stereo PTV setup.
The well known continuity equation of hydrodynamics (conservation of mass with
→
−
density of mass ρ instead of g) is ∂ρ
∂t + ∇(ρ · u ) = 0 , which is the same as equation
→
−
(5.7) if ∇ · u = 0. Integrating the grey-value
I over the area Ω of a streak and
R ∂g
10
→
−
applying the Gauß-integral
dx dy + u
g ds(x, y) = 0 leads to a constant
Ω ∂t
| ∂Ω {z
=0
grey value for the streak
G=
Z
Ω
}
g dx dy = const.
The sum of all pixel grey-values is therefore constant; the required assumptions are
a homogeneity of the illumination over time and space, a boundary of the particle
where the grey-value is zero and the particle shape is constant in time.
In figure 5.10 the grey-value sum and the grey-value sum per area over time of a
streak is shown. This illustrates the typical case for the stereo PTV setup where
a particle becomes defocused, the illumination intensity changes and therefore the
grey-value is not constant (left part of figure 5.10). As defocusing enlarges the
streak, the grey-value per area should be constant in the first order11 (see Geissler
and Scholz [1999]) if the illumination is constant. The grey-value per area is not
constant as seen in the right part of figure 5.10 and therefore the illumination is not
constant either. If the variation of the grey-value and the area of a streak is smooth
or shows small changes from one image to the next, a threshold for the variation
can be set as a criterion of correspondence.
10
Z Z
Ω
∇g dx dy =
Z
g ds(x, y)
∂Ω
On the boundary of the particle g |∂Ω = 0
a simple single lens system and a first order approximation the blurring radius ε of an object
moved out of the focal plane by the distance ∆d is linear: ε ∼ ∆d.
11 In
5.2 The particle tracking velocimetry (PTV) algorithm
5.2.3.2
Velocity estimation
If further ambiguities of corresponding streaks exist, a velocity estimation can be
a good criterion for solving the correspondence search. The available information
−
at time tN is the streak position (section 5.2.2) →
x (tN ) and all the previous positions
→
−
x (ti ), i = 0, . . . , N at time ti . This information is used to estimate the position at
time tN+1 and a χ2 -test is used to obtain the most probable corresponding candidate
(see Hering et al. [1996]).
The choice of the model for the velocity estimation depends in particular on the
velocity range of the particles. The simplest models are linear, such as
→
−
−
−
x (tN+1 ) = →
x (tN ) + →
u (t)∆t
→
−
→
−
= 2 x (tN ) − x (tN−1 )
−
where →
u (tN ) =
−
−
x (tN )−→
x (tN−1 ))
(→
∆t
and ∆t = 1
or if acceleration is taken into account
→
−
−
−
−
x (tN+1 ) = →
x (tN ) + →
u (t)∆t + 21 →
a (t)∆t 2
−
−
−
= 12 (5→
x (tN ) − 4→
x (tN−1 ) + →
x (tN−2 ))
−
where →
a (tN ) =
−
−
u (tN )−→
u (tN−1 ))
(→
∆t
.
If the velocity changes are high then a more sophisticated velocity estimator is
required. The Kalman-filter is a recursive linear optimal estimator (Jähne [1997])
which can be used.
The simple estimators are sufficient in most cases. Further improvements, of less
computational costs a Kalman-filter would cause, can be obtained with an improved stereoscopic correspondence match (see section 7).
5.2.3.3
PTV post processing
The list of streaks constituting a particles trajectory is already constructed. The
final step is to remove ‘outlier’ streaks from a trajectory. The ‘outlier’ elements
of the trajectory are defined by a smoothness criterion for the trajectory using an
interpolation technique. A streak element of a trajectory is considered as an outlier,
if the velocity vector of the streaks exceed a certain threshold value from an interpolated value. The PTV algorithm in this work uses an adaptive Gauß-windowing
technique12 proposed by Agüí and Jiminéz [1987] and implemented by Hering
12 Interpolated
Kernel:
−
−
−
velocity →
v (→
x i ) for streak number i at location →
x i and σ the with of the Gauß →
−
n
x −→
x |
|−
−
−
u j (→
x j ) exp − iσ2 j
∑ →
j=1
→
−
−
→
v (→
x i) =
−
n
x i −→
x j|
|−
∑ exp −
σ2
j=1
75
76
Image sequence analysis for stereo PTV
Geometric constraints
location of epipolar lines
geometric similarity
uniqueness
Object properties
compatibility of object features
continuity of disparities
figural continuity
coherence principle
disparity limit
local disparity limit
disparity gradient
spatial order of objects or pixel
temporal order of objects
connectivity of borders
Table 5.1: Stereoscopic constraints.
[1996], or simply polynomial interpolation. A spline interpolation method was described by Spedding and Rignot [1990] which is a more elaborate technique, but
without significant enhancement. In addition, the AGW interpolation technique is
used for interpolation of the vector flow field on a regular grid.
5.3
Stereo correspondence solving
The problem to be solved for the stereoscopic imaging system is to identify an
object uniquely in the stereoscopic views. This is done by solving the stereo correspondence problem. Usually the stereoscopic correspondence problem is solved
for images of two (or more) different views in a static scene where the objects are
not changing. In this work image sequences over a time interval were considered.
The ‘objects’ are trajectories which consist of a list of temporal subsequent ordered
streaks.
There is no general method to solve this correspondence problem uniquely. Each
method takes a certain amount of constraining assumptions concerning the setup of
the cameras and the properties of the observed object such as geometry or movement. The constraints which are relevant to stereo PTV will be discussed in the
next sections. Table 5.1 lists some stereoscopic constraints.
5.3 Stereo correspondence solving
77
Probing Volume
P=(X,Y,Z)
Epipolar line
O
O’
(x,y)
Retinal plane
Camera 1
(x’,y’)
Epipole 1
Epipole 2
Retinal plane
camera 2
Figure 5.11: Construction of an epipolar line in the retinal plane of camera 2 from
an image point (x,y) on the retinal plane of camera 1.
5.3.1
Geometric constraints
The epipolar constraint
The knowledge of the camera arrangement and the geometry of the object - the
stereoscopic geometric setup of the system - allows to reduce the correspondence
possibilities considerably. The epipolar constraint is derived from the stereoscopic
geometry and is one of the strongest constraints which reduces the computational
effort significantly.
Figure 5.11 shows the geometry which constitutes the epipolar line. The object
(point) is located at the world coordinate P = (X,Y, Z) and imaged through the
center of the lens O onto the retinal plane of camera 1 to the camera coordinate
(x, y). The box marks (symbolically13 ) the probing volume, where this volume
is defined as the intersecting common volume of observation for the two cameras
and the finite depth range. To find the corresponding point in the retinal plane of
camera 2 the geometry states that the point P has to be found on the outgoing ray
OP of camera 1. This ray is depicted by the camera 2 and called epipolar line.
The endpoints of the line are called epipoles and are the projections of limits of the
13 The real probing volume is constructed from two intersecting cylinders. The cylinders have an
angle of the opening of the lens system of each camera.
78
Image sequence analysis for stereo PTV
Probing Volume
(X,Y,Z)
n
Water
Glass
Air
O
1
n
n
2
3
Epipolar line
O’
2 ε Window
(x,y)
(x’,y’)
Retinal plane
Camera 1
Epipole 1
Epipole 2
Retinal plane
camera 2
Figure 5.12: Epipolar constraint including multiple media.
probing volume. The epipolar plane is given by the plane through the center of
the lenses O, O0 and object point P - the intersection of the plane with the retinal
planes results in the epipolar lines in retinal plane 1 and 2, respectively.
The search of the projected point P in the retinal plane of camera 2 (x0 , y0 ) is therefore reduced to a single line. In the special case where the optical axes of the
cameras are parallel, the epipolar lines match the scan-lines of the cameras. In this
case no explicit calculation of the epipolar lines has to be performed and is called
standard-stereo geometry.
The relevance for the stereoscopic correspondence search is summarized:
• A pixel in the image of retinal plane of camera 1 has to correspond to exactly
one pixel in the image of the second camera plane, where the pixel must be
found on the epipolar line.
• For a given image point the correspondence search is reduced from a twodimensional to a one-dimensional problem.
In reality this is not a straight line but curved. This is due to deviations in the optical
system and small errors in the camera parameters. Therefore a narrow window of
width 2 ε takes these deviations into account.
5.3 Stereo correspondence solving
79
For multiple media geometry the epipolar line is slightly bent as shown in figure 5.12. In practice this deviation from a line and the lens distortions proved
to be small enough to be accounted for by the tolerance window of size 2 ε, which
means the search of point (x0 , y0 ) on a line is replaced by a search inside an epipolar
‘window’.
The reduction of the computational effort can be easily shown by the number of
comparisons which have to be performed for each trajectory to find the corresponding trajectory. If the epipolar search area is A = 2 ε · l, given by its width 2 ε and
length l, and the entire area of the image is Ao , the number of possible matches
is n · AAo where n is the number of correspondence candidates. The number of
comparisons which have to be performed for each trajectory is therefore
N1 = K · n ·
A
+1
Ao
where K is a threshold factor (K is the ratio of number of correlating to number of
non correlating elements of a trajectory, K = 4 as shown in section 5.3.4.2) and one
match has to be done in any case with the corresponding trajectories (presupposing
no overlap of trajectories). The number of total comparisons is
N = n · N1 = K ·
A 2
·n +n
Ao
(5.8)
which is significantly lower if performing comparisons for all trajectories (n! comparisons).
The geometric similarity constraint
The images of the object in the two retinal planes of the cameras have to be similar.
For instance lines need to have a similar orientation and a similar length in both
images (considering the geometry of the cameras).
5.3.2
Object properties constraints
Uniqueness constraint
One pixel of image 1 has to correspond to exactly one pixel in the second image this constraint was already shown for the epipolar constraint.
