Volumetric Measurements of the Transitional
Backward Facing Step Flow

Der Fakultät für Maschinenbau, Verfahrens- und Energietechnik
der Technischen Universität Bergakademie Freiberg
eingereichte

DISSERTATION

zur Erlangung des akademischen Grades
Doktor Ingenieur(Dr.-Ing.)

vorgelegt

von Dipl.-Ing. Jens Kitzhofer
geboren am 22.03.1981 in Moers

Freiberg, den 14.6.2011

Versicherung
Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter undohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremdenQuellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht.
Bei der Auswahl und Auswertung des Materials sowie bei der Herstellung des Manuskripts
habe ich Unterstützung von folgenden Personen erhalten:
Prof. Dr.- Ing. habil. Christoph Brücker
Weitere Personen waren an der Abfassung der vorliegenden Arbeit nicht beteiligt. Die
Hilfe eines Promotionsberaters habe ich nicht in Anspruch genommen. Weitere Personen
haben von mir keine geldwerten Leistungen für Arbeiten erhalten, die nicht als solche
kenntlich gemacht worden sind.
Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Formeiner anderen Prüfungsbehörde vorgelegt.

Abstract
The thesis aims to extend the knowledge of separated and reattaching shear layers in thetransitional regime (Reh=4440) in case of the benchmark experiment “Backward Facing
Step Flow (BFSF)”. The final goal is a better physical understanding of complex
dynamical interactions of coherent flow structures and their influence on reattachment
position. Furthermore, the thesis tries to fill up the lack of missing three-dimensional andtime resolved data regarding the BFSF. The recorded measurement volume is roughly
90x60x15 mm³ (LxBxH) in size and is positioned in a gravity driven water tunnel. The
first part of the thesis focuses on the selection and implementation of a sufficientmeasurement technique for the identification and characterization of volumetric coherent
flow structures in case of the present BFSF configuration. A detailed literature review
regarding volumetric measurement techniques is given. The development andimplementation of the measurement technique of choice “3D Scanning Particle Tracking
Velocimetry” and its optimizations for measuring the present BFSF configuration arehighlighted. The second part of the thesis studies the BFSF in a volumetric as well as in a
temporal manner. The volumetric measurements give deeper insight into three-dimensional
shape of coherent structures of the BFSF, i.e. shear layer, roller, hairpin like structure andrecirculation region. Complex interactions of single coherent structures and influences on
the reattachment position are highlighted. A feedback mechanism is detected showing acyclic change from line to finger configuration of reattachment. This cyclic change isfound to be the result of vortex pair interactions (roller - hairpin like structure) in vicinityof reattachment.

Zusammenfassung
Das Ziel der vorliegenden Dissertation ist es, das Wissen über abgelöste undwiederanlegende Strömungen im transitionellen Strömungsbereich (Reh=4440) einer„zurückspringenden Stufe (BFSF)“ zu erweitern. Der Fokus liegt hier auf dem
physikalischen Verständnis komplexer dynamischer Interaktionen von kohärenten
Strukturen und deren Einfluss auf die Position der Wiederanlegung. Darüber hinaus soll
die Lücke in fehlenden volumetrischen und zeitaufgelösten Datensätzen bezüglich desBFSF aufgefüllt werden. Die Größe des interessierten Messvolumens beträgt ca. 90x60x15
mm³ (LxBxH). Alle Messungen sind in einem Schwerkraft getriebenen Wasserkanal
durchgeführt worden. Der erste Teil der Arbeit konzentriert sich auf die Auswahl und
Implementierung einer geeigneten volumetrischen Messmethode zur Untersuchung vonvolumetrischen kohärenten Strömungsstrukturen in Bezug zu der vorliegenden BFSF
Konfiguration. Neben einer detaillierten Literaturrecherche, wird die Entwicklung und
Implementierung der ausgewählten Messmethode “3D Scanning Particle Tracking
Velocimetry” im Detail erläutert. Der zweite Teil der Arbeit konzentriert sich auf dievolumetrische und zeitaufgelöste Auswertung der rekonstruierten Geschwindigkeitsfelder.
Die volumetrische Untersuchung erlaubt einen tieferen Einblick in die Form und
Entwicklung von kohärenten Strukturen, d.h. Scherschichten, Wirbelrollen, HarrnadelWirbel und Rezirkulationsgebiet. Komplexe dynamische Interaktionen und deren Einfluss
auf die Position der Strömungswiederanlegung werden untersucht. Ein Rückkopplungsmechanismus ist detektiert worden, der eine zyklische Schwankung von Linien- zu
Fingerkonfiguration der Wiederanlegelinie zeigt. Diese Arbeit zeigt, dass die zyklische
Schwankung das Resultat von Wirbelpaar-Interaktionen (Wirbelrollen-Haarnadel-Wirbel)
im Bereich der Wiederanlegung ist.

Contents
Nomenclature………………………………………………………………………………I
List of Figures…………………………………………………………………………….IV
Tables…………………………………………………………………………………….VII
1 Introduction.......................................................................................................................1
1.1 Overview......................................................................................................................1
1.2 Aim of the Thesis.........................................................................................................5
1.3 Structure of the Thesis .................................................................................................6
2 Measurement Techniques ................................................................................................8
2.1 Planar Measurement Techniques .................................................................................92.1.1 Laser Induced Fluorescence................................................................................10
2.1.2 Particle Image Velocimetry ................................................................................10
2.2 Volumetric Measurement Techniques .......................................................................122.2.1 Optical Registration ............................................................................................132.2.2 Hardware.............................................................................................................16
2.2.3 3D Particle Reconstruction Methods ..................................................................18
2.2.4 3D Velocity Field Generation.............................................................................29
2.3 3D Scanning Particle Tracking Velocimetry .............................................................412.3.1 Experimental set-up ............................................................................................432.3.2 Tracking Algorithm and Point Reconstruction...................................................54
2.3.3 Post Processing ...................................................................................................58
3 Backward Facing Step Flow ..........................................................................................62
3.1 Inlet Flow Conditions ................................................................................................62
3.2 Mean Flow Characteristics ........................................................................................65
3.3 Dynamical Flow Features ..........................................................................................69
4 Results and Discussion....................................................................................................75
4.1 Region I: Shear Layer Roll Up ..................................................................................75
4.2 Region II: Vortex Convection....................................................................................94
4.3 Region III: Reattachment...........................................................................................98
4.4 Region IV: Recirculation Region ............................................................................110
4.5 Region V and VI: Step Corner and Recovery Zone ................................................114
4.6 Roller - Hairpin Vortex Interaction .........................................................................115
5 Conclusion and Outlook...............................................................................................118
5.1 Conclusion ...............................................................................................................118
5.2 Outlook ....................................................................................................................126
Bibliography .....................................................................................................................128

I

NomenclatureA

filled profile [mm²], intern transformation matrix, universal boundary layer
constant, amplitudeAR

aspect ratio (B/h)=57

B

step span, solution matrix 3x3 (3D point)

BFSF

backward facing step flowCCcross correlation

CCD

charge coupled device

Crshearshear layer criterionDextern transformation matrix

DCT-PLSdiscrete cosines transform - penalized least squares

DNS

direct numerical simulationE

solution matrix 4x3 (3D point)ER

expansion ratio (h2/h1=1.25)

F

fundamental matrix

FFT

fast fourier transformationFR

flow reattachment

FSflow separationJjacobianLE

entry length

Lyimage heightLSlight sheet

LSM

least squares matchingMmagnificationNN

nearest neighbour

P’perspective transformation matrixPprojection matrix

PDF

probability density function

PIV

particle image velocimetry

POD

proper orthogonal decomposition

PTV

particle tracking velocimetry

IIQ

solution matrix (calibration P), vortex detection criterionRrotational matrix

Reh

Reynolds number based on step height

ROI

region of interest

P,Q,R

invariants

S

rate of strain tensor

S’deviatoric rate of strainTux

turbulence intensity in streamwise direction (= (u’²)1/2)

U

velocity (u,v,w)TU

instantaneous velocity (u,v,w)T [mm/s] ,wetted circumference [mm]U*normalized velocity: U*=U/ U∞

Ubulk

free-stream/bulk velocityW

object point

X

image coordinates [pix]XC

camera coordinates [mm]XSsensor coordinates [mm]XW

object coordinates [mm]csolution vector 1x3 (3D point), camera centerdmax

particle displacementedifficulty in tracking single particles

ffocal length

fps

frames per secondhstep height [mm]h1channel height upstream of expansion

h2

channel height downstream of expansionkscaling factor [pix/mm]

ld,eq

equivalent diameter: 4A/U [mm]nrefractive index

ppp

particles per pixelsmedian particle distancettranslational vector, normalized time t*=t·U∞/h

tL
u

+

temporal length of a trajectorysublayer scaled velocity u+ =u/uτ [-]

III
uτ

wall shear velocity [mm/s]

xr1

primary point of reattachment

xr2

secondary point of reattachmenty+sublayer scaled distance from the wall y+ = y· uτ /υ [-]Φ

quality criteriaΩrate of rotationαinclination angle of vortex roll upδtotal shear layer thickness (δ=δ++δ-) [mm]δ+

upper shear layer thickness [mm]δ-lower shear layer thickness [mm]∂δ/∂xgrowth rateΔt

separation timeθ

momentum thickness θ*= θ/hκuniversal boundary layer constantΔ

vortex detection criterion

λeigenvalue, wave length

λ2

vortex detection criterion

ρP

density of seeding particles [kg/m³]

υkinematic viscosity [m²/s]

ωvorticity

IV

List of Figures
Fig. 1.1

Sketch of flow separation and flow reattachment

2

Fig. 1.2

Definition of geometry and main parameters

4Fig. 1.3

Flow facility

5Fig. 2.1

Definition of planes recorded via LIF and PIV

9Fig. 2.2

Snapshot of the BFSF visualized via LIF

10Fig. 2.3

Pointwise and voxelwise representation of particles

13

Fig. 2.4

Sketch of geometrical and wave optics

14

Fig. 2.5

Schematic of perspective projection

14

Fig. 2.6

Three Aperture defocusing arrangement

19

Fig. 2.7

Experimental set-up of in-line holography

20

Fig. 2.8

Experimental multi-camera set-up

21

Fig. 2.9

Schematic of stereoscopic camera set-up

24

Fig. 2.10

Patented V3V system

24

Fig. 2.11

Schematic of multi-camera algebraic reconstruction technique25

Fig. 2.12

Schematic of depth layer reconstruction

26

Fig. 2.13

Principal sketch of multiple pair wise tomographic reconstruction

27

Fig. 2.14

Comparison of Eulerian frame and Lagrangian frame

29

Fig. 2.15

Theoretical limit in PTV

31

Fig. 2.16

Particle matching yield for different PTV methods

32

Fig. 2.17

Reconstructed trajectory and environment

33

Fig. 2.18

Direct 3D visualization of Lagrangian trajectories

34

Fig. 2.19

Schematic of cube transformation

35

Fig. 2.20

Visualization of results obtained by LSM

38

Fig. 2.21

Definition of volume of interest

42

Fig. 2.22

Experimental set-up of backward facing step flow

43

Fig. 2.23

Influence of the water filled prism on the quality of particle images

44

Fig. 2.24

Parallel projection due to telecentric imaging

45

Fig. 2.25

Influence of the number of calibration points on projection

48

Fig. 2.26

Magnitude and direction of deviation in 3D position

49

Fig. 2.27

Influence of deviation in 2D position on 3D position

50

Fig. 2.28

Principal of rotating mirror drum

50

V
Fig. 2.29

Positions of light sheets in the side view51

Fig. 2.30

Intensity distribution of an exemplarily chosen image section

52

Fig. 2.31

Synthetic generated volume filled with particles

52

Fig. 2.32

Reduction of the search area along the epipolar line

53

Fig. 2.33

Schematic of two-dimensional tracking

55

Fig. 2.34

2D trajectories of light sheet 1 to light sheet 4

57

Fig. 2.35

Schematic of gaps and ambiguities in the proposed PTV algorithm

59Fig. 2.36

Three-dimensional trajectory tracked over all light sheets

59

Fig. 2.37

Enhanced trajectory

60

Fig. 2.38

Direct 3D visualization of Lagrangian trajectories

61

Fig. 2.39

Instantaneous Eulerian velocity field visualization via LIC

61

Fig. 3.1

Mean BFSF between -12h<x<12h63Fig. 3.2

Inlet flow conditions

64

Fig. 3.3

Sublayer scaled velocity profile

65

Fig. 3.4

Definition of reattachment line

66

Fig. 3.5

Influence of Re and ER on mean reattachment length

67

Fig. 3.6

Parameter space of Re and boundary layer thickness

68

Fig. 3.7

Mean flow field in the x-y-plane visualized via LIC

69

Fig. 3.8

Principal sketch of the unsteady flow phenomena

71

Fig. 4.1

Categorization in regions

75

Fig. 4.2

Comparison of mixing layer and BFSF

76

Fig. 4.3

Flow visualization of a free mixing layer

76

Fig. 4.4

Post processed LIF visualization

77

Fig. 4.5

Definition of du/dzmax-criterion

78

Fig. 4.6

Extracted shear surface

79

Fig. 4.7

Shear Layer thickness

80

Fig. 4.8

Definition of spatial characteristics

81

Fig. 4.9

Shear Layer properties in x- and y-direction

82

Fig. 4.10

Convection of shear layer perturbation

83

Fig. 4.11

Comparison o vortex detection criteria

85

Fig. 4.12

Characteristics of a roller

86

Fig. 4.13

Reconstruction of a vortex center line

87

VI
Fig. 4.14

Definitions of the vortex center line

88

Fig. 4.15

Properties of vortex center line

89

Fig. 4.16

Model of change of spatial structure of vortex center line

89

Fig. 4.17

Roll up configurations

90

Fig. 4.18

PDF of the inclination angle

91

Fig. 4.19

Isometric view of vortex line configuration

92

Fig. 4.20

FFT of w-velocity signal

92

Fig. 4.21

Interaction of shear layer and vortex center line

93

Fig. 4.22

Development of rollers without external disturbances

95

Fig. 4.23

Model of change of spatial structure of vortex center line

96

Fig. 4.24

Biot-Savart law applied to a curved vortex center line

97

Fig. 4.25

Instantaneous vector field in the symmetry plane

98

Fig. 4.26

Definitions of recirculation region

99

Fig. 4.27

Time signal of reattachment length

100

Fig. 4.28

Comparison reattachment length obtained by 2D PIV and 3D SPTV

101

Fig. 4.29

Sequence of large scale reattachment break in obtained by 2D PIV

102

Fig. 4.30

Finger and line configuration of reattachment

104

Fig. 4.31

Mean reattachment length as a function of time

105

Fig. 4.32

Finger to line configuration of reattachment

106

Fig. 4.33

Iso view of recirculation region and corresponding rollers

108

Fig. 4.34

Schematic of the generation of fingers

109

Fig. 4.35

LIF-visualization of a mushroom-like vortex

110

Fig. 4.36

Detection of hairpin-like vortex

111

Fig. 4.37

Schematic proposal for generation of hairpin like vortex

113

Fig. 4.38

Schematic flow situation in vicinity of the step corner114

Fig. 4.39

Schematic of hairpin-roller interaction

115

Fig. 4.40

Instantaneous visualization of hairpin-roller interaction

116

Fig. 5.1

Principal sketch of line to finger configuration

125

VII

TablesTab. 2.1
Tab. 2.2

Accuracy and processing time of 2D centroid estimation
Mean accuracy of multi-camera calibration

30
48

Volumetric Measurements of the Transitional Backward Facing Step Flow

1

1 Introduction

1.1 Overview
Flow Separation (FS) and Flow Reattachment (FR) have been in focus of investigation for
more than hundred years since the famous studies of Ludwig Prandtl at the beginning of
the 19th century in Göttingen. Understanding the way flow separates and reattaches on asurface is critical in engineering applications. Today, one of the major tasks is designing ina drag reducing manner for efficiently reducing fuel consumption in transportation. Here,drag is highly influenced by separation and reattachment of the flow. It is obvious thatsome kind of boundary layer separation occurs on vehicles, airplanes or in combustors.
The most hazardous consequence of FS is stall in aerodynamics. The boundary layer
separates on the surface and the airfoil looses lift.
First studies about separated flows are dated back to 1868 (Helmholtz) and 1869
(Kirchhoff) in case of bluff bodies. From theoretical, experimental and numerical point ofview much effort has been made in understanding the physical background of the flow
phenomena. No matter if laminar FS or turbulent FS occurs, the FR is characterized byturbulent flow behaviour. The basic for the analysis of such highly fluctuating flows is theneed “to separate the random processes from the non-random processes“ as stated by
Dryden (1948) [29]. This statement implies the existence of ordered structures (nonrandom processes) in turbulence. Via qualitative flow visualization some of the “orderly
elements” began “to be recognized and described” as discussed by Roshko (1976) [111].
Nevertheless, three-dimensional coherent structures were hidden for quantitative
measurements due to insufficient measurement techniques. Here, recent developments involumetric measurement techniques allow the analysis of the “orderly elements”. In fact,
recent results concerning three-dimensional coherent structures in a turbulent boundary
layer show the importance for investigation, because these structures seem to form theskeleton of turbulent flow [126].
The investigation of FS and FR is connected with some difficulties since both are linked
together. Figure 1.1 on top shows a schematic representation of FS-FR on an airfoil. The
boundary layer separates at xs, develops into a shear layer and reattaches at xr. In between,
the separated boundary layer rolls up into vortices with the main axis of rotation in
spanwise direction (Kelvin Helmholtz instability). The size of the rollers is amplified in2downstream direction. Underneath the shear layer a separation bubble (recirculation
region) is formed. The recirculation region is characterized by a low speed, but highly
fluctuating flow state. Furthermore, these fluctuations result in fluctuations of xs and xr.
This is a complex feedback system, as the fluctuation of the separation bubble results in avariable point of reattachment.
A more controlled situation is in the Backward Facing Step Flow (BFSF) as shown infigure 1.1 on the bottom. Herein, the point of separation is spatially fixed. The boundary
layer separates at the step edge due to the sudden geometrical expansion resulting in apositive pressure gradient in flow direction. Due to the forced separation, the flow
configuration is also called forced shear layer. The separated shear layer is characterizedby high, positive velocities on the free stream side and low positive or negative velocities
on the other side. Behind the separation edge, the shear layer rolls up into discrete vortices
due to Kelvin-Helmholtz instabilities. The shear layer further develops in streamwise
direction including complex vortex interactions such as vortex pairing and vortex merging.
The separated boundary layer finally reattaches at a certain distance downstream of thestep on the bottom wall at xr and forms a recirculation region. The point of reattachment ishighly unsteady as the flow in vicinity of reattachment is characterized by turbulent shear
layer impinging.

Fig 1.1 Sketch of Flow Separation (FS) and Flow Reattachment (FR) on an airfoil (top) and at a Backward
Facing Step Flow (bottom)

Volumetric Measurements of the Transitional Backward Facing Step Flow3

Although FS and FR have been the subject of many investigations, both are not fully
understood at high Reynolds-numbers, where transitional or turbulent effects start todominate. Due to the spatially fixed point of separation, the BFSF is a frequently applied
flow configuration for the analysis of reattachment. The motivation is in a deeper
understanding of fundamental events concerning this type of flow for possibilities of active
control. Especially, the influence of three-dimensional coherent flow structures is not fully
disclosed.
Former quantitative studies on the BFSF can be divided into the use of pointwise
measurement techniques such as hot-wire anemometer or Laser Doppler anemometer andinto the use of planar measurement techniques like Particle Image Velocimetry (PIV). The
pointwise techniques were primarily applied for statistical analysis delivering deep inside
into overall flow behaviour like characteristic regions, flow reattachment or mean velocity
profiles. Planar measurement techniques like PIV supported the evaluation of coherent
flow structures. Limits are reached with these classical techniques beyond a Reynolds
number, where three-dimensionality occurs. Thus, three-dimensional datasets orinvestigations at high Reynolds numbers are rare in literature, limited to three-dimensional
Direct Numerical Simulations (DNS) up to low transitional Re-numbers [119]. Volumetricmeasurements on the BFSF were performed by Kasagi and Matsunaga (1995) [61] via 3D
Particle Tracking Velocimetry and more recently by Schröder et al. (2011) [127] via
Tomographic PIV. The authors show the necessity for consideration of threedimensionality in the analysis of the BFSF as three-dimensional vortical structures areomni-present in the BFSF.
Due to the availability of newest, high quality hardware at rather low cost, a rush isrecognized in further development of existing techniques and development of newtechniques for the measurement of volumetric velocity fields. Differences of the techniques
can be found in applied hardware and software, in temporal and spatial resolution, incomplexity, in optical registration and in flow description (Lagrangian or Eulerian).
However, all up to date techniques for measuring velocities use tracer particles for marking
the flow. The particles are illuminated by a light source and recorded on a sensor or an
arrangement of sensors. By proper algorithmic reconstruction techniques the velocity fieldis calculated. Once the velocity field is generated, it is described by velocity vectors at
random or structured positions in a defined volume.
The availability of all three velocity components in a time resolved manner gives rise to
the analysis of the velocity gradient tensor. The tensor consists of the spatial derivatives of

4

the velocity field and has become the most famous mathematical description for the
extraction of coherent flow structures.
For that purpose, this thesis applies a state of the art volumetric measurement technique,time resolved 3D Scanning Particle Tracking Velocimetry, for the investigation of thenatural BFSF in the transitional to turbulent regime. Here, ‘natural’ means that the flow
structure is generated by self-organization contrary to the ‘manipulated’ BFSF, where theflow dynamics is somehow forced into a regular pattern by certain excitation.

Fig 1.2 Definition of geometry, major parameters and coordinate system: B = 570 mm, LE = 1600 mm, Ubulk =0.52 m/s, ld,eq = 92 mm, l1=60 mm, l2=60 mm, h1 = 40 mm, h2 = 50 mm, h = 10 mm, δ = momentum thickness,
xR = reattachment length

The BFSF is studied herein in a rectangular (50mm x 570mm) closed circuit water channel(υ=10-6 m²/s) with canonical channel geometry as sketched in figure 1.2 and the flow
facility is shown in figure 1.3. The distance between the duct entrance and the volume ofinterest, called entry length LE is about 17 time the equivalent diameter ld,eq. The channel
flow is driven by gravity and the mean flow is regulated via the hydrostatic head. All
experiments were carried out for a defined height of the hydrostatic head generating a bulk
velocity upstream of the step of U∞=0.47m/s. The backward facing step is positioned on
the bottom of the channel with respect to Beaudoin et al. (2004) [11]. The step height h ischosen to 10mm, resulting in a Reynolds-number Reh = 4440 based on the step height. The
length of the ramp as well as the length of the plateau is 60 mm. The aspect ratio AR (B/h)
results in 57 and the expansion ratio ER (h2/h1) results in 1.25.

Volumetric Measurements of the Transitional Backward Facing Step Flow5

Fig 1.3 Experimental flow facility

All calculations regarding development and testing are performed via Matlab R2008b onan Intel Core2 Quad CPU9450 @ 2.66 GHz, 3 GB RAM. The final analysis of in total 80
GB raw image data is performed on a machine with 2 Six-Core AMD Opteron (tm)processors 2435 @ 2.6 GHz, 32 GB RAM.

1.2 Aim of the Thesis
The thesis aims to extend the knowledge of separated and reattaching shear layers in thetransitional/turbulent regime. The final goal is a better physical understanding of complex
dynamical interactions of coherent flow structures. Furthermore, the thesis tries to fill up
the lack of missing three-dimensional and time resolved data regarding the BFSF. For that
purpose, the thesis is structured into three partial goals.
Goal one is the development and implementation of a three-dimensional measurement
technique (3D Scanning Particle Tracking Velocimetry) operating at high spatial andtemporal resolution. The technique is developed and adapted in a manner to be optimallysuited for the analysis of the present, natural BFSF in the transitional regime.
Goal two is the analysis of the time resolved and three-dimensional datasets of the BFSF.
Here, the thesis focuses on the extraction of shape and influence of three-dimensional
coherent structures like vortices or shear layers.
Goal three is the fluid mechanical interpretation of the achieved results. For that purpose,the present test case is categorized with respect to the already existing large amount ofpointwise, planar and joint measurements and numerical simulations available in literature.
Proposals for mechanisms are presented trying to explain the observed unsteadycharacteristics of the BFSF.

6

As shown in figure 1.1 on the bottom, unsteady phenomena in the BFSF take place in thevicinity of the step corner. Thus, the region of interest (ROI) is positioned about thesymmetry plane immediately downstream of the step. The ROI is sized to about 90 x 90 x
15 mm³ allowing the observation of the unsteady flow phenomena. The region contains theshear layer and the turbulent reattachment.
All characteristic events disclosed herein will be described in context with earlier findings
published in literature, via qualitative flow visualization and via results obtained byquantitative visualization (3D Scanning PTV). Parts of the thesis are funded by the German
Research Foundation (DFG) in the SPP 1147: Bildgebende Messverfahren (1494/11-1),
which are greatly acknowledged here, and supported by Dantec Dynamics A/S.

1.3 Structure of the Thesis
The thesis is organized in two main parts. Part one describes the applied measurement
techniques (chapter 2). Part two focuses on the categorization (chapter 3) and analysis(chapter 4) of the BFSF.
Chapter 2 describes the applied measurement techniques. Beside the volumetric technique,planar time resolved PIV is used in multiple planes for the characterization of the inlet
flow conditions and for the validation of the results obtained by the volumetric technique.
The boundary conditions of the measurement and the estimated accuracy is pointed out.
The volumetric measurement technique is discussed in detail in the second part. Available
volumetric measurement techniques like tomographic PIV, digital holographic PIV ordefocusing PIV to study three-dimensional unsteady flow in complex flow fields arepresented. Advantages and drawbacks will be pointed out and discussed. The third part ofthe chapter explains the measurement technique of choice, 3D Scanning Particle Tracking
Velocimetry. The principle of the technique is pointed out in detail. Further developments
for increasing the capability, especially the temporal and spatial resolution, are discussed.
The necessary hardware devices will be presented and the integration of the hardware to
the experimental set-up is explained. Parts of the chapter have already been published inarchived journal papers [64][65][66].
Chapter 3 focuses on the categorization of the BFSF in terms of existing literature. For that
purpose, characteristic geometrical properties are pointed out. Mean kinematic propertiesare calculated via the results obtained by the PIV measurements. The obtained propertiesare compared in a detailed literature review including qualitative flow visualizations,

Volumetric Measurements of the Transitional Backward Facing Step Flow

7

quantitative pointwise measurements and planar optical or pressure based measurements.
Review of CFD results focuses on recently published papers. Furthermore, unsteadyphenomena concerning the BFSF are presented.
Chapter 4 provides detailed interpretation of the measured velocity field. The velocity field
is separated into characteristic regions. Unsteady events inside the single regions aredetected and discussed. The focus is in the extraction of large scale coherent structures
influencing the reattachment position.
Chapter 5 summarizes the obtained results and draws a conclusion. The outlook provides
recommendations for future research regarding the backward facing step flow andvolumetric measurement techniques.