This is not always given for all the pixels: If two points lie on the same visible
ray of camera 1 they are seen as separate points for camera 2. This shows that the
similarity constraint can suffer from this - a line in image plane 1 can be divided
into parts in the second image. Thus the length as contrasted to the orientation of
the lines is not a good criterion without special consideration.
80
Image sequence analysis for stereo PTV
Compatibility constraint
Characteristics in the two stereo images correspondent to each other only if they
have the same physical cause based in the object of the 3D-world. Characteristics
which depend on the viewpoint are rejected. One of this characteristics of an object
could be for instance a border line of the object and its orientation or the intensity
(grey-value) variation due to illumination. A counterexample is the reflection on
the surface of the object, which depends on the point of view and is therefore not a
characteristic of the object.
Continuity constraint
The surface of an opaque object is assumed to be continuous almost everywhere where ’almost everywhere’ means excluding the border lines. This implies that for
visible points on the surface of the object the distance from a point to the observer
and the intensity (grey value) varies continuously.
Figural continuity constraint
This is more strict than the previous continuity constraint. It is not always guaranteed that a region in the stereo images can be identified with an object surface.
Therefore the continuity can be restricted to contour lines of the object. The disparities of the object vary continuously along the contours of the object.
Coherence principle
If the shape (3D volume) and physical properties (opacity, transparency etc.) of the
object are known, this information can be used to identify the object surfaces. The
continuity constraint is a special case of the more general coherence constraint:
Local discontinuities in disparity fields may result from a number of superimposed
objects (continuous disparity fields) corresponding to a smooth surface. The coherence principle states, that neighboring discontinuities have to be similar in the
stereo images if they belong to the same (3D-) object.
Disparity limit
The continuity principle is not very precise in respect to the term continuity. Due
to discretization and disturbances in the optical system the characteristics do not
differ continuously but rather up to a certain threshold value. This threshold value
is accounted for by the disparity limit which states that the characteristics do not
exceed a certain threshold. An example is the grey value (intensity) of an object
which can differ in the two images up to a certain threshold. As another example
5.3 Stereo correspondence solving
image plane 1
81
image plane 2
A
1
superposed images
A
2
1
∆ y’
∆y
S
B2
B
1
∆x
∆ x’
Figure 5.13: Disparity gradient. A1 , B1 and A2 , B2 are corresponding points in
image plane 1 and 2. The superimposed stereo image is on the right side.
the precision of the matching can differ up to certain limit, if a block matching
method is used.
Local disparity limit
The previous disparity limit is a global threshold. A more specific threshold is the
local disparity limit where the threshold is determined in the local vicinity of a
pixel point. This means a specific threshold (maximum disparity value) is given
for the characteristics of an object.
Disparity gradient limit
The disparity gradient is defined by the difference in disparity of neighboring
points.
Two neighboring points in image plane 1 and image plane 2 correspond to each
other if the disparity gradient does not exceed a certain value. The disparity gradient Γd is defined as the distance S between the central points on the line of the
corresponding points on the superposed image by
S=
q
((∆x + ∆x0 )/2)2 + ((∆y + ∆y0 )/2)2
with
Γd =
∆x − ∆x0
∆y − ∆y0
/S
The constraint is not satisfied, if |Γd | exceeds a certain threshold value.
82
Image sequence analysis for stereo PTV
B
A
B
Center of lens 2
Center of lens 1
A
1
B
1
C
1
C
2
B A
2 2
Figure 5.14: Order of the points A, B, C on the epipolar line of image plane 1 is
sustained in the corresponding epipolar line on image plane 2.
Ordering constraint
This is a very strong and therefore important constraint which can reduce the computing effort enormously. This constraint has a spatial and a temporal aspect:
• Pixel points located on a epipolar line of the first image plane have to be
found on the corresponding epipolar line in the second image plane in exactly the same order. If occlusions occur in one of the image planes, the
corresponding point is missing in the other plane - but still the order is sustained.
• If there is a temporal information attached to the object, this temporal order
has to be sustained in both stereo images (or image sequences).
Connectivity constraint
An object which can be described through connected elements (such as the border
of an object described by connected lines or pixels) in the image plane has to be
connected in the second image plane. This is often not the case because parts of
an object can be hidden in one camera view and visible in the second, or another
hidden object can be visible in the second camera view - therefore the connectivity
constraint is broken.
5.3 Stereo correspondence solving
5.3.3
Applied constraints for 3D PTV
The stereo PTV is based on the classical 2D-PTV. The available information from
the 2D-PTV is the following:
• A list of all trajectories,
• where a trajectory consists of a list of streaks which are sorted in chronological order
• and each streak is characterized by
– a number of the image where it was detected,
– the area (size in pixels) of the segmented streak
– and an average grey value in this area.
– The (sub-pixel precise) position of the streak is determined by the center of mass calculated from the grey-value of the pixels on the detected
area of the streak.
From the list of stereoscopic constraints listed in table 5.1, some constraints proved
to be important for the 3D-PTV stereo correlation.
In a first step a list of possibly corresponding candidates is constructed. The following constraints are applied to build the correspondence list (index 1 stands for
image sequence of camera one, 2 for image sequence of camera two):
• Ordering constraint: Temporal order.
Because each trajectory consists of a list of subsequent, timely ordered streaks,
this constraint requires the correspondence of all streaks of a trajectory T1 of
image sequence one with trajectory T2 of the image sequence two. If only
parts of the trajectories correspondent (‘holes’ in the trajectory), it results in
a splitting up in two or more trajectories.
• Geometric constraint: Epipolar constraint.
The corresponding trajectory T2 to trajectory T1 was found within the epipolar window of width ε.
• Object property constraint (section 5.3.4.2):
– If the relation of correlating to non correlating streaks of trajectories T1
and T2 exceeds a threshold, the trajectory is considered as not corresponding.
83
84
Image sequence analysis for stereo PTV
– If the two trajectories T1 and T2 correspond to each other, the distance
d between the streaks varies solely within the epipolar window of size
ε and is roughly the same for all streaks. If T1 and T2 move relative
to each other, d is proportional to this relative motion and T1 and T2
therefore do not correspond.
The ordering (temporal) and the geometric (epipolar) constraints are the strongest
constraints. Depending on particle densities these two constraints may suffice to
allow a unique solution of the correspondence problem. This may not always be
the case as high particle densities are beneficial for high spatial resolutions in visualizing flows.
5.3.4
Stereo correlation algorithm
The purpose of the stereo correlation is to find corresponding trajectories in the
two image sequences (1 and 2). A list of correlating trajectories (from 2D-PTV) is
constructed and from this list the 3D (world-) coordinates can be readily calculated
(see section 5.9).
The two strongest constraints are considered at the begin. The first step is to find
the correspondence from the geometric (i.e. epipolar) constraint. The next step
is to apply the ordering constraint in the time domain (section 5.3.4.1). If the
correspondence is not resolved, further constraints are applied (section 5.3.4.2).
5.3.4.1
Application of the geometric and ordering constraint
There are four possible outcomes of the correspondence search:
1. No correspondence:
This is the trivial case where to a trajectory T1 in the retinal plane of camera
one no corresponding trajectory in the retinal plane of camera two was found.
2. One-to-one correspondence (figure 5.15):
Exactly one trajectory T2 was found as correlating to trajectory T1 .
3. One-to-many correspondence (figure 5.16 a and 5.16 b):
For the trajectory T1 two or more trajectories T2,i ( i = 1 ... n, a number of n
trajectories) were found which in turn correspond to trajectory T1 .
4. Many-to-many correspondence (figure 5.17):
A multiple of trajectories T2,i were found for trajectory T1,1 whereby T2,i
corresponds to multiple trajectories T1,i .
5.3 Stereo correspondence solving
85
Retinal plane of camera 1
T
Retinal plane of camera 2
T
1
2
Figure 5.15: One-to-one correlation.
Retinal plane of camera 1
T
Retinal plane of camera 2
T
1
2,1
T
2,2
(a)
Retinal plane of camera 1
T
Retinal plane of camera 2
T
1
2,1
T
2,2
(b)
Figure 5.16: One-to-many correlation (a), (b) with temporal overlap.
Retinal plane of camera 1
T
Retinal plane of camera 2
T
1,1
2,1
T
T1,2
Figure 5.17: Many-to-many correlation.
2,2
86
Image sequence analysis for stereo PTV
A reason for non corresponding trajectories (case 1) could be if a particle trajectory
is imaged by one camera only and if it is out of the image plane of the second camera. To minimize this occurrence, the intersecting volume (volume of observation)
of the two cameras has to be maximized. Furthermore an imprecise calibration
(especially in the outer regions of the image) might fail the correspondence match.
Another reason could be an incorrect segmentation of a trajectory. This can be
caused by grey-value of the streaks being to small due to restricted illumination or
if the particle (streak) is out of focus.
Therefore a homogeneous illumination, a maximum of the observation volume and
of the depth of field, and a good calibration are crucial for obtaining a high number
of correspondence matches.
For case 2 (figure 5.15) a unique correspondence is found and the world coordinates
can be reconstructed.
With increasing particle density the probability of crossing trajectories and therefore failing correspondences increases. In this case the one-to-many and the manyto-many correspondence search increases the number of corresponding trajectories.
For case 3 (figure 5.16) two distinct outcomes occur:
The first (figure 5.16 a) one occurs if the trajectory T1 has a multiple of corresponding trajectories T2,i which do not overlap in time. This may happen if streaks were
not segmented correctly over the whole time interval of T1 . The resulting trajectory
T2,i is thus generated by the same particle and it corresponds therefore to T1 .
The second outcome (figure 5.16 b) occurs if there is a temporal overlap of T2,i
which are found as corresponding to T1 from the geometric constraint. This violates the boundary condition that trajectories (and the streaks) have to be unique.
Therefore they must belong to n different particles and their related trajectories:
T2,i −→ T2 , T3 . . . Tn .
For case 4 (figure 5.17) multiple trajectories T2,i were found for T1 , where T2,i
in turn correspond to different trajectories T1, j . In this case further criteria for
resolving these ambiguities are required.
5.3.4.2
Object property constraint
A strong constraint in resolving the remaining ambiguities is the uniqueness constraint. This constraint states that an object point viewed in the first image should
not match more than one image point in the other view. This constraint does not
hold for transparent objects exceeding a certain size. A bubble or hollow glass
sphere (in water) can be viewed as a single image point in one view and, due to
reflections, as two image points in the second view. Therefore the shape has to be
modeled in order to resolve the ambiguities.
5.3 Stereo correspondence solving
In flow visualization small and rotationally symmetric particles are frequently used
(10 to 500 µm). The particles may thus be viewed as point objects and the uniqueness constraint can be applied. Two criteria are applied for the 3D-PTV:
• The trajectory Tk is constructed from the streaks Sk,i . Even if not all streaks
of the first image sequence S1,i of trajectory T1 correlate to streaks S2, j of
trajectory T2 from the second image sequence, they may nevertheless be corresponding. This is accounted for by a heuristic threshold value K that is
determined by the ratio of the number of correlated streaks k to the non correlated streaks n. If the trajectory T1 consists of streaks S1,i , i = 1 . . . I, and
the corresponding trajectory T2 consists of streaks S2, j , j = 1 . . . J, then the
number of correlated streaks k are given by k = min(I, J). Therefore the
threshold K is written as
min(I, J)
K=
|I − J|
• Even if the streaks of trajectories T1 and T2 are found within the epipolar
windows (of size ε), T1 and T2 do not correspond if the particle streaks are
moving relative to each other. Thus a criterion to resolve this ambiguity is
of a statistical nature taking the distance between each other as a matching
criterion (figure 5.18). The standard deviation σd is a criterion to distinguish
between crossing and non crossing trajectories; in the case of crossing trajectories σd is larger than in the case of non crossing ones. The standard
deviation is written as:
v
u
−
→ 2
u ∑ ∆→
−
x
−
∆x
∑ i
i
t
σd =
N(N − 1)
−
→
where ∆xi is the distance between the trajectories (perpendicular lines in figure 5.18), i = 1 . . . N for N the number of points (streaks). A threshold value
of σd = 0.5 pixels was found to be suitable (Netsch [1995]). If the threshold
value σd is exceeded, the trajectories show a relative motion towards each
other and do thus not correspond.
5.3.5
Stereo coordinate reconstruction
The stereo reconstruction requires at least two perspective views different from
each other or a model of the scene. The reconstruction methods of a stereoscopic object from two-dimensional images of the object is called triangulation14 .
There are several common stereo triangulation methods available (Schwarte et al.
[1999]):
14 Definition
triangulation: Determination of an unknown visual point (in world coordinates)
within a triangle of the optical basis (O, O’) and the related lateral angle pointing to the unknown
point.
87
88
Image sequence analysis for stereo PTV
T
1
Trajectories ‘parallel’
T
2
Epipolar window
σ
1
σ1< σ2
T
1
Trajectories crossing
T
2
σ2
Figure 5.18: Rejecting ‘false’ correspondences of two crossing trajectories T1 and
T2 .
• Shape from shading:
This method requires the knowledge of the light source (light intensity distribution) and the reflectance properties of the object surface but needs only
one image to reconstruct the object. The local steepness of the object surface
is taken to reconstruct the surface shape. The steepness is determined by the
intensity (grey-value) of each pixel.
• Focus techniques:
– Confocal microscopy: The CCD-sensor detects only illuminated points
at the focal (X,Y) plane. The third dimension is obtained by scanning
the depth (Z)
– Depth from focus: A single, two dimensional image is taken from the
object. The center of the object (X,Y) is taken from the grey-value center (multiplied by the magnification of the optics). The depth distance
(Z) of the object can be inferred from the defocus.
• Active triangulation: The light source projects a light intensity structure (by
a light beam scanner, a light sheet or a mask with a certain structure) onto
the object. With the knowledge of the angle of the light projecting optics,
the projected intensity pattern and the angle of the receiving optics of the
5.3 Stereo correspondence solving
89
CCD, the depth map of the object can readily be calculated. A light intensity
which varies in time and space according to a certain pattern can increase
the resolution (Wolf [1996]).
• Passive triangulation: Contrary to the active triangulation method this method
does not consider the geometrical arrangement of the illumination. Different
views of the object are required to determine its 3D position. This can be
achieved either by using multiple cameras (static stereo analysis), by using a
moving camera and taking images in several positions or by tracking a moving object (structure from motion). The positions of the cameras have to be
known (calibrated) or self-calibrating methods have to be used.
The purpose of the stereo coordinate reconstruction is to obtain the real, physical
→
−
location of the object in the world coordinates X , the so called triangulation. The
simplest triangulation method in static stereo analysis (figure 5.19, for details see
Klette et al. [1996]) considers only the projective transform P (equation (4.1)).
The disparity of the correlating object points is given by the distance between the
→
−
correlating object points d in the superimposed images of the two cameras (figure
−
−
5.19 ). The image coordinates →
x and →
x 0 in the retinal planes of camera 1 and
camera 2 are known.
The disparity is defined as
→
− →
−
d =−
x −→
x0
with
0 d1
x1
x1
→
−
0
→
−
→
−
d =
, x =
, x =
.
d2
x2
x20
Using the projective transform P :
→
−
x =
x1
x2
f
=−
X3
X1
X2
(5.9)
where f is the camera constant (for both cameras).


b1
→
−
The baseline distance ( OO0 ) b =  b2  is given by the distance of the center
b3
of the lenses.
The equation system (5.9) can be solved:
X3 = f
⇔
X1 (
X1
(X1 + b1 )
=f
x1
x10
1
b1
1
− 0 )= 0
x1 x1
x1
90
Image sequence analysis for stereo PTV
ray 2
ray 1
U1
U
3
X=(X , X , X )
1 2 3
O
f
O’
b
U
2
x´=(x´
1 , x´2 )
x=(x , x )
1 2
P=(0,0)
Retinal plane
of camera 2
Retinal plane
of camera 1
d
Overlay image
Figure 5.19: Stereo reconstruction. The disparity d~ is illustrated in the bottom of
the figure.
5.3 Stereo correspondence solving
91
⇔
X1 =
x1
b1
0
(x1 − x1 )
⇔
X1 =
x1
b1
d1
X2 =
x2
b2
d2
X3 =
f
b1 .
d1
and the same with
→
−
→
−
If the world point X is close to the cameras the disparity d is large and the camera
→
−
coordinates can be determined relatively precise compared to the case where X is
further away from the cameras.
A more sophisticated and more precise method uses the complete camera model.
If the camera parameters are known (section 4.1) and the stereo correlation problem is solved (section 5.3), the 3D world coordinates can be determined. For this
purpose the camera model has to be inverted. In the case of the pinhole camera
transform F = B · R · T (equation (4.7)) the inverse transform is
U = F −1 u = T −1 R−1 B−1 u.
The function to be minimized is:
|u − F(U)|Z=Zo → min
(5.10)
where U is the coordinate of the object in homogeneous coordinates (U ∈ IR4 ), u
the image coordinate in homogeneous coordinates and F the transform matrix of
the camera model. As the function to be minimized is convex15 , a method based
on conjugate gradient minimization, the Powell’s method (Vetterling et al. [1992])
which rapidly converges, is used.
The solution is not unique in the physical space because the equation is written in
homogeneous coordinates which are determined up to a scaling factor λ (section
4.2.1)
F .−1 : {U = (U1 , U2 , U3 , 1)T ε IR4 } → {u = (u1 , u2 , 1)T ε IR4 }.
15 No
local minima exist.
92
Image sequence analysis for stereo PTV
This means, the solution in the physical space is an optical ray. This procedure
is applied to the coordinate points of the first and second camera image plane.
Ideally the two rays intersect and the intersection point is the world point U soughtafter. In praxis the two rays do not intersect due to noise and deviations in the
camera parameters. The simplest way is to take the middle of the distance of the
two rays. If the distance between the two rays is small compared to the distance
between the image plane and the object plane this method is sufficiently precise.
For uncalibrated cameras there are several methods better suited to find the best
match for the object point U (see Schwarte et al. [1999]).
Chapter 6
Analysis of data and discussion of
results
Results of the stereo particle tracking velocimetry (stereo PTV) are shown and
discussed in section 6.3. It contains experimental investigations of flow fields in
a gas-liquid reactor and investigations of flow field visualization and analysis in
wind-wave flumes. The main interest of this work is the influence of wind induced
waves on the liquid flow field. Since the large AEOLOTRON wind-wave facility was in the scope of this work newly constructed, the investigations served to
validate the conditions and to test the new experimental and analysis methods.
For quantitative results the quality of the calibration and the accuracy and resolution of the used methods is essential and discussed in section 6.1. Since the stereo
correlation method is a fundamental part of the stereo PTV, the limits and capabilities of the stereo correlation are summarized in section 6.2.
6.1 Calibration and resolution
To obtain quantitative results in digital image processing the calibration procedure
is an essential part of the measuring procedure and evaluation. It consist of several
parts: The 2D calibration as described in section 5.1 and the camera calibration for
the stereo setup as discussed in section 4.2.5. The calibration data obtained from
the 2D calibration are required for the camera calibration. Therefore the precision
of the 2D calibration is crucial.