8

2 Measurement Techniques
The fundamental theorem of Helmholtz says that every infinitesimal motion of a fluid
element can be decomposed into translation, rotation and deformation. In the last decades
several investigations have been performed to experimentally describe these fluid motions.
The importance of measurement is stated by Kähler (2004) [60]. The mathematicaldescription of fluid motion assumes pointwise description. This is valid by detailed
knowledge of local properties of all neighbouring fluid elements in a three-dimensional
space. Thus, the kinematic state of a fluid element is described by twelve parameters: threetranslational velocities and nine spatial derivatives of the velocity field in that point.
Standard pointwise measurement techniques like Hot Wire anemometry are limited to themeasurement of one velocity component. More sophisticated sensor configurations allow
measuring all three velocity components at approximately a single measurement location.
Nevertheless, evaluation of spatial derivatives or simultaneous measurements of multiple
points in a sufficient spatial resolution is not possible. In the last decades, PIV andStereoscopic PIV have become standard techniques for measuring two or three velocity
components in a planar field providing the 3 translational and 6 spatial derivatives in acertain plane. Recently developed fully volumetric measurement techniques with (under
distinct restrictions) sufficient temporal and spatial resolution are able to measure thewhole set of fluid motion. Thus, the new rush in volumetric measurement techniques
allows a deeper inside into the three-dimensional nature of unsteady flow phenomena.
For that purpose, this chapter describes the applied techniques for measuring the velocity
field of the BFSF shown in figure 1.2. The focus of the first part of the chapter is in planarmeasurement techniques, namely two-dimensional, two-component Particle Image
Velocimetry (2D2C PIV) and Laser Induced Fluorescence (LIF). The techniques arenowadays standard [108]. Thus, the implementation and an estimation of the achieved
quality will be briefly discussed. The second part contains a detailed literature review ofexisting volumetric measurement techniques based on particle imaging for measuring
three-dimensional (3D) and three-component (3C) velocity vector fields of a fluid flow.
Principles and hard- and software requirements are described for each techniquehighlighting the differences in hardware and software, temporal and spatial resolution,complexity, optical description and flow description. Techniques will be preferably
described, whose results are directly 3D3C, (without applying e.g. continuity equation

Volumetric Measurements of the Transitional Backward Facing Step Flow9[110] or out of plane correlations [106]) and are based on illuminated particle images. The
chapter finishes with a detailed description of the measurement technique of choice, 3D
Scanning Particle Tracking Velocimetry.

2.1 Planar Measurement Techniques
LIF is used herein for qualitative flow visualization and PIV is used for quantitativedescription. Flow visualization via LIF provides a first impression of the structure anddynamic of coherent flow structures. This is a kind of visual extraction of dominant events
being further analyzed in the quantitative measurements via PIV and 3D Scanning Particle
Tracking Velocimetry.
Figure 2.1 shows the investigated planes. PIV is performed in each of the streamwise
symmetry planes (P1-P4: overlap 20 mm; P6) and in a plane parallel to the bottom wall at
y/h = 0.1 (P5). The LIF measurements were exclusively performed in P3. A digital highspeed camera (Photron APX RS, 1024 x 1024 Pixel², 3000 fps) is used for both types ofrecording. The camera is adjusted perpendicular to the light sheet and variable instreamwise direction. For the measurements of the flow filed in P1-P5, the camera isequipped with a telecentric lens system resulting in a magnification of 0.2, which means an
object format of 90 x 90 mm² in physical space. The physical size of the single recorded
images is then defined by the image width and the channels height resulting in 90 x 50
mm² for P1-P4 and 90 x 90 mm² for P5. P6 is recorded with a long distance microscope(LDM) to achieve a high spatial resolution in the region near the wall for the calculation of
accurate inlet flow conditions. The size in physical space is 3 x 3 mm².

Fig 2.1 Definition of symmetry planes recorded via PIV and LIF: plane P1-P6 PIV, plane P3 LIF

10

2.1.1 Laser Induced Fluorescence
In principal, it is possible to analyze LIF measurements in a qualitative and in aquantitative manner. Herein, the technique will be exclusively used for time resolved
qualitative flow visualization as shown in figure 2.2 in an instantaneous snapshot of theflow region in P3. For that purpose, fluid with fluorescent properties (Rhodamin B) is
injected upstream of the step. The fluid is excited with a high speed laser (Coherent
Innova, 532 nm) and the emitted light is recorded by the high speed cameras at 1000 fps.

Fig 2.2 Snapshot of the BFSF visualized via Laser Induced Fluorescence. The size of recording region isapproximately W x H = 90 mm x 12 mm (W: horizontal width, H: vertical height).

2.1.2 Particle Image Velocimetry
Planar PIV is a non-intrusive, optical measurement technique. Seeding particles are
injected into the flow field. A light source is expanded to a light sheet via proper lenssystem for slicing the flow volume. The illuminated particles inside the plane are recorded
by a proper sensor, usually a CCD camera. The particles are registered as gray values on
the image plane. Thus, the analysis of double frames, which are separated by a distinct
time, allows the estimation of the planar velocity field. For that purpose, cross correlationon the gray value images is frequently applied estimating the particle displacement inside asmall interrogation area. The assumption is that the flow field is smooth and not
significantly influenced by rotational or shear displacements, thus yielding the zero-ordertranslational displacement field with an additional uncertainty in measurement due toneglecting the higher-order terms. The mean translational velocity of an interrogation area
is calculated by knowledge of the exact separation time. The result is a planar velocity field
on a regular structured grid (Eulerian). An overview of the method is given by Raffel et al.
(2007) [108].

Volumetric Measurements of the Transitional Backward Facing Step Flow11

Drawbacks of classical, planar PIV appear in three-dimensional or shear flows. Threedimensionality results in particles moving with an angle relative to the light sheet plane
resulting in the “out of plane” problem and consequently erroneous velocity vectors. High
shear flows are sensitive to cross correlation approaches as particles inside theinterrogation region represent different velocities. This type of measurement uncertainty
can be reduced by window deformation techniques as presented by Scarano (2002) [115],
but for larger computational cost.
The present quantitative PIV measurements are performed with homogenous sized (70 μm)
Vestosint particles (ρP≈ 1kg/m³) of approximately spherical shape. The neutrally buoyant
seeding particles are injected directly upstream of the duct of the water channel. The
particles are illuminated by a 10 mJ high speed laser with a maximum repetition rate of 10
kHz. The outlet of the laser light sheet is installed on top of the observed volume. The
particle images are recorded by the mentioned high speed camera (8 GB) equipped with atelecentric lens or a long distance microscope. The side of the volume opposite to theentrance side of the laser is coloured black and the face opposite to the cameras is covered
with a light absorbing mat that reduces stray reflections, thus giving a perfectly black
background. The camera and the laser light sheet equipment are installed on a positioning
device allowing the simultaneous shifting of laser and camera.
The system is run by Dynamic Studio 3.0. The software enables the control of therecording mode (time resolved or double pulsed) and assures the synchronization ofcamera and laser. The estimation of mean velocity profiles of the instantaneous flow field
is performed in the double pulse mode at a repetition rate of 25 Hz resulting in 160 s
recording time to assume a statistical independence in the time domain. The time resolved
measurements are made at 1000 frames/s. The memory of 8GB allows measuring asequence of about 8s.
The planar PIV images are post processed by standard software (Dynamic Studio 3.0) for
calculating the velocity vector fields. The recordings are analyzed with an adaptive crosscorrelation meaning a window refinement of three iterations for P1-P5 (128 x128 Pix; 64 x
64 Pix; 32 x 32 Pix) with an overlap of 50% and a refinement of two iterations for P6 (128x 128 Pix; 64 x 64 Pix) with an overlap of 50%. Outliers of velocity vectors are identified
by a correlation peak validation and a velocity range validation. The invalid data yield forthe symmetry planes (P1-P4) is about 1 %, the invalid data yield for P5 is 9 % and theinvalid data yield for P6 is 4%. The higher invalid data of P5 results from the difficulty toseed the plane directly above the bottom floor. Rejected erroneous velocity vectors are12

replaced by an averaging interpolation. The grid spacing of the velocity vectors results in a
resolution of 1.4 x 1.4 mm² for P1-P5 and 0.09 x 0.09 mm² for P6.
The calculated vector fields of the time resolved measurements (1000 fps) are postprocessed by a temporal ensemble averaging of three vector fields (Δt*=0.14). The numberof three vector fields has been carefully chosen for two aspects: on the one hand to smooththe vector field and on the other hand for not averaging the dynamical behaviour of flow
structures of interest. This has been decided by qualitative inspection. Small fluctuations
corresponding to random processes are removed by the smoothing and large scale
fluctuations are highlighted.

2.2 Volumetric Measurement Techniques
The measurement of a volumetric velocity field is similar to that of a planar velocity field.
Differences are found in the three-dimensional representation of particle fields. Similarities
are found in algorithms for the calculation of velocity fields out of recorded particle
displacements. The algorithms are expanded by the third dimension allowing thevolumetric description of fluid motion.
An important difference in volumetric measurement techniques is the description ofparticles in object space. The representation of reconstructed particles is pointwise orvolumetric as shown in figure 2.3 [125]. In pointwise representation the position of aparticle is described by the center of mass coordinates determined with subpixel accuracy.
Hence, the information of the intensity distribution of a particle is lost. This is not the casein a volumetric representation of particles, the so called voxel-model. A voxel is the threedimensional expansion of a pixel. The 3D measurement volume is discretized by voxelelements in all three directions. The voxel space has a certain intensity distribution(lightfield) and particles are identified as clusters of coherent intensity distributions, e.g.
Gaussian intensity distributions. The displacement of particles in a defined time interval(separation time) yields the velocity vector field similar to classical, planar PIV.

Volumetric Measurements of the Transitional Backward Facing Step Flow13

Fig 2.3 a) Pointwise representation of particles in a volumetric space b) Voxelwise representation of particles
in a volumetric space

2.2.1 Optical Registration
A major classification of volumetric measurement techniques is the optical registration of
the projection of 3D object space onto a 2D image plane. One class of techniques is basedon wave optics and the other class is based on the approximation of wave optics bygeometrical optics.
Wave optics is used in holography based approaches. In contrast to geometrical optics, therecorded 2D image contains information about the phase and the intensity of light (objectwave). For that purpose, the sensor is illuminated by light scattered from an object (lightscattering particles) and a reference wave as shown in figure 2.4 on the right. The lightscattered by particles is a spherical wave. The reference wave is a plane wave. The spatial
superposition of a spherical with a plane wave results in an interference pattern on thesensor, if the conditions for interference are fulfilled (coherence, monochromatic light,
same polarization and spatial superposition). The advantage of holography is the nearly
infinite depth of field, which is limited in techniques based on geometrical optics.
Furthermore, the recorded intensity distribution contains depth information.
In geometrical optics, the intensity of the incoming light is integrated on the sensor chip. In
case of PIV, the intensity distribution results from a lens projection of the illuminatedparticles onto the 2D sensor chip as shown in figure 2.4 on the left. A basic mathematicalmodel for the calculation of geometrical optics is the pinhole model [125]. The pinholemodel is a frequently applied model in image processing, which has been investigated andapplied in hundreds of publications. Most important principles will be briefly discussed for14

understanding the projection of objects space onto the image plane. For a detailed
description the reader is referred to Schreer (2005) [125].
(a)

(b)

Fig 2.4 Schematically sketch of (a) geometrical optics (b) wave optics

The mapping of three-dimensional object space onto a two-dimensional image plane iscalled a projection. Mathematically, the projection of an object point Xw onto an image
point X in pixel coordinates is described by an extern D, a perspective P’ and an interntransformation A as shown in figure 2.5. The superposition of the single transformations
results in the projection matrix P. The projection matrix is built in homogeneous
coordinates. Thus, inhomogeneous systems of equations become homogeneous allowing
simple and efficient solutions.

Fig 2.5 Schematic of perspective projection; D: extern transformation; P’: perspective transformation, A: interntransformation, c: camera center

The extern transformation D describes the conversion of the object space coordinate
system into the camera coordinate system. For that purpose, rotation and transformation isapplied. The scaling factor s describes a camera specific parameter.

Volumetric Measurements of the Transitional Backward Facing Step Flow

⎡ xc ⎤⎡ xw ⎤⎢y ⎥⎢y ⎥s⎢ c ⎥ = D ⋅ ⎢ w ⎥ ,⎢ zc ⎥⎢ zw ⎥
⎢ ⎥
⎢ ⎥
⎣1⎦
⎣1⎦

15

⎡R
with D = ⎢ T
⎣0 3

t⎤
1⎥⎦

(2.1)

The 3x3 rotational matrice R is defined as the multiplications of rotation matrices about thesingle directions and the translational vector t is defined as translations in the single
directions.

[

R = Rx ⋅ R y ⋅ Rz ,t = tx ,t y ,tz

]

T

(2.2)

The perspective transformation P’ describes the conversion of the camera coordinate
system into the sensor coordinate system. The perspective transformation with a defined
focal length f is defined by the intercept theorem in pinhole optics:
xs y sf==
xc y c z c(2.3)

Thus, the projection from camera coordinates into sensor coordinates is defined by:⎡ xc ⎤⎡ xs ⎤⎢y ⎥⎢⎥s ⋅ ⎢ y s ⎥ = P'⋅⎢ c ⎥ ,⎢ zc ⎥⎢⎣ 1 ⎥⎦⎢ ⎥⎣1⎦⎡fwith P ' = ⎢⎢ 0
⎢⎣ 0

0
f
0

0 0⎤
0 0⎥⎥
1 0⎥⎦

(2.4)

The intern transformation describes the conversion via a scaling factor k of the metricsensor coordinates, e.g. [mm], into discrete sensor coordinates [pix]. Furthermore, atranslation (xs0, ys0)T on the sensor ship is considered.

⎡ xc ⎤⎡ x⎤⎢ ⎥⎢ y ⎥ = A ⋅ ⎢ yc ⎥ ,⎢ ⎥⎢ zc ⎥⎢⎣ 1 ⎥⎦
⎢ ⎥
⎣1⎦

⎡k xs
with A = ⎢⎢ 0
⎢⎣ 0

0
k ys
0

xs0 ⎤
y s 0 ⎥⎥
1 ⎥⎦

(2.5)

16

The superposition of the single transformations results in the projection matrice P, which
completely describes the projection of object space onto the image plane.

⎡ xw ⎤
⎡ x⎤
⎢y ⎥
⎢
⎥
s ⋅ ⎢ y⎥ = P ⋅ ⎢ w ⎥ ,⎢ zw ⎥⎢⎣ 1 ⎥⎦⎢ ⎥⎣1⎦with P = A ⋅ P'⋅D

(2.6)

The focus of this study is in geometrical optics, as the applied measurement technique inthis thesis is referred to geometrical optics. A recent overview of holographic techniques
can be found in Arroyo et al. (2008) [5].

2.2.2 Hardware
The current state of the art in hardware for acquisition is described in this chapter formeasuring image based data, which means a large amount of image based data. The
processing of the data results in an even larger amount of 3D velocity field data. Thus notmandatory but preferable are advanced processing and storage units. As this part is under
rapid development, the current status of development in hardware requirements foradvanced data processing will be pointed out.
Hardware Requirements for Acquisition
The state of the art recording device in geometrical optics is a CCD or CMOS digital
camera [41]. The development of cameras with higher resolutions (4872x3248 pix², 3 fps,
Prosillica GE4900) and with higher frame rates (1280x800 pix², 6242 fps, Phantom V12.1)
offers the possibility to analyze unsteady flow fields in a time resolved manner at highspatial resolutions. Nevertheless, a compromise has to be made between high resolution
and high frame rates. Furthermore, appropriate lenses are necessary for a large depth offield. Once the measurement volume has been appropriately mapped onto the image plane,
the volume has to be illuminated. Volume illumination is a crucial part in geometrical
optics, it should be homogenous and of high intensity. Very large volumes, see Illmenauer
Fass, are usually illuminated by high energetic continuous sources or strobe sources, where
strobe sources deliver best results for illuminating large particles (1mm) [74]. In smallerVolumetric Measurements of the Transitional Backward Facing Step Flow

17

volumes the preferred illumination is a pulsed (up to several kHz) and largely expanded
laser beam due to its high pulse energy (up to several mJ). Due to the expansion of thebeam, the intensity distribution and homogeneity decrease. Thus, the volume, which can berecorded with high-quality particle images, is limited in its spatial extension. This problem
is partially overcome in scanning or multi layer illumination devices that deliver highquality particle images at the cost of lower temporal resolution [17], [107]. A new trend forillumination is the use of pulsed LEDs as presented by Willert et al. (2010) [144]. Recent
developments in LED technology result in high power LEDs with continuous radiant flux
levels and even higher intensities in pulsed operation at much lower cost making LEDs
suitable for PIV applications.
In traditional holography the recording medium is e.g. silver halide holographic film. Thismedium offers high resolution (3000 lines/mm) and large formats (100 x 125mm³), but is
not suitable for time resolved measurements. A promising material for non-digitalholography is Bacteriorhodopsin, which allows superior resolution (5000 lines/mm) [10].
Nevertheless, a drawback is the erasure of the hologram at room temperature during
readout. Furthermore, drawback of non-digital holography is the wet-process (chemical
processing), which makes the technique unpractible for a widespread use. This is overcome
in digital holography at the cost of lower resolution (100 lines/mm). Here, the film isreplaced by a digital camera allowing a fast digital recording with the possibilities of image
processing and numerical reconstruction [5]. In the present paper, the description ofholographic techniques will be limited to digital imaging, as this technique is promising ina wide field of applications.
Hardware Requirements for Data Processing
Whilst the development of hardware paces reasonably well with the acquisition demandplaced by current experiments, the ability of computers and current algorithms to transfer,process and further reduce the data within a reasonable period has fallen dramaticallybehind. As pointed out in 2.2.3, multi camera approaches are preferable in volumetricmeasurements. A typical three-camera set-up results in 6GB raw data for a recording
period of 2s by assumption of the recording system applied in this thesis. Already thetransfer process of the images to the computer last some tens of minutes. Processing thisdata on high performance computers will often take many hours or even days to complete.
Originally designed for games and visualization applications, graphical processing units(GPU’s) are recognized today as necessary tools for the processing of large amounts of

18

data in parallel. Computation is carried out in kernels, small fragment programs that
operate independently of one another on separate cores. A single GPU is made up ofmultiprocessors, where each multiprocessor contains a number of cores. Nevertheless, up
to now the coding of GPU is rather difficult and is only optimally performed by specialists.
The striking point is memory read out. The flow of data is determined ultimately by the
programmer, who must design the code to minimise memory conflicts. Examples of GPU
implementations for PIV data processing can be found in Champagnat et al. (2009) [21] orin Schiewitz et al. [121].

2.2.3 3D Particle Reconstruction Methods
The three-dimensional reconstruction of particles within a measurement volume can bedivided into single camera approaches and multi camera approaches. The general task of
all reconstruction techniques is the extraction of three-dimensional information (object
space) from two-dimensional information (camera sensor).
Single Camera Approaches
The advantage of single camera approaches is the simple experimental setup compared tomulti camera approaches, where a precise multi camera calibration is mandatory.
Furthermore, the costs for such a set-up are much lower. A drawback is the lower
permissible number particle density, which strongly limits the spatial resolution.
To the authors knowledge there are two approaches using a single camera based ongeometrical optics. The first approach is presented by Willert and Gharib (1992) [145], thedefocusing technique. The technique is based on a single camera equipped with a threehole aperture. A single particle is imaged on the sensor chip as three particles forming anequilateral triangle as shown in figure 2.6. By analyzing the positions and the lengths ofthe triangles, the information of depth position can be extracted. Originally, the result is apointwise reconstructed particle field. Pereira et al. (2002) blow up the infinitesimal small
point to a voluminous particle with a Gaussian intensity distribution to have voxel based
description of the measurement volume [99]. A drawback in this defocusing technique isthe need for a high intensity illumination source due to the small aperture of the holes
reducing largely the light amount integrated on the sensor. Furthermore, the spatial
resolution is rather coarse. An increasing number particle density results in overlapping
triangles and consequently in problems of detection.

Volumetric Measurements of the Transitional Backward Facing Step Flow19

Fig 2.6 Three Aperture defocusing arrangement; triangle on image plane contains relative position to areference plane at Z=L [36]The second approach is presented by Hain et al. (2009) [40]. The authors propose theanalysis of defined optical aberrations, namely astigmatism, to include depth information
beside the 2D sensor information (wave front deformation). The particle images are
deformed in an elliptical manner depending on the depth position, thus delivering thepossibility to analyze the shapes of the particle images and to extract depth information.
Up to now the technique has been limited to the reconstruction of very small measurement
volumes. The result is a pointwise reconstructed particle field.
A wave optics based single camera approach is digital holography [123]. In contrast to
afore mentioned single camera techniques the result is a 3D intensity distribution of theparticle field. Here, different kind of optical set-ups have been investigated in the last
decade, especially in photographic holography. Nevertheless, the advantages anddrawbacks are adaptable to digital holography. Typical methods applied to PIV are in-line
holography [136], [85], off-axis holography [103], [85] and phase shifting holography [8].
In in-line holography illumination beam and reference beam are identical. A schematic of atypical set-up is shown in figure 2.7. A coherent light source is arranged in-line with thesensor chip. A lens system generates a collimated beam, which is used to illuminate tracer
particles. The scattered light is used as object wave and the undisturbed illumination beam,passing through the particles, represents the reference beam resulting in a simple and stableset-up. However, if particle density is increased, there won’t be enough undisturbed lightfor the reference beam. This is overcome in off-axis holography, where illumination beamand reference beam are decoupled. This is obtained with the loss of inherent interference
conditions resulting in a more complicated optical set-up. Up to now, commonly used indigital holographic PIV applications is the in-line holography, probably due to its robust
and simple set-up compared to other holographic set-ups [75], [97].

20

Fig 2.7 Experimental set-up of in-line holography

Once the interference patterns are recorded on the CCD-Chip, the digital hologram can be
numerically reconstructed in object space resulting in an intensity distribution indiscretized object space. All algorithms for numerical reconstruction of the voluminous
intensity field are based on the Rayleigh-Sommerfeld diffraction theory [5]. An overview
ofnumerical

reconstructions(Fresnel-Kirchoff

integral,Fresnelapproximation,

convolution approach) is described in [122]. Other reconstruction techniques are thewavelet transformation [19] and fractional-order Fourier transformation [25]. As
mentioned, the result is a light intensity field. Thus, when looking at the 3D image of aparticle, the original shape of a spherical particle is not reconstructed. The particles are
stretched in depth direction, while maintaining the circular shape in the cross section plane[85]. The problem is referred to as depth-of-field problem [98]. As a consequence, the 3D
position of a particle has lower accuracy in depth than in the in-plane [75]. Up to now, themeasurement volumes in digital holographic PIV are limited to small volumes. In [83] ameasurement volume of 15x15x64 mm³ with a seeding density of roughly 4 particles/mm³
is reconstructed. The advantages of digital holographic PIV are found in three-dimensional
information stored on the 2D camera sensor and in the large depth of field. Drawbacks are
the small achievable volumes (dependent on the size of the chip), the resolution of digitalsensors and the poor accuracy in depth position [75].
Multi-Camera Approaches
Multi-camera approaches are manifold in literature. The idea of multi-camera approaches
is the combination of multiple 2D information to obtain 3D information. For that purpose,the cameras record the measurement volume from different angular directions. Voxelwise
as well as pointwise reconstruction techniques take advantage of intersecting line of sights.
Visually speaking, lines of sight are shot out of the image planes and intersect in a 3D
point/voxel in object space. In principal, the 3D reconstruction of a point or voxel needs at

Volumetric Measurements of the Transitional Backward Facing Step Flow

21

least two cameras recording the object scene from different angular directions. However,
due to a large number of particle images, a multi camera set-up is preferable to avoid
ambiguities or ghost particles [78]. Typically, a three or four camera set-up is used, see
figure 2.8 for a typical volumetric three-camera set-up. A precise calibration procedure ismandatory for all multi-camera techniques to obtain the orientations of the cameras. For
that purpose, a three-dimensional calibration target with known control points is necessary.
The control points are recorded from each camera. After the extraction of the twodimensional centroid position of the control markers in the image planes, the known 3D
positions are compared with the calculated 2D sensor positions. For a finite number ofcontrol points, the observations result in an over-determined system of equations, which
can be solved by e.g. iterative least squares procedures.

Fig 2.8 Experimental set-up: a) Ar-Ion-laser, b) lens system, c) rotating mirror drum, d) high speed camera,e) octagon, f) light sheets, g) vortex generator [64]The pointwise reconstruction, so-called stereo-triangulation, is primarily performed inclassical 3D PTV [78]. By knowledge of the 2D centroid positions of a homologue particle
in the single image planes and by knowledge of the single projection matrices, the 3D point
position is reconstructed by proper reconstruction techniques (homogeneous orinhomogeneous method [125]). In detail, lines of sight are shot out of the image planes andthe intersection point of these lines defines the volumetric position. Due to improper
calibration, improper estimation of 2D centroid positions and optical errors, the lines ofsight are usually skewed. Thus, the center between the shortest distances of the lines ischosen. In the following, the inhomogeneous method for pointwise reconstruction is
described. In a stereoscopic set-up, where subscript 1 indicates the first camera andsubscript 2 the second camera, equation 2.6 is written as:

22⎡ xw ⎤⎡ x1 ⎤⎢y ⎥⎢⎥s1 ⋅ ⎢ y1 ⎥ = P1 ⋅ ⎢ w ⎥⎢ zw ⎥⎢⎣ 1 ⎥⎦⎢ ⎥⎣1⎦⎡ xw ⎤⎡ x2 ⎤⎢y ⎥⎢⎥s 2 ⋅ ⎢ y 2 ⎥ = P2 ⋅ ⎢ w ⎥⎢ zw ⎥⎢⎣ 1 ⎥⎦
⎢ ⎥
⎣1⎦

(2.7)

By assumption of identical cameras (s1=s2), the vector product results in an overdetermined system of equation in the form EX=0:⎡ p111 − x1 p131
⎢ 21
p1 − y1 p131⎢
EX W = 11⎢ p 2 − x 2 p 231⎢ 21
31⎣⎢ p 2 − y 2 p 2p112 − x1 p132
p122 − y1 p13232
p 12
2 − x2 p2
p 222 − y 2 p 232

p113 − x1 p133
p123 − y1 p13333
p 13
2 − x2 p2
p 223 − y 2 p 233

p114 − x1 p134 ⎤ ⎡ X w ⎤⎥⎢ ⎥p124 − y1 p134 ⎥ ⎢ Yw ⎥
=0
34 ⎥
⎢Zw ⎥
p 14
2 − x2 p2⎥⎢ ⎥p 224 − y 2 p 234 ⎦⎥ ⎣ 1 ⎦(2.8)

The decomposition of E into:⎡ e11⎢e
B = ⎢ 21
⎢e31
⎢
⎣e41

e12
e22
e32
e42

e13 ⎤
⎡ e14 ⎤
⎥
⎢e ⎥
e23 ⎥
and c = ⎢ 24 ⎥⎢e34 ⎥
e33 ⎥
⎥
⎢ ⎥
e43 ⎦
⎣e44 ⎦(2.9)

results in:B ⋅ ( xw , y w , z w )T = c(2.10)

If BTB is invertible, the solution for the three-dimensional position Xw is:
X W = BT B −1BT c(2.11)For the identification of homologue particles in different camera views a rather strong tool
is described in literature, the epipolar line [125]. It describes, that a point, which is found in
one camera, lies along a line in the second camera as shown in figure 2.9. The twodimensional search area is reduced to one dimension, namely along the epipolar line.
Mathematically, the relation between both cameras is described by the fundamental matrix
F. The relation of both 2D point positions (x1, y1), (x2, y2) of the same 3D point is called
epipolar equation and the line l2 is called epipolar line:

Volumetric Measurements of the Transitional Backward Facing Step Flow

(x2

y2

⎛ x1 ⎞⎜ ⎟1) ⋅ F ⋅ ⎜ y1 ⎟ = 0⎜1⎟⎝ ⎠⎛ x1 ⎞⎜ ⎟, l 2 = F ⋅ ⎜ y1 ⎟
⎜1⎟
⎝ ⎠

23

(2.12)The evaluation of F is made within the calibration of the projection matrices. The
fundamental matrix for each camera combination is determined by comparing homologue
centroid positions in the images. If the 2D positions of the target markers in the camera
images have been localized, F is evaluated e.g. with the 8-point algorithm expanded by the
Frobenius-norm [125], which needs 8 homologue 2D positions. In practice, it is common
to use more than several hundred positions to determine F.
The difficulty in identification of homologue particles in the image planes increases withincreasing number particle density. A particle image usually contains >10³ particles, which
do not strongly vary in shape, size or intensity. Thus, a reduction of spatial ambiguities due
to imaging characteristics is not sufficient. Here, the geometrical condition epipolar line
intersection is more appropriate to identify homologue particles [78]. The method needs atleast three cameras, whereas further viewing direction will reduce ambiguities. By proper
arrangement of the cameras, the epipolar lines between all camera combinations willintersect near the homologue 2D point positions. Nevertheless, as particle density is
increased, a new problem arises due to overlapping particle images, which cannot be
identified as separate particle images on the sensor plane. Thus, the number of particles islimited to the order of 10³ for a four camera set-up in pointwise reconstruction techniques
for a typical sensor size of 1000x1000 pix² [48]. The size of the measurement volume orsize of the reconstructed volume can range up to several cubic meters depending on the
mapping system [14]. However, the larger the measurement volume, the lower becomes
the spatial resolution.

24

Fig 2.9 Schematic of stereoscopic camera set-up: stereo-triangulation and reconstructed epipolar line

Another, rather elegant, system for pointwise reconstruction is the patented V3V-system[101]. The single body system takes advantage of multi-sensors in combination with thedefocusing concept described above [99]. Three sensors are arranged in a coplanar triangle
pattern each illuminated by a one-hole aperture. Thus, homologue particle identification on
the three sensor chips is performed, to the author’s point of view, by intersecting epipolarlines, although not mentioned by [101]. Due to the arrangement of the chips and apertures,the identified homologue particles build a triangle as in [145]. The center of the triangle
identifies the in-plane coordinate and the size of the triangle the depth position. The multisensors allow higher number particle densities compared with the single camera three-hole
aperture system, because the sensor configuration is extended from one sensor to three
sensors.

Fig 2.10 Patented V3V system; three-camera system in a single body (left); Calibration triplets at multiple
planes through the measurement volume [148] (right)

Volumetric reconstruction - in contrast to pointwise reconstruction - doesn’t need theidentification of homologue particles in different camera views, thus allowing higher

Volumetric Measurements of the Transitional Backward Facing Step Flow25

seeding densities for the cost of ghost particles. Ghost particles are reconstructedambiguities decreasing the reconstruction quality. All volumetric reconstruction techniques
are based on the shape from silhouette method [70]. Each pixel of a particle image, definedby a finite number of pixels, is projected into 3D space via the line of sight. If this is donefor each camera, the lines of sights intersect in object space and generate a visual hull of
the 3D particle. It is obvious that the more cameras are used, the more precise thedescription of the particle will be. For that purpose, all volumetric reconstruction
techniques use the voxel-model. The characteristics of volumetric reconstruction of
intensity distributions within the measurement volume in PIV are on the one hand thediscretized particle distributions from a limited number of viewing directions and, on theother hand, high spatial frequencies. Thus, algebraic reconstruction methods are more
appropriate than Fourier or back projection methods [33]. A schematic of the algebraic
reconstruction techniques is shown in figure 2.11. The figure shows the top view of the
pre-defined voxel space. Filled voxel elements represent intensity values greater than acertain threshold.

Fig 2.11 Schematic of multi-camera algebraic reconstruction technique.

Since Elsinga et al. (2006) [34], a number of algorithms and algorithmic refinements forthe application in PIV are proposed in literature. The conventional line of sightreconstruction is found in multiplicative algebraic reconstruction technique (MART) [45],in adaptive algebraic reconstruction technique (AART) [76], in improved iterativealgorithm for sparse object reconstruction (IIASOR) [72] and in simultaneous iterative
reconstruction technique (SIRT) [45]. The basic principle is that the recorded intensity oneach pixel of the camera is the result of an integration of intensity along the line of sight.
The integral description of pixel intensity can be discretized due to the voxel-model and26

expressed as the sum of a weighting matrix multiplied (MART) by the voxel intensity
along the line of sight. The weighting matrix includes the camera parameters that connect avoxel in object space and a pixel in the image plane. This results in an under-determined
system of linear equations. An iterative correction of the voxel intensity is performed until
the intensity distribution of the pixels in each projection is satisfied within defined error
limits.
The reconstruction algorithms mentioned afore have been tested with respect to PIV by [7].
The results show that MART and AART deliver the highest quality of reconstruction. The
main problems of MART and AART are the large computational processing overhead dueto the iterations and the fine voxel grid as well as the intensive cost for memory in order tostore the weighting matrix for each voxel. Improvements in accelerating the iterative
reconstruction are published by Atkinson and Soria (2009) [6]. Possible particle locationsin the voxel volume are determined by intersecting line-of-sights. This method circumvents
the use of memory intensive weighting matrices in regions without particles.
Furthermore, non-iterative methods for 3D particle reconstruction have been presented. In
Putze (2008) [105] a method is described, which is based on a multiple projective
transformation of each camera image into depth layers of the object space usingtransformation parameters derived from the camera orientation and calibration parameters.
For that purpose, the contents of each camera are transformed depth-layer wise into thepredefined voxel intensity field. A schematic sketch of multiple projective transformations
is shown in figure 2.12 for one exemplary camera.

Fig 2.12 Schematic of depth layer reconstruction

In the course of this thesis, an alternative reconstruction technique is developed, which is
described in detail in Kitzhofer and Brücker (2010) [64]. A simplified reconstruction
method is proposed, which is based on telecentric imaging. The reconstruction isVolumetric Measurements of the Transitional Backward Facing Step Flow27

performed as a multiple combination of pair-wise reconstructed voxel spaces using the
epipolar geometry pixelwise in a set-up with three or more cameras. The principal
processing steps are indicated in figure 2.13. A multi camera set-up is given (in this case a
3 camera set-up) recording particles in the object space. The camera set-up is virtually
separated into all possible camera pairs. Gray value weighted voxel spaces are thenreconstructed for each camera pair. The pair wise tomographic reconstruction of the single
voxel spaces is realized by using the epipolar geometry. For that purpose, the epipolargeometry is applied pixel wise. Thereafter, the voxel spaces are transformed into one
common voxel space. An up to now basic condition of the described method is the parallel
projection resulting from telecentric imaging. Advantages can be seen in the non-iterative
process and a simplified understanding due to the underlying parallel projection.

Fig 2.13 Principal sketch of multiple pair wise tomographic reconstruction; blue rectangle: reconstructed voxel
space cam1/2, red rectangle: reconstructed voxel space cam 1/3, green rectangle: reconstructed voxel spacecam 2/3, black rectangle: final combined voxel space

A multi-camera approach belonging to holographic recording technique is presented by
Soria and Atkinson (2008) [132]. The motivation is the elimination of the depth-of-fieldproblem in holographic reconstruction of particles. The principle is the multiplicative
combination of several reconstructed intensity fields, which are recorded from different
directions. The single intensity fields are reconstructed via wave optics and the final
intensity field is combined by multiplication of the single volumes.

28

A detailed literature review of existing techniques for three-dimensional reconstruction ofa particle space is presented. One and multi-camera approaches, pointwise and volumetricreconstruction techniques as well as optical and wave optics approaches are presented. The
review shows that holographic techniques are, up to now, not practicable for analysis oflarge datasets. Furthermore, possible sizes of measurement volumes are far away from thevolume measured in this thesis (90x60x15 mm³). Nevertheless, digital holography will beof further value for PIV due to recording all necessary information for three-dimensional
reconstruction in a single record.
Volumetric reconstruction has become a key issue in three-dimensional reconstruction ofparticle fields. Striking is the achievable number particle density for the cost of ghost
particles. Nevertheless, processing times are much larger than that of pointwisereconstruction. Beside the iterative process performed in volumetric reconstructions, thenumber of operations is larger. Take one single particle of finite size 3x3pix. The
reconstruction results in 27 voxel elements in object space. Thus, 27 operations are
performed in volumetric reconstruction. In pointwise description, the 3x3 particle isreduced to a single point resulting in one operation for three-dimensional reconstruction.

Volumetric Measurements of the Transitional Backward Facing Step Flow29

2.2.4 3D Velocity Field Generation
Once the measurement volume is reconstructed at consecutive time steps (volumetric orpointwise) the volume can be analyzed for fluid motion. The description of fluid motion is
performed in the Eulerian or in the Lagrangian frame. Both differ in necessary algorithms,
particle densities, spatial structure and information. The spatial data representation in the
Eulerian frame is characterized by a fixed structured grid. In case of PIV, the calculated
velocity vector represents time varying median particle displacements within theinterrogation areas. The velocity field in the Lagrangian frame is described on scattered,spatially and temporally varying positions as shown in figure 2.14.

Fig 2.14 Comparison of Eulerian frame and Lagrangian frame

Lagrangian Description

The velocity field in the Lagrangian frame determined from PTV-methods results fromindividual particle detection schemes and analysis of single particle displacements andpaths. For that purpose, accurate particle detection algorithms and suitable particle tracking
algorithms are mandatory.
First step in the evaluation of particle trajectories is the detection of single particles. This isperformed via segmentation. Segmentation is the grouping of connected pixels to anindividual particle. Methods frequently applied are region growing and thresholding. In
region growing the particle is detected by successively increasing the structure, started
with an elementary pixel. The boundary is defined by proper thresholding. Here, a global
or local threshold is applicable, which is rather dependent on the quality of particle images.
A local threshold is preferable in lower quality particle images, e.g. if a strong backgroundnoise is apparent.
Once clusters of pixels are identified to belong to an individual particle, the centroid
position is determined with sub-pixel accuracy. A comparison of several algorithms isnumerically performed by Ponitz (2010) [100]. The influence of noise, size and subpixel
position is analyzed. Table 2.1 summarizes median deviation and processing time of the30single analyzed methods. The 2D Gauß-fit delivers highest quality for the cost of longprocessing time. The calculation is performed with Matlab R2008b on a single core at 2.66
GHz. Of slightly less accuracy, but with shorter processing times is the gray-value
weighted center of area.
MethodDeviation [pix]Processing time [ms]Pixel maximum intensity

0.36

0.01

Bounding box

0.24

0.009

Center of area

0.09

0.015

Gray-value weighted center of area

0.02

0.006

Gauß-fit

0.01

1.934

Table 2.1 Comparison in median accuracy and processing time of methods for evaluation of centroid
positions of particles

Next step in PTV is the tracking of single particle positions. Here, tracking is performed in
2D, in 3D or as a combination of both, depending on the flow problem. Since [82], several
proposals for tracking particles in a spatial-time domain are considered in literature. The
proposals are based on kinematic analysis [95], analysis of probabilities [9], conservationof energy [93], on binary image cross correlations [151], or on the analysis of the velocity
gradient tensor [56], to mention only a few. Willneff (2003) presents a spatio-temporal
algorithm, which combines image based and object space based information of particles atconsecutive time steps [147].
A generally accepted criterion for the difficulty in tracking single particles is given by
Malik et al. (1993) [82]. The criterion compares mean particle-to-particle distances withmean flow displacements.

e=Δsu ⋅ Δt(2.13)The value e can be interpreted as the difficulty to track particles correctly. If the mean
particle-to-particle distance Δs in one single frame in image plane or voxel space is muchhigher than the mean particle displacement from one frame to the next frame, meaninge>>1, the tracking is easy. If e<1 the tracking is getting nearly impossible.
The principal limit of PTV for an idealized, homogenous distributed 2D particle image
resulting from the difficulty in tracking is shown in figure 2.15. The figure shows theVolumetric Measurements of the Transitional Backward Facing Step Flow

31

particle density ppp as a function of the maximum particle displacement dmax = v·Δt fordifferent difficulties e of tracking in a 2D image. The figure has been constructed in thefollowing manner: A given particle density ppp results in a mean particle-to-particle
distance in the 2D image for homogenous distributed particles. The maximum particle
displacement results from a given value e for difficulty in tracking. Although the graphshows the limit in 2D tracking, which is more difficult than 3D tracking, the principal
tendencies are the same in 2D and 3D.

Fig 2.15 Theoretical limits in PTV (tomographic and classical), limit of number particle density in classical PTV
resulting from a 3 camera set-up (dotted line), estimated limit of number particle density in Tomo PTV resulting
from a 3 camera set-up (slashed dotted line); shaded area: tracking of particles is possible [64]The proposed criterion identifies difficulties with the assumption that all necessary particle
positions are reconstructed correctly. However, in practical applications the tracking is
made more difficult due to insufficient reconstructed particle positions resulting fromambiguities and overlapping particle images. In pointwise reconstruction techniques thenumber particle density in 3D PTV is limited to about 0.005 ppp due to afore mentioned
ambiguities and overlapping particle images [78]. The problem is overcome in volumetricreconstruction techniques (Tomo PTV) shifting the number particle densities to about 0.01
ppp [64]. Here, centroid positions of the particles are determined in voxel space due to theintensity distribution, similar to Holographic PTV. The higher number particle density isachieved at the cost of generating ghost particles. The influence of ghost particles in theyield, which is defined as the estimated links divided by the real number of links, is clearlyvisible in figure 2.16. The figure shows the yield as a function of number particle density.

32

The figure is generated with respect to a simulation of a 3D velocity field with an aspectratio of 1 (100x100x100 voxel). The instantaneous turbulent flow field is generated with
Open FOAM V1.5 (Tool: BoxTurb). For that purpose, the flow field is filled withhomogenous sized particles (7 voxel) at random positions with a Gaussian 3D intensity
distribution and parallel projected onto three 100 x 100 pixel² image planes as shown infigure 2.13. The separation time between the frames has been chosen to have a medianparticle displacement of 3 pix resulting in e ≈ 6 for 0.005 ppp and e ≈ 4 for 0.01 ppp. For
both densities, e is much bigger than 1 making the tracking itself easy. The results show
the main influence of particle density on the quality of pointwise and of volumetricreconstruction techniques.

Fig 2.16 Particle matching yield for different PTV methods, classical approach: yield < 100% because ofproblems in stereo matching and overlapping particle images, tomographic approach: yield > 100% because
yield includes ghost particles [64]A characteristic difference between classical 3D PTV and Tomo PTV is expressed in thecalculated value of the yield. In classical 3D PTV methods, the yield is smaller than 100%and decreases with an increasing number of particles. The reason is an increase ofoverlapping particle images and growing difficulties in stereo matching with increasing
seeding density. In Tomo PTV the yield is higher than 100% due to the reconstructed ghost
particles forming ghost trajectories. In contrast to classical 3D PTV the necessary
optimization of Tomo PTV is reducing the yield to a value of 100 %, whereas the
optimization in classical PTV is increasing the yield to a value of 100 %.
In classical PTV with a 4 camera setup the practical limit of particle density is 0.005 ppp.
With a further increase of the particle density up to 0.01 ppp, the 3 camera setup is able toVolumetric Measurements of the Transitional Backward Facing Step Flow33

link only 60 % of the particles in a correct way. In Tomo PTV, the yield of the 3 camera
set-up is 137 % at 0.01 ppp. This value can be reduced down to 109 % using further
criteria to remove ghost particles from the voxel space. A size based criterion (particles are
represented by connected voxels of certain intensity) for detecting ghost particles reduces
the yield about 15 % leading to an absolute yield of 115 %. Furthermore, an intensity based
detection criterion reduces the yield to 109 %, meaning that 9 % of the links are ghost
links. The remaining 9 % can be further reduced within the tracking procedure using theinformation of temporal behaviour within the voxel space. For that purpose, the detectionof ghost particles changes to the detection of ghost trajectories.
Figure 2.17 exemplarily shows a typical experimental result for a volumetric reconstructed
single particle trajectory and its local neighbours. The true particle path is visible as a longspiral track surrounded by some fragments of ghost trajectories.
The criteria to remove erroneous ghost trajectories can be interpreted as a kind of safe
threshold. Most of the ghost trajectories are of small lengths less than 3 - 4 frames. In
general, the length of the short ghost trajectories depends on the size and the velocity of theparticle images. At constant velocity, the length of a ghost trajectory produced by a small
sized particle image will be shorter than that produced by a big sized particle image. In
addition, particle paths containing only three frames don’t contain much valuable
information for a fluid mechanical interpretation. Thus a cut off is made, which has to bechosen carefully for not deleting true particle paths.

Fig 2.17 Reconstructed trajectory and environment in temporal development (short and long ghost
trajectories); the trajectories are colour coded with time steps. The reconstructed path contains 205 frames.

For the detection of long ghost trajectory, a first guess proves similarities of trajectories in
3D space. This is a check for frame identity, speed, direction and change in direction.

34

Trajectories fitting these conditions are compared. Long parallel ghost trajectories have acharacteristic blinking. The particle path has breaks. Furthermore, a characteristic of thiskind of ghost trajectory is the particle size of the particle building a trajectory. The particle
size in true trajectories is nearly constant. The particle size of ghost trajectories changes
while moving along the particle path. If these criteria fit a selected trajectory of the first
guess, the trajectory is detected to be a ghost trajectory and is deleted. Figure 2.18 shows
the result of Tomo PTV of the benchmark experiment of a vortex ring.

Fig 2.18 Direct 3D visualization of Lagrangian trajectories obtained by Tomographic Particle Tracking
Velocimetry in case of the vortex ring experiment [64]. The measurement volume is approximately 90 x 90 x
90 mm³. The result contains 30 frames.

Regarding computing performance, particle tracking techniques in themselves do not apply
well to large scale multiprocessor architectures due to the inherent nature of tracking
particle histories. Typically, a large queue of detected particles over a time are recorded
and then - in very much a serialised manner - tracks are determined for those particles
whose positions most likely fulfil a set of criteria. A considerable amount of data sorting isnecessary to reduce repetition. Some performance improvement is still possible throughmultithreading, though true parallel processing is not possible.
Eulerian Description

The velocity field in the Eulerian frame is represented by velocity vectors located on a
structured grid in flow space. In case of PIV, the calculated velocity vector representsmean particle displacements within the interrogation areas, indicated in figure 2.14.
Advantage of the representation of the flow by the Eulerian velocity field is the frequent
application in fluid mechanical investigations in the last decades. Thus, several tools forfurther analysis of the Eulerian velocity field are already available in literature resulting inseveral detection schemes for coherent structures. Some examples are vortices,Volumetric Measurements of the Transitional Backward Facing Step Flow35

separation/reattachment lines, shear layers, recirculation regions, stable/unstable manifolds
or topology based structures.
Algorithms for the calculation of mean particle displacements are 3D Cross Correlationand 3D Least Squares Matching. Both methods need a light intensity field, thus volumetricreconstruction or blow up of the pointwise description is mandatory.
To the author’s knowledge, 3D cross correlation was first applied in [99]. Here, pointwise
particle positions were blown up by a Gaussian intensity distribution in voxel description.
Meanwhile, 3D cross correlation in intensity filled voxel spaces is standard [34]. The threevelocity vectors are obtained by cross correlation of two 3D intensity maps corresponding
to subsequent exposures of the particle images. The accuracy in velocity vectors isincreased by volume deformation techniques analogue to the 2D counterpart of window
deformation [114].

Fig 2.19 Schematic of cube transformation; dots: volumetric particles; solid line: measurement volume,
dashed line: interrogation volume. The gray values of the voxel spaces are replaced by pointwise particles
positions for visual clarity.

3-D Least Squares Matching (3D LSM) is a volumetric tracking technique, which isadaptive to deformation and rotation of the interrogation volume. It was first introduced in[79] for tracking structures in 3D Laser Induced Fluorescence data. It is an iterative leastsquares adjustment procedure, where the gray values of the interrogation volume at time
step one are transformed into the gray values of the interrogation volume at time step two
as indicated in figure 2.19. This results in estimations for translation and deformations of a
fluid element. Up to now, the applied geometric transformation between the two volumes
is a 3-D affine transformation, which could also be replaced by non-linear cubictransformations [80]:36x 2 = a 0 + a1 ⋅ dx1 + a 2 ⋅ dy1 + a3 ⋅ dz1
y 2 = b0 + b1 ⋅ dx1 + b2 ⋅ dy1 + b3 ⋅ dz1(2.14)

z 2 = c0 + c1 ⋅ dx1 + c 2 ⋅ dy1 + c3 ⋅ dz1 E

The variables ai, bi, ci (i = 0,1,2,3) are the parameters of the affine transformation and canbe estimated within a Gauss-Markov Model. The parameters a0, b0 and c0 are
displacements valid for each point within the infinitesimal cuboid E. The remaining nineparameters describe shear, rotation and strain.
In addition to the three translational velocity vectors, the parameters of the 3-D affine
transformation contain the information of strength and direction of deformation. Thus, thedirect calculation of the velocity gradient tensor is possible [67], which is a basic elementin the extraction of coherent structures. The velocity gradient tensor or Jacobian of thevelocity field is defined as:
⎡ ∂u
⎢ ∂x
⎢
∂v
J = ∇u = ⎢
⎢ ∂x
⎢ ∂w
⎢
⎢⎣ ∂x

∂u
∂y
∂v
∂y
∂w
∂y

∂u ⎤
∂z ⎥⎥
∂v ⎥
∂z ⎥
∂w ⎥
⎥
∂z ⎥⎦

(2.15)The Jacobian can be split into a symmetric and an anti-symmetric part, namely the rate-ofrotation tensor Ω = ½(J - JT) and the rate-of-strain tensor S = ½(J + JT).
J =Ω+S(2.16)The deformation rate tensor describes strain εij, i=j and shear εij, i≠j of the infinitesimal
volume element resulting from velocity gradients in the inner of the volume.

⎛
∂u
⎜
∂x
⎜
⎜ 1 ⎛ ∂v ∂u ⎞S = ⎜ ⎜⎜ + ⎟⎟⎜ 2 ⎝ ∂x ∂y ⎠⎜ 1 ⎛ ∂u ∂w ⎞⎜⎜ ⎜ +
⎟
2
z
x
∂
∂
⎝
⎠
⎝

1 ⎛ ∂v ∂u ⎞⎜ + ⎟2 ⎜⎝ ∂x ∂y ⎟⎠∂v∂y1 ⎛ ∂w ∂v ⎞⎜+ ⎟2 ⎜⎝ ∂y ∂z ⎟⎠1 ⎛ ∂u ∂w ⎞ ⎞⎜ +⎟⎟2 ⎝ ∂z ∂x ⎠ ⎟⎡ε xx
1 ⎛ ∂w ∂v ⎞ ⎟ ⎢⎜+ ⎟ ⎟ = ε yx
2 ⎜⎝ ∂y ∂z ⎟⎠ ⎟ ⎢⎢⎟ ⎣ε zx∂w⎟⎟∂z⎠

ε xy ε xz ⎤⎥ε yy ε yz ⎥ε zy ε zz ⎥⎦(2.17)

Volumetric Measurements of the Transitional Backward Facing Step Flow37

The rotation rate tensor describes the vorticity ω in each spatial direction x, y, z.
⎡
1 ⎛ ∂v ∂u ⎞ 1 ⎛ ∂u ∂w ⎞ ⎤0− ⎜⎜ − ⎟⎟
⎜ −
⎟⎥
⎢
2
∂
2
∂
x
y
⎝ ∂z ∂x ⎠ ⎥ ⎛ 0⎝⎠⎢⎢ 1 ⎛ ∂v ∂u ⎞1 ⎛ ∂w ∂v ⎞⎥ ⎜0
Ω = ⎢ ⎜⎜ − ⎟⎟− ⎜⎜− ⎟⎥ = ⎜ ω z
2 ⎝ ∂y ∂z ⎟⎠⎥ ⎜⎢ 2 ⎝ ∂x ∂y ⎠−ωy⎥ ⎝⎢ 1 ⎛ ∂u ∂w ⎞ 1 ⎛ ∂w ∂v ⎞0⎥⎢− 2 ⎜ ∂z − ∂x ⎟ 2 ⎜⎜ ∂y − ∂z ⎟⎟
⎝
⎠
⎝
⎠
⎦
⎣

− ωz
0

ωx

ωy ⎞
⎟
− ω x ⎟ (2.18)
0 ⎟⎠It is obvious that calculation of the deformation rate and rotation rate tensor for fully
describing fluid motion needs the evaluation of the velocity gradient tensor. Kitzhofer et al.
calculate the velocity gradient tensor via results obtained by LSM [67] without applicationof central difference schemes to calculate the velocity gradient tensor. A numerical as wellas an experimental result for the extraction of coherent structures via results obtained by
LSM is shown in figure 2.20. Figure 2.20 on top shows results of LSM applied to asimulated hill type vortex ring. The figure shows the velocity vectors in the symmetry
plane x-z, 3D streamlines starting above the upstream stagnation point, and isosurfaces ofthe vorticity magnitude (gray) and positive (orange) and negative (blue) strain.
Furthermore, some numerical results of the deformation rate and rotation rate tensor are
included. The generated streamlines clearly follow the exact solution of the hill typevortex. The streamlines form a spherical shape. The velocity vectors in the symmetry plane
identify the velocity distribution of a hill type vortex. No outliers are recognizable. The
isosurfaces of vorticity and strain excluded from the transformation matrix derived by
LSM identify the main features of the hill type vortex without any further post processing
steps. The upper and lower stagnation points are highlighted. The volume elements in theupper stagnation point are compressed in z-direction and expanded in x- and y-direction(εzz<0, εxx>0, εyy>0). In the lower stagnation point the scene is inverted. The isosurface ofthe vorticity magnitude identifies the donut like vortex core. The numerical result for therotation rate tensor in the symmetry plane x-z at the vortex core for x<0 and x>0 is given.
The prefixes of the entries for vorticity ωy agree with the physical interpretation. Figure
2.20 on bottom shows results of LSM applied to the benchmark experiment of a vortex
ring. The volumetric reconstruction as well as LSM is performed by Dantec Dynamic
Studio. The volumetric reconstruction bases on a multiple projective transformation ofeach camera view into the depth layer of the observed volume and a minstore algebraic
reconstruction technique [143]. A regular grid of 278x1112x944 voxel with gray value

38

information is generated for each time step. LSM is applied on double frames yielding theresults visualized in figure 2.20 on the bottom. The vortex ring is clearly recognizable.
Beside the velocity vectors, the isosurface of vorticity magnitude is included. To point out,
the vorticity magnitude is not calculated via central difference schemes, but via theparameters of the affine transformation.

Fig 2.20 top Visualization of results obtained by LSM applied to the simulated hill type vortex ring; isosurfaceof vorticity magnitude (blue) identifies the donut like vortex core, isosurface of positive (red) and negative(blue) strain rates identify stagnation points; 3D streamlines show a spherical shape and the velocity vectors inthe symmetry plane y-z characterize the velocity field of the hill type vortex bottom Experimental vortex ring:
the velocity vectors have a uniform length; the blue isosurface represents the vorticity magnitude showing thevortex coreVolumetric Measurements of the Transitional Backward Facing Step Flow

39

Beside the splitting into a symmetric and an anti-symmetric part, the characteristic
equation of the Jacobian J can be analyzed for detection of coherent structures:λ3 + P ⋅ λ2 + Qλ + R = 0(2.19)

The first invariant is P = - ∇ · u (equivalent to zero in incompressible flows due to massconservation). The second invariant for incompressible flow is Q = ½ (||Ω||2 - ||S||2), where||·|| denotes the Euclidean norm. The third invariant is R = -det (J).