The 2D calibration procedure finds the line crossings with a precision of 1/10 up to
1/100 of a pixel, depending on the manufacturing precision of the calibration target
and the distortions due to the lens system and CCD sensor. Extensive studies on
the dependency of noise added to the calibration points on the camera parameters
have been reported in a diploma thesis of Garbe [1998]. If the nonlinear distortion
94
Analysis of data and discussion of results
Figure 6.1: A rotating grid as a calibration target to validate the calibration algorithm, measured by Garbe [1998]. The display software was developed by Bentele
[1998]. The ‘missing area’ of the rotating disc is caused by the limited volume of
intersection of the two cameras.
parameters are small (in the order of 10−5 ), the deviation of the camera parameters
increases linearly with noise level (normal distribution) added to the calibration
points. The linear behavior is a consequence of the camera model of small linear
√
distortions. The error also decreases with n where n is the number of calibration
points.
Furthermore the dependency of the deviation of reconstructed 3D world coordinates X = (x1 , x2 , x3 ) on the deviation of the calibration points is of interest. The
grid points lie in the x1 , x2 -plane. The deviation of the x3 component of the world
point is found to be about three times larger than the deviation of the x1 and x2
component. The dependency of the deviation of the world coordinates on the deviation of the calibration points is linear. The same linear dependency is found
for the deviation in the internal and external camera parameters. However, its dependency on the internal parameters is by a factor of the order 108 larger than the
dependency on the external parameters. The precise determination of the internal parameters is therefore much more important than the precision of the external
camera parameters.
For the validation of the stereo PTV algorithm a rotating grid (with an inclination
angle of 27.8◦ toward the z-axis) was used. Figure 6.1 shows qualitatively the
result - all the trajectories are within a plane and the movement of the grid points
is circular. Taking the given radial velocity the deviation of the measured velocity
can be determined and was found to be less than 5%. The average displacement
error of the grid points is calculated from the standard deviation of each trajectory
and amounts to 3%.
If a Scheimpflug camera setup is chosen, the optimal angle for the best suited depth
6.2 Stereo correspondence
95
sharpness
4150.0
3650.0
3150.0
2650.0
0
4
8
angle [step units]
12
Figure 6.2: The sharpness measure is defined as the variance over the whole image
area for each adjusted angle (for details see text).
of field (see section C) has to be found. This is accomplished by shifting the angle
stepwise and determining for each step a sharpness measure (calculated from the
global variance of the image). The used lens system included three degrees of
freedom (distance between lens). It turned out that a one degree of freedom lens
system would be much better for calibration. The dependency of the depth of field
(as determined by the sharpness measure) is shown in figure 6.2.
6.2 Stereo correspondence
The resolution of ambiguities in the correspondence search proved to be a critical
point in respect to the particle density. For low particle density (<200 particles)
most ambiguities can be resolved. The number of unresolvable ambiguities increases with the density of particles. It is therefore not possible just to increase the
spatial resolution by increasing the density of particles. The same argument applies
for the 2D-particle tracking velocimetry (PTV). Because the stereo correspondence
is based on the 2D-PTV, it is important to know the range of particle density where
the 2D-PTV fails.
To test the stereo correspondence search the system can be simulated. The model
should be simple but close to real conditions. As the physical model the linearized
wave theory was taken (see section B).
u
→
−
The two dimensional velocity field u =
of a surface water wave in the
w
linearized wave theory is written in two dimensions as
u(t, z) = A ω ekz cos(kx − ωt) + uo (z)
w(t, z) =
A ω ekz sin(kx − ωt)
(6.1)
where A is the amplitude, ω the frequency and k the wave number. This model
96
Analysis of data and discussion of results
ε
8
4
2
8
4
2
ideal PTV
Sk
K = Sn /Sk
1100
0.5
1500
0.4
>1900
0.13
real PTV
800
1.0
1200
0.7
>1750
0.3
Table 6.1: Stereo correlation simulation according to Netsch [1995]. Ratio of corresponding trajectories Sk to the number of multiple corresponding (unresolved
ambiguities) trajectories Sn as a function of the epipolar window size ε and Sk for
‘ideal’ and ‘real’ PTV (see text for details). For ε=2 the limit of the algorithm is
not yet achieved.
describes orbital particle paths (equation (B.8)) of frequency ω decaying exponentially in amplitude with depth z (z = 0 at the surface, z becoming negative with
increase in water depth). A surface drift velocity uo (z) is added as a linear component to the horizontal velocity u.
The simulation is done in the following way: The particles are randomly placed
in the volume of observation and moving according to equation (6.1). At each
time step (time of exposure) the particle (of a Gauß function shape) movement is
projected on the two image planes and stored. The particle is imaged as a streak
and its grey-value obeys the continuity of optical flow (slow movement means a
large grey-value and a small streak, fast movement a small grey-value and a large
streak). If the streaks overlap in the image plane, the trajectory is divided into
different parts. Therefore the number of trajectories in the 2D-plane is larger than
the number of actual trajectories.
Extensive simulation studies using the described model have been done by
Netsch [1995] and are summarized in the following. The applied camera model
is the pinhole camera (section 4.2.3) where the translational method camera setup
was used (figure 3.13(a)).
The efficiency of the stereo correlation algorithm is determined by the ratio
K = Sn /Sk of the number of corresponding trajectories Sk to the number of multiple
corresponding (unresolved ambiguities) trajectories Sn , and by the computational
costs given by the number of trajectory comparisons. The simulation is a sequence
of 40 images and the depth range z is 5cm. Two cases are considered, first the ideal
PTV where the 2D-trajectories have the same length as the generated trajectories
and second the real PTV where the 2D-trajectories are interrupted (by a probability
of 36%). The results are shown in table 6.1.
With a decreasing epipolar window size ε the ratio K is decreasing (more cor-
6.2 Stereo correspondence
respondences found) for a given Sk . In the case of the real PTV ratio K is significantly lower as compared to the ideal PTV. This is due to the devision of the
trajectories in real PTV and the reduced average length of the trajectories because
of interruptions.
The computational costs are decreasing with a reduction of the epipolar window
size ε (this is a consequence of equation (5.8)). The number of comparisons does
not depend on ε.
Increasing the depth z (volume) improves K due to a smaller density of particles,
but for the same density of particles K does not differ significantly.
The quality of the 2D-PTV and the choice of the epipolar window size ε turned out
to be important for the quality of the stereo correlation algorithm.
97
98
Analysis of data and discussion of results
Figure 6.3: The stereo PTV algorithm applied for a gas-liquid reactor (bubble column) at the BASF AG (from diploma thesis of Stöhr [1998]). Left: Fluid trajectories of seeding particles with aeration on the right side. Right: Aeration of the
bubble column with ascending bubbles in the liquid.
6.3
Stereo particle tracking velocimetry
The stereo PTV was developed and optimized to operate for flow field measurements. It was applied for investigations in a gas-liquid reactor bubble column
where the movement of the ascending bubbles was obtained as their trajectories.
Here the main interest was flow field investigations in wind-wave flumes. The
stereo PTV was applied in the Heidelberg wind-wave facility for the first time
(Engelmann et al. [1998], Engelmann et al. [1999a]). This facility is now dismantled and replaced by a roughly three times larger wind-wave facility
(AEOLOTRON) at the Institute of environmental Physics, Heidelberg. Preliminary measurements have been performed in the new facility. Experiences from the
previous facility have lead to several improvements for the camera setup (section
3.3.3) for the visualization techniques (illumination and seeding particles/hydrogen
bubbles, see sections 3.2.2 and 3.2.1) and for the experimental arrangement. These
improvements were included into the new setup. Data processing methods have
been developed and applied for the PTV by several authors (see for instance
Hering [1996]) and are well established. Since the stereo PTV is a new method,
it has been extended to three dimensions, but further techniques concerning the
3D data processing can be applied for image processing of the stereo flow field
evaluations.
6.3 Stereo particle tracking velocimetry
(a)
Figure 6.4: Three dimensional trajectories visualizing the flow as measured at the
Heidelberg wind/wave facility in two different perspectives: (a) Perpendicular to
the water surface (top) and parallel to the main direction of the flow (left, horizontal); (b) perpendicular to the main direction of flow and in a slight angle to the
water surface.
The stereo PTV algorithm applied for investigations in a gas-liquid reactor (bubble
column) at the BASF AG is shown in figure 6.3 (Stöhr [1998]). Flow properties
in different aeration states were examined. The trajectories of air bubbles and
the trajectories of seeding particles (representing the liquid flow) were obtained
separately by using different color spectra of seeding particles and air bubbles.
The air bubbles in 6.3 ascend in the liquid tank. If the aeration is on the left side
of the tank, the liquid flow is rather uniform in the z-direction on the left side and
more turbulent on the opposite side in descending z-direction. The results were in
good agreement with LDA and ultrasonic flow measurements.
First stereo PTV investigations have been performed at the Heidelberg wind-wave
facility. Figure 6.4 shows the 3D trajectory of seeding particles ((30 ± 15) µm diameter) representing the Lagrange flow field beneath a free wind-driven wavy water
surface. The sequence of 500 images covers a time interval of 8.3s. The spirally
shaped trajectories show the orbital motion of ‘fluid elements’ where the waves
amplitude was less than 5mm and of little steepness (<5%).
After the Heidelberg wind/wave facility was dismantled and substituted by the
large Heidelberg wind/wave facility (AEOLOTRON), first stereo PTV measure-
99
100
Analysis of data and discussion of results
Y-Position
400
15
200
10
5
0
0
200
400
X-Position
0
600
Y-Position
400
15
200
10
5
0
0
200
400
X-Position
600
0
Figure 6.5: Trajectory image of camera one (top) and two (bottom). The velocity
is coded in colors from 0-15 pixels/∆t, where the frame rate ∆t is 1/60s.
ments were carried out with the experimental setup described in section 3.3.2.2.
The results are shown and discussed in the following.
A sample of trajectories from the image sequence of camera one and camera two
is shown in figure 6.5. The arrangement of the cameras is according to figure 3.7
and 3.8 in an angle of about 27◦ from the inner (camera one) and the outer (camera two) side from the walls towards the top of the water surface. The trajectory
image differs from the classical two dimensional PTV: Here an extended volume
of about 80 × 80 × 50 mm3 (X,Y and Z coordinates where Z is perpendicular to the
-still- water surface) was used instead of a light sheet of about 10 mm thickness
(X-direction).
As tracer ‘particles’ hydrogen and oxygen bubbles were used (section 3.2.2) which
implies the buoyancy velocity as the upward velocity of the bubbles (section 3.2.1.2).
The buoyancy velocity u p∞ and the associated time constant τ is related to the
6.3 Stereo particle tracking velocimetry
Medium
Tracer
ρ(20◦ C)
u p∞ /d 2
τ/d 2
101
H2 O
H2
8.99 · 10−2 kg/m3
5.5174 · 104 1/sm
2.759 · 104 s/m2
O2
1.4289 kg/m3
5.5100 · 104 1/sm
2.767 · 104 s/m2
Table 6.2: Buoyancy velocity and time constant for hydrogen bubbles in water.
buoyancy velocity 10E-02[m/s]
3.5
2.8
2.1
75.0
1.4
56.7
0.7
0.0
38.4
0
20
40
area [pixel]
60
80
20.0
Figure 6.6: Buoyancy velocity as a function of size (area in pixel). The color
represents the grey value. Some ‘outliers’ or other moving particles (dirt) can be
identified (and neglected) by the grey value (red marks).
square diameter d 2 , listed in table 6.2 and is determined by equations (3.16) and
(3.14). The time constant is determined from the velocity u p = u p∞ (1 − e−t/τ ) at
t = τ, which is less then 3 ms for an upper limit diameter of 100 µm. This means,
velocity u p∞ is almost immediately achieved (within millimeters of movement).
The buoyancy velocity u p∞ is, according to equation (3.16), a function of the diameter d with u p∞ ∼ d 2 , where d 2 is the size of the particle which is imaged on the
CCD as an area of pixels. Figure 6.6 shows the dependency and the solid line is the
linear regression of equation (3.16). The area of the imaged bubbles depends on
their velocities and is generally not imaged as a circular object but as a streak (figure 5.4). Because of the small buoyancy velocity range of the bubbles (figure 6.6)
the streak form can be neglected. The dependency of the buoyancy velocity on the
size (imaged area in pixels) is shown in figure 6.6. The mean velocity (z-velocity
component) is 0.013 m/s ±50%. The mean area is 23±50% pixels, but the imaged
area is much larger than the actual size of the bubbles. Some size calibration would
102
Analysis of data and discussion of results
rate of occurence
300.0
200.0
100.0
0.0
0
10
20
30
area [pixel^2]
40
50
Figure 6.7: Size distribution of bubbles. The solid line is the sum of two normal
distributions, representing the H2 and the O2 bubbles.
be necessary to determine the real size, but this is not of much interest here. Taking
the buoyancy velocity u p∞ /d 2 from table 6.2, the (mean) diameter of the bubbles is
50 µm. This is a quite reasonable size according to Oertel and Oertel [1989], who
reported a size range one to two times the size of the diameter of the used wire.
The size distribution from figure 6.6 is shown in figure 6.7. The mean size is
23±50% where the distribution shows two peaks at 12.5 and 21.9 pixels. This
distribution is approximated by two superposed normal distributions (blue, solid
line in figure 6.7) and can be associated with the two bubble types (H2 and O2 )
which show a different rate of occurrence, where O2 has a larger mean bubble size.
The size distribution in the AEOLOTRON wind-wave flume is shown in figure 6.8.
The small inset in this figure shows the ‘pollution’ with dirt which appears in the
images as ‘particles’. The behavior of this ‘pollution’ is different from the bubbles
as can be seen in the deviation of bubbles (see tables in section A).
The conditions for the stereo PTV flow measurements are listed in section A. At
this time, only qualitative conditions could be adjusted. Devices such as the wind
velocity measuring devices were not yet installed. Basically two conditions were of
interest: A wave field with wind stress and a wave field without wind stress, where
in both cases the wave amplitudes are less than 1 cm. The wave field without wind
stress was obtained from the decaying wave field from the previously generated
wind waves. The volume of observation was about 30 mm in z-direction and had
an extend of 100×100 mm2 in the x,y-plane.
Figure 6.9 shows 3D-trajectories of a wave field without wind stress. The velocity is represented in colors from 0.3 m/s to 1.2 m/s. The same view as in figure
6.9 is shown in figure 6.10, but now the time is represented in color. In this way
the character of the trajectories changing with time can easily be distinguished.
6.3 Stereo particle tracking velocimetry
103
100.0
rate of occurence
rate of occurence
800.0
600.0
80.0
60.0
40.0
20.0
0.0
400.0
0
10
20
30
40
50
area [pixel^2]
60
70
200.0
0.0
0
10
20
30
40
50
area [pixel^2]
60
70
Figure 6.8: Bubble size distribution from the AEOLOTRON wind-wave flume.
Partly H2 and O2 bubbles, partly pollution by ‘dirt’ (inset).
The analysis of the velocity components in x,y and z- direction is shown in figure 6.11. The dominating frequency in all velocity components is (1.4 ± 0.1) Hz
where a superposed frequency is (0.25 ± 0.05) Hz. In section A the properties
of different series of measurements are tabulated. The low frequency component
(0.25 ± 0.05) Hz is found in the same range for all series (with and without wind
stress), and are most probably related to a resonant frequency due to the circular
shape of the channel. If the decay of the wave field is progressing and therefore the
average wave amplitude decreasing (wind generation off), the frequency is slightly
increasing (qualitatively from 1.4 Hz to 1.8 Hz). In all conditions with wind stress
the main frequency (for all velocity components) is in the range from 2.1 Hz to
2.5 Hz and is roughly 3/2 higher than without wind stress.
Due to the buoyancy the z-component w is slightly positive and the component
in the wind direction (along wind) is the main flow component u (x-direction). It
shifts the oscillation to the negative mean velocity. The w-component (cross wind
direction) is significantly smaller and oscillates around zero. The phase shift of u
and w are nearly 90 degrees - which is a characteristic orbital (circular) movement
of fluid ‘elements’ and the velocity components v and w are almost in phase. It will
be discussed in more detail later.
The particle tracking velocimetry allows to obtain the trajectories (paths) of the
−
−
−
−
−
particles →
x =→
x (→
x o ,t − to ) and the velocity →
u (→
x o ,t − to ) (section 2.1). The
Lagrange-representation can be transformed to the Euler-representation of the flow
field (equation (2.3)), but the Lagrange-representation can generally not be calculated from the Euler-vector field. This is one of the advantages of the PTV method.
An example is the orbital movement of ‘fluid elements’ which can be analyzed
only from the trajectories of the seeding particles or bubbles. Properties such as
the penetration depth of gravitational waves, determined by the curvature Kxi ,x j (or
Analysis of data and discussion of results
1.2
0.4
ve l o c i t y [ m / s ]
0.8
0.0
(a)
1.2
0.0
(b)
Figure 6.9: Stereo trajectories of a ‘wavy surface’ state from lateral view (a) and
the ‘classical’ 2-D PTV view in the x,z-plane (flow direction is x). The view from
the upper top is show in (b). The wave amplitude is partly higher than the volume
of observation (3cm height).
0.4
ve l o c i t y [ m / s ]
0.8
104
6.3 Stereo particle tracking velocimetry
105
487.0
t i me s t e p
325.0
163.0
1.0
Figure 6.10: Trajectories of figure 6.9 coded in time with color (time step in 1/60
s).
0.100
component
u
v
w
velocity in [m/s]
0.050
0.000
-0.050
-0.100
0
100
200
300
time in 1/60 [s]
400
Figure 6.11: Velocity components as a function of time. All components are averaged over a time of 10 frames (0.17 s).
106
Analysis of data and discussion of results
the curvature radius Rxi ,x j = 1/Kxi ,x j , where xi , x j = {(x, y), (y, z), (z, x)} provide
the plane of the 3D-Cartesian coordinate system) of orbital trajectories
Kxi ,x j , =
ẋi ẍ j − ẋi ẍi
(ẋi2 + ẋ2j )3/2
(6.2)
can be calculated (which can not be obtained from the Euler-velocity field). For
the 2D-PTV this was done by Hering [1996]. For the stereo PTV data this was
not possible up to now due to a still to small amount of trajectories (less than 300
trajectories per image).
A consequence of the Lagrange-representation concerns the distribution of particles at a certain time t1 > to which determines the density of the velocity vectors
in a given volume. This density of vectors can vary within the volume and time of
observation. The Euler-representation of the flow field is a ‘snapshot’ of the flow.
To determine the vector field the ‘snapshot’ is ideally an instant of time where in
the real case a time interval is required to determine the vector field. The Eulervelocity field needs the vector from (at least) two subsequent image frames. If the
density of velocity vectors is too low, it can be increased by increasing the time
interval (more than two image frames) and by averaging over the time.
Many other measurement techniques such as LDA or PIV (section 3.1) obtain
Euler-vector fields and for comparison of the results the Euler-velocity vector field
must be calculated. The kinetic energy or the vorticity of the flow field are examples where the calculation of the Euler-velocity field is also required. For further
−
−
−
processing of the data an interpolation of this velocity field →
u =→
u (→
x ,t1 ) at time
t1 - which is the Euler-representation, equation (2.1) - on a regular grid is of advantage.