Data Enhancement

Once the velocity field is generated, whether in Lagrangian or Eulerian description, it isprerequisite to enhance the derived data in a spatial as well as in a temporal manner.
Experimental datasets, however, are characterized by Gaussian background noise, outliers
and in worst case missing data. Before any analysis (e.g. calculation of spatial or temporal
derivatives) is performed, it is crucial to detect outliers, to remove background noise and to
fill up lacking regions, where erroneous vectors have been removed.
The resulting dataset in the Lagrangian frame is characterized by Lagrangian trajectories.
The velocities and derivatives in the Lagrangian frame are prone to improper calculatedparticle positions. Lüthi et al.(2005) propose a low pass filter combined with a moving
cubic spline interpolation on the position signal to smooth the dataset and to reduce
uncertainties [77]. The velocities and accelerations are calculated by differentiating the
cubic spline representation. Another enhancement of PTV based results is the connectionof trajectories, which are separated by several time steps [150]. Here, segments oftrajectories are tracked in a six-dimensional position-velocity space.
Furthermore, the Lagrangian description allows simple interpolation on the Eulerian frame[134], e.g. by adaptive Gaussian window averaging [3], polynomial interpolations [81] orinverse distance interpolation [128]. Algorithms, originally developed for 2D dataenhancement in the Eulerian frame, can be adapted to 3D velocity vector fields.
In data validation or outlier detection, the important aspect is the balance between removalof spurious vectors and valid data [108]. Furthermore, an automated algorithm is required
without pre-knowledge or visual inspection. Commonly applied criteria are global
histogram operator [108], dynamic mean value operator [108] or normalized median test

40[142]. Other methods, up to now time consuming, are cellular neural network [73] orbootstrapping [104].
The detected outliers are then replaced by median, bilinear or spline interpolations [108].
Noise reduction is achieved by ordinary smoothing using average kernels.
Another trend in data enhancement is the combination of experimental data with numerical
data. More recently, Gunes and Rist (2007) [37] compare Kriging interpolation [38] with
DNS-based Proper Orthogonal Decomposition (POD) mode interpolation. In case of lowresolution experimental data, DNS-based POD mode interpolation outperforms Kriging
interpolation. Here, POD is applied to DNS data to extract the DNS-based POD modes.
These modes are then projected onto the experimental data to obtain coefficients of thehybrid model. The experimental data is interpolated and smoothed with respect to thehybrid model. In case of high resolution PIV data, Kriging interpolation works well in
eliminating noise and increasing the spatial resolution. The authors claim to use Kriging
interpolation based on a Gaussian correlation function. However, afore mentioned
detection criteria request a user specified input. Garcia (2010) presents an automatedmethod without user specified input [35]. Here, outlier detection, replacement of missing
data and smoothing is performed by a method called DCT-PLS. The method is based on
penalized least squares approach combined with the use of the discrete cosines transformand the generalized cross-validation.
In comparison with spatial enhancement, temporal enhancement has achieved lessattention. Durault et al. (2005) propose a POD-based method [30]. POD is applied on time
resolved dataset to calculate temporal POD coefficients. The temporal POD coefficientsare then cubic spline interpolated to enhance the temporal resolution. However, as pointed
out by Bouhoubeiny and Druault (2009) the method is only valid for large scale flow
structures [15]. Another approach deals with the application of the advection equation,
invoking Taylor’s hypothesis of frozen turbulence [117]. The authors successfully show
the effectiveness of the approach increasing the temporal resolution of 3D data obtained bytomographic PIV.

Volumetric Measurements of the Transitional Backward Facing Step Flow41

2.3 3D Scanning Particle Tracking Velocimetry
The choice of a suitable volumetric measurement technique is mainly dependent on thespecific experimental set-up, the flow configuration as well as on the available hardware.
In case of this thesis, the flow phenomenon is tested in a rectangular water channel. Opticalaccess is ensured from the top and from the side. The volume of interest is positioned
about the symmetry plane and is sized to about B x L x H = 90 x 90 x 15 mm³ as shown infigure 2.21 The measurement volume is characterized by a high dynamical range of
normalized velocity, yielding from u*≈-0.3 up to u*≈1. The free stream region has mostly
constant direction, whereas the recirculation region is highly fluctuating with suddenchanges of flow direction. For the volumetric flow measurement, three digital high speed
cameras equipped with telecentric lenses are available. As the measurement volume is toolarge for holographic techniques, a technique based on geometrical optics is more
appropriate. Here, classical 3D Particle Tracking Velocimetry is the most convenienttechnique. This is for the following reasons:a) highly turbulent flow with large spatial and temporal gradientsb) Lagrangian and Eulerian information is desiredc) high particle seeding densities are critical to achieve in the set-upd) comparative low processing cost
Large spatial and temporal velocity gradients are critical to PIV based algorithms like CC
or LSM, because of significant changes in intensity distributions due to turbulent
displacements of particles in the inner of interrogation areas. A solution to reduce these
gradients is a decrease of the separation time to yield smaller displacements between theframes for the cost of reduced dynamic bandwidth.
A rather strong advantage of 3D PTV is the availability of the Lagrangian and of theEulerian velocity field. The condition is a high spatial resolution in the Lagrangian frame,
which is defined by the number of tracked particles. Furthermore, high-speed recording
allows the evaluation of long Lagrangian trajectories, giving rise to the calculation of
Lagrangian accelerations [88].
The direct calculation of the Eulerian velocity field via CC or LSM needs high particle
densities. A defined number of gray-value voxel information is necessary for a good
estimation of the local velocities. This is not possible in the present test case. The total
volume of the channel is too large for constant seeding. Furthermore, the volume of42interest is surrounded by fluid. If the channel volume is homogeneously seeded, themultiple scatter along the path to the volume of interest results in a low contrast image ofthe volume of interest.

Fig 2.21 Volume of interest and physical dimensions of the experimental set-up

As mentioned above, the spatial resolution in classical 3D PTV is limited to about 0.005
ppp using pointwise reconstruction. The particle density can be increased, on the one handby the chosen method (volumetric or pointwise) and on the other hand by a clever choiceof illumination. Hoyer et al. (2005) choose a scanning illumination for classical Particle
Tracking Velocimetry [48]. The authors show an increase of the number particle density
compared to classical PTV. The implementation of a scanning illumination connected withhigh-speed recording is demonstrated by Kitzhofer et al. (2009) [66]. The number particle
density is increased about a factor proportional to the number of generated light sheets.
This is due to simplifications in identification of homologue particles and due tosimplifications in tracking single particles.
In the following, the implementation of 3D Scanning Particle Tracking Velocimetry is
described. The first part describes the experimental set-up and the boundary conditions of
the set-up. The recording system consists of digital high speed cameras equipped with
telecentric lenses. The illuminating system consists of a scanning device. Thus,
assumptions and simplifications of both will be pointed out in detail. The second partdescribes algorithmic developments regarding the measurement technique to calculate theLagrangian as well as the Eulerian velocity field. The developed algorithm is called hybrid
PIV/PTV 4-frame method.

Volumetric Measurements of the Transitional Backward Facing Step Flow43

2.3.1 Experimental set-up
An iso-view of the experimental set-up is shown in figure 2.22. The laser beam of acontinuous Argon-Ion laser (8) (Coherent Innova 70, 2 W) passes an optical lens system(9) to adjust the desired thickness of the light sheets. A rotating mirror drum (7) reflects thelaser beam into the direction of the measurement volume and generates successively 4parallel light sheet planes (4). The particle images are recorded with an imaging system
consisting of three digital high-speed cameras (Photron APX RS) (3) and telecentric lenses(Sill Optics)(2). The three camera system is installed above the water channel with anangular displacement of about 0°, 45°, -45°. The f-number of the lenses is adjusted to f# =16 to reach a sufficient depth of focus. A water filled prism reduces astigmatism in theimage planes (3). The side of the water channel (10) opposite to the entrance side of thelaser is covered with a light absorbing mat that reduces stray reflections, thus giving a
black background. The neutrally buoyant seeding particles (6) of roughly 100 μm are
injected upstream of the BFS with respect to [11].

Fig 2.22 Experimental set-up of backward facing step flow: 1) high speed cameras (3000 fps) 2) telecentriclenses 3) water filled prism 4) voluminous illumination 5) backward facing step 6) seeding injection 7) mirror
drum 8) Ar-Ion laser 9) tilted mirror 10) turbulent water channel at Reh=4440, ER=1.25, AR=57

High speed cameras Photron APX RS (1024x1024 pix, 3000fps, 2GB storage) are installed
above the channel in the arrangement shown in figure 2.22. The cameras record the scene
with 1000 fps allowing time resolved measurement of the BFSF.
As the cameras are installed outside the channel, the optical rays pass through varying
indices of refraction. By assumption of thin (2mm) glass walls of the water tunnel, theoptical rays pass through water and air. For minimizing optical refraction in the outercameras, a water filled prism is installed above the tunnel (wall thickness 1mm). Thus, all44

optical rays enter perpendicular into the boundary surface between water and air.
Furthermore, the water filled prism reduces optical aberrations resulting from theinstallation with an inclined angle. Figure 2.23 shows the influence of the water filledprism on the optical path and on the quality of the particle images.

Fig 2.23 Influence of the water filled prism on the quality of particle images. Left image shows distorted
particles. Right image shows regular shaped particles due to the installation of the water filled prism.

Imaging system

The characteristics of telecentric lenses and the resulting parallel projection are wellknown and discussed in detail in e.g. Schreer (2005) [125]. This section summarizes themain points for understanding the application of telecentric imaging in the course of threedimensional reconstruction. The telecentric lenses used in this application are telecentric in
object space. The advantages of using telecentric lenses are in a large depth of field, a
constant magnification over the whole depth of field, high accuracy and negligible optical
distortions.
In perspective transformations the focal point has a defined distance to the image plane.
This results in a variable magnification in the object space. In case of telecentric imaging
the distance between focal point and image plane is at infinity resulting in an
approximately parallel projection in the object space within the range of telecentry andconsequently in a constant magnification as sketched in figure 2.24. The constantmagnification yields a defined measurement volume. It is defined by the width and theheight of the image plane, the magnification of the lenses system and by the depth of focus.

Volumetric Measurements of the Transitional Backward Facing Step Flow45

The measured distortions are less than 0.1 % of the image diagonal in the outer image
areas making nonlinear lens distortions negligible [66]. The telecentric range withdeviations of less than 0.1% has been evaluated with the given lenses to be about 90 mm
with a scaling factor of 11.4 ± 0.02 Pix/mm at an f# = 16. This is equivalent to amagnification M = 0.2.

Fig 2.24 Parallel projection due to telecentric imaging

In parallel projection, the focal point is at infinity. This results in several simplifications in
the mathematical description of the projection from object space to image plane. The
simplification in the extern transformation is the loss of depth information, resulting in tz=0
(equation 2.3). The extern transformation matrice results in:⎡ r11⎢r
Dorth = ⎢ 21
⎢ r31
⎢
⎣0

r12
r22
r32
0

r13
r23
r33
0

tx ⎤
t y ⎥⎥
0⎥⎥1⎦(2.20)The perspective transformation is simplified due to the transition from intercept theorem tothe orthographic equation of projection:x S = xc , y S = y c .

(2.21)The perspective transformation matrice results in:

'
orth

P

⎡1 0 0 0⎤= ⎢⎢0 1 0 0⎥⎥⎢⎣0 0 0 1⎥⎦(2.22)

46

Finally, the scaling parameter in the intern transformation is identical in x- and y-direction
for parallel projection. The intern transformation matrice for parallel projection reads:⎡k s
Aorth = ⎢⎢ 0
⎢⎣ 0

0
ks
0

xs0 ⎤
y s 0 ⎥⎥
1 ⎥⎦(2.23)

Thus, the complete projection matrice for parallel projection results in:

Porth = Aorth ⋅ P

'
ortzh

⎡ k s ⋅ r11⋅ Dorth = ⎢⎢k s ⋅ r21⎢⎣ 0

k s ⋅ r12k s ⋅ r13k s ⋅ r220

k s ⋅ r230

k s ⋅ t x + x0 ⎤k s ⋅ t y + y 0 ⎥⎥⎥⎦1

(2.24)The simplified projection matrice results in massive reduction of steps in the calculation ofa three-dimensional position. By consideration of p31=0, p32=0 and p33=0 due to theorthographic projection, equation 2.8 is simplified to:⎡ p111⎢ 21p
E orth X W = ⎢ 111
⎢ p2
⎢ 21
⎣⎢ p 2

p112

p113

p122
p 12
2

p123
p 13
2

p 222

p 223

p114 − x1 ⎤ ⎡ X w ⎤⎥⎢ ⎥p124 − y1 ⎥ ⎢ Yw ⎥=0⎥⎢ Z w ⎥p 14
2 − x2⎥⎢ ⎥p 224 − y 2 ⎦⎥ ⎣ 1 ⎦(2.25)The decomposition of Eorth into constant Borth and variable, but simplified corth results in:⎡ e11⎢eBorth = ⎢ 21
⎢e31
⎢
⎣e41

e12
e22
e32
e42

e13 ⎤
⎡ e14 ⎤
⎥
⎢e ⎥
e23 ⎥
= const and corth = ⎢ 24 ⎥⎢e34 ⎥
e33 ⎥
⎥
⎢ ⎥
e43 ⎦
⎣e44 ⎦(2.26)

A comparison of equation 2.8 and 2.23 shows the reduction in the number of steps forcalculation of a single three-dimensional point. In total, fifteen steps for calculation are
saved due to telecentric imaging.

Volumetric Measurements of the Transitional Backward Facing Step Flow47

The calibration of the imaging system, i.e. the estimation of the projection andfundamental matrice is done via a planar target at two positions in the volume of interest.
The target consists of 1369 black dots and a white background guaranteeing high contrast,
see figure 2.25. Visible by all cameras are 832 markers (overlapping area). Thus, a total
amount of 1664 markers are available for calibration.
The two dimensional image coordinates of the markers are calculated and compared withthe known tree-dimensional positions for the estimation of the projection matrice of thesingle cameras. The observations result in systems of equations in the following form:⎡ xw ⎤
⎡ x⎤
⎢y ⎥
⎢
⎥
s ⋅ ⎢ y⎥ = Q ⋅ ⎢ w ⎥ ,⎢ zw ⎥⎢⎣ 1 ⎥⎦
⎢ ⎥
⎣1⎦

⎡ q11
⎢q
with Q = ⎢ 21
⎢q31
⎢
⎣q 41

q12
q 22

q13
q 23

q32

q33

q 42

q 43

q14 ⎤
q 24 ⎥⎥
q 34 ⎥⎥
q 44 ⎦(2.27)For a number of N observations, 2N homogeneous linear equations will result. A
comparison between equation 2.22 and 2.25 directly yields q41,42,43=0 and q44=1. The
remaining parameters are estimated by a classical non-linear Gauss-Newton method. The
first guess of the parameters is calculated by singular value decomposition.
The influence of the number of control points on the absolute deviation of the projection isshown in figure 2.25. The absolute deviation is the mean value of the difference between
the re-projected and the recorded two-dimensional positions of the markers. Above a
number of about 20 control points, no more increase in accuracy is expected. Nevertheless,to yield a homogeneous and high quality solution in the total volume, 1000 homogeneously
distributed calibration points are considered. The remaining points are considered for theestimation of the achieved accuracy. The median deviation for the estimated projection
matrices ranges from 0.21 pix up to 0.28 pix, which is summarized in table 2.2.
The fundamental matrice is calculated via the 8-point algorithm expanded by the
Frobenius-norm [125], which needs 8 homologue 2D positions. The mean deviation, which
is defined as the shortest distance between the two-dimensional marker position and the
epipolar line, is calculated to be in between 0.15 and 0.2 pix.

48

Fig 2.25 Influence of the number of calibration points on the median deviation in projection

calibration

Deviation [pix]Fundamental matrice camera 2-1

0.17

Fundamental matrice camera 2-3

0.20

Fundamental matrice camera 1-3

0.15

Projection matrice camera 1

0.23

Projection matrice camera 2

0.28

Projection matrice camera 3

0.21

Table 2.2 Mean accuracy of multi-camera calibration

The accuracy of the imaging system in the estimation of a three-dimensional position isvalidated two-fold. On the one hand, calculated three-dimensional marker positions arecompared with known values. On the other hand, theoretically deviation in the threedimensional positions due to erroneously calculated 2D positions of particles are
considered.
The 3D reconstruction of the marker positions via equation 2.11 showed a systematicallydeviation of the reconstructed position. The deviation is calculated in the following
manner: Homologue control point images used for the estimation of the projection matrices
are reconstructed via equation 2.11 in physical space. The reconstructed positions arecompared with the known values, delivering a deviation in each spatial direction Δ3D,x
Δ3D,y Δ3D,z [mm] in a certain point (x,y,z) in physical space. Figure 2.26 shows themagnitude Δ3D of all directional deviations averaged over the z-direction. The deviations
range from 0.2 mm to 0.5 mm. The reasons for the deviation are manifold, but mainVolumetric Measurements of the Transitional Backward Facing Step Flow49

influences are seen in the separation planes between prism and water tunnel and in the not
perfectly planar planes of the prism. The single deviations are then used as nodes tolinearly interpolate onto a “deviation space”. Figure 2.26 on the right shows a single vector
slice through the deviation space at z/h=0.8. Vectors show the structured deviation in aplane at z/h=8 parallel to the bottom wall. The deviation space is used to remove theobserved systematically deviation from the measurement results and to increase theaccuracy. The accuracy is estimated by quality check of the marker positions, which are
not used for the calibration procedure after post-processing via the deviation space. The
RMSmm,cal value resulting from the calibration evaluates to RMSmm,cal = 0.014 mm, whichcorresponds to RMSpix,cal ≈ 0.16 pix.

Fig 2.26 (left) Magnitude of deviation [mm] between reconstructed and known three-dimensional positions;(right) direction of deviation at z/h=8 parallel to the bottom wall

The influence of erroneous calculated positions of 2D particle images on the calculation of
a 3D position shows figure 2.27. As basic for the estimation of deviation in threedimensional positions due to erroneously calculated 2D positions of a particle image is thetheoretically calculated deviation from table 2.1. All particle positions in this thesis arecalculated by the gray-value weighted center of area, resulting in a theortical deviation of
0.02 pix. The worst case RMSmm,2D value resulting from the detection of single particle
positions is calculated to RMSmm,2D = 0.007 mm from a theoretical point of view. In total,
the measurement uncertainty is estimated to RMSmm = RMSmm,cal + RMSmm,2D = 0.021
mm, which corresponds to RMSpix ≈ 0.2 pix.

50

Fig 2.27 Influence of the deviation in estimated two-dimensional centroid position of a particle image on thethree-dimensional deviation

Volumetric illumination

Volumetric illumination is mandatory in 3D Particle Tracking Velocimetry. Primarily, theillumination directly affects the quality of the particle images. In 3D PTV, a particle image
with high-contrast is preferable, as single particle detection schemes are applied. For that
purpose, a scanning illumination device is applied, i.e. the mirror drum presented by
Brücker (1996) [17]. The principal of the technique is shown in figure 2.28. A continuous
laser beam is reflected on a 45° tilted mirror, which is installed on a drum. Due to therotation of the drum, the laser beam is deflected and sweeps through the volume,
generating a kinematic light sheet. The cameras are synchronized to record the scene
during the time interval of one complete sweep of the laser beam. In this manner, thecamera integrates the scattered light of the beam during one scan and only records theilluminated plane. In total, a number of twenty mirrors are installed on the circumferentialof the drum allowing a total amount of twenty light sheets.

Fig 2.28 Principle of the rotating mirror drum (left) iso view (right) top view

Volumetric Measurements of the Transitional Backward Facing Step Flow51

The positions of the mirrors are variable allowing the adjustment of the number light sheets
and of the position of the light sheets. The thickness is adjustable by changing the laser
beam thickness. The arrangement of the mirrors on the drum defines the recording mode.
Here, double imaging or time resolved imaging is possible. In this thesis, time resolved
measurements are performed. Thus, the mirrors are installed in an arrangement to generate
light sheets in a way that is sketched in figure 2.29. Four light sheets with a thickness of
5mm are generated. The mirrors are installed to yield an overlap of the light sheets ofroughly 1.5mm. This results in a thickness of 15.5mm of the illuminated volume.
Furthermore, the sketch shows the scanning direction, from the top to the bottom. The
separation time between the single light sheets is 1ms, resulting in separation time of 4 ms
for a homologue light sheet. The separation time is chosen with respect to particle
displacement and intensity of the particle images. A compromise is chosen between short
separation times for small particle displacements and long separation times for highintensity particle images. The camera is recording with the same frequency of 1000 Hz as
the sweep frequency of the laser beam.

Fig 2.29 Positions of light sheets in the side view

The quality of the recorded particle images is shown in figure 2.30 by means of arepresentative particle image. One exemplary image section is shown in a threedimensional plot. The image section shows two particles surrounded by background noise.
The chosen image section is exemplary for all recorded images. The images arecharacterized by homogeneous gray-value distribution and a median background noise ofroughly 2 out of 255. Thus, a global threshold of 5 is chosen for segmentation.

52

Fig 2.30 Intensity distribution of an exemplarily chosen image section. The image shows two particles
surrounded by a low level background noise giving evidence to the quality of illumination.

Beside the high quality illumination, the scanning illumination results in a higher number
particle density traceable with PTV. This is for two reasons: First, the slicing of the volume
results in a smaller number of projected particles compared to a full volume illumination.
This is shown in figure 2.31. The left figure shows a synthetic generated volume filled withparticles. The volumetric particles with a Gaussian intensity distribution are represented bydots and fill up the whole volume. The corresponding parallel projection of the particles isshown in the sub-image. Here, no single particles can be detected due to a high number
particle density, which results in an enormous overlap of particle images. The figure on theright shows the same particle distribution, but only a sub-volume is illuminated. In thecorresponding parallel projection, single particles are detectable, allowing 3D PTV.
Furthermore, the decrease of particles per pixel (ppp) in the image plane reduces thedifficulty in tracking single particles, see figure 2.15. The scanning illumination results in
simplifications of identifying single particles and in simplification of the tracking
procedure, as the difficulty in tracking (equation 2.13) decreases.

Fig 2.31 Synthetic generated volume filled with particles. The figure on the left represents a full volume
illumination and the figure on the right a sub-volume illumination as evident in scanning illumination.

Volumetric Measurements of the Transitional Backward Facing Step Flow53

The second reason for a higher number particle density traceable with 3D Scanning PTV is
the reduction of the search area along the epipolar line for finding homologue particles, see
figure 2.32. A particle in the image plane of one camera is along the corresponding
epipolar line in the second image plane. Here, the slicing of the volume results in a further
reduction of the search area along the epipolar line. The length of the search area isdependent on the thickness of the sub-volume and on the angle between light sheet andimage planes. The reduction of the epipolar line results in considerable reduction of
ambiguities in finding homologue particles.

Fig 2.32 Reduction of the search area along the epipolar line due to slicing of the volume

To summarize, the scanning illumination is applied for the following reasons:- the quality of the particle images is high, allowing efficient detection of particles- the volume is sliced, resulting in a higher spatial resolution, which is limited byoverlapping particle images- the correspondence problem in finding homologue particles via the epipolar geometry isdiminished- the difficulty in tracking decreases.

54

2.3.2 Tracking Algorithm and Point Reconstruction
In the following, the implemented algorithm for 3D Scanning Particle Tracking
Velocimetry is described. The algorithm is a hybrid PIV/PTV 4-frame method. Hybrid
PIV/PTV means that the first guess in which direction a particle has moved is estimated via
cross correlation in an interrogation area. 4-frame means that if a particle has been tracked
for several time steps, the next position is estimated via the kinematic history of theparticle.
Basically, tracking algorithms are discriminated in three ansätze [105]:a) tracking in image plane (2D)b) tracking in object space (3D)c) combination of a) and b)
In this thesis, ansatz c) is chosen. The tracking of particles is first performed in one image
plane and afterwards performed in object space for reduction of ambiguities. For that
purpose, a large amount of methods are described in literature. The methods calculatequality criteria, which are defined manifold: gray value distribution, shape, position, size,
probability or kinematic conditions, to mention only a few. Nevertheless, common in all
methods is the definition of a search area, where a particle is expected in the next frame.
The proper choice of a search area is crucial in PTV algorithms for the reduction of
ambiguities.
Search area

The choice of the search area depends on the temporal length tL of a trajectory. A trajectorywith tL=1 is a single particle (case 1) and a trajectory with tL=5 is a particle, which istracked for 5 time steps (case 2). It is obvious that the proper estimation of expected
position in the next frame is more precise in the latter case.
Case 1 occurs at the beginning of processing or if a particle enters the measurement
volume. If no pre-information of the velocity field is available, a proper choice of thesearch area is not possible. Thus, it is mandatory to estimate the center of the search area
via the maximum particle displacement. If a pre-information is available, e.g. local mean
flow direction, the choice of a search area is getting more precise. One possibility for theestimation is the application of correlation based methods. The local mean velocity is firstestimated by PIV to define the location of the search area. A necessary condition is a highparticle density, which is prerequisite for correlation. Then, single particle positions aretracked. The method is successful, if the depth extension of the volume is not too large. At

Volumetric Measurements of the Transitional Backward Facing Step Flow

55

high spatial gradients, a large depth extension will result in erroneous estimated searchareas and consequently erroneous trajectories. In this thesis, the depth direction isdecreased by the scanning illumination, allowing the estimation of the search area via
correlation. This is due to the reduction of the dynamical range of velocities in the single
sub-volumes.
Case 2 occurs if a single particle could be tracked for several time steps, as shown in figure
2.33. If a particle is tracked for at least two time steps, the center of the search area iscalculated by the length and the direction of the velocity vector. The diameter of the searcharea is chosen to be 6 pix.

Fig 2.33 Schematic of two-dimensional tracking; definition of search area for case 2

Tracking

All particles inside the search area are possible candidates for the trajectory (ambiguities).
A choice is made by consideration of the kinematic history of the trajectory, namely the 4frame method [82]. The aim of the method is the evaluation of a quality criterion for eachpossible connection of a particle from one time step to the other. The results are quality
values of evaluated trajectories. The one with the smallest quality value is chosen to be thecorrect path.
The algorithms used herein are classified into the temporal length of a particle path. If aparticle path has no history, it is unavoidable to use the nearest neighbour criterion. The
quality function for the nearest neighbour criterion Φij, NN reads to:Φ ij,t NN = x tj+1 − xit(2.28)

If a trajectory is observed for two time steps, the three frame method is chosen. The
prediction of the particle position in the next time step is estimated due to the velocity of

56

the particle resulting from the two defined positions. The quality function for minimum
acceleration Φij, 3 reads to:

Φ

t
ij,3

=

x tj+1 − 2 xit + xit −1(2.29)

2Δt 2

If a trajectory is observed for more than two time steps, the four frame method is chosen.
The prediction of the particle position in the following time step is estimated due to thevelocity of the particle resulting from the two last defined positions and due to theacceleration of the particle calculated from the four time steps. It is assumed that theparticle moves locally with a minimum change in acceleration, thus one can define a
quality factor Φij, 4, which determines the probability of a correct particle path:tΦ ij,4=(

1
x tj+ 2 − 2 xit +1 + xit − x tj+1 − 2 xit + xit −1
2Δt 2

)

(2.30)In the case of conflicts occurring from the quality criteria, other criteria have to be defined.
A conflict occurs, if the quality function for two possible candidates is equal in the rangeof tolerance. Thus, a second criterion is used, which identifies corresponding particle
images in the successive time steps because of the number of occupied pixel. The particle
with the smallest change in image size is chosen as the most probable candidate. The
quality function reads to:Φ tij , size = Ν tij+1 − Ν tij(2.31)

If this criterion provides no solution, finally the intensity distribution of the possible
candidates is compared with the intensity distribution of the particle already being part of
the trajectory. The one with the best fit is chosen.
Exemplary results of the two-dimensional tracking procedure in each of the recorded
planes are shown in figure 2.34 by means of calculated trajectories. In total, a sequence of
60 time steps is shown. Trajectories are plotted having at least a temporal length of 30 time
steps. The different flow states (dependent on the distance from the bottom wall) areclearly recognizable: regular direction in the upper fat light sheet and increasing irregular
directions with decreasing distance to the bottom wall.