A frequently applied interpolation is the adaptive Gaussian windowing technique,
introduced in the flow field visualization by Agüí and Jiminéz [1987]. The interpolated velocity field vi is written as

−
n−1
−
−
u j (→
x j )e
∑ →
−
vi (→
x i) =

→
−
−
x i −→
x j σ2
j=0

n−1
∑ e
−

−
→
−
x i −→
x j σ2

(6.3)

j=0
where n is the number of particles and σ the width of the convolution kernel which
was found by Agüí and Jiminéz [1987] to be proportional to the mean distance of
the particles δ by the constant c = 1.24:
σ = c·δ
6.3 Stereo particle tracking velocimetry
where in two dimensions δ =
q
107
A
πn ,
A is the area of observation. In three di1
3V 3
mensions the mean distance is δ = 4π
where V is the volume containing the
n
particles.
This adaptive Gaussian windowing is a special case of the more general normalized
convolution technique (see Knutsson and Westin [1993]). The generalized form of
convolution is defined as
→
−
→
− −
→
− −
−
−
x )B(→
x ) c( ξ − →
U( ξ ) = ∑ a(→
x )T ( ξ − →
x)
→
−
x
(6.4)
U = {aB cT }
(6.5)
or in short from
where denotes some multi-linear operation (in standard convolution this is the
→
−
−
scalar multiplication), ξ is the global spatial coordinate, →
x the local spatial coor→
−
→
−
dinate, T ( ξ ) is a tensor representing the input signal, c( ξ ) is a scalar function
→
−
−
representing the certainty of T ( ξ ), B(→
x ) is a tensor representing the operator fil→
−
ter basis and a( x ) is a positive scalar function representing the applicability of
−
B(→
x ) (equivalent to the certainty).
→
−
The normalization factor for U( ξ ) is
→
−
→
− −
−
−
−
N( ξ ) = ∑ a(→
x )B(→
x ) B∗ (→
x )c( ξ − →
x)
→
−
x
(6.6)
and the normalized convolution follows from equation (6.4) as
UN =
→
−
U( ξ )
→
− .
N( ξ )
(6.7)
The adaptive Gaussian windowing is obtained if the following operators are chosen:
The operator filter basis consists of only one positive invariant basis function with
→
− −
→
− −
B = 1, the certainty is the delta function c( ξ − →
x ) = δ( ξ − →
x ), the input signal is
→
−
→
−
→
−
→
−
the velocity
T (ξ − x ) = u ( ξ ), and the applicability function is
→
− −
→
x − ξ −
−
σ2
a(→
x)=e
.
This interpolation method is applied for a ‘strong’ wind-wave field shown in figure
6.12 (top). Because the density of trajectories is too low to determine the velocity
field within a single time step (two subsequent image frames), the interpolation is
extended over a time interval of 20 frames (0.3 s). The velocity as a function of time
is shown in figure 6.12 (bottom). The case without wind stress is shown in figure
6.13. The wave amplitude (of gravitational waves) was approximately the same as
108
Analysis of data and discussion of results
0.8
0.3
ve l o c i t y [ m / s ]
0.6
0.1
u
v
w
velocity in [m/s]
0.020
0.010
0.000
-0.010
-0.020
0
100
200
300
time in 1/60 [s]
400
Figure 6.12: Top: Stereo trajectories of a ‘strong wind-wave’ field. Bottom: The
velocity components as a function of time. The velocity is averaged over 10 frames
(0.17 s). Euler-velocity fields are calculated for the marked (with boxes) time intervals (20 frames) and shown in figure 6.14.
6.3 Stereo particle tracking velocimetry
109
0.8
0.3
ve l o c i t y [ m / s ]
0.6
0.1
Figure 6.13: ‘Wavy surface’: Without wind generation and with decaying wave
field from a series shown in figure 6.12, but same state as before.
in figure 6.12, as this wave field was obtained right after switching off the wind
stress and decaying. Even if the number of obtained trajectories was lower in the
case without wind stress as compared to the case with wind stress, the trajectories
length is larger and the trajectory ‘character’ is rather straight as compared to the
shorter, highly varying trajectories in figure 6.12. This is discussed quantitatively
in the following.
The interpolated velocity field is shown in figure 6.14 for several time intervals
marked in figure 6.12 (bottom) where the mean velocity component in x- and zdirection was subtracted. Interpretations for this velocity have to be done with care
because of the temporal averaging which is around the period length of the main
frequency component. But nevertheless some orbital movement can be noticed as
a circular structure in the z,x-plane.
To characterize the flow field of the measuring series listed in section A over the
whole time domain, the temporal averaged velocity components (table A.1) and
the deviation from the mean velocity (table A.2) were calculated. The deviation
from the mean velocity represents the mean turbulent1 energy
(‘velocity fluctup
ations’) in direction of the velocity components Ed (dV ) = ∑i (ui,d − ui,d )2 where
the sum covers all velocity components in direction d = x, y, z in the volume dV .
The volume dV is divided according to figure 6.15 into stripes in the horizontal
1 ‘turbulent‘
in the sense of the deviation from the mean.
110
Analysis of data and discussion of results
0.7
0.5
0.3
0.0
0.8
0.6
0.3
0.0
1.3
0.9
0.5
0.0
1.0
0.7
0.4
0.0
0.7
0.5
0.3
0.0
Figure 6.14: Interpolated velocity field (mean velocity subtracted) from trajectories
of figure 6.12 (top) for the time intervals marked in 6.12 (bottom). The velocity is
represented by the arrow length and for a better view also by the color.
velocity 10E-2 [m/s]
1.0
velocity 10E-2 [m/s]
1.3
velocity 10E-2 [m/s]
1.8
velocity 10E-2 [m/s]
1.1
velocity 10E-2 [m/s]
0.9
6.3 Stereo particle tracking velocimetry
111
z
dz
y
dy
x
Figure 6.15: Volume of observation divided into ‘boxes’ from a grid. The volume is
divided into slices along the z-axis (blue) and into stripes along the y-axis (black).
direction (division of the volume along the z-axis) and into slices along the y-axis
(division of the volume along the y-axis). The averaging due to the wave amplitude
has to be taken into account; while the volume division is constant, the wave height
changes during the time of measurement. Therefore the averaging is done over the
wave amplitude which was about 0.5 to 1.0 cm.
The results of table A.1 and table A.2 are summarized:
• The x-velocity component is decreasing with z (see figure 6.16 (1a) and
(1b)),
• the z-velocity component is approximately the buoyancy velocity in the case
of the wind generation being on,
• without wind stress on the water surface the z-velocity component is reduced
by about 50 %, which means there must effectively be a downward flow.
• For the y-velocity component there are no significant differences. The main
component is of course the one in the wind direction (x-direction). The main
components related to the wave movement are in z- and x-direction. This
can be seen if the velocity fluctuations are considered (figure 6.16 (2a) and
(2b)). In y-direction the fluctuations are roughly 10 times smaller than in
z-direction. Velocity fluctuations in x-direction are decreasing with depth in
the case of wind stress, otherwise no tendency is observed.
Even if these results are obtained from preliminary measurements only, the effect of
wind shear stress is clearly observed: There is a velocity gradient of the x-velocity
component. If the surface would have been within the volume of observation and
the position of the surface would have been known, the friction velocity u∗ (section
2.2) could be determined. Furthermore a gradient of the velocity fluctuations Ex (z)
112
Analysis of data and discussion of results
depth z [mm]
0
depth z [mm]
-1
-12
-12
-22
-33
-0.0242
-23
-0.0235 -0.0227 -0.0219
velocity in x [m/s]
-35
-0.0242
(1a)
(1b)
depth z [mm]
0
depth z [mm]
0
-12
-11
-23
-23
-35
0.1488
-0.0225 -0.0207 -0.0189
velocity in x [m/s]
0.2081 0.2674 0.3267
velocity fluctuations
(2a)
-34
0.1795
0.2019 0.2243 0.2468
velocity fluctuations
(2b)
Figure 6.16: (1a), (1b): Dependency of velocity in x-direction on depth z (samples,
number 4 and 5 of table A.1), (2a), (2b): Dependency of the velocity fluctuations
Ex on z.
6.3 Stereo particle tracking velocimetry
gives evidence for enhanced velocity fluctuations closer to the water surface for
a wind induced wave field (with shear stress). The difference in the dominating
frequency of the velocity between the two states with and without wind shear stress
can not be explained simply by a higher mean velocity in the x-direction (alongwind component) causing the increased frequency. In a range within 10% there is
no difference in the mean velocity in x-direction.
The influence of wind shear stress on the velocity field is qualitatively demonstrated. The next step is obviously an investigation aiming at quantitative results.
Furthermore the combination of different techniques to obtain different physical
properties in the same volume of observation and the same time is another step
which will now be possible to take. Thus, the determination of the water surface
shape (topology) is of importance for the stereo-PTV in the case of wave fields. The
imaging slope gauge technique (ISG, see Balschbach [2000]) obtains this surface
shape by the steepness of the surface. This combined measurement was already
carried out in the predecessor of the newly constructed wind-wave flume for the
2D-PTV and should be extended to the third spatial dimension.
The stereo Particle Tracking Velocimetry was proven to be a valuable method for
the three-dimensional flow visualization which allows to obtain a high spatial resolution of a Lagrange-velocity flow field. The fundamental techniques of stereoPTV in wind-wave flumes have been evaluated and allow in the close future to
increase the number and density of trajectories which is one of the most crucial
aims.
113
114
Analysis of data and discussion of results
Chapter 7
Outlook
A major task is to increase the trajectory density (number and length of trajectories). As this technique is new, improvements can be achieved almost in all parts
such as experimental setup (in particular the illumination setup), the calibration
procedure (in particular the Scheimpflug correction), the particle tracking procedure and the post processing of stereoscopic trajectories to obtain stereoscopic flow
fields.
First measurements were performed at the AEOLOTRON wind/wave facility by
applying new techniques such as the Scheimpflug stereo setup. The used Scheimpflug lens system lacked precise adjustment possibilities, thus complicating the
search for an optimal depth of field range and also for the calibration procedure. A
better lens system will be applied in the future.
Due to light refraction at the water surface (in the case of illumination from the
top) the intensity of light in the volume of observation can vary depending on
the curvature of the surface. Steep water waves, such as capillary waves, induce
therefore a large variation in light intensity. If a color CCD camera would be
used, a colored light source could improve the stereo correspondence search. The
volume of observation would then be structured by the colors and particles could
be matched in the correspondence search algorithm also by the color.
The calibration procedure is sub-pixel precise but depends on a calibration target
with known position of world coordinates (world coordinates of landmarks such
as cross middle points). A calibration procedure which does not require the knowledge of the world coordinates would simplify the procedure. Techniques such as
described by Sturm and Maybank [1999] and Zhang [1998] requires the camera to
observe a planar pattern shown at a few (at least two) different orientations.
A main improvement can be achieved with the extension of the stereo correlation
procedure to the streaks. The stereo correlation of single streaks yields another very
powerful criterion for the particle tracking algorithm. Streaks which are hidden
by another particle for camera one can be visible separately for camera two and
116
Outlook
therefore the trajectories can be reconstructed for camera one. This method is
especially well suited if multiple cameras are applied. Ambiguities due to hidden
streaks can thus be resolved.
For flow-field measurements close to the water surface it is necessary to determine
the position of the surface boundary layer. The position of the particles can be obtained by the stereo-PTV, but if there are not enough particles visible in the surface
boundary layer the position can not be determined. This is especially critical under
the conditions of a wavy water surface. The position of the water surface could be
obtained by imaging slope gauge techniques (Balschbach [2000]) or in a simpler
setup by a capacity-wire method.
Finally the amount of data transfered from the frame grabber to the computer and
to the hard disc storage is enormous for high frame rates and two cameras (120 Hz
.
frame rate and a resolution, 512×512 pixels per image = 62 MByte per second).
An online compression using an FPGA (Free Programmable Gate Array) frame
grabber can be applied to reduce the data by a factor of 100-200 (depending on the
density of particles/streaks). This would allow to observe the flow field through a
longer time interval.
Appendix A
Data tables
Several series of measurements have been taken. The analysis was done for the
T
−
mean velocities (→
u = u v w
in x,y,z-components) and its deviation and
tendency (inclining or declining with water depth) and are listed in table A.1. Furthermore the deviation from the mean velocity have been determined and are listed
in table A.2. It represents
p the mean ‘turbulent’ energy in direction of the velocity
components Ed (dV ) = ∑i (ui,d − ui,d )2 where the sum is over all velocity components in direction d = x, y, z in the volume dV . The volume dV is divided according
to figure 6.15 into stripes in the horizontal direction (division of the volume along
the z-axis) and into slices along the y-axis (division of the volume along the y-axis).
In table A.1 the number of the measuring series and the direction of the column
division (z,y) is found in the first row. The second row tells the wind state: ‘on’
means wind generator on (3-5m/s) and ‘on+’ a higher velocity (4-8m/s), ‘on-’ is
the lowest wind velocity (2-4 m/s). In the third to the fifth row the velocity components and the deviations are listed. The tendency of the velocity is given only
qualitatively: ‘- -’ decreasing with depth z, ‘-’ decreasing slightly, ‘++’ increasing,
‘+’ slightly increasing (samples are shown in figure 6.16).
Table A.2 lists only the variability and the tendency (same convention as A.1) of
Ed (dV ) where d = x , y , z and dV = dz, dy.
Comments:
• From series 1 to 5 the wind generation was turned on and slightly increased
from 1 to 5.
• In series 6 the wind generation was turned off and the wave field decaying
(from series 5).
• Series 7 was a test measurement without hydrogen bubbles to determine the
behavior of ‘pollution’ (particles of different size).
118
Data tables
series number/
direction dV
1/z
1/y
2/z
2/y
3/z
3/y
4/z
4/y
5/z
5/y
6/z
6/y
7/z
7/y
8/z
8/y
9/z
9/y
10 / z
10 / y
11 / z
11 / y
wind
state
on
on
on
on on
off
on +
off
off
off
off
u in 10−2 [m/s]
u
∆u
-0.82 -1.0/-0.6
-1.0/-0.2
-1.75 -1.9/-1.7
-1.8/-1.6
-2.25 -2.3/-2.2
-2.5/-2.1
-2.28 -2.4/-2.2
-2.5/-2.1
-2.16 -2.3/-2.0
-2.4/-2.0
-2.86 -3.0/-2.7
-3.0/-2.7
-1.00 -1.1/-0.8
-1.0/-0.8
-2.5 -4.8/-1.8
-3.0/-2.0
-2.8 -2.9/-2.8
-2.9/-2.8
-2.02 -2.1/-1.9
-2.1/-1.9
-2.16 -2.2/-2.1
-2.3/-2.1
------
--
v in 10−2 [m/s]
v
∆v
-0.07 -0.1/-0.0
-0.1/-0.0
-0.15 -0.2/-0.1
-0.2/-0.1
-0.17 -0.2/-0.1
-0.3/-0.1
-0.12 -0.1/0.1
-0.2/-0.1
-0.10 -0.2/-0.1
-0.2/-0.0
-0.10 -0.2/-0.0
-0.2/-0.0
0.35
0.1/0.4
0.2/0.6
-0.20 -0.5/-0.1
-0.6/0.1
-0.27 -0.3/-0.2
-0.4/-0.1
-0.02 -0.1/0.0
-0.1/0.0
-0.11 -0.2/-0.1
-0.3/-0.0
+
+
+
+
w 10−2 [m/s]
w
∆w
1.11 0.8/1.3 ++
1.0/1.3
1.06 0.8/1.2
0.5/1.1
1.06 0.8/1.3
0.7/1.4 ++
1.37 1.2/1.5
0.6/1.7 ++
1.24 1.0/1.4
0.6/1.3
1.67 1.3/1.9
0.5/2.0 ++
0.48 0.4/0.6
0.1/0.8
0.71 0.4/1.0
0.1/1.0
0.56 0.5/0.6
0.3/0.6
0.28 0.2/0.3
0.2/0.3
0.36 0.3/0.4
0.1/0.5 +
Table A.1: Average velocity of in y and z direction and its variation. The averaging
is done according to figure 6.15.
119
series number/
direction dV
1/z
1/y
2/z
2/y
3/z
3/y
4/z
4/y
5/z
5/y
6/z
6/y
7/z
7/y
8/z
8/y
9/z
9/y
10 / z
10 / y
11 / z
11 / y
Ex
[m/s]
0.03/0.07
0.01/0.04
0.07/0.11
0.07/0.22
0.12/0.16
0.12/0.17
0.13/0.18
0.12/0.3
0.12/0.19
0.18/0.46
0.19/0.26
0.18/0.46
0.03/0.05
0.30/0.46
0.23/0.78
0.22/0.43
0.20/0.21
0.19/0.21
0.11/0.12
0.11/0.13
0.11/0.13
0.11/0.13
++
++
++
+
++
++
++
++
++
++
++
Ey
[m/s]
0.003/0.08
0.003/0.007
0.04/0.07
0.004/0.007
0.002/0.008
0.004/0.007
0.003/0.007
0.003/0.006
0.007/0.013
0.007/0.01
0.008/0.01
0.007/0.012
0.015/0.027
0.012/0.026
0.04/0.06
0.04/0.06
0.007/0.01
0.006/0.011
0.008/0.014
0.008/0.012
0.004/ 0.006
0.003/0.005
-
Ez
[m/s]
0.04/0.06
0.04/0.07
0.05/0.06
0.03/0.06
0.03/0.06
0.03/0.06
0.06/0.08
0.03/0.09
0.06/0.09
0.05/0.16
0.08/0.14
0.05/0.16
0.04/0.08
0.04/0.08
0.10/0.35
0.18/0.33
0.03/0.04
0.02/0.04
0.03/0.06
0.02/0.05
0.01/0.03
0.01/0.02
++
++
++
++
++
++
-
Esum
[m/s]
0.07/0.14
0.07/0.11
0.12/0.17
0.12/0.27
0.17/0.20
0.16/0.23
0.18/0.3
0.16/0.4
0.19/0.27
0.27/0.63
0.30/0.39
0.27/0.63
0.10/0.14
0.09/0.13
0.37/1.00
0.60/0.80
0.23/0.25
0.23/0.25
0.16/0.18
0.16/0.18
0.14/0.16
0.13/0.16
Table A.2: Velocity and fluctuations in ‘boxed’ volume according to figure 6.15.
+
+
+
++
++
+
++
+
+
++
120
Data tables
• Series 8 to 9 was a decaying wave field (wind state off) from series 7.
Appendix B
Linearized wave equation
A simple case of fluid flow should be discussed. The three dimensional case is
already far from being simple and therefore the two dimensional case is considered.
−
−
If an irrotational flow of velocity →
u = [u(x, y,t), v(x, y,t)] is considered (∇× →
u =0
⇔
∂v
∂x
− ∂u
∂y = 0), there exists a velocity potential
→
−
u = ∇φ.
−
By virtue of the incompressibility condition ∇ · →
u = 0 the velocity potential satisfy
the Laplace’s equation
∆φ = 0.
(B.1)
−
The velocity potential is constant along the streamline ψ ((→
u · ∇)ψ = 0), where
the velocity can be expressed with the Cauchy-Riemann equations (of complex
variable theory) as
∂φ ∂ψ
∂φ
∂ψ
u=
=
, v=
=− .
(B.2)
∂x
∂y
∂y
∂x
A further simplification is the assumption of small-amplitude waves where the water surface displacement y = η(x,t) and the associated fluid velocity u, v are small1 .
∂η
The kinematic boundary condition2 at the free surface ∂η
∂t + u ∂x = v then simplifies
to
∂η(x,t)
v(x,t) =
.