Volumetric Measurements of the Transitional Backward Facing Step Flow57

Fig 2.34 2D trajectories of light sheet 1 (a) to light sheet 4 (d) tracked for at least 30 time steps in a sequenceof 60 time steps

3D Reconstruction

The 3D reconstruction is performed for each two-dimensional trajectory following
equation 2.11. For a given two-dimensional trajectory, homologue particles are searched in
the other image planes. For that purpose, the epipolar geometry is applied. The tolerance
around the epipolar line is chosen to be 0.5 pix with respect to table 2.1. Possible
candidates are then reconstructed in three-dimensional space and chosen to be part of the
trajectory following the quality criteria similar to the pre-described two-dimensional case.
All particle positions along the trajectory, which can unambiguously be identified, arechosen to be part of the three-dimensional trajectory. The remaining points are studied in a
spatio-temporal manner. Here, the temporal length of the pieces of the trajectorycharacterizes the quality function, nearest neighbour, 3-frame and 4-frame quality function.
The three-dimensional position with the smallest deviation from the calculated position ischosen. If the deviation is higher than a certain threshold, the 3D position is not58

reconstructed resulting in a gap in the trajectory. If a candidate is identified to be part of
the trajectory, the three-dimensional point is reconstructed and the corresponding twodimensional image points are deleted from the database. This is repeated for each twodimensional trajectory resulting in a dataset consisting of three-dimensional trajectories foreach light-sheet.
In total, 13 experimental recordings of the BFSF are processed, yielding 13 Lagrangian
velocity fields, each 2s in physical time (2000 time steps). A mean number vector density
is calculated to about 42 vectors/cm³. Nevertheless, it should be noted that there is adistinct decrease in number vector density from LS1 to LS4. This is due to difficulties inbringing particles to the wall, which manifests in a lower local seeding density in LS1 incomparison to LS4. The particle displacements vary largely in the single light sheets
having displacements of 20 pix in the upper light sheets and 1-2 pix in the step corner ofthe lower light sheets. Thus, the difficulty in tracking varies largely. The planar number
particle density is calculated to 0.005 particles/pix, resulting in a difficulty in tracking
e=1.25 (see figure 2.15). The yield (number of 2D centroid positions/number of threedimensional vectors) is estimated via the mean number of linked particles with respect tothe center camera to 58%.

2.3.3 Post Processing
The post-processing includes the manipulation of the raw trajectories and the extraction of
fluid-mechanical parameters thereafter. The manipulation of the raw data is the connecting
and smoothing of trajectories and the calculation of velocities and accelerations from thesmoothed particle path.
The connection of trajectories is for two reasons:-gaps in the evaluated three-dimensional trajectories and-use of the scanning illumination and therefore overlapping trajectories.

Gaps occur due to the “drop out” behaviour of particles and from ambiguities. The “dropout” behaviour of particles is the disappearance of single particles in one of the image
planes resulting in lacks in the three-dimensional trajectory [150] as shown in figure 2.35.
The disappearance results from inhomogeneous illumination, angle-dependent illuminationor simply image noise, which is strongly reduced due to the scanning illumination, see
figure 2.30. Nevertheless, lacks due to ambiguities still exist. Usually, segments oftrajectories need to be identified to be part of the same trajectory. Xu (2008) [150]Volumetric Measurements of the Transitional Backward Facing Step Flow

59

proposed a “tracking in six dimensions” for the connection of segments of trajectories inobject space. The main idea is the extension of tracking single particles to the tracking ofsegments of trajectories. Beside the information of position and time, the information ofvelocities and accelerations is used. In our case, segments of trajectories are alreadyidentified to be part of the same trajectory due to the calculated two-dimensional trajectory.

Fig 2.35 Schematic of gaps and ambiguities in the proposed PTV algorithm

The identification of homologue trajectories between the light sheets is possible due to theoverlapping light sheets. This is done via a simple nearest neighbour criterion in object
space as the separation time between the single light sheets is small (1ms) resulting insmall particle displacements. Figure 2.36 shows an exemplarily chosen trajectory tracked
over all light sheet volumes. Furthermore, one overlapping area within two successive lightsheets is shown in the magnification. Due to the scanning illumination, the particle
positions need to be corrected to represent the same instant in time with equal separation
times [48].

Fig 2.36 Three-dimensional trajectory tracked over all light sheets (black: LS1; blue: LS2, green: LS3; red:LS4). The magnification shows the overlapping area of LS3 and LS4.

60

The connection of the trajectories due to gaps, ambiguities or scanning illumination is
performed via moving cubic spline interpolation with respect to Lüthi et al. (2005) [77].
The authors correct the three-dimensional positions of particles resulting from uncertainty
in measurement. For that purpose, a moving cubic spline interpolation is applied.
xi (t ) = ci , 0 + ci ,1t + ci , 2 t 2 + ci ,3 t 3(2.32)

The coefficients ci,j are calculated piece-wise on the trajectory with a number of supporting
points depending on the length of a trajectory. The minimum number is set to 6 supporting
points and the maximum is set to 21. This results in an over estimated system of at least 6
equations or a maximum of 21 equations for 4 coefficients. In this manner, the newpositions are evaluated for a separation time of 1ms. The enhancement is shown in figure
2.37 for the exemplarily calculated trajectory from figure 2.36. The first and second
derivative of equation 2.30 results in the velocity and acceleration of the trajectory.

Fig 2.37 Enhanced trajectory: moving cubic spline interpolation with piece-wise 21 supporting points

A time sequence of 200 ms is shown in figure 2.38 with particles tracked for at least 150
time steps. The BFSF is clearly recognizable. The trajectories are colour-coded withnormalized streamwise velocity. The region of shear layer roll up, development andimpinging, as well as the recirculation region and the redeveloping boundary layer isrecorded by the imaging system. A median number of 42 particles/cm³ are reconstructed.

Volumetric Measurements of the Transitional Backward Facing Step Flow61

Fig 2.38 Direct 3D visualization of Lagrangian trajectories obtained by 3D Scanning PTV. The mean number
particle density is 30 particles/cm³. The image shows a sequence of 200 ms and particles tracked for at least
150 ms.

The transformation of the Lagrangian velocity field into the Eulerian velocity field isevaluated by an inverse distance interpolation. Each single time step is interpolated on agrid of 16x90x60 mm³ with 1 mm spacing. In a next step, erroneous vectors are deleted bya simple threshold of magnitude in velocity (0.5 m/s). The resulting velocity field is mean
averaged over a 3x3 kernel and further interpolated in a temporal manner by averaging
three time steps. The interesting time scale in this thesis is far below 1 kHz, see section 3.
The resulting Eulerian velocity field is shown in figure 2.39. The voluminous velocity fieldis represented by slices, colour-coded with normalized streamwise velocity.

Fig 2.39 Instantaneous Eulerian velocity field calculated by inverse distance interpolation. vector field isvisualized via slices through the volume. Slices are colour coded by Line Integral Convolution (integration
length: 15 frames)

62

3 Backward Facing Step Flow
In the last decades, several extensive investigations (experimentally and theoretically) havebeen performed to analyse the characteristics of the backward facing step flow, eventhough most of the results deal with the statistical description of the flow phenomena.
Major part of the experimental studies used pointwise measurement techniques such as
Laser-Doppler Velocimetry (LDV) or Hot Wire Anemometry. The pointwise measurement
techniques yield high temporal resolution and high accuracy at low spatial resolution. The
techniques were primarily used to characterize integral flow behaviour of the BFSF and its
sensitivity to changes of e.g. aspect ratio, expansion ration, Re-number and boundary layer
thickness.
This chapter gives an overview of the state of the art regarding the BFSF and gives adetailed description of the implemented BFSF geometry. The literature is reviewed
twofold. On the one hand influences on the mean behaviour of the BFSF are described. On
the other hand unsteady phenomena regarding the BFSF are reviewed in detail. The
integration of the present flow configuration into the large amount of available literature is
documented by the integral results obtained by the planar measurement techniques.

3.1 Inlet Flow Conditions
The mean velocity field in the symmetry plane is shown in figure 3.1. The flow is colour
coded with the normalized magnitude in velocity. Furthermore, streamlines are plotted forvisualization of the flow structure in the recirculation region. Three independently
performed planar measurement realizations are used for the calculation of the mean
velocity field resulting in a total amount of 12000 vector fields in 480 s for each plane. The
bulk velocity Ubulk is calculated at x/h = -0.5 to Ubulk = 0.47 m/s. The corresponding
Reynolds number based on the equivalent diameter of the rectangular cross section of theflow tunnel is Red = 40000 (at 292.15 K). The Reynolds number Reh based on the stepheight is 4440 (at 292.15 K).
The magnitude in velocity increases in streamwise direction between -12<x/h<-6, isconstant in between -6<x/h<0 and decreases due to the sudden expansion between
0<h<12h. Two structures are visible by the streamlines in the low speed recirculation area
(0<x/h<6), a primary recirculation vortex (clockwise) and a secondary corner vortex(counter-clockwise).

Volumetric Measurements of the Transitional Backward Facing Step Flow63

Fig 3.1 Mean BFSF between -12h<x<12h. The flow is colour-coded with the dimensionless magnitude invelocity. Streamlines are used for visualization of flow topology.

At x/h = -6 the flow exhibits a strong directional change. Here, the boundary layer
separates and influences the tunnel flow. A shear layer develops, reattaches and recovers.
The length of the recovery region is calculated via the deviation of several velocity profiles
between -5<x/h<0.5 from the mean velocity profile at -0.5h. Figure 3.2 on the left shows
the influence of the angled edge, which ranges up to -2h. In the range -2<x/h<0 thedeviation is nearly constant. Thus, the flow is considered to have reached a fully developed
state again at the BFS.
The normalized boundary layer thickness is defined by the momentum thickness withrespect to the step height:

θ* = θ / h =1 u( z) ⎛u( z) ⎞⎜⎟dz−
1
h ∫ u ∞ ⎜⎝
u∞ ⎟⎠

(3.1)

Figure 3.2 on the right shows the boundary layer thickness (momentum thickness) as afunction of distance from the step. The edge from inclination to plateau triggers theboundary layer thickness resulting in a thickening of the boundary layer up to x/h=-2. In
the range -2<x/h<0 the boundary layer thickness is nearly constant (a kind of engaging)validating the fully developed flow regime. The mean turbulence intensity in streamwise
direction Tux is calculated to 4,32%, whereas Tux has a maximum of roughly 8% near the
wall.

64

Fig 3.2 (left) Deviation of several velocity profiles between -5h<x/h<0.5 from the mean velocity profile at 0.5h; (right) boundary layer thickness in the range -5<x/h<0

The mean velocity profile scaled with the wall shear velocity at y/h = -0.5 is shown infigure 3.3. The outer region (z+ > 65) is obtained by recordings of the telecentric lenses andthe inner region (z+ <= 65) by recordings with a larger magnification using the LDM. The
velocity profiles are composed and cubic interpolated. The wall shear velocity uτ isestimated by the conventional Clauser chart method [24]. The method assumes a universal
boundary layer profile. The wall shear velocity uτ is estimated by a least squares fitting for
optimally suiting the universal turbulent velocity profile. The wall shear velocity results inuτ = (τW /ρ)1/2 = 25 mm/s ≈ 1/20Ubulk. The universal constant values used for that purposeare κ = 0.41 and A = 5.3. The small lack between universal boundary layer profile andcalculated velocity profile for z+ > 50 is due to the positive pressure gradient [130]. The
universal boundary law assumes a flow with zero pressure gradients. Despite high spatial
resolution of the LDM (SR=3μm/pix), it is not possible to obtain correct data for z+ < 25.
Reasons are the difficulty of near wall PIV measurements. The near wall region ischaracterized by reflections, stray light of particles settled at the wall and insufficientseeding. Near the wall, particles tend to move away from the wall. Nevertheless, the scaled
velocity profile at x/h = -0.5 clearly indicates a turbulent flow regime.

Volumetric Measurements of the Transitional Backward Facing Step Flow65

Fig 3.3 Sublayer scaled velocity profile in the symmetry plane at x/h = -0.5 (left) logarithmic (right) linear

3.2 Mean Flow Characteristics
A large amount of high quality datasets and publications exist describing the mean flow
characteristics of the BFSF. This allows a general classification of the present BFSF
geometry. Basic for the general classification is the integral parameter “reattachment
length” xR. It is defined by the streamwise distance between separation and the time

averaged reattachment point and is determined in three different ways in literature:intersection of zero-velocity line with bottom wall, intersection of dividing streamline withbottom wall or point upon the bottom wall with 50% positive and negative streamwise
velocity

components

in

all

measurement

ensembles(especiallyinpointwisemeasurements). All methods deliver similar results with low deviation. In this study, thezero velocity line is chosen due to direct availability. For that purpose, the change of signin streamwise velocity downstream of the step at a constant level of y/h=0.1 is calculated.
Figure 3.4 on the left shows the mean streamwise velocity plotted along x/h downstream of
the BFS at y/h=0.1. Two zero crossings are apparent. The first zero crossing at xr2 is thesecondary reattachment point resulting from the corner eddy. The further downstream zerocrossing at xr1 is the primary reattachment point. Figure 3.4 on the right shows thenormalized instantaneous reattachment lengths xr1/h and xr2/h as a function of time for aperiod of 10s. The sampling frequency is 25 Hz. Furthermore, the mean values of xri mean
are plotted as dashed lines. The reattachment length of the recirculation area is 5.65h +/1.5h, that of the secondary recirculation length 1.24h +/- 1h. A clear frequency is notobserved.

66

Fig 3.4 (left) Definition of reattachment (right) reattachment as function of time, sampling frequency is 25 Hz

The main influences on the reattachment point/line are Reynolds-number Reh, expansion
ratio ER, aspect ratio AR and boundary layer thickness θ*. One of the first authors

describing the quantitative behaviour of the BFSF for varying Reh is Armaly et al. (1983)[4]. They studied the BFSF (ER = 1.94, AR = 18) by measuring the downstream velocity
distribution and the mean reattachment length via Laser-Doppler measurements. Assuming
the flow to be fully developed and two-dimensional, they observed an on average twodimensional flow only at low (Reh < 200) and high Reynolds numbers (Reh > 3000)
downstream of the BFS. In between, the mean flow is strongly three-dimensional. In thelaminar regime the reattachment length increases with Re to a maximum of 17h. In thetransitional regime the reattachment length decreases down to 6h, and approaches aconstant value of 7h in the turbulent regime. The authors postulate increasing turbulent
entrainment of the shear layer while convecting above the recirculation region, which
results in a decrease of fluid volume in the inner of the recirculation region. Thus, the pointof reattachment is shifted closer towards the step. The authors note an influence of therather high expansion ratio as they observed a second recirculation zone on the top wall.
Durst and Tropea (1981) [31] and references herein measured several expansion ratios forvarying Re-numbers. The results are shown in figure 3.5 on the left. The authors claim thatwhen defining the Re-number scales with the step height, all measurements approach thesame curve for different expansion ratios, which is not the case for a Re-number definedwith momentum thickness upstream of the step. Furthermore, figure 3.5 shows the
behaviour of the present test case, which is in good agreement with literature. The Rehnumber of 4440 describes the sate with a minimum of reattachment length. Figure 3.5 on
the right shows the reattachment length as a function of ER. At low expansion ratios, themean reattachment length increases, whereas the gradient decreases beyond a distinct

Volumetric Measurements of the Transitional Backward Facing Step Flow67

expansion ratio. Controversially, Ötügen (1991) [94] observed a decrease with increasing
expansion ratio. The installed BFSF differs largely from the curve.

Fig 3.5 (left) Influence of Re-number on mean reattachment length for variable expansion ratios (right)
influence of expansion ratio on median reattachment length [31]These deviations are not new in different BFSF geometries. Deviations are already
documented for identically experimental conditions of the BFSF by different authors,which are by far beyond the accuracy of the applied measurement techniques as shown infigure 3.6 on the left, especially at high Re-numbers. Motivated by the deviations, theinfluence of boundary layer thickness is studied by Adams and Johnston (1988) [1]. The
boundary layer thickness is manipulated using a combination of upstream wall suction to
reduce the boundary layer thickness and trips to thicken it. The boundary layer profile ismeasured by detailed hot-wire measurements. Results are presented in figure 3.6 on theright. The authors generated a parameter space of Re-number and boundary layer thickness
and characterized important regions, ranging from laminar separation and reattachment toturbulent separation and reattachment as well as mixings of both regimes. From theseresults, the authors concluded that there is a strong dependency on initial boundary layer
state, especially in the transitional regime, where the re-increase of reattachment length isobserved, see figure 3.6 on the left. To the author’s point of view, the state of the shear
layer (laminar/transitional/turbulent) is responsible for the reattachment length asconcluded by others, too. The length required for a shear layer to reach a fully developed
turbulent state is much longer than the reattachment length. Thus, the shear layer is not
fully developed at reattachment and carries the footprint of the initial conditions. With
respect to Adams and Johnston (1988) [1], the present BFSF configuration with δ/h=0.116
and Reh=4440 is in the mixing state of transitional separation and turbulent reattachment.

68

Fig 3.6 (left) Influence of Re-number on mean reattachment length for several experimental investigations.
(right) The location of transition regions for the BFSF in the parameter space of Reh and δ/h at ER=1.25 and
AR=11 [1]Most of the investigations regarding the BFSF assume on average two-dimensional flow inthe symmetry plane for aspect ratios larger than AR>10. Brederode (1972) [16] defined acriterion for the aspect ratio AR>10 for assurance. Nevertheless, the width of the channel
is not infinitely large since it ends abruptly at the sidewalls. Numerical simulations at
rather low Re-numbers showed that the side walls result in the creation of a wall jet
directed from wall towards the symmetry plane [146], [13]. Chiang and Sheu (1999) [22]extend the necessary AR to 50 to assure two-dimensionality. The numerical findings arevalidated by Tylli et al. [138] experimentally via PIV for an AR=20 (ER=2). PIV andstereoscopic PIV measurements showed an increasing intensity of the wall jets withincreasing Re at Re<1000 and a decreasing intensity of the wall jets with increasing Re at
Re>3000. The present AR of 57 clearly shows sidewall jets as described in literature, see
figure 3.7. A cross flow is generated immediately in the vicinity of the step directing fromsidewalls to the symmetry plane.

Volumetric Measurements of the Transitional Backward Facing Step Flow69

Fig 3.7 Mean flow field in the x-y- plane visualized via Line Integral Convolution. The mean reattachment line
is at roughly x/h=5.2 Sidewall jets are apparent in the range 0<x/h<1. The white arrows indicate flow direction.

3.3 Dynamical Flow Features
Beside the analysis of the mean flow field via pointwise measurement techniques, unsteadyflow features are apparent in the transitional and turbulent regime. In principal, there are
five large scale effects: a) shear layer roll up, b) vortex growing and merging, c) flappingof the shear layer, d) fluctuating position of the reattachment line and e) fluctuatingrecirculation region. It is obvious that each unsteady feature influences the other and viceversa. Thus, the understanding of the physical character and behaviour of the dynamical
process needs the extraction of each feature simultaneously. Furthermore, to resolve thedynamics of the unsteady features, the whole field needs to be measured successively witha high temporal resolution. These conditions are not achievable with conventional
pointwise measurement techniques. More sophisticated experiments were carried out using
PIV or surface pressure measurement via microphone arrays. The main advantage of PIV
is the possibility to combine topological analysis with quantitative velocity measurements
giving hope to extract coherent flow structures [116]. Footprints of coherent structures aremeasured by surface pressure measurements, if the probes are arranged in a planar set-up[53]. Furthermore, sophisticated combinations of pointwise and surface pressure
measurements allow the estimation of the volumetric velocity field [51] under certain
restrictions and assumptions, e.g. Taylor’s frozen hypothesis. Numerical simulations with
sufficient temporal and spatial resolution are nowadays limited to Reh = 3000 [118]. The

70

authors complain a lack in three-dimensional and time resolved data corresponding tohigher Re-numbers.
Shear layer roll up

Following the path of a fluid element started immediately upstream of the step, the firstfluctuation appears as the shear layer rolls up and generates discrete vortices being shedinto downstream direction, see figure 3.8. Troutt et al. (1984) [137] measured the shear
layer via multiple hot-wire technique. Their results indicate a spanwise vortical structure
similar to that of a free mixing layer, named as rollers in the following. Similarities to afree mixing layer and the corresponding rollers are also observed by other authors [71],[118], [116]. Scarano et al (1999) [116] performed instantaneous, planar PIV
measurements in the pointwise symmetry plane with a high spatial resolution (Reh = 5000).
They applied a pattern recognition technique to detect large scale coherent structuresappearing downstream of the BFS flow in the shear layer. The vortical structures in thesymmetry plane of the turbulent shear layer were sized between 0.12h and 0.44h. Up tox/h=3.5, small vortices (< 0.22h) in the shear layer show a quasi-linear growth resulting inhigh kinetic energy budgets. From linear instability analysis it is widely accepted that thischaracteristic formation is the consequence of Kelvin-Helmholtz instability. The authors
support the hypothesis of early 3D breakdown of these primary rollers. It is commonly
accepted from measured velocity fluctuation in this region that the shedding frequency
downstream of the step is in the order of Sth ≈ 2·10-1 [87].
Shear layer development

Further downstream is the region of shear layer development and convection of rollers
within the shear layer. Trout et al. (1984) [137] showed an increase of the shear layer
thickness with increasing distance from the step. The authors postulate that the growth of
the shear layer is strongly connected to vortex pairing mechanisms as omnipresent in freemixing layers. The convection velocity of the spanwise vortical structure is estimated to
0.6U0 [54]. Snapshots of vortex pairing are qualitatively observed via LIF measurements
by Kostas et al. (2002) [69]. Nevertheless, the sampling frequency is too low for resolving
pairing events. Furthermore, the authors performed detailed instantaneous 2D-PIV
measurements in several x-y and x-z planes downstream of the BFS (Reh = 4660). Besidethe calculation of vorticity, Reynolds stress and turbulent kinetic energy production, Proper
Orthogonal Decomposition (POD) has been performed trying to extract high energetic

Volumetric Measurements of the Transitional Backward Facing Step Flow

71

flow structures. Vortex trains and pairing are observed in the separated shear layer
increasing the turbulent stresses and turbulent kinetic energy production. The Reynolds
stress peaks were associated with vortex pairing and vortex growth in the shear layer. The
largest amount of turbulent kinetic energy was located at regions of small but intense
vortices. The POD showed mostly uniform vortical structures near the step andincreasingly irregular structures downstream of the step. The authors suggest an increase of
three dimensional turbulent vortex interactions with increasing distance from the step.
Delcayre and Lesieur (1997) [27] observed Λ-vortices resulting from three-dimensional
distortions of the rollers. Hudy et al. (2006) [51] studied the temporal evolution of coherent
structures via surface pressure measurements (Reh=8000). They synchronized the time
resolved surface pressure measurements with 2D PIV measurements at 10 Hz. A large
scale coherent structure was found developing at half of reattachment length inside therecirculation region. The structure grows in size to about the step height and then sheds andaccelerates downstream. This temporal growth of the flow structure is compared with thedevelopment of the vortex structures in the wake of a bluff body (wake mode – temporal
growth, shear layer mode – spatial growth).

Fig 3.8 Principal sketch of the unsteady flow phenomena

Shear layer reattachment

Next highly fluctuating and dynamical process is the shear layer impinging. The impinging
of the separated shear layer on the wall generates positive streamwise velocities
downstream of reattachment and negative streamwise velocities upstream of reattachment.
Instantaneous measurements show a fluctuation of the reattachment point of roughly
ΔxR=2h in the transitional and turbulent regime. Huang and Fiedler (1997) [50] measured
the fluctuation of the reattachment length via PIV to fR = 0.4 Hz corresponding to Sth=0.04(ER = 1.25, AR = 2.5, Reh = 4300). This is close to the results obtained by the direct

72

numerical simulation performed by Le et al. (1997) [71], which shows a quasi-periodic
fluctuation of Sth=0.06. Furthermore, DNS show a sawtooth pattern of the temporal
reattachment location [71], [118]. Starting from a short reattachment length, thereattachment point moves downstream in a continuous way. Suddenly, the reattachment
point breaks up in two parts, indicating shrinkage of the recirculation region and sheddingof a part of the recirculation region. This is confirmed by Lee et al. (2002) [54] via
pressure measurements. Furthermore, the reattachment position varies also in spanwise
direction. This is shown in the DNS by Le et al. (1997) [71] and confirmed by Jovic andDriver (1995) [59]. Nevertheless, the authors used a rather low AR of 4, probably having alarge influence of the side walls on reattachment position. In Beaudoin et al. (2004) [11],an approximately side wall independent wave length is measured for AR=15 at a rather
low Reh=300. The independent wave length is positioned near the streamwise symmetry
plane indicating the decreased influence of the sidewalls. To the authors point of view
there is a lack of results showing the spanwise structure of the reattachment line.
Recirculation region

As three dimensional coherent vortex structures in the separated shear layer and in therecirculation region seem to influence the behaviour of the unsteady flow, Lee at al. (2002)[55] studied the large scale vortical structures of the BFS flow three-dimensional via
extensive

coupledpressure-velocity

measurements.

The

conditionallyaveragedmeasurements showed “hairpin-like” coherent vortex pairs in the recirculation area withlength scales from 2.5h to 4h. They indicated the growing of the vortices moving
downstream. Recent results indicate that hairpin like structures are generated at shear layer
impinging. Schröder et al. (2011) [127] study a manipulated BFSF with laminar separation
via tomographic PIV. The authors show the generation of hairpin like structures in bothflow directions with respect to the reattachment line, when the generated rollers impingeon the wall. Highest negative velocities appear in the region 3<x/h<4.
Step corner and recovery region

The unsteady behaviour of the BFS flow in the vicinity of the step corner has been
investigated by Spazzini et al. [9] via skin friction measurements and flow visualisation.
They observed a cyclic process of growing and successive breakdown of the secondary
corner eddy. As the region of the step corner is of lower interest in the present study, thereader is referred to Spazzini et al. [9] and references herein for more detailed information.

Volumetric Measurements of the Transitional Backward Facing Step Flow73

Downstream of reattachment is the recovery region, where the shear layer redevelops to ausual turbulent boundary layer [69]. Organizations of vortex packets are found comparable
to that observed in the turbulent boundary layer measurements by [2], [27].
Global unsteadiness

Beside the locally fluctuating features described above, a global unsteadiness of low
frequency exists in the BFSF. Driver et al. (1987) [28] among others measured the power
spectrum of velocity fluctuations near the reattachment line with LDV. The streamlines in
the conditionally averaged mean flow show a global change in flow pattern. The
corresponding frequency has been estimated to be one order of magnitude lower than theshedding frequency [133]. The origin of this unsteadiness is controversial. Different
proposals are described in literature ranging from flapping of the shear layer to vortex
breakdown at reattachment.
In [32], Eaton and Johnston (1980) suggest that the low frequency fluctuation corresponds
to the flapping of the shear layer. The flapping of the shear layer is caused by an oscillating
volume of recirculating flow. They suggest that the unsteady recirculation volume is a
result of imbalance between shear layer entrainment from the recirculation volume andinjection of fluid near reattachment. The imbalance occurs, when the spanwise rollers
suffer a rapid breakdown. The consequence of rapid breakdown is a decrease ofentrainment rate and thus a growing of the recirculation volume. Due to the short term
breakdown, the entrainment rate decreases and the recirculation volume grows. Similar to
Eaton and Johnston (1980) [32], the dynamical behaviour of the shear layer is maderesponsible for the low frequency flapping in Driver et al. (1987) [28]. Differences arefound in the disorder of the shear layer that alters the rate of reverse flow. It is understoodthat a vortical structure with a locally higher convection velocity than the neighbours can“escape” the recirculation region. The reduction of reverse flow mass results in collapse of
the separation bubble. Schäfer et al. (2009) [118] link quasi-periodic rollers with thefluctuation of the recirculation region. The rollers carry reverse flow mass from therecirculation region while convecting downstream and passing the reattachment location.
The passing of the rollers is followed by a sudden shrinkage of recirculating region. The
observations are based on numerical flow visualization and quantitative study of thetemporal reattachment location. In contrast to other authors, who see the shear layer mode
as basic motion for fluctuation, Hudy et al. [51] suggest a scenario, where the dynamics are
driven by the wake mode nature of separated flows. The proposal is concluded via

74

combined surface pressure and planar PIV measurements. The authors observe a temporal
growth of a large scale structure located in the recirculation region at half the reattachment
location. At distinct size, the structure is transported downstream carrying reverse flow.
The starting flow behind a backward facing step is studied by Brücker (1999) [18] in a
volumetric manner via 3D Scanning PIV. Generation of large primary vortices areobserved. In addition, a secondary vortex is generated within the corner which displaces
the primary vortex and tilts-up the flow between both vortices. This “burst” can partly
affect the main flow up to the center. To this moment, the behaviour of the position of thereattachment point changes from a square-root function to a linear function with time.
To summarize, most of the unsteady frequencies and resulting proposals are based on
measured velocity fluctuations in a single point or instantaneous, but not time-resolved,
planar or volumetric velocity maps. Up to now, temporal and spatial resolution has only
been achieved by numerical simulations, see Le et al (1997) [71] or Schäfer et al. (2009)[118] at rather low Re.