(B.3)
∂t
Applying a Taylor expansion of v around η = 0 yields v(x, 0,t) = ∂η/∂t. The
same linearization is applied to the dynamic boundary condition3 and simplifies
1 u ∂η
∂x
is small compared to v (as in the kinematic boundary condition) and u2 + v2 small compared
to gη (as in the dynamic boundary condition).
2 Fluid ‘elements’ at the surface must remain on the surface.
3 ‘pressure condition’: if the fluid is inviscid, the pressure p is equal to the atmospheric pressure
at y = η(x,t).
122
Linearized wave equation
∂φ
∂t
+ 12 (u2 + v2 ) + gη = 0 to
∂φ
+ g η = 0 on y = 0
∂t
(B.4)
(where g is the gravitational force).
A sinusoidal traveling wave at the surface is
η(x,t) = A cos(kx − ωt)
(B.5)
where A is the amplitude of the surface displacement, k the wave number and ω the
frequency.
The velocity potential
φ = f (y) sin(kx − ωt)
is consistent with the boundary conditions B.3 and B.4. It has to satisfy the Laplace
equation (B.1) which leads to
∂2 f
− k2 f = 0
∂y2
and its solution
f = C eky + D e−ky .
If the water is of infinite depth it must be chosen D = 0 and it follows φ = C eky sin(kx−
ωt). The constant C can be obtained by substituting φ and η (equation. B.5) into
the boundary conditions B.3 and B.4 and the result is
φ=
A ω ky
e cos(kx − ωt)
k
(B.6)
with the dispersion relation ω2 = gk.
The particle paths can be obtained from equations (B.2) and (B.6)
u = A ω eky cos(kx − ωt), v = A ω eky sin(kx − ωt).
(B.7)
Assuming a small depart (x0 , y0 ) of the particle from its mean position (x, y) the
velocity can be approximated by
dx0
dy0
= A ω eky cos(kx − ωt),
= A ω eky sin(kx − ωt)
dt
dt
and results in
x0 = −A eky cos(kx − ωt), y0 = A eky sin(kx − ωt).
(B.8)
The particles describe circular paths and the radius decreases exponentially with
depth y. As the fluid velocity also decreases exponentially, the energy of the surface
water wave is mainly contained within half a wavelength below the surface.
Appendix C
Depth of field, depth of focus
The equation for par-axial lenses of focal length f is
1
1
1
= +
f
do di
where do is the distance of the object and di the distance of the image from the
lens plane. Only objects at location do are imaged focused on the image plane
at distance di , if the object is moved from its location by ∆do to location do0 =
do + ∆do the object is out of focus. In a zero order approximation this out of focus
is described by a blurring circle of radius ε. The f-number O characterizes the depth
of field through the ratio of focal length f to the diameter R of the lens, O = f /2R.
From geometry the following relation is valid
di ∆do
di
ε
= 0 −1 =
R di
do do,
where the magnification M = di /do and
di, −di
di,
(C.1)
,
o
,
= M dod−d
.
o
The depth of field ∆do is determined through an acceptable blur radius εc :
∆do =
do
Mf
2 O εc
−1
.
If 2 O εc M f then the depth of field is
∆do =
2 O εc
do .
Mf
Moving the image plane instead of the object plane results also in defocusing. In
the same way as for the depth of field, the depth of focus is determined through
124
Depth of field, depth of focus
an acceptable blur radius εc on the image plane. Taking equation (C.1) and if
2 O εc f the depth of focus is defined as
∆di =
2O
di εc .
f
The relation between the depth field of and the depth of focus is
∆do = M 2 ∆di .
Appendix D
Basics of the finite element
method
The finite element method (FEM) offers systematic rules to produce stable numerical schemes. This method is well suited to handle two- and three-dimensional
structures even with complex structures and boundaries.
The starting point of the finite element method (FEM) is the variational equation1
(see figure D.1):
Find u ∈ V so that
∀v ∈ V
a(u, v) = f (v)
where V is a given function space.
A simple example is the heat conductivity equation
−∆T = q
T =0
in Ω
auf Γ.
(D.1)
where T is the temperature and Γ = ∂Ω. The body is restricted to the area Ω.
If the solution of equation (D.1) for T is u and v ∈ V where V is a set of all differentiable functions in Ω, the integral equation can be written as
Z
(∆u)vdΩ =
Ω | {z }
a(u,v)
Z
qv dΩ
Ω |{z}
f (v)
The boundary value problem is transformed into a symmetric bi-linear form, the
variational equation2
a(u, v) = f (v)
∀v ∈ V
(D.2)
1 Variational problem: J(v) → min! ∀v ∈ V ,
Ritz method: J(vh ) → min! ∀vh ∈ Vh , Galerkin-Ritz: a(uh , vh = f (vh ) ∀vh ∈ Vh .
2 considering the boundary conditions on Γ and applying the Gauß integral theorem.
126
Basics of the finite element method
variational principle
Ritzmethod
boundary value problem
variational
calculations
integral equations
residue method/
integral equations
variational equation
Ritz
Galerkin
discrete problem
Figure D.1: Discretization methods. For the finite element method the variational
equation is the starting point.
where u ∈ V is searched for. This functional space V is discretized by the Galerkin
discretization:
The functional space V is discretized in N-dimensional sub-spaces.
Vh ⊂ V
The approximate solution for uh ∈ Vh can be expressed as
N
uh = ∑ ui wi
i=1
with the base functions wi ∈ V for Vh and the constants ui . The discrete problem of
equation (D.2) is then written as the so called weak formulation
a(uh , vh ) = f (vh )
with base function wi j (xk , yl ) =
∀vh ∈ Vh
1 for i = k, j = l
0
otherwise
The fundamentals of the finite element method are summarized as follows:
• Choose wi with i = 1 . . . N which deviate from zero only inside Ωi not equal
to zero, and Ωi , Ω j with as few as possible common points.
127
y
ϕ
Ω
y
i
x
x
ij
ϕ
i,j
1 y y
j y
Ω i,j
j-1
0
x
i
i-1
x
x
i
Figure D.2: Illustration of the construction of the base function. Left: Each square
is divided into two triangles. The discretized space Vh is characterized by functions
inside the triangle which are linear in x and y (right figure). Outside of Ωi j the base
functions wi j (x, y) are zero.
• Construct a(wi , w j ), f j , and solve ∑Ni=1 a(wi , w j )ui = f j ,
j = 1, · · · , N.
• Approximate the solution
N
uh = ∑ ui wi
i=1
The ‘weak formulation’ of the Navier-Stokes equation is formulated in the following:
Find velocity u ∈ V and pressure p ∈ Q so that
∂u
, v + (u · ∇u, v) − ν (∇u, ∇v) + (p, ∇v) = ( f , v)
∀v ∈ V,
∂t
(∇ · u, v) = 0
∀q ∈ Q,
with V = H01 (Ω) and Q = L2 (Ω).
The Galerkin discretization step is then applied: Vh ⊂ V and Qh ⊂ Q with grid
parameter h.
To achieve a stable discretization, so called stabilization terms have to be added.
128
Basics of the finite element method
Functional space for FEM
The variability in the choice of the functional space entails the flexibility of the
finite element method which is superior to other methods such as finite differences
or finite volume methods. General properties of the discrete space Vh for FEM are
summarized in the following:
The weighting functions wi j are continuous, but not continuously differentiable
(for example on the border of Ωi j or on the sides of the triangles, see figure D.2).
A concept of derivation operator is therefore introduced which allows to partly
differentiate partly differentiable functions. Depending on the variational equation
and the associated boundary conditions the functional space V is constructed. In
the previous sample the functional space is V = H01 (Ω) with the boundary condition
u|∂Ω = 0 (Dirichlet) where V = H 1 (Ω) is the Sobolev-space of first order, and the
lower index of H01 symbolizes that the Dirichlet boundary condition is applied3 .
Definition of Sobolev-space: A function v belongs to H 1 (Ω), if it belongs to L2 (Ω)
∂v ∂v
, ∂y ) does also belong to the L2 (Ω).
and the generalized derivative ( ∂x
The space H01 (Ω) is of importance for solving elliptic differential equations of
second order and Dirichlet boundary condition.
3 The boundary conditions are partly enclosed in the functional space and partly enclosed in the
variational equation, which is a characteristic of the FEM.
Acknowledgements
This work would not have been possible without the help and support of many
people.
I thank Prof. Dr. Bernd Jähne for giving me the opportunity to do this work in his
image processing group at the Interdisciplinary Center for Scientific Computing
and the Institute for Environmental Physics and for his advice and support. I appreciated his ideas of combining different fields concerning numerical calculations
of fluid flow, image processing and experimental investigations. This lead me to a
very interesting work and experience in many different fields.
I thank Prof. Kurt Roth for his effort in reviewing this thesis in a rather short time.
For the numerical mathematics used in this work I thank Prof. Dr. Rolf Rannacher
and many members of his group, especially Dr. S. Turek who helped a lot in the
very difficult field of finite element methods for fluid flows, numerical solvers with
software related problems and discussions.
Furthermore I thank all the members of the image processing group for a good
atmosphere and fruitful discussions. An effective team work was especially necessary for the construction of the new AEOLOTRON wind-wave facility and I thank
all the helpers and especially Reinhard Kalkenings. In many aspects of the work,
image processing problems and experimental constructions, the members of the
group supported me. The works of Christoph Garbe and Michael Stöhr were of
fundamental importance for this work.
The enormous effort for the construction of the AEOLOTRON wind-wave flume
and all the experimental constructions would certainly not have been possible without the help and assistance of the machine shop. I would like to thank all the staff
of the machine shop at the Environmental Physics.
A scholarship in the ‘Graduiertenkolleg Modellierung und wissenschaftliches Rechnen in Mathematik und Naturwissenschaften’ is gratefully acknowledged.
I also thank my parents, who supported me during my studies.
I address special thanks to my wife Agnieszka for her support and patience.
130
Basics of the finite element method
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