Volumetric Measurements of the Transitional Backward Facing Step Flow75

4 Results and Discussion
The investigation of the BFSF is discussed with respect to the categorization shown infigure 4.1. The categorization is close to Rinioe et al. [109] in a slightly changed manner.
Region I is positioned immediately downstream of the edge centred at y/h=1, where shear
layer roll up and vortex shedding appears. The Kelvin-Helmholtz instabilities amplify
while

convecting

inregion

II.

Here,

shear

layer

development

andvortexconvection/interaction is observed. Due to the amplification of the instabilities and theimposed pressure gradient, the shear layer finally impinges downstream on the wall.
Region III is characterized by a highly fluctuating and three-dimensional state generating
intense velocities in up- and downstream direction. The interface between recirculation
region (IV) and shear layer (I, II) is characterized by a large turbulent entrainment. Region
V contains the flow in the vicinity of the corner. The recovery region (VI) is positioned
downstream of reattachment. Here, the separated boundary layer redevelops.

Fig 4.1 Categorization in regions: I: shear layer roll up; 0<x/h<3, 0.7<y/h<1.5; II: vortex convection, 3<x/h<8,0.5<y/h<1.5; III: reattachment zone, 4<x/h< 7, 0<y/h<0.7; IV: recirculation zone 1<x/h<4, 0<y/h<0.7; V:
sidewall jets and corner eddy, 0<x/h<1, 0<y/h<0.7; VI: recovery zone 7<x/h<9, 0<y/h<0.7

4.1 Region I: Shear Layer Roll Up
The sudden geometrical expansion forces the boundary layer on the bottom wall toseparate at the step edge. The separated boundary layer is influenced by high shear
gradients δu/δz immediately downstream of the step. In region I, the properties of theseparated shear layer are similar to that of a free mixing layer. Differences are found in thestate of the velocity profile of the low speed side as shown in figure 4.2. The velocity ratio
in the BFS-geometry is negative and rather high. Furthermore, the lower region is

76

dominated by unsteady events resulting in temporal and spatially differing velocity
profiles.

Fig 4.2 Comparison of mixing layer generated at a flat plate (left) and a shear layer generated at a backward
facing step (right)

The transition from laminar state into turbulent state in case of a mixing layer behind asplitter plate is studied by Meiburg et al. (1988) via a periodically perturbed laminar flow
along the span [84]. Stunning flow visualization is presented by the authors showing theskeleton of the shear layer as a wavy structure with cells. Figure 4.3 on the left shows thecase if perturbations on upper and lower side are out of phase and the figure on the rightshows the in phase configuration. The in phase mode leads to nearly regular distributed
rollers, which are amplified in downstream direction. The rollers are connected via
streamwise vortex tubes generating three-dimensionality. The authors indicate a strong
influence of magnitudes in perturbation and consequently a strong influence of theinstantaneous velocity ratio on the state of the shear layer.

Fig 4.3 Flow visualization of a free mixing layer generated downstream of a splitter plate via periodic
perturbations: (left) in phase perturbation on upper and lower side; (right) out of phase perturbations on
upper and lower side [84]In the present BFSF, the perturbations are natural without periodicity, depending on the
one hand on the initial state of the inlet flow and on the other hand on the unsteady
perturbations generated in the recirculation region. Figure 4.4 shows exemplarily LIF
visualizations of region I. The boundary layer separates at the step edge and vortex

Volumetric Measurements of the Transitional Backward Facing Step Flow77

shedding begins downstream of the step. Three different states of the shear layer roll up areshown, in the following damped state, regular state and amplified state. The states are
categorized by visual inspection. The damped state is characterized by a long distance
between vortex roll up and step edge. In the regular and amplified state, the roll up of theshear layer already appears at roughly x/h=1. The KH-type instability leads to growing of
the vortices while convecting downstream. The streamwise wave length of the rollers isdefined by the distance between the rollers. With increasing distance from the step, thewave length increases. The amplification of the rollers size with increasing distance fromthe step is rather moderate in the regular state, while high in the amplified state. The
amplified state is characterized by loss of coherent structures within three step heights
downstream of the step. However, the spatial structure of the rollers is not clear from thevisualization. Here, the analysis of the volumetric velocity field gives a further inside into
the spatial structure of the shear layer and allows a look at the reasons forcing the different
states.

Fig 4.4 Post processed LIF visualization of shear layer formation in region I (0<x/h<3.5)

Shear layer extraction

The extraction of shear layers is a crucial part in the analysis of coherent structures asstated by Schafhitzel et al. (2010) [120]. Shear layers are formed long before a vortex isborn. This is especially true in the case of the BFSF as the rollers are built due to the shear
layer as shown in figure 4.4. Thus, the quantification of the shear layer is essential toextract and visualize shear layers. However, compared to vortex detection criteria, there
are only few approaches for identifying shear layers. One approach is the inverted Qcriterion (Q<0), see equation 4.5, delivering regions of high shear stress, as the Euclidean

78of the rate-of-strain dominates. Haimes and Kenwright (1999) [39] suggest the calculationof a scalar value for identifying regions of high shear. They propose an eigenvalue analysisof the deviatoric rate-of-strain tensor S’. From continuum mechanics it is known that1
S ' = S − ⋅ I ⋅ trace(S ) .
3

(4.1)

S’ is symmetric and positive delivering three real eigenvalues λS’i. The Galilean invariant
and objective criteria for a shear layer Crshear is then defined by:

Crshear =(λ S '1 − λ S '2 )2 + (λS '1 − λ S '3 )2 + (λ S '2 − λ S '3 )2
6

>0

(4.2)

High values of Crshear define regions of high shear. Based on this criterion, Schafhitzel et al.
(2010) [120] introduce 2D shear sheets for the presentation of volumetric shear layers.
These sheets are defined as 2D height ridges of the criterion Crshear. Shear sheets represent
the skeleton of shear layers, analogous to vortex center lines in the case of vortices. In thisthesis, a simplified approach is used in region I, which identifies the shear layer by the
analysis of the spatial derivative δu/δz. This is possible due to the geometrical simplicity ofthe backward facing step as the shear layer is primarily dominated by shear resulting fromstreamwise velocity gradients perpendicular to the bottom wall, see figure 4.5 on the left.
Thus, the center line of the shear layer is defined in approximation by the local maximumof δu/δz for each x- and y-position.

Fig 4.5 (left) Definition of du/dzmax-criterion; (right) Comparison of Cr-criterion (upper) and du/dzmax-criterion(lower). The plot shows the symmetry plane at y/h=0 and is colour coded by intensity of the single criteria.
White means a high intensity and black a low intensity. White dots represent extracted local maxima of thecriteria showing the shape of the shear layer.

Volumetric Measurements of the Transitional Backward Facing Step Flow

79

Figure 4.5 on the right shows the comparison of both criteria for an instantaneous flow
situation in the streamwise symmetry plane. Both criteria deliver similar results, especiallyin region I. Here, both criteria identify the shear layer as a wavy line. With increasing
distance from the step (x/h>4), the single shear layer line is increasingly distorted.
Furthermore, regions of high shear layers are generated due to the shear layer impinging on
the wall.
The spatial connection of the single data points into a surface for geometricalrepresentation of the shear layer is performed by a nearest neighbour and intensity
criterion. For that purpose, a representative point is chosen as initial guess. The initial
guess for the position of the shear layer in region I is X/h = (0,y,1)T. The mean and an
instantaneous shear surfaces are shown in figure 4.6. The shear layer is formed over thewhole span ranging from z/h≈1 at the step to z/h=0.8 at roughly x/h=6. Below the main
shear layer, a second region of high shear is formed located in the recovery region. In
contrast, figure 4.6 on the right shows the instantaneous shear layer in the range 0<x/h<4.
The shear layer is represented by a wavy structure with heights and ridges. The deviationof the single instantaneous shear planes from the mean is described by superposition of atranslational shift and a position dependent wave function. The wavy deviation will beinterpreted as perturbations of the shear layer.

Fig 4.6 Extracted shear surfaces; (left) temporal averaged within 2s in the range 0<x/h<9 (right) temporal
averaged within 3 ms in the range 0<x/h<4

Shear layer thickness

Starting from the shear layer center line, a positive and negative shear layer thickness δ+and δ- is defined by the positive and negative momentum thickness. Main parameters areshown in figure 4.7 on the right. The local maxima of velocities in positive and negative
direction are chosen for normalization. The sum of positive and negative thickness is thetotal thickness δ, respectively δ/h. As stated by Ho and Huerre (1984) [47] the shear layer
thickness as well as the growth rate ∂δ/∂x is influenced by the velocity ratio of upper to80lower velocity. Here, the splitting of shear layer thickness in upper and lower thickness
allows a statement about influences from the upper and lower velocity profile on the shear
layer state.
Figure 4.7 on the left shows the pointwise and temporally averaged shear layer thickness asa function of streamwise distance. Several areas are extractable, where a change in shear
layer state happens. In the range 0<x/h<1, the shear layer thickness slightly decreases
having a minimum at x/h=1. The event is dominated by the change of the momentum
thickness in the lower part. Thus, the reason forcing the thinning must lie in region V. At
roughly the maximum of the shear strength, the beginning of vortex roll up is observed.
From x/h=1 up to x/h=3 the shear layer performs an approximately linear growth aspreviously reported by e.g. Troutt et al. (1984) [137], whereas the upper thickness grows
faster. Between 3<x/h<4 the shear layer thickness forms a plateau, which is characteristic
for free mixing layers [47]. Here, vortex merging events are observed in free shear layers.
It should be noted, that the plateau is forced by the lower boundary layer thickness
indicating an influence from region IV. At roughly x/h=5 the shear layer interacts with thebottom wall resulting in decrease of shear layer thickness with increasing distance fromreattachment.

Fig 4.7 Shear layer thickness; (left) spatially and temporally averaged shear layer thickness (normalized total
thickness δ/h, normalized negative thickness δ-/h, normalized positive thickness δ+/h) (right) sketch ofimportant parameters

Shear layer structure

Beside the shear layer thickness and strength, other characteristics are necessary todescribe the structure of the instantaneous shear layer. Here, the wave lengths of theperturbations on the shear layer are calculated as shown in figure 4.8 on the left. The figure

Volumetric Measurements of the Transitional Backward Facing Step Flow

81

shows the magnification of a part of the shear surface immediately downstream of the step.
The calculation of local maxima and minima in z-direction delivers positions of height andridges on the surface. The distance between local maxima and minima is chosen as half the
wave length λx and λy. Furthermore, the amplitude of the perturbation is extracted in the
single directions. The probability density function (PDF) of the measured wave lengths inregion I is shown in figure 4.8 on the right. The most probable wave lengths in x-direction
is λx/h = 1.1 and in y-direction λy/h = 1.2. Furthermore, the PDF is slightly skewed tolarger wave lengths. A rather wide range of wave lengths is observed indicating several
roll up configurations as shown in figure 4.4.

Fig 4.8 (left) Definition of characteristic wave length of the shear layer; (right) Probability density function(PDF) of the wave length in x- and y-direction of perturbations on the shear surface.

The shear layer properties in region I as a function of distance is shown in figure 4.9 for atime period of Δt*Ubulk/h = 100 (Δt = 2s). The figure shows the mean values of theproperties calculated in four sections (0≤x/h<1.0, 1≤x/h<2, 2≤x/h<3, 3≤x/h<4). The valuesof the wave lengths λx/h and λy/h are represented by the maximum of the PDF. The
amplitudes Ax/h and Ay/h are given as mean values. The ratio is defined by Ai/λi. For
reasons of better visibility in the plot, the amplitude and the ratio is multiplied by 10.
Figure 4.9 on the left shows the streamwise properties as a function of streamwise distance.
The wave length of the perturbations increases only slightly from 1.2h to roughly 1.4h in
between 0<x/h<3. In contrast, the amplitude of the perturbations increases about roughly
300% from 0.05h to 0.14h in between 0<x/h<3.
Figure 4.9 on the right shows the spanwise properties of the shear layer as a function of
streamwise distance. The wave length slightly decreases up to x/h=3. Here, the wave
length in x-and y-direction is nearly identical. This is also true for the amplification and theratio. In the range 0<x/h<2, perturbations in x- and y-direction converge to a common

82

value. The streamwise perturbations are amplified and the spanwise perturbations aredamped. In the range 2<x/h<4 the rate of amplification is identical for both directions.

Fig 4.9 (left) Shear layer properties in x-direction for region I; (right) Shear layer properties in y-direction forregion I; solid line: amplitude, dashed line: wave length, dash-dot line: ration of amplitude and wavelength; forplotting, the amplitude and the ration are multiplied by 10

Shear layer convection

The temporal resolution of the volumetric measurements allows a look at the developmentof single shear layer perturbations. Figure 4.10 on the left shows the projection of the shear
layer height on a plane surface for two instants in time separated by 8 ms (ΔtUbulk/h=0.4).
The convection of the perturbations is clearly visible. The streamwise frequency of theperturbations is measured by pointwise analysis of the instantaneous shear layer position in
z-direction. An exemplarily time signal at (x/h=2.5, y/h=0) and (x/h=3, y/h=0) is shown infigure 4.10 on the right. The displacement Δxp of the large scale perturbations instreamwise direction is indicated. The mean convection velocity in streamwise direction is
calculated via cross correlation to 0.6Ubulk. This convection velocity of the perturbations isin the same range as the roller velocity estimated by e.g. Schäfer et al. [118]. The authors
observed regularly distributed rollers convecting downstream. Nevertheless, in the present
study the convection velocity of the instantaneous perturbations varies largely as a functionof time. As shown in figure 4.10 on the right, the velocities of the perturbations vary in therange ±0.2Ubulk, thus making the estimation of a clear frequency inaccurate. A trend isobserved showing that the frequency of fluctuations is slightly increasing in the range
1<x/h<3. The increase of the frequency indicates an acceleration of perturbations instreamwise direction.

Volumetric Measurements of the Transitional Backward Facing Step Flow83

Fig 4.10 (left) contour plot of instantaneous shear layer height in region I; the separation time is 8 ms (t*=0.4);(right) time plot of shear layer height; solid line: point 1, dashed line: point 2

The previous analysis extracted main spatial and temporal properties of the shear layer inregion I. The shear layer is quantified in terms of position, thickness, strength, wave
lengths and amplitudes. The results showed a wavy structure, which is identically
amplified in x- and y-direction beyond a distinct point positioned around x/h=2.
Furthermore, the analysis determined the convection of disturbances in downstream
direction with a transport velocity of roughly 0.6U0±0.2U0.
Vortex extraction

In the following, the neighbourhood of the shear surface is analyzed. This is a detection ofvortical structures in stream- and spanwise direction. Basic principle of multiple vortex
criteria is the velocity gradient matrix or the Jacobian of the velocity field. In the Euleriansense, the Jacobian is defined by applying the gradient operator on the velocity field, seeequation 2.15.
Most famous detection criteria are the Q-criterion, the Δ-criterion, the λ2-criterion and theswirling strength criterion. For a detailed review of these criteria, the reader is referred to
Chakraborty et al. (2005) [20]. Common in all detection criteria is the calculation of ascalar field, where large values mean high intensity.
The Q-criterion is introduced by Hunt et al. (1988) [52]. The detection criterion is basedon the local analysis of the velocity gradient tensor delivering a Galilean invariant criterion
for extracting vortical regions. For incompressible flows the Q-criterion is defined as:

84

Q=

(

)

1
2
2
Ω − S >02

(4.3)

A vortex region is identified, if the Euclidean norm of the rate of rotation tensor is larger
than the Euclidean norm of the rate of strain tensor.
The Δ-criterion, introduced by Chong et al. (1990) [23], uses the analysis of thediscriminate of the characteristic polynomial from equation (2.18).
2

3⎛1 ⎞ ⎛1 ⎞Δ = ⎜ R⎟ + ⎜ Q⎟ > 0⎝2 ⎠ ⎝3 ⎠(4.4)

If Δ>0, the Jacobian has complex eigenvalues. Topological theory says that if the Jacobianin a point has a pair of complex eigenvalues, the streamline pattern in the neighbourhoodof this point is spiralling.
The λ2-criterion, introduced by Jeong and Hussain (1995) [57], uses the pressure minimum
criterion in the center of a vortex derived from the two-dimensional Navier Stokes
equation under adiabatique assumption. A local pressure minimum exists, if theintermediate eigenvalue

λ2 (S 2 + Ω 2 ) < 0 .

(4.5)

The swirling strength criterion λci is proposed by Zhou et al. (1999) [153]. The criterion isan extension of the Δ-criterion by analysing the imaginary part of the eigenvalues resulting
in a quantity for the swirling strength and the local plane of swirling.
Figure 4.11 shows the application of the mentioned criteria to an arbitrarily chosen
instantaneous, three-dimensional velocity field of the BFSF. For that purpose, the original
90x65x15 grid (filled with scalar values of the criteria) is sub-sampled on a finer grid about
the factor 10 via moving spline interpolation. The figure shows the streamwise symmetry
plane. The images are normalized by the maximum value of the single criteria. At first, it
should be noted, that all detection criteria deliver similar results. Nevertheless, a difference
is found in the contrast of the single criteria. Here, contrast means the strength of a local
peak in comparison to its neighbourhood. For that reason, the λci-criterion (3rd from thetop) is chosen for vortex center estimation. In the range 0<x/h<3 organized vortex roll up
is detected by the criteria. Increasingly unorganized vortical structures are apparent with

Volumetric Measurements of the Transitional Backward Facing Step Flow

85

increasingly streamwise distance from the step as observed by several authors, see e.g
Kostas et al. (2002) [69], Scarano et al. (1999) [116] validating the present threedimensional calculation of the implemented criteria.

Fig 4.11 Comparison of vortex detection criteria in streamwise symmetry plane; (top to bottom) Δ-criterion,
Q-criterion, λci-criterion, λ2-criterion

The three-dimensional center (x/h, y/h, z/h)T of the vortices in the single planes is
estimated in the following manner. The spanwise position (y/h) is defined by the analyzed
plane in y-direction. The height (z/h) and streamwise position (x/h) is estimated via theanalysis of the swirling strength value and following analysis of the velocity vector field.
For that purpose, each plane coded by the swirling strength criterion is analyzed for local
maxima, which are used for a first guess in the vortex center estimation. The exact position
results from the w-velocity profile in the vicinity of the local maxima. A characteristic ofthe developing rollers is that the main axis of rotation is in spanwise direction and that thedirection of rotation is clockwise. The condition for a clockwise rotation is w1>w2 asshown schematically in figure 4.12 on the left. Furthermore, the figure shows important
properties describing a roller. The vortex diameter Dvortex is defined as the spatial distance
between local velocity maxima and minima. The center of the vortex is then defined at half
the diameter. Furthermore, the velocity at the center position describes the convection
velocity of the vortex. A characteristic instantaneous velocity signal from region I in thevicinity of a detected vortex is shown in figure 4.12 on the right. The characteristic
velocity signal clearly describes a rotating vortex.

86

Fig 4.12 (left) Characteristics of a roller and (right) v-velocity profile along x/h positioned in the center ofdetected vortices estimated by the swirling strength criterion; vortices have streamwise displacement ∆xroller

The spatial connection of vortex centers to a vortex center line is performed via plane-wiseconnection of local maxima following the definition that the vortex center line is theconnection of plane-wise local maxima with similar properties. Beside the spatial position,
properties are e.g. vortex diameter, rotational velocity or rotational direction. The principal
is shown in figure 4.13 on the left. The spanwise displacement of the planes is 0.1h. The
connection of center positions follows quality criteria similar to PTV [66]. Starting with avortex center in planey, the next position is searched in planey+1. In a first step, the nearest
neighbour is chosen. If ambiguities arise, the vortex diameter is chosen for quality criteria.
Two reconstructed vortex lines at an arbitrarily chosen instant in time are shown in figure
4.13 on the right. Two corresponding planes colour coded with the swirling strength
criterion are included for orientation. It should be noted, that the chosen vortex lines arerather long (≈ 4h) and all other detected lines are removed. The raw dataset after vortex
line detection is full of short vortex lines, which may be other vortices or artefacts as thevortex line detection is rather sensitive. This is obvious, as the center line results from local
maxima of data, which is derived from the gradient of the velocity field. Thus, there aremany operations, where uncertainties are introduced or amplified. Thus, for further
investigation a threshold of 0.5h is chosen for the length of a vortex line trying to exclude
noise (or maybe other vortices). Vortex lines with less than the threshold are removed fromthe dataset. The threshold is chosen to be slightly lower than half the expected wave length(see figure 4.8). The last step is the spatial smoothing of vortex lines via moving splineinterpolation.
A first interpretation can directly be drawn from figure 4.13 on the right. The vortex lines
are not characterized by straight lines as valid in a laminar flow, but by a wavy structure

Volumetric Measurements of the Transitional Backward Facing Step Flow87

similar to that of the investigated shear layer (see figure 4.8). Furthermore, the first roll up
happens at varying x-positions in y-direction generating three-dimensionality immediately
downstream of the step.

Fig 4.13 (left) Conceptional reconstruction of a three-dimensional vortex tube (roller) (right) Reconstructed
vortex center lines. Vortex line in the range 0<x/h<4. Streamwise planes are colour coded by swirling strength

Roller Structure

The further investigation of the rollers is performed similar to that of the shear layer. The
classification is performed in terms of wave length, amplitude and convection velocity.
Important definitions regarding the spatial structure of the vortex center line are shown infigure 4.14 on the left. The vortex center line is shown in an isometric view. Furthermore,the 3D center line is projected into the xy-, xz-, and yz-plane. For further analysis, the 2Dlines are smoothed using a rather high smoothing parameter (dashed, red line). This isnecessary to exclude small scale distortions on the line for extracting only large scale
length scales higher than 0.5h. Nevertheless, relevant parameters can be estimated from thesmoothed line. As the vortex center line of the rollers is oriented in y-direction, a wave
length in x- and z-direction is calculated. The corresponding amplitudes for a guess of thestrength of distortion are calculated as shown in figure 4.14 on the left. The probability
density functions of the wave lengths λy-x , λy-z of the rollers in region I are shown in figure
4.14 on the right. In comparison to the PDF of the wave lengths of the shear layer, themeasured values show a large scatter. A Gauß-fit is chosen to estimate the most probably
wave lengths resulting in λy-x/h=0.75 and λy-z/h=0.85. Furthermore, the PDFs show thatlarger wave lengths are more probable in z-direction than in x-direction showing an
asymmetry in the single directions. Compared to the wave lengths of the shear layer (seefigure 4.8), the most probable wave lengths of the rollers are slightly lower than that of theshear layer.

88

Fig 4.14 (left) Definitions for classification of vortex center line in the range 0<x/h<4; solid black line: 3D
vortex center line, dotted lines: projection of 3D line into single planes, red dotted: smoothed 2D vortex centerline (right) probability density function of single wave lengths of the roller in y-direction. Black dots: measured
points, black lines: Gauß-fit

The wave lengths and amplitudes as a function of streamwise distance are shown in figure
4.15 for a temporal average of Δt*=100 (Δt =2s). The figures are generated by mean valuesof the properties calculated in ten sections (starting positions: x/h=0.75, section length:0.5h, step width: 0.5h) and following moving cubic spline interpolation.
Region I (0≤x/h≤3.5) is characterized by increasing wave lengths and amplitudes.
Immediately after vortex roll up (x/h≈1) the roller already describes a wavy line. Here,wave lengths in x- and z-direction are in the same range. In contrast, the amplitude in xdirection is larger than the amplitude in z-direction. With increasing downstream distance,
the wave lengths in both directions similarly increase to about the step height. In case of
the amplitudes a difference is observed. The amplitude in x-direction is largely amplified
up to x/h≈3.5 being about 20 % of the corresponding wave length. The amplitude in ydirection increases only uo to x/h≈2 being nearly constant with further downstream
distance up to x/h≈3.

Volumetric Measurements of the Transitional Backward Facing Step Flow89

Fig 4.15 (left) Vortex center line properties in yz-plane as a function of x-direction for 0.5≤x/h≤5.5; (right)
Vortex center line properties in yx-plane as a function of x-direction for 0.5≤x/h≤5.5; solid line: amplitude,
dashed line: wave length; for plotting, the amplitude is multiplied by 10

The following conclusion is drawn and sketched schematically in figure 4.16.: the roller isdistorted immediately after genesis. The wave lengths of the rollers are similar in both
directions, whereas the amplitude in x-direction is larger. With increasing downstream
distance, the wave lengths in both directions increase, thus showing that the properties of
the wave lengths are similar in both direction. A large difference is found in the amplitudes
in the single directions. Immediately after birth, the amplitude in x-direction is larger thanthat in z-direction. This discrepancy grows further with downstream distance. The physicalinterpretation is that the streamwise vortex loops of the rollers are stretched (increasing
amplitude) in streamwise direction due to the main flow in streamwise direction. The
vertically aligned vortex loops (amplitude in z-direction) is bounded by the separation
bubble resulting in a constant amplitude of the vortex loops in z-direction in the range
2<x/h<3.

Fig 4.16 Model of change of spatial structure of vortex center line as a function of streamwise distance,
deducted from the results presented in figure 4.15

90Roller Orientation
In course of the analysis, a dominant feature of roll up is observed: inclined vortex

formation. Figure 4.17 on the left schematically shows observed configurations in the top
view. All following descriptions are with respect to the orientation of the step edge. Thus,
the configuration top left describes parallel roll up, the middle and bottom leftconfiguration inclined roll up, top and middle right convex and concave roll up. The
configuration bottom right is omnipresent. It describes multiple piecewise roll up, where
the single pieces vary largely from single to several step heights. In the following, we will
focus on linear configurations, i.e. parallel and inclined vortex roll up. For that purpose, thefigure on the right defines the inclination angle of the rollers. The figure shows the
projection of three independent vortex lines in different colours. The dashed lines show theapproximated orientation of the rollers resulting in the inclination angle α with respect tothe step. The calculation of the inclination angle is performed via a linear least squares fit.
Vortex lines with lengths below 2h are excluded from the calculation. This is for the
reduction of erroneous calculated angles as short vortex lines are sensitive to inaccurately
calculated positions in x-direction.

Fig 4.17 (left) Definition of roll up configurations: left top: parallel roll up, middle/bottom left: inclined roll up,
top/middle right: convex/concave roll up, bottom right: piecewise roll up (right) Definition of inclination angle:
blue: negative inclination, red: zero inclination, black: positive inclination

The PDF of the inclination angle is shown in figure 4.18. The figure shows the measured
points for a time sequence of ∆t* = 100 (∆t = 2s). The curve describes a Gaussian fit,showing that the gradient of the Gaussian function fits pretty well, but not the maximum
peak. The distribution is centred on 0° being the most probable inclination angle.
Nevertheless, inclination angles of ± 20° are observed with a probability of 25%. The PDF

Volumetric Measurements of the Transitional Backward Facing Step Flow91

indicates the necessity of spatial analysis for the understanding and interpretation of the
BFSF, which was not covered by former measurement techniques (pointwise or planar).

Fig 4.18 Probability density function of the inclination angle of the rollers; dots: measured points, black line:
Gaussian fit

Roller convection

The convection of the rollers in downstream direction is shown in figure 4.19 via two
instants in time separated by normalized time Δt*=1.41. The black lines represent instant 1
and the gray lines instant two. Two rollers are captured by the vortex criteria at roughly
x1(t1)/h=1 and x2(t1)/h =2.2. Furthermore, missing pieces of vortex lines are shown infigure 4.19 on the right making the analysis rather difficult as a choice is necessary, if
missing pieces result from inaccurate data or from fluid-mechanical events influencing thevortex line.
Nevertheless, the convection of the rollers is clearly visible. The rollers travelled a variable
distance with respect to the streamwise and spanwise direction. A mean convection
velocity averaged over the width is calculated for the present configuration tov(x1)/U∞=0.28 and v(x2) /U∞=0.42 showing the acceleration of the rollers in downstream
direction. This is validated by detected rollers at about x/h=4 having even larger
convection velocities of 0.56U∞. The vortex line at x1(t1)/h=1 describes an approximately
parallel roll up of a wavy shear layer. At time instant 2, the vortex line seems to havestayed at the same position at about y/h=0 while travelling in downstream direction for
y/h<0. Thus, the vortex line describes the development of a parallel to an inclined roller.

92

Fig 4.19 (left) Isometric view of vortex line convection; black line: t1*, gray line: t2*, solid line: 3D line, dotted
line: 2D projections of 3D vortex center lines. (right) top view of the situation left

The shedding frequency of the rollers is calculated via an FFT of the w-velocity signal at aconstant (x/h =1.2, y/h=0, z/h=1.1). The w-velocity signal is chosen as the characteristic
property of the vortex roll up (see figure 4.12). The position is chosen to be in the region ofinterest, where vortex roll up occurs. In total, a period of t* = 600 (t = 12s) of the time
resolved measurements is analyzed. Figure 4.20 shows the FFT of the signal. The peak in
the FFT is at 11.4 Hz, which corresponds to a Strouhal number for vortex shedding of
StVS=0.24 based on the step height. Beside Metais et al. [87], several authors calculated aStrouhal number of roughly 0.2, which is in good agreement with the results obtained here.

Figure 4.20 Fast Fourier Transformation (FFT) of w-velocity signal at (x/h =1.2, y/h=0, z/h=1.1) to extract thevortex shedding frequency

Volumetric Measurements of the Transitional Backward Facing Step Flow

93

Shear layer structure vs. roller structure

Detected shear layer and detected rollers are shown simultaneously in figure 4.21 for one
instant in time. Immediately downstream of the step, the shear layer describes a wavy
structure. Vortex roll up happens due to high shear gradients at roughly x/h=1. The vortex
line is already distorted. The second vortex line describes a convected roller. By visualinspection it is obvious that distorted vortex lines are primarily connected to the distorted
shear surface. The vortex lines follow heights and ridges of the shear surface.
With respect to the statement that vortices are born, where high shear is apparent as statedby Schafhitzel et al. (2010) [120], this thesis proposes that the distorted vortex line is theresults of the wavy shear surface immediately downstream of the step. The common
opinion about the principal process regarding the vortex roll up is that the roll up occurs
along a linear line [129]. The condition for such a regular roll up is a smooth and straight
shear surface. This is probably true in the laminar separation regime (see figure 3.6 on theright), but not in the transitional to turbulent separation regime. As shown in figure 4.9 theshear layer has a wavy structure immediately downstream of the step. Thus, as the rollers
develop due to high shear, it is rather natural that the spatial structure of the roll up is adistorted line.

Fig 4.21 Interaction of shear layer (surface) and vortex center line (black line) in the range 0<x/h<4

The physical reason for the waviness of the shear layer is not yet clear. Two possible
explanations can be given: The waviness may be generated due to the unsteady flow below
the shear surface in the vicinity of the step (region V) and/or the waviness may result fromperturbations originated in the boundary layer upstream of the step. As shown later, the94intensity of the recirculation flow in region V has only velocities of 5% of the free stream
velocity. Thus, higher influence is expected from the turbulent boundary layer, which isconvected in free stream direction. Furthermore, the initial state of the boundary layer isstated in literature as a main influence on the characteristic behaviour of the BFSF (seefigure 3.6) supporting the hypothesis that the waviness in y-direction results from theinitial state of the boundary layer. An analysis of spanwise planes shows secondary flow
structures as investigated by Scarano et al. (2001) [112]. The authors connected thesesecondary vortices with the appearance of Λ-vortices resulting from the turbulent inflow.
Further investigations about the separation of coherent structures from the upstream
boundary layer are necessary.

4.2 Region II: Vortex Convection
Region II (3<x/h<9) is characterized by a highly unsteady flow regime full of local
stagnation points and complex vortex dynamics. The flow evolution is shown in figure
4.22 via LIF for a period t*≈9.4. The red dotted line represents the tracking of one
exemplary roller. The sequence shows an “undisturbed” development of a roller without
large external perturbations. In terms of the categorization shown in figure 4.4, the flow isdamped at the beginning of the sequence and becomes regular at the end of the sequence.
The shear layer roll up happens in the range 0<x/h<3 (t0). Later in the sequence, the roller
starts to develop and is transported in downstream direction. At t5 the roller builds a large
scale structure positioned around x/h=4. Here, the roller starts to interact with the bottom
wall resulting in a breakdown of the regular structure within about three time steps (t5-t8).
The breakdown is finally documented by the loss of concentrated rhodamin. Sudden loss ofconcentration is due to increased turbulent mixing as omnipresent in structure breakdown.
The spatial hull of the roller is still observable at t10. The analysis of the time resolved
measurements showed that the rollers have lost most of their rotational energy at t10.

Volumetric Measurements of the Transitional Backward Facing Step Flow95

Roller development

In region II, the distortions of the vortex lines (in the following loops) are highly amplified
as extractable from figure 4.15. The principal three-dimensional structure of the rollers isshown in figure 4.23 via a schematic vortex center line for varying streamwise positions.
The basic of the following description is found in figure 4.15. Immediately after vortex rollup the vortex center line is slightly distorted due to the wavy shear layer. WithFig 4.22 Development of rollers without external disturbances; separation time: Δt=20 ms (Δt*=0.94); total
length of sequence: t≈200ms (Δt*≈9.4)

96

increasing distance, the vortex loops are stretched in downstream direction resulting inroughly 30% higher amplitudes of the vortex loops in x-direction compared to the zdirection at x/h=3.5. This is concluded to be the result of the primary streamwise directed
shear velocities. In the range 4<x/h<5, the vortex center line performs a rapid deformation
to a structure, where the vortex loops are primarily oriented in z-direction (Ay-z ≈ 1.5Ay-x).
The vertical alignment of the vortex loops again vanishes in the range 5<x/h<6. Here therollers start to interact with the bottom wall. Thus, the vertical aligned vortex loops areagain re-oriented in horizontal direction forced by the presence of the bottom wall.

Fig 4.23 Model of change of spatial structure of vortex center line as a function of streamwise distance,
deducted from the results presented in figure 4.15

Influences on roller development

Several mechanisms might explain the re-orientation of the dominant vortex loops of therollers: roller pairing or merging events (a), roller induction effects (b) and/or interactions
with recirculation region (c).
a) In literature, the range 3<x/h<6 is closely connected to complex vortex pairing
events as observed in free mixing layers [131]. As already discussed, the plateau andfollowing increase in shear layer thickness (figure 4.7) between 3<x/h<6 is comparable tothat of a free mixing layer [47]. The development of coherent structures in a turbulent
mixing layer is observed by Roshko (1976) [111]. A picture is drawn, where neighboured(in streamwise direction) rollers in the fully developed shear layer start to rotate aroundeach other and merge into a single but larger vortex. The vortex pairing/merging events
result in large thickening of the boundary layer thickness. As discussed earlier, the shear
layer in the present BFSF is not fully developed. This was concluded with respect to therequired length for a shear layer to reach a fully developed state [1]. Vortex pairing events
as described in [111] are not observed. Nevertheless, as the distance between the rollers is

Volumetric Measurements of the Transitional Backward Facing Step Flow97

in the range ∆x/h=1 +/-0.5 (see figure 4.19), there might be a distinct interaction between
neighboured rollers.
b) Muller et al (1982) [90] connect complex merging events with respect to the Biot
Savart law as shown in figure 4.24. The curved vortex tube generates three-dimensionality
by inducing velocities on itself resulting in distortions of the vortex line. Furthermore, avortex loop bent in upstream direction is affected by induced velocities of downstream
directed loops of a successive roller in such a way that the upstream vortex loop istransported in vertical direction. It should be noted that this scenario has been drawn via
flow visualization experiments.

Fig 4.24 Biot-Savart law applied to a curved vortex line; solid line: original vortex center line, dashed line:
distorted vortex center line due to induced velocity vi by vorticity ωvortex at position s along the vortex centerline.

c) Small scale turbulent zones resulting from impingement are made responsible forturbulence in the recirculation flow by Pronchick (1983) [102]. These zones influence theoncoming rollers and lead to structural roller breakdown events in the range 3<x/h<6. A
typical scenario of large scale coherent structures in the recirculation area influencing
oncoming rollers is shown in figure 4.25. The figure shows the vector plot of the symmetry
plane from the PIV measurements. The vectors have uniform length to focus on the
direction of the flow. The magnification shows a vortex interaction scene resulting in a
vertical perturbation originating from the recirculation region. The convection direction ofroller 1 is in downstream direction. Roller 2 generates upstream directed velocity resulting
in a local saddle point marked by a cross-style marker. In this point three-dimensionality is
generated resulting in vortex center line distortions and consequently in instabilities.
From the own results it is concluded that vertical perturbations from the recirculation zone
are the primary mechanism leading to a change of orientation of the vortex loops from98

horizontal to vertical orientation. This is for two reasons: a) If the Biot Savart law had adominant influence, the re-orientation should be a continuous process and should also be
observable at positions in upstream direction. b) An indicator for the influence of therecirculation area is observed by the development of the lower shear layer thickness (figure4.7). In fact, chapter 4.4 will show that the vortex loops generate vortices, which aretransported into the recirculation area and interact with the oncoming rollers. Thus, it canbe concluded that roller breakdown doesn’t happen naturally due to long term instability,
but due to short term perturbations. With respect to figure 4.7, the perturbations
influencing the shear layer development originate from the recirculation area. This isconcluded due to the increasing lower shear layer thickness in the range 4<x/h<6.

Fig 4.25 (top) Instantaneous vector field in the symmetry plane; vectors have constant length to indicate flow
direction. (bottom) Zoom into of short term perturbation. Scenario is sketched for clarity.

4.3 Region III: Reattachment
The spatial expansion as well as the temporal behaviour of the integral length scale
reattachment line is in the focus of this chapter. Open questions are discussed as: what is
the spatial structure of the reattachment line, what is the reason for the spatial structure andwhat causes the fluctuation of the reattachment line? First, the reattachment position interms of time resolved planar measurements is discussed and compared with literature.
Next, the spatial structure of the reattachment line will be pointed out showing that theplanar investigation is not sufficient for the analysis of the reattachment location and forthe interpretation what forces the fluctuations of the reattachment line. At the end, theinteraction of shear layer rollers with the reattachment line is investigated. This is possible
for the first time due to the volumetric measurement technique.

Volumetric Measurements of the Transitional Backward Facing Step Flow

99

Reattachment: Characteristics of xz-plane, y/h=0

As discussed earlier, measurements of the BFSF were primary performed in streamwise
symmetry planes or via wall pressure measurements. Here, a sawtooth pattern in thetemporal progress of the reattachment position is observed. This is that the reattachment
length constantly grows in size and suddenly shrinks as discussed in chapter 3.
The analysis of the time resolved measurements showed that the definition of thereattachment point described in figure 3.4 is not sufficient for the description of the
dynamic behaviour of the reattachment point. Figure 4.26 shows a normalized
instantaneous velocity profile in the streamwise symmetry plane (y/h=0) at z/h=0.1. There
are two recirculation regions, a regular one and a separated one. Thus, two reattachment
points are introduced. The one defined by the regular recirculation region xr11 and the one
defined by the separated recirculation region xr12.

Fig 4.26 Definition of recirculation region and separated recirculation region of the instantaneous flow field

The calculated reattachment length of the reattachment points xr11 (black line) and xr12 (redline) obtained by the time resolved planar PIV measurements is shown in figure 4.27 for aperiod of ∆t=6s (∆t*=282). Furthermore, exemplary vector fields colour coded withstreamwise velocities are shown describing characteristic scenarios. The negative
streamwise velocity is blanked. Thus, white areas represent recirculating flow.
The progress of the regular reattachment point shows a quasi-periodic characteristic. The
point constantly moves downstream and suddenly breaks up in two parts. A quasi periodic
frequency is obtained by counting the events resulting in 4 events/s. This corresponds toStre,2D=0.09. The reattachment point of the separated recirculation region differs slightly
from the regular one. At positions, where xr11 suddenly jumps upstream, xr12 constantly

100

moves further downstream followed by a sudden jump back to the streamwise position of
xr11. The evolution of both reattachment points indicates the cyclic release of parts of therecirculation region in downstream direction. A classification of the separated regions is
performed via the distance between xr11 and xr12. For that purpose, large gray shaded areas
represent large scale separation of low frequency and small gray shaded areas represent
small scale separation of high frequency. Snapshot a) shows the visual representation of
the flow field for large scale separations. It is characterized by separation of large
recirculation region ranging up to several step heights. Snapshot b) shows the scenario,
where the recirculation region is continuous. Here, both reattachment points areoverlapping. Snapshot c) shows the characteristic situation of small scale separation. Here,small parts with backflow are (<1h) separated from the recirculation region.

Fig 4.27 Regular reattachment point xr11 (black line) and separated reattachment point xr12 (red line) as afunction of normalized time t*. Vector fields are colour coded with streamwise velocity. Negative streamwise
velocities are blanked. a) large scale separation of recirculation region b) continuous recirculation region c)small scale separation of recirculation region

The corresponding characteristic sawtooth pattern is highlighted in figure 4.28 by red lines.
The figure shows a time sequence of Δt*=100 of the regular reattachment point. The figureon the left shows the reattachment point obtained by the PIV measurements and the figureon the right shows the results obtained by 3D Scanning PTV. Both figures show thereattachment point xr11 in the symmetry plane as a function of time. The reattachment point
is characterized by a constant motion in downstream direction and a sudden jump backvalidating that both techniques are sufficient to resolve the temporal behaviour of theVolumetric Measurements of the Transitional Backward Facing Step Flow101

fluctuating reattachment point. Nevertheless, it should be noted that small zones ofrecirculating flow downstream of reattachment with extensions less than 0.1h cannot be
captured by the measurements (see chapter 2).

Fig 4.28 Reattachment point as a function of time. (top) Time Resolved PIV measurements (bottom) 3D
Scanning PTV; the red line highlights the sawtooth pattern.

In the following we will focus on the first large scale separation between t*=40 and t*=60.
This sequence shows a significant scenario, where zone of recirculation flow is released
from the regular recirculation area. Figure 4.29 on the top shows the magnification of theregion of interest from figure 4.27. Ten positions are marked numbered from t1 to t10. The
corresponding representations in terms of the vector field are shown below. The vector
fields are colour coded by streamwise velocity. Areas of backflow are blanked.
Furthermore, sectional streamlines are plotted for visualisation. The sectional streamlines
are calculated following a Runge-Kutta scheme.

102

1)

6)

2)

7)

3)

8)

4)

9)

5)

10)

Figure 4.29 Large scale break in (mechanism 1) lengthening of recirculation region (1-3), separation of large
scale recirculation region (3), downstream movement of separated recirculation region (4-5), cut in of
separated recirculation region (6) and connection with main recirculation region (7-10)

Volumetric Measurements of the Transitional Backward Facing Step Flow103

The underlying vector field topology allows a qualitative detection of dominant vorticalstructures. Nevertheless, the early deformations of the rollers makes a vortex based
analysis via planar measurements difficult as the main axis of rotation is not aligned
normal to the measurement plane.
At time step t1 the recirculation area is continuous. Both reattachment points are at thesame position. Several vortices are located along the layer between forward and backward
flow, whereas the aft part of the recirculation region is fixed by a vortical structure.
Between t1 and t3 is constant growing in length of the recirculation region. This goes within
the convection of the roller train in downstream direction. Connected with the lengthening
is a narrowing of the recirculation region in z-direction. The reason for the lengthening isin the large oncoming vertical structure positioned about x/h=3, which generates negative
velocities in z-direction resulting in a separation of the recirculation area in two parts. In t5
and t6 the released region moves downstream with the aft vortex. While moving
downstream, the negative velocities dissipate resulting in a reduction of the size of theseparated recirculation region. Furthermore, the release of the aft region forces a flappingof the streamline (t3-t6). In t6 a second small separated region appears at x/h=6. This isbelieved to be the result of three-dimensionality as will be shown later. From t6) to t7) the
negative velocities in the separated regions are diminished, which is simultaneous to thejump of xr12. The convective velocity of the rollers downstream of reattachment has
become larger than the induced negative velocities of the rollers. While convecting out of
the image plane, the rollers influence the streamline pattern resulting in a large scale
flapping (t7-t9). Furthermore, sequence t7 to t9 shows merging of recirculation regions. In t7
the recirculation region starts to split. Nevertheless, the region doesn’t shed downstream,
but again connects with the regular recirculation. Again, the outer boundary of therecirculation region is fixed by a vortical structure.
From the time resolved PIV measurements, the following statements are concluded: The
sawtooth pattern of the reattachment point is clearly linked to the interaction of the rollers
with the wall while moving downstream. Rollers build the outer boundary of therecirculation area. Thus, while a roller is transported in downstream direction, thereattachment point moves in downstream direction, too. Backflow regions below therollers are generated by convecting rollers. The negative velocities of the released regions
are exhausted within a few step heights. The quasi-periodic release of the aft part of theseparation region downstream follows Stre=0.09.

104

Limits of the planar PIV measurements are reached near the reattachment region. On the
one hand zones of recirculation suddenly appear downstream of reattachment and on theother hand vortical patterns appear and disappear while convecting in downstream
direction. This is thought of to be the result of the three-dimensionality.
Reattachment: Characteristics of xy-plane, z/h=0.1

The three-dimensionality about reattachment is getting clear by the investigation of thereattachment line. For that purpose, figure 4.29 shows a comparison of two instantaneous
results obtained by the planar PIV measurements. The figure shows results ofmeasurements in P5 (see figure 2.1) visualized via a vector plot and colour coding bystreamwise velocity. Negative velocities in streamwise direction are blanked, thus
highlighting recirculating flow. Two characteristic states are shown. The left figure
describes the reattachment line as an organized structure of “fingers” (finger configuration)and the right figure describes the reattachment line as an approximately linear line (lineconfiguration). The fingers have a spacing ranging from 1h to 2h in spanwise directions
and length scales of 2h in streamwise direction. The streamwise velocities of the fingers
successively change from forward flow to backward flow in spanwise direction. Beside thereattachment line, regions of backflow are observed downstream of reattachment.

Fig 4.30 Normalized planar streamwise velocity field at z/h=0.1 obtained via Planar PIV (backflow is blanked;(left) finger configuration (right) line configuration

The characterization of the fingers in terms of dominant shape is rather difficult as there is
no dominant structure, which holds for several time steps. This is more a continuous
process. Due to the fingers, the reattachment length varies largely up to 3h in spanwise

Volumetric Measurements of the Transitional Backward Facing Step Flow105

direction. Thus, we define a mean reattachment line xr11,mean as the linear fit of the single
reattachment points in spanwise direction as shown in figure 4.31 on the left. From here,
the root mean square value (RMS) Δxr11,mean is defined. Thus, high values of Δxr11,mean
indicate a finger configuration and low values of Δxr11,mean indicate an approximately linear
configuration. The normalized values as a function of normalized time is shown in figure
4.31 on the right. The progress of the mean reattachment position (solid line) is describedby a continuous increase and continuous decrease in time. The sawtooth pattern(continuous growing and sudden jump back) observed in the planar measurements is not
replicated here (see figure 4.27). The reattachment location varies about 10% in time.

Fig 4.31 (left) Mean reattachment over the span and definitions of important parameters for detection of line
and finger configuration. (right) Temporal signal of mean values over the span defined on the left

The comparison of the mean reattachment position with the RMS value shows a clear
linkage of both. At low values of the amplitudes (line configuration), the reattachment
position moves in downstream direction resulting in lengthening of the recirculation
region. At a certain reattachment length, the finger configuration starts to become
dominant resulting in a stop of lengthening and a subsequent shrinking of the reattachment
length. The transfer from line configuration into finger configuration is rather sharp andclearly visible in figure 4.31 on the right. A quasi periodic frequency is estimated to 4 Hz
(Stfingers=0.09) by counting the total amount of single finger configurations in certain timeintervals. Stfingers is similar to Stre,2D,(corresponding St-number obtained by the planar time
resolved measurements) from where the conclusion is drawn that the observed frequency

106of the sawtooth pattern is linked to the presence of the finger configuration. The reason for
the finger configuration is searched in the presence of the convecting rollers. Nevertheless,the St-number of vortex shedding (Stshedding=0.2) is approximately twice the number of thefinger configuration.

Fig 4.32 Time resolved visualization of formation of small zones of recirculating flow due to merging of fingers

The time resolved measurements show a continuous process from finger configuration intoline configuration and vice versa. An exemplarily time resolved sequence is shown infigure 4.32. The figure is colour coded with streamwise velocity and backflow is blanked.
A finger configuration is given at t0. The qualitative reattachment line is sketched black
highlighting the finger configuration. The single fingers are numbered 1 to 4. In thefollowing, we will focus on finger 3. Once the fingers are generated, they immediately startVolumetric Measurements of the Transitional Backward Facing Step Flow107

to deform at the single endings. In course of the deformation, the fingers incline withrespect to the main orientation in stream wise direction and lengthen and shorten showing
the instability of the finger configuration (t2). If the fingers have a certain degree of
deformation, the fingers start to merge (t3). Due to the merging at the ending of the fingers,
small zones of recirculation are engulfed by positive flow (finger 3 at t4). Once the zone ofrecirculating flow is separated from the main recirculation region, the small zones aretransported in downstream direction. At the end (t5), the reattachment line is described by
an approximately line configuration.
Reattachment: volumetric characterization

In the following, the volumetric measurements are used to investigate the interaction of
oncoming rollers and reattachment line. The volumetric expansion of the fingers is shownfor one instant in time in figure 4.33 via isosurface of the zero-velocity. The correspondingtime step in figure 4.31 is about t*=18. The height of the fingers ranges up to 0.5h. Besidethe main fingers, separated zones of recirculation flow are observed with similar heights.
The approximated reattachment line is sketched as a black line for visualization. The
rollers in the range of reattachment are represented by vortex center lines (red). The
vortices are positioned around the single fingers with the main axis of rotation in spanwise
direction. The vortex lines are superimposed by long scale wave lengths (1h-3h) and small
scale wave lengths(<1h). The small scale wave lengths (also observed in figure 4.15) arehypothesized to be the result of vortex center line instability and are interpreted as
perturbations only locally influencing the vortex center line. The large sale wave lengths
describe dominant vortex loops of the rollers. The visual inspection shows that thevertically aligned vortex loops are primary positioned over the single fingers, thus
engulfing the recirculation fingers.

108

Fig 4.33 Zero-velocity surface showing the heights of the fingers and the connection with rollers

Furthermore, figure 4.33 allows the extraction of another characteristic. Along one single
finger at least two in streamwise direction successive rollers are positioned. The reason isin the spatial dimensions of the fingers. The length of the fingers ranges up to 3h instreamwise direction. The rollers have an approximately spatial spacing of 1.2h (see figure4.18). Thus, two roller formations fit along a finger in streamwise direction. The
comparison of rollers belonging to a common finger showed that the large scale wave
length is valid for a number of successive rollers indicating a global perturbation.
From here, the following hypothesis is drawn trying to explain the reason for the finger
shape of the reattachment line. A schematic description is shown in figure 4.34. A
characteristic finger is engulfed by a roller travelling above the finger. Thus, due to thevertically aligned vortex loop, the roller has different distances from the bottom wall in
spanwise direction. The figure on the bottom shows three exemplary xz-slices through andbeside the finger. From here it is clear that the part of the roller having larger distance fromthe wall (y0) will generate more backflow than that part having smaller distance from thewall (y-, y+). This local imbalance causes the generation of the characteristic finger
configuration.

Volumetric Measurements of the Transitional Backward Facing Step Flow109

Fig 4.34 Schematic of the generation of fingers; (top) isometric view of finger configuration and correspondingroller; (bottom) slices through and beside the finger showing variable distances of roller center from bottom
wall

The investigation around reattachment validated the influence of the oncoming rollers on
the reattachment position as stated by several authors [54][50][118]. The present test caseidentified a “finger configuration” around reattachment with Stfinger=0.09. The fingers ofrecirculating flow are the result of deformed rollers with large length scales. The
investigation showed that the fingers are hulled in by the vortex center line. Varying
distances of the vortex center lines from the bottom wall result in different amounts of
induced backflow. Furthermore, we identify at least two vortices to be part of a finger.
Thus, the reason for the halving of the Strouhal number from vortex shedding toreattachment line fluctuation is assumed to be the result of the finger configuration, where
two roller lines belong to the formation of one finger. Furthermore, we observe small zones
of recirculation downstream of reattachment. These zones are found to be the result offinger merging as shown in figure 4.32.

110

4.4 Region IV: Recirculation Region
The recirculation region is located upstream of reattachment and below the shear layer.
The outer boundary of the region is defined by the zero-velocity iso-surface as shown infigure 4.33. The inner of the recirculation region cannot be classified as a “dead water”
region as stated by several authors, e.g. Simpson (1989) [131]. Largest backflow velocities
are measured in the range 3<x/h<4 to about 20% of the bulk velocity. Nevertheless, thezone of recirculation gained less attention.
An exemplary LIF visualization for one instantaneous snapshot is shown in figure 4.35. A
large scale vortical structure is positioned in the inner of the recirculation region at roughly
x/h=2.5. The vortical structure induces vertically directed velocities and thus interacts withthe shear layer. The perturbation results in a triggering of the shear layer in the range of thevortical structure. The time resolved LIF visualization as well as previous investigations
show a strong interaction between the recirculation region and the shear layer, as alreadydiscussed in chapter 4.2. Thus, the aim of this chapter is the extraction of dominant
structures inside the recirculation region influencing the developing shear layer.

Fig 4.35 LIF visualization showing a mushroom-like vortical structure in the inner of the recirculation region,
directed vertically and moving away from the wall, thus disturbing the free shear layer.

Detection of vortical structures

The measurement inside the recirculation region is rather difficult. The flow is highly
three-dimensional and unsteady with a large amount of spatial and temporal length scales.
Nevertheless, previous investigations observed vortical structures inside the recirculation
region. Hairpin-like vortex structures with length scales from 2.5h to 4h in streamwise
direction are observed in a conditional averaging of pressure-velocity joint measurements
by Lee et al. (2004) [55]. Schröder et al. (2011) visualized hairpins like vortices in amanipulated BFSF via Tomo PIV [127]. The roller formation is triggered via external
acoustic excitation. While the rollers impinge on the wall, hairpin like vortices are
generated in both flow directions. Characteristic of hairpin like vortical structures is theVolumetric Measurements of the Transitional Backward Facing Step Flow111

lift-up effect and the corresponding large wall normal velocities at the front of the head andbetween the legs, thus defining a possible detection criterion. Furthermore, the structures
are usually found in vicinity of reattachment, thus the region of interest is located, too.
Figure 4.36 shows the zero velocity iso surface and the +0.12w iso surface of an
instantaneous velocity field in the region of interest at t*=27 (figure 4.31). Four coherent
areas of upward directed velocities (blobs) are positioned about reattachment in a certain
arrangement. First of all, all blobs are ranging out of the recirculation region, thusinteracting with the shear layer and the oncoming rollers. Blob a) and b) are positioned
upstream of reattachment and c) and d) are positioned downstream of reattachment.
Furthermore, the blobs are aligned in streamwise direction showing a linkage between
blobs and fingers.

blob a)Fig 4.36 Instantaneous visualization of region of interest positioned about reattachment at t*=27 from figure
4.31 (top) gray: iso-surface of zero velocity, red: iso-surface of w/U∞=0.12; black line: approximatedreattachment line (bottom) Blob a) visualized via Line Integral Convolution (LIC) on slices through the hairpin
like vortical structure. LIC is superimposed by in plane velocity vectors colour coded by normalized z-velocity

112

The visualization of the hairpin like structure positioned at (x/h, y/h) = (4, -1.5) is shown infigure 4.36 on the right. The visualization is performed via Line Integral Convolution(LIC) on slices through the volume of interest. The integration length for LIC is chosen byqualitative inspection to t*=0.75. The LIC visualization is superimposed by in plane
velocity vectors colour coded by normalized z-velocity. The stagnation point as well as theinfluence of the surrounding fluid on the hairpin like structure is visible. The downstream
convecting fluid above the recirculation region collides with the upward directed velocitiesof the hairpin like structure. The structure has an approximately inclination angle of 45° at
t*=27. Following its evolution in time, we can see from the measurements a change ofinclination from initially 30° to finally 90° and a following sweeping away withdownstream convecting outer fluid. The lifetime is approximated by the study of single
events to t*≈10 depending on the strength of the structure. A breakup is defined here as theloss of upward induced velocity. Further investigations are necessary to study the lifecycle
and breakup of these structures in a statistically meaningful manner.
Mechanism of hairpin vortex generation

A basic feature of the travelling rollers is the generation of backflow while travelling withthe shear layer in downstream direction. The amount of backflow is dependent on thestrength of the roller, the convection velocity of the rollers and on the distance between
roller and wall. From the present measurements, it is known that the structure of the rollers
in the region of reattachment is described by a wavy line with main orientation in spanwise
direction (perturbation). The wavy vortex center line is defined by differing distances fromthe wall in spanwise direction. Thus, differing velocity profiles are generated along the
span forming the characteristic backflow fingers. Due to the generated backflow in thefingers, a hairpin like vortex is generated pointing with its head against the mean flow
direction, see figure 4.36.
A proposal for the generation of a hairpin like structure in case of the BFSF is shown infigure 4.37. The roller passes the bottom wall with the convection velocity uroller at t0.
Negative velocities are induced between the roller and the bottom wall due to thecirculation around the vortex center line. As the rollers are spatially finite and pass by
successively, the temporal behaviour of the backflow is interpreted as sudden bursts withlocally higher negative velocity than the surrounding fluid (t1). Here, similarities can beseen in a jet flow. The injection of fluid with large velocities into a low speed region
results in a stagnation point at the front of the jet and in high shear gradients at its border.

Volumetric Measurements of the Transitional Backward Facing Step Flow113

In case of the BFSF, the lower boundary of the jet is defined by the wall. In vicinity of thestagnation point the fluid feels a negative change of velocity in streamwise direction. This
results in positive change of velocity in y- and z-direction with respect to continuity. Thus,
fluid is deflected in positive and negative y-direction, as well as into positive z-direction.
The redirection of fluid as well as the shear gradients at the jet boundary generates pairs ofvortical structures in the x-y planes generating the legs of the hairpin vortex. The
developed hairpin like vortical structure is shown at t2. The legs of the hairpin vortex are
inclined against the bottom wall. Due to the orientation of the vortex loop, down-washvelocities are induced between the legs of the structure, while up-wash velocities occur infront of the head. A comparison can be drawn to hairpin vortices in a turbulent boundary.
Here, the inclination angle is typically about 45° and followed by natural break up. In caseof the BFSF, the lifetime seems to be shorter due to rapid vortex interaction and thus heavy
manipulation of the hairpin like structure. Beside the large scale hairpin like vorticalstructure, several small scale structures are observed, ranging from arbitrary oriented shear
vortices to secondary counter rotating vortical structures as observed in e.g. vortex ring
impingement. Chapter 4.6 will focus on the influence of the observed hairpin structure
onto the oncoming rollers and the resulting shape of reattachment.

Fig 4.37 Schematic proposal for generation of hairpin vortices in case of the BFSF (mechanism: jet in a crossflow): t0: roller-wall interaction and corresponding velocity profile (gray), dash dotted black line: reattachment
line, t1: gray: corresponding velocity field in xy-plane, black: dominant velocity directions, x: stagnation point,
dashed black line: developing hairpin vortex; t2: black line: vortex center line of hairpin like vortex structure,
black: induced velocity

114

4.5 Region V and VI: Step Corner and Recovery Zone
The analysis of region V and VI is beyond the scope of the present research as of lower
importance for the dominant characteristics at reattachment. A brief overview will be given
for completeness highlighting characteristic events in the single regions.
Region VI is positioned downstream of reattachment and is called recovery region. This is
for the redevelopment of the separated boundary layer into a turbulent boundary layer. The
redevelopment of the boundary layer is visible in figure 4.6 on the left by the median shear
sheet in this area. Coherent areas of recirculating flow are observed travelling with theimpinged rollers in downstream direction. Furthermore, hairpin like structures are observed
downstream of reattachment (see figure 4.36).
Region V is positioned in the vicinity of the step corner and is characterized by a low
speed, but fluctuating state. Side wall jets are observed in literature and herein, too.
Although, the present test case has a large AR, the influence of the side wall jets is not
negligible as shown in figure 3.7. Characteristic of region V is the differing boundary
condition in comparison with the other regions. Upstream and bottom boundaries are solid
walls. The spanwise boundary of the measurement volume in region V is defined by theside wall jets, the upper boundary by the separated shear layer and the downstream
boundary by backflow. Thus, multiple stagnation points are observed. Figure 4.38 shows aschematic scenario in the vicinity of the step corner. Due to the sidewall jets a stagnation
point is formed upstream of x/h=1, which induces velocities in streamwise and verticaldirection. The collision of the streamwise induced velocity with the backflow generates asecond stagnation point downstream of x/h=1 again inducing upward directed velocities.

Fig 4.38 Schematic flow situation in vicinity of the step corner; gray: in plane velocity vectors immediately
upon the wall; black: resulting upward directed velocity due to stagnation points

Volumetric Measurements of the Transitional Backward Facing Step Flow115

The influence of the side wall jets on the developing shear layer is not clarified by thepresent investigation. The influence of the upward directed velocities and generated
vortices due to the stagnation points need more investigation. Similar influences as in caseof the generated bursts are expected.

4.6 Roller - Hairpin Vortex Interaction
The influence of the hairpin like structure on the oncoming rollers is shown in figure 4.39
schematically. At approximately x/h =1 the separated boundary layer rolls up with aspanwise wave perturbation of the axis of rotation in wall-normal direction. Let’s assume
that at the same time a hairpin like structure is generated at a distinct distance downstreamof the roller. The structure should be regarded to be spatially stable for a certain time spanof Δt*=10-20 (gray). While travelling in the direction of the hairpin like structure, theroller starts to deform in the xy-plane. The hairpin like structure acts like a local bump.
Locally, the roller tumbles over the bump. In between 3<x/h<4, the vortex loops are
therefore locally in the outer region of the shear layer at larger streamwise velocities.
Compared to the neighbouring downside directed loops, the locally upside directed rollers
reattach later on the bottom wall, generating the finger shape reattachment. Thus, vortex
stretching and tilting due to the hairpin structure is the cause of the characteristic ‘finger
shape’ near the wall between 5<x/h<6. It should be noted that the orientation of the vortex
loop axes is always a mixture of horizontal and vertical components. Nevertheless, theevents are dominated in the pre-described states.

Fig 4.39 Schematic of roller-hairpin interaction; black: roller convecting in downstream direction (t0 – t4); gray:hairpin like vortical structure (frozen for the observed time sequence)

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Figure 4.40 shows an exemplary interaction of a hairpin like structure on an oncoming
roller and the resulting reattachment line. The event is described in between t*=22 andt*=30 from figure 4.31 on the right. One identical roller visualized via the vortex centerline is tracked for the sequence while influenced by the structure (t0*≈22, t1*≈24 t2*≈26,
t3*≈28, t4*≈30). The hairpin like structure is visualized via the isosurface of 0.12w/U∞ att*≈27. Furthermore, a slice at z/h=0.1 is colour coded with backward (blue) and forward(white) flow direction identifying the reattachment position at t4. It should be mentionedthat three-dimensional visualization is rather difficult on a two-dimensional image. In aninteractive session, the process described in figure 4.39 is clearly visible in figure 4.40.
Nevertheless, the chosen perspective shows the deformation of the roller in verticaldirection from t1 to t3 while passing over the upward directed velocity. From t3 to t4 thevertical aligned vortex loop is reoriented to an inclination angle of roughly 30° withrespect to the bottom wall. In between t2 and t4 the distance between vortex and bottom
wall is rather large and consequently results in a locally large amount of backflow
generating the backflow finger.

Fig 4.40 Instantaneous visualization of region of interest positioned about reattachment: rollers are visualized
via vortex center line (t0≈22, t1≈24 t2≈26, t3≈28, t4≈30); red: iso-surface of w/U∞=0.12; slice at z/h=0.1 shows
reattachment line (blue: backward flow, white: forward flow)

This chapter focussed on the influence of large scale vortical structures in the inner of therecirculation region (hairpin vortices) on the oncoming rollers. The investigation showedmechanism, where rollers induce a large amount of backflow inside the backflow fingers.
These backflow jets result in stagnation points in the inner of the recirculation region. Due

Volumetric Measurements of the Transitional Backward Facing Step Flow117

to the stagnation point, a hairpin like vortical structure is generated, which induces wallnormal velocities ranging out of the recirculation region (bursts). The interaction of thesebursts with the oncoming roller results in a deformation of the roller. Due to the larger
distance of the deformed vortex loop, a larger amount of backflow at a somehow later
instant in time is generated. Furthermore, the deformed roller is made responsible for the
shape of reattachment.

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5 Conclusion and Outlook
The thesis studies the dynamical and volumetric behaviour of the Backward Facing Step
Flow. The first part of the thesis (chapter 2) focuses on the selection and implementation ofa sufficient measurement technique for the identification and characterization of
volumetric coherent flow structures in case of the present BFSF configuration. A detailed
literature review regarding volumetric measurement techniques is given. The development
and implementation of the measurement technique of choice “3D Scanning PTV” and its
optimizations for measuring the present BFSF configuration are highlighted. This chapter
summarizes the results of development and shows limits of the implemented technique. An
outlook of further optimizations and future trends regarding volumetric velocimetry
measurement techniques is given. The second part of this thesis (chapter 3 and 4) studies
the BFSF in a volumetric as well as in a temporal manner. The volumetric measurements
give deeper insight into three-dimensional shape of coherent structures of the BFSF, i.e.
shear layer, roller, hairpin like structure and recirculation region. Complex interactions of
the single coherent structures are highlighted. This chapter summarizes main results,
develops a feedback mechanism and gives an outlook of future work as the new knowledge
results in new questions.

5.1 Conclusion
Measurement Technique

The experimental study of the BFSF in the turbulent regime is performed via 2D2C time
resolved Particle Image Velocimetry, 2D time resolved Laser Induced Fluorescence and
3D Scanning Particle Tracking Velocimetry.
The planar PIV technique is primary applied to study the categorization of the test case interms of literature and to calculate accurate inlet flow conditions. It should be noted thatthe implementation and the processing of the two-dimensional raw data into velocity fields
is automated in an efficiently manner with high quality by commercial software. The
analysis of time resolved data used for the investigation of unsteady phenomena is still notwithout complications. Data processing, data storage as well as data visualization is criticaldue to the large amount of data. One time resolved series of the PIV arrangement results in
8 GB raw data producing 2 GB velocity field information. The storage as well as anVolumetric Measurements of the Transitional Backward Facing Step Flow

119

automated quality check is a challenging step. Commercial software still fails under thesestrict conditions.
The planar LIF measurements give great inside into the appearance and dynamical
behaviour of coherent structures in the BFSF. Furthermore, the visualization acts as initial
guess for the description and detection of flow structures in the three-dimensional data set.
In some cases, the flow visualization already gives much information for the physicalunderstanding of the complex processes.
In the present test case, 3D Scanning PTV is selected to be the volumetric measurement
technique of choice. The selection is done with respect to a detailed literature review,experimental testing, available hardware and tested flow configuration:
In course of the thesis, pointwise as well as volumetric particle reconstruction is
implemented and tested for the analysis in a Lagrangian sense, termed classical PTV andtomographic PTV. For that purpose, a simplified volumetric reconstruction technique isdeveloped using the advantages of telecentric imaging. At least a doubling of the spatial
resolution is achievable in tomo PTV compared to classical PTV. Nevertheless, drawbacks
are the need for the detection of ghost particles and the large processing overhead. Here,
processing on a GPU is not implemented up to now. Thus, the applied reconstruction
technique is pointwise reconstruction with respect to the analysis of in total 80GB raw
image data.
Lagrangian particle tracking is selected for the calculation of the velocity field. This is forseveral reasons. A high particle density (as necessary in Eulerian approaches) is critical in
the present experimental set-up. Thus, a technique capable of lower particle densities isfavoured. Furthermore, a highly fluctuating flow state is apparent in the inner of therecirculation region making a mean translational velocity estimation using correlation
based techniques in distinct interrogation areas rather difficult.
The scanning illumination is used for increasing the quality of the particle images as well
as for the simplification in the three-dimensional reconstruction. Thus, the scanning
illumination increases the number particle density about the factor of applied light sheets.
This is for two reasons: The number of projected particle images is reduced due to theslicing of the volume and the search area along the epipolar line is reduced with respect tothe thickness of the light sheet.
Advantages of telecentric imaging in terms of image quality and simplification of thepointwise reconstruction are highlighted. Telecentric imaging is characterized by a large
depth of field, a constant magnification, high accuracy and negligible distortions. In total,

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fifteen steps in calculation of the stereo triangulation are saved due to telecentric imaging,speeding up the pointwise reconstruction.
An estimation of accuracy of the pointwise reconstruction is performed two-fold: On the
one hand, calculated three-dimensional marker positions in physical space are comparedwith the known values, resulting in a mean deviation of 0.014 mm (≈ 0.15 pix). On theother hand, deviation in the three-dimensional positions due to erroneously calculated 2D
positions of particles are considered. Here, a theoretical position uncertainty of roughly
0.007 mm is calculated.
The implemented tracking algorithm is a hybrid PIV/PTV 4-frame method. Hybrid
PIV/PTV means that the first guess in which direction a particle has moved is estimated via

cross correlation in an interrogation area. 4-frame means that if a particle has been tracked
for several time steps, the next position is estimated via the kinematic history of theparticle. The method is applied on the two-dimensional image plane. Homologue particles
for three-dimensional point reconstruction are searched via epipolar line intersection andquality check in object space. For that purpose, the 3D reconstruction is performed
trajectory wise in a spatio-temporal manner.
In total, a number of 13 experimental realizations of the BFSF have been recorded with 3D
Scanning PTV. Each of the Lagrangian velocity fields contains the same measurement
volume (90x60x15 mm³) and a measurement period of 2s in physical time at a recording
rate of 1000 fps.
A mean number vector density is calculated to about 42 vectors/cm³. Nevertheless, it
should be noted that there is a distinct decrease in number vector density from LS1 to LS4.
This is due to difficulties in seeding close to the channel wall. A larger amount of injectedparticles is transported in downstream direction than is caught by the recirculation region.
The difficulty in tracking is about e=1.25 (see figure 2.15). The yield (number of 2D
centroid positions/number of three-dimensional vectors) is estimated via the median
number of linked particles with respect to the center camera to 58%. Each Lagrangian
velocity field is interpolated onto an Eulerian velocity field (grid spacing: 1mm) via
inverse distance interpolation. Here, difficulties in non constant seeding are observed. The
local number particle density defines the size of the local spatial structure to be measured.

Volumetric Measurements of the Transitional Backward Facing Step Flow

121

Backward Facing Step Flow

The measurements of the BFSF include three major parts, the inlet flow conditions(chapter 3), the categorization of the present configuration with respect to the large amountof available data in literature (chapter 3) and the volumetric investigation of coherent flow
structures (chapter 4). The investigation of volumetric and unsteady properties of the BFSF
is focussing on characteristic regions, namely the region of shear layer formation (regionI), the region of shear layer development (region II), the region of shear layer impinging(region III) and the region of recirculation (region IV). The analysis of the single regions
showed a distinct crosstalk, where the single regions are interacting with each other. The
identification of coherent structures focussed on shear layer extraction in region I and large
scale vortex detection in region I, II, III and IV. The influence of the region in vicinity ofthe step corner and of the region of recovery (region V and VI) are assumed to be
negligible for shear layer reattachment.
Inlet Flow Conditions

The inlet flow conditions are calculated via planar PIV measurements upstream of the
expansion. The installed backward facing step of height h=10mm is characterized by awedge 6h long and a following plateau of 6h resulting in a contraction from 5h in height to
4h in height. Thus, the expansion ration is ER=1.25. The wedge triggers the boundary
layer thickness resulting in a thickening of the boundary layer. The velocity profile isvalidated to be fully developed again at x/h=-2. The wall shear velocity is calculated via
the conventional Clauser chart method to uτ = (τW/ρ)1/2 = 25mm/s = 0.05·Ubulk. The
sublayer scaled velocity profile indicates a turbulent flow state prior of separation. The Renumber defined with the step height is Reh = 4440, the momentum thickness prior toseparation is θ*=0.114 and the turbulence intensity is Tux=8%.
Categorization

The categorization of the BFSF is primarily performed with respect to the integral lengthscale reattachment length. The reattachment length is influenced by the aspect ratio, theexpansion ratio, the boundary layer thickness prior to separation, the turbulence intensity
and the Reynolds number. The reattachment length is calculated to about 5.65 h +/- 1.5h inthe center of the flow channel. The large aspect ratio of AR = 57 minimizes influencesfrom the side wall on the velocity field about the symmetry plane. Nevertheless, side wall
jets are observed in vicinity of the step corner. The low expansion ratio minimizes

122

influences from the top wall on the region of interest. Secondary recirculation areas above
the region of interest are not observed in the planar PIV measurement. The Reynolds
number indicates a transitional to turbulent state of the separated boundary layer. The
parameter space of Reh and θ* positioned the present BFSF configuration into the region oflaminar to turbulent separation and turbulent reattachment.
Shear layer characteristics

The shear layer is extracted as the surface of maximum shear, identified via a simplified
analysis of the wall-normal velocity gradient ∂u/∂z. A comparison is drawn with a more
complex approach analysing the deviatoric rate of strain tensor. Both approaches deliver
similar results, especially in region I. The surface of maximum shear is characterized by awavy structure with heights and ridges immediately downstream of the step. The shear
layer is divided in an upper and lower shear layer thickness, defined as the momentum
thickness above and below the maximum shear. As expected, the temporal and spanwise
averaged shear layer thickness increases up to the point of reattachment. The progress ofshear layer thickness shows similarities with a free mixing layer indicating complex vortex
dynamics in the range 2<x/h<4. Here, a dominant influence of the lower shear layer
thickness is observed indicating an influence from the recirculation region. The shear layershows a wavy structure in streamwise and spanwise direction. The amplitude and wave
length as well as the ratio perform a kind of balancing in between 0<x/h<2, while bothincrease identically in stream- and spanwise direction beyond a distinct point positioned
around x/h=2. Immediately downstream of the step the amplitude in spanwise direction isroughly 50% larger than that in streamwise direction. Furthermore, the analysis described
the convection of perturbations in downstream direction with a convection velocity ofroughly 0.6U0±0.2U0.. In irregular distances large scale perturbations travel along the shear
layer showing the transitional flow state.
Roller characteristics

Originating from the shear layer instability, coherent vortical structures are formed withmajor axis of rotation in spanwise direction. Several Eulerian vortex detection criteria aretested delivering similar results for the detection of the mentioned structure. The swirlingstrength criterion is chosen due to the highest signal to noise ratio. The 3D profile of thevortex center line is estimated via pointwise connection of local maxima.

Volumetric Measurements of the Transitional Backward Facing Step Flow123

Several roll up configurations of the shear layer are identified, i.e. parallel roll up, inclinedroll up, convex/concave roll up and piecewise roll up. The most dominant configuration isparallel roll up, but inclined roll ups are observed with inclination angles up to 30° withrespect to the step edge. The Strouhal number for vortex formation is calculated to
StVS=0.24. The center line of the roller is identified to be wavy immediately after roller
genesis generating three-dimensionality immediately downstream of the step. This ispostulated to be the result of the roll up of the wavy shear layer. The roller wavinessshowed distinct specifications depending on the streamwise distance. Thus, the roller
performs a stretching in streamwise direction in between 0<x/h<3. At x/h=3, the roller
loops are primarily oriented in streamwise direction. Further downstream, the rollers
perform a tumbling motion resulting in a re-orientation of the loops in a vertical alignment
and following tilting to a somehow streamwise alignment.
Reattachment characteristics

The reattachment line is defined as the intersection of the zero velocity iso-surface with thebottom wall. The volumetric visualizations clearly identified a cyclic change between two
states, a finger and a line configuration in the instantaneous reattachment line and viceversa. The non-dimensional frequency is calculated to Stfingers=0.09. That Strouhal number
describes the instantaneous low frequency flapping described in several publications.
The characteristic sawtooth pattern in the temporal progress of reattachment point is foundto be the result of a pointwise analysis of reattachment. The sawtooth pattern appears atchanges from line to finger configuration. The consideration of all reattachment positions
along the span results in a constant decrease and constant increase of reattachment lengthfrom line to finger configuration removing the sawtooth pattern. Nevertheless a successive
increase and decrease of reattachment length is still observed.
Recirculation characteristics

The analysis of the recirculation region focussed on the extraction of coherent vortex
structures generated inside the recirculation region, which influence the oncoming rollers.
For that purpose, iso-surfaces of upward directed velocities in vicinity of the streamwise
zero-velocity iso-surface are detected. The extraction of the bursts shows a hairpin vortex
like shape. The hairpin vortices are generated at the endings of the backflow fingers in
upstream and downstream direction. They are concluded to be the result of locally higher
backflow velocities induced by the convecting rollers. The unsteady backflow results in the124generation of stagnation points from where the hairpin vortices are born. The lifetime isapproximated by the study of single events to t*≈10 depending on the strength of thestructure.
Global unsteadiness: line-to-finger configuration

A global unsteadiness with Stfingers=0.09 is observed, defined by the cyclic change of finger
to line configuration and vice versa. Furthermore, a local unsteadiness of shear layer rollup with StVS=0.24 is observed.
This thesis proposes the following scenario for the cyclic change from line to finger
configuration. A schematic proposal is shown in figure 5.1. A roller formation is given,
which travels in downstream direction. Roller 1 is undisturbed, thus generating anapproximately straight reattachment line (t1). While roller 1 convects in downstream
direction, the reattachment length increases (t2). As the vortex lines are not characterized
by straight lines immediately downstream of roll up as valid in a laminar flow, but by awavy line similar to that of the investigated shear layer, the rollers have varying distancesfrom the bottom wall in spanwise direction. This is roller 2 in figure 5.1 starting to
generate the finger configuration of the reattachment line. The roller travels in downstream
direction locally inducing larger backflow, thus generating a hairpin like structure inside
the recirculation region (t3). The coherent hairpin structure acts as a kind of bump on thelower wall and forces the oncoming roller (roller 4) to tumble over the bump (t4). Due tothe vortex stretching and tilting in the range of the backflow finger, the characteristic‘finger shape’ of reattachment is supported (t5). Furthermore, the hairpin vortex isdissipated as a result of pairwise vortex interaction. The disturbed roller travels further
downstream, while the next oncoming roller (roller 5) is undisturbed and thus generates anapproximately line configuration of reattachment (t5). This process is repeated quasiperiodic with Stfingers=0.09.
The approximately halving of the frequency from vortex shedding to reattachment line
fluctuation is postulated to be the result of the spatial structure of the fingers. The
volumetric visualization of rollers and zero velocity iso-surface shows at least two rollers
positioned above the backflow fingers. This becomes clear by knowledge that the backflow
fingers have an approximated length of 2h to 3h. The streamwise spacing between therollers is in the range 1h. Thus, the fingers are built by at least two roller formations
resulting of at least a halving of the St-number.

Volumetric Measurements of the Transitional Backward Facing Step Flow

125

Roller 1: undisturbed
Æ generation of straight reattachment line
Roller 2: undisturbed
Æ convection in downstream direction

Roller 1: undisturbed
Æ increase of reattachment length
Roller 2: initial perturbation
Æ varying distances from the wall

Roller 2: varying distances from the wall
Æ generates pointwise varying backflow
Roller 3: influenced by local backflow of roller 2
Æ generation of local perturbation

Roller 2+3: induce large amount of backflow andgenerate backflow finger Æ genesis of hairpin
vortexRoller 4: influenced by upwash of hairpin vortex
Æ amplification of local perturbation
Roller 4: turns over the hairpin vortex
Æ loop amplification
Hairpin vortex: is influenced by roller 4 anddissipates
Roller 5: undisturbed
Roller 4: local perturbation (vortex loop)Æ stabilizing backflow finger
Roller5: undisturbed

Roller 5+6: undisturbed
Æ generation of straight reattachment line

Fig 5.1 Principal sketch of line to finger configuration of reattachment and corresponding rollers; red line:
reattachment line; black lines: roller center line, hairpin vortex

126

The volumetric analysis showed a clear feedback mechanism generating the dynamics ofthe BFSF. The measurements showed that the consideration of three-dimensionality is
mandatory in understanding unsteady flow separation and flow reattachment in detail.

5.2 Outlook
First of all, it should be mentioned that the presentation of a three-dimensional scenario
varying in time on a two-dimensional paper is difficult and by no means trivial. Especiallythe three-dimensional representation of structures varies largely in time and space andmakes the visualization from a given perspective subjective. Therefore, some informationis only obtainable in an interactive session allowing the change of perspective and zoom ina time varying manner and is presented herein in form of simplified sketches.
Next striking point is the long processing time for the calculation of three-dimensional
velocity fields out of image data. Here, GPU implementation is probably the key issue. If
the processing for the volumetric reconstruction is getting faster, new approaches (using
advantages of volumetric reconstruction) are applicable. A hybrid tomo and classical PTV
is conceivable, which combines the advantage of quick processing in classical PTV andhigher achievable number particle density in tomo PTV.
The applied volumetric measurement technique is a multi camera approach, where highaccuracy calibration is mandatory. Here, single camera approaches outperform the multi
camera approaches for a lower number particle density. For that purpose, new trends arerecognized in μ-lens systems. Here, a virtual multi camera system is generated by a properlens combination in front of a single recording sensor. The advantage of a single camera
system and the advantage of a multi camera system are combined. Prerequisite is a highspatial resolution of the sensor chip.
The generated volumetric velocity field showed several new aspects regarding a fluidmechanical interpretation. Nevertheless, this thesis analyzed a small amount of extractable
data. The focus is on coherent structures in proximity of the shear layer. The region in thevicinity of the step corner needs more attention as well as the recovery region and vortex
pairing events. Further investigations in other directions (statistical analysis, stability
analysis or event tracking) are necessary to understand the simplification of the BFSF interms of flow separation and reattachment.
As there is no possibility for a single person or a single working group to investigate all
facets of the flow configuration, the volumetric dataset of an exemplary measurement inVolumetric Measurements of the Transitional Backward Facing Step Flow127

the Lagrangian and Eulerian frame can be downloaded on the homepage of the Institute for
Mechanics and Fluid-Dynamics of the University of Freiberg.
The major outcome of the thesis (with respect to fluid mechanics) is the cyclic interchangeof the shape of reattachment between line and finger configuration. The volumetricmeasurements allowed the identification of mechanisms forcing the different
configurations. Recent investigations at the Institute for Mechanics and Fluid-Dynamics
test the influence of flexible flaps on the suction side of an airfoil. The flaps are positionedin spanwise lines on the suction side of the airfoil at differing downstream positions. Flow
separation is forced by pitching and thus a recirculation region is induced. The results
show that the motion of flaps varies in a wavy manner in spanwise direction andconsequently indicates that the reattachment location varies in spanwise direction, too.
Further investigations are planned to measure the flow details on the airfoil with the same
method of 3D Scanning PTV developed in this thesis for finding links between the resultspresented herein and the reattachment characteristics of an airfoil.

128

